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Mathematiques & Applications 40 Bernard Bonnard Monique Chyba

Singular Trajectories and their Role in Control Theory

ra,

I

Springer

El

MATHEMATIQUES

APPLICATIONS Directeurs de la collection: X. Guyon et J: M. Thomas

40

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Bernard Bonnard Monique Chyba

Singular Trajectories and their Role in Control Theory

Springer

Bernard Bonnard Dept. of Mathematics, Laboratoire d'Analyse Appliqude et d'Optimisation, LAAO University de Bourgogne BP 47870 Dijon, France E-mail: [email protected] Monique Chyba Dept. of Mathematics, University of Hawaii Honolulu, HI 96822, USA E-mail: [email protected]

Mathematics Subject Classification 2000: 93B06, 93B27, 49KI5

ISBN 3-540-00838-1 Springer-Verlag Berlin Heidelberg New York

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Preface

Consider a smooth control system

dt (t) = f (x(t), u(t)) where x(t) belongs to a n-dimensional manifold M (the state space) and the control u(t) belongs to the control domain taken as RP, p E N. We assume the set U of admissible controls to be the set of bounded measurable mapRI', T(u) > 0. Such a map u(.) can be extended on pings u : [0,T(u)[ [0, +oo[ by setting u(t) = 0 for t > T (u) and the set of controls is endowed with the LO0 norm. Let x(., xo, u) be the trajectory solution of (0.1) corre-

sponding to u(.) E U and starting at x(0) = xo. Fix T > 0 and consider the end-point mapping E : u(.) x(T, xo, u). This mapping has singularities which are called singular trajectories. One can show that they are parametrized by the Pontryagin Maximum Principle applied to the time minimal control problem of system (0.1) as follows. Introduce the pseudoHamiltonian H(x, p, u), defined in local coordinates on T*M\{0} x RP by H(x, p, u) = (p, f (x, u)), where p E T,, M\{0} and (,) denotes the standard inner product. The singular trajectories are the projection on the state space

of the solutions of the constrained Hamiltonian equations: (t) = aH (x(t), p(t), u(t)), dt ap dx

dp

dt (t)

OH

ax (x(t), p(t), u(t)),

(0.2)

aH(x(t), p(t), u(t)) = 0.

(0.3)

The objective of this book is to provide a geometric framework to analyze the singular trajectories and to explain their role in nonlinear control theory. This role is mainly threefolds. 1. First of all, consider the time minimum control problem for a system (0.1),

the control set being now a given subset f2 of R. The singular trajectories associated to controls u(.), u(t) E .R are candidate as minimizers for the time optimal control problem. In particular for a control system (0.1), a time optimal control is not in general bang-bang contrary to the linear situation. The role of singular trajectories in this phenomenon will be explained in this volume in two ways. First, we shall describe the time

V1

Preface

minimal synthesis for systems (0.1) in the generic situations. Secondly we shall apply this analysis to solve a nonacademic optimal control problem which is the time optimal control problem of a batch reactor for a sequence of two consecutive reactions A -' B - C. 2. Secondly, the importance of singular trajectories was recognized since a long time in the Lagrange problem in classical calculus of variations. It can be restated in the optimal control framework as: T

L(x(t), u(t))dt

min u(.)EUJ

(0.4)

subject to the constraint (0.1): i(t) = f (x(t), u(t)). Indeed, in this discipline these trajectories are called abnormal extremals and they are solutions of the maximum principle corresponding to the Hamiltonian H(x, p, u) = (p, f (x, u)) + AoL(x, u) where the multiplier A0 is taken as 0. In our geometric framework their role is clear. We must minimize a functional fo L(x(t), u(t))dt on a set of curves satisfying the constraint (0.1) and the singular trajectories are the singularities of this set. Hence we are faced to an infinite dimensional generalization of the following problem: mini, f (y) where y E S, with S C Rq, f : Rq --+ R. If S is a manifold the situation is standard but if S has singularities a preliminary step is to analyze the singularities of S. Hence one main objective of this book is to initialize a classification of the singularities of the end-point mapping. When trying to solve real applications, the system (0.1) is often of

the form of an affine control system: i(t) = F°(x(t))+E' ui(t)Fi(x(t)). 1

We shall study in details this important case. Our main result is to show

that when p = 1, a singular trajectory is C°-isolated in the set of trajectories if its time domain is small enough. Hence the situation is similar to the finite dimensional situation where S is given by {y E R, y2 = 0}.

The property of being C°-isolated for singular trajectories has an important consequence in sub-Riemannian geometry where we consider the length minimization problem: T

m

min fo (> u?(t)) Idt

u(.)EU

(0.5)

i=1

subject to the constraint: i(t) = E' l ui(t)Fi(x(t)) and where we assume m = 2. In particular, we explain the role of singular trajectories when computing the sub-Riemannian sphere and distance. Our main result is to show that due to singular minimizers the sphere of small radius and

the distance function are not in general sub-analytic. 3. Besides the importance of the singular trajectories in optimal control theory, these trajectories play an important role in the classification of systems (0.1) for the action of the feedback group induced by the following transformations:

Preface

x = 0(y),

u = tb(x, v)

VII

(0.6)

called feedback transformations. Indeed, since the singular trajectories correspond to the singularities of the end-point mapping they are feedback invariants. In particular they can be used to generate a complete set of invariants for various control problems. An important geometric application is to use singular trajectories to compute invariants to classify

distributions. This book is mainly based on geometric methods and is oriented towards applications. Therefore, we have decided to make some regularity assumptions which are twofolds. First of all, the systems and the cost functions are taken CO° or even real analytic. Secondly, the class of admissible controls is the set of bounded measurable functions and hence the trajectories have bounded derivatives almost everywhere. Both assumptions are in our opinion reasonable in a book devoted to singular trajectories and do not avoid complexity met in the synthesis problem. On the other hand, a reader interested by dealing with general absolutely continous curves associated to unbounded control problem and a maximum principle under minimal hypothesis can consult the book by L. Cesari [38] and recent articles from H.J. Sussmann [101],[100] about a generalized maximum principle. To conclude this preface, we must point out the following fact. Most of the content of this volume has its origin into a research program started in 1981 by B.Bonnard jointly with I. Kupka and presents mainly their results. Hence

it is not an exhaustive presentation of the subject. Nevertheless, its aim is to unify results published in various journals and show the interconnection between all those results. It is an important topic, because its seems that singular trajectories are some universal objects in various optimal control problems and classification problems as singularity theory when analysing functions. In particular, in this framework both applied problems like the optimization of batch reactors and geometric problems like sub-Riemannian geometry are interconnected and use similar tools to be handled. They are indeed both sides of a similar problem. Warning to the reader. Due to space restrictions, the proofs of standard results are often sketchy. For more details, the reader is invited to use the given references. Exercices have been added at the end of each chapter, some are inspired by articles refereed in the bibliography and are highly nontrivial. The authors are indebted to the "Conseil regional de Bourgogne" for his financial support for the project "Commande en temps minimal des reacteurs batch" in which most of the results concerning sub-Riemannian geometry and time optimal control of batch reactors presented in this volume were obtained. Dijon, September 2002 Hawaii

Bernard Bonnard Monique Chyba

Table of Contents

1

Linear Systems and the Time Optimal Control Problem ..........................................

1

1.1

Linear Systems .........................................

1

1.2 1.3

Accessibility Set and Controllability Controllability and Feedback Classification in the Autonomous Case 1.3.1 Controllability

1

1.4

...................... .................................................. ................................... 1.3.2 Linear Classification .............................. 1.3.3 Feedback Classification ............................ 1.3.4 Stabilization ..................................... Controllability in the Nonautonomous Case ............... 1.4.1

Application ......................................

1.5

Time Optimal Control for Linear Systems .................

1.6

Feedback Equivalence to a Linear System ..................

....................................... 1.6.1 Problem Statement ............................... Time Minimal Synthesis ................................. 1.7.1 Notion of Synthesis ............................... 1.7.2 The General Algorithm in the Plane ................ 1.5.1

1.7

Notations

2 2 3 5

6 7 8 9 9 15 15 17 17 21

Exercises .................................................. 24

2

Optimal Control for Nonlinear Systems ...................

27

2.1 A Short Visit into the Classical Calculus of Variations ...... 27

.....

27 2.1.1 Statement of the Problem in the Holonomic Case 29 2.1.2 Hamiltonian Equations 2.1.3 Hamilton-Jacobi-Bellman Equation ................. 30 2.1.4 Euler-Lagrange Equations and Characteristics of the

............................

HJB Equation 2.1.5 2.1.6 2.1.7 2.1.8 2.1.9 2.2

...................................

Second Order Conditions ..........................

31 31

The Accessory Problem and the Jacobi Equation ..... 32 33 Conjugate Point and Local Morse Theory 34 Scalar Riccati Equation

........... ...........................

Local CO Minimizer - Extremal Field - Hilbert Invari-

ant Integral ......................................

Optimal Control and the Calculus of Variations ............

35 36

X

Table of Contents 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.2.9

............................... ........................... ................................

Problem Statement 36 The Augmented System 37 38 Related Problems Optimal Control and the Classical Calculus of Variations 38 Singular Trajectories and the Weak Maximum Principle 39 First and Second Variations of Exo,T 39 Geometric Interpretation of the Adjoint Vector 42

...............

.......

The Weak Maximum Principle .....................

42

Abnormality ..................................... 43

2.2.10 The Weak Maximum Principle and Euler-Lagrange

Equation ........................................ 43

2.2.11 Comparison with the Calculus of Variations ......... 44 2.2.12 LQ-Control and the Weak Maximum Principle ....... 44

Pontryagin's Maximum Principle (PMP) .................. Comments about the Existence Theorem ............ Dynamic Programming and the Maximum Principle ........

45 52 52 53

Exercises ..................................................

56

Geometric Optimal Control ...............................

65 65

2.3 2.4

Filippov Existence Theorem ............................. 2.4.1

2.5

3

3.1

Introduction to Symplectic Geometry .....................

3.2

Geometric Classification of Extremals in one Dimensional

Problems of Calculus of Variations ........................ 3.2.1 Problem Statement and Preliminaries ...............

69 69 71 3.2.2 Singularity Analysis 72 3.2.3 Generic Classification near El 3.2.4 Classification and Existence of Optimal Solutions ..... 75 Time Minimum Control Problem ......................... 76 Determination of the Singular Extremals .................. 77 3.4.1 Hamiltonian Formalism and Singular Extremals ...... 78 Geometric Classification of Extremals near E; .............. 79 3.5.1 Normal Switching Points .......................... 79 80 3.5.2 The Fold Case

.............................. .....................

3.3 3.4 3.5

...................................

3.6

Comparaison Between the Calculus of Variations and the

Time Minimum Problem for Affine Systems. The Role of Singular Extremals ........................................ 3.7

The Fuller Phenomenon .................................

3.7.1

3.8

Fuller Example

...................................

82 82

82

Geometry of the Time Optimal Control in the Plane ........ 84 3.8.1 3.8.2 3.8.3 3.8.4

Preliminaries .................................... 84 Generic points ................................... 85 Singular arc

.....................................

Elliptic Case - The Concept of Conjugate Point ......

86 88

Exercises .................................................. 94

Table of Contents

4

Singular Trajectories and Feedback Classification

.........

........................... .................

Classification of Affine Systems 4.1.1 Computations of Singular Controls 4.1.2 Singular Trajectories and Feedback Classification 4.1.3 Time Optimality and Feedback Classification 4.2 Singular Trajectories and the Problem of Classification of Dis4.1

.....

XI

97 99 99 102

........ 105 tributions ............................................. 108 4.2.1 Preliminaries .................................... 108 4.2.2

Local Classification in Dimension 3, with rank D = 2 . 109

4.3 Feedback Classification and Analytic Geometry 4.3.1 Preliminaries 4.3.2 Critical Hamiltonians and Symbols

............ 111

.................................... 111

................. 112

Exercises .................................................. 113

5

Controllability - Higher Order Maximum Principle - LegendreClebsch and Goh Necessary Optimality Conditions ....... 117 Some Notations and Formulas from Differential Geometry ... 117 5.1.1 Controllability with Piecewise Constant Controls ..... 119 5.2 Integrating Distributions 5.2.1 Preliminaries 5.2.2 Flobenius Theorem 5.1

................................ 120 .................................... 120 ............................... 120

5.2.3 Nagano-Sussmann 5.2.4 C°°-Counter Example 5.3 5.4 5.5 5.6

Theorem ....................... 121

............................. 122 Nonlinear Controllability and Chow Theorem .............. 122 Poisson Stability and Controllability ...................... 124 5.4.1 Application ...................................... 125 Controllability and Enlargement Technique ................ 125 5.5.1 Problem Statement ............................... 125 Evaluation of the Accessibility Set ........................ 127 5.6.1

Legendre-Clebsch Condition

....................... 129

5.7 The Multi-Inputs Case: Goh Condition .................... 131 5.8 The Concept of Rigidity - Strong Legendre Clebsch and Goh Conditions as Necessary Conditions for Rigidity 5.8.1

Exercises

6

............ ..................................................

132

Intrinsic Second Variation - Morse Index ............ 133 134

The Concept of Conjugate Points in the Time Minimal Control Problem for Singular Trajectories, C°-Optimality .... 143 6.1

Single-Input Case 6.1.1

......................................

143

Preliminaries .................................... 143

6.1.2 Feedback Semi-Normal Forms in the Hyperbolic and Elliptic Cases .................................... 146

....................................... 147 LQ-Model in the Elliptic Case ..................... 148

6.1.3 LQ-Model 6.1.4 Approximation of the End-Point Mapping Using the

XII

Table of Contents

...................................... 153 153 ...................................... 155

6.1.5 Conclusion 6.1.6 The Exceptional Case ............................. 6.1.7 Conclusion 6.1.8 Comparison of the Elliptic-Hyperbolic Case and the 6.2

Exceptional Case ................................. 156 Applications ........................................... 157

6.2.1 6.2.2

Time Optimal Synthesis for Planar System .......... 157 Example in Dimension 3 and Connection with the

Hamilton-Jacobi-Bellman Equation ................. 157 6.3 The Case in 1R3. Intrinsic Computations of Conjugate Points

- Connection with the Time Minimal Synthesis Problem. The Concept of Curvature ................................... 160

6.3.1 6.3.2 6.3.3 6.3.4 6.3.5

6.4

Conjugate Points and Time Minimal Synthesis ....... 163

Connection with the Liu-Sussmann Example ............... 164 6.4.1 6.4.2 6.4.3

7

Euler-Lagrange Equation .......................... 160 Geometric Interpretation .......................... 161 Intrinsic Computation ............................ 162 The Concept of Curvature ......................... 162

Preliminaries .................................... 164 Conclusion

......................................

166

Statement and Proof of Liu-Sussmann Result ........ 166

6.4.4

Conclusion ...................................... 167

6.4.5

Connection with the Sub-Riemannian Geometry ...... 168

Time Minimal Control of Chemical Batch Reactors and Singular Trajectories ......................................... 171 7.1

Introduction ........................................... 171

7.2

Mathematical Model of Chemical Batch Reactors and De-

scription of the Control Problem ......................... 172 7.2.1 7.2.2 7.2.3 7.2.4

7.3

The Optimal Control Problem ..................... 175

Projected Problem ............................... 175

Singular Extremals - Curvature - Conjugate Points ......... 176 7.3.1

7.4

Chemical Kinetics ................................ 172 Control Device ................................... 174

Projected System .................................

177

Time Minimal Synthesis for Planar Systems in the Neighbor-

hood of a Terminal Manifold of Codimension One .......... 180 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.4.6 7.4.7 7.4.8

Problem Statement ............................... 180 Assumption C1 .................................. 181 The Generic C1 Case ............................. 181 The Generic C1 Flat Case ......................... 181

The Case C1 of Codimension One .................. 182

Generic Hyperbolic Cases ......................... 185 Generic Exceptional Case ......................... 188 Generic Flat Exceptional Case ..................... 189

Table of Contents 7.5

.......................... 191 .................................... 191

Global Time Minimal Synthesis 7.5.1 Preliminaries 7.5.2 7.5.3

Singular Arc ..................................... 191 Regular Arcs .................................... 192

........ 193 .................................. 193 ................................ 194

Optimal Synthesis in the Neighborhood of N Switching Rules Optimal Synthesis 7.6 State Constraints due to the Temperature ................. 7.5.4 7.5.5 7.5.6

7.6.1 7.6.2

7.7

8

8.2 8.3 8.4 8.5

8.6 8.7

8.8

195

Preliminaries .................................... 195

................................ 198 ............................. 199 .................................... 199 ............................. 200 .................................. 201 ..................................... 206

Optimal Synthesis The Problem in Dimension 3 7.7.1 Preliminaries 7.7.2 Stratification of N by the Optimal Feedback Synthesis 199 7.7.3 Orientation Principle 7.7.4 Switching Rules 7.7.5 Local Classification near the Target ................. 203 7.7.6 Focal Points

Generic Properties of Singular Trajectories 8.1

XIII

............... 209

.............................. 209

Introduction and Notations Determination of the Singular Extremals with Minimal Order 210 Statement of the First Generic Property ................... 211 Geometric Interpretation and the General Concept of Order . 211 Proof of Theorem 25 8.5.1 Partially Algebraic and Semi-Algebraic Fiber Bundles. 217 8.5.2 Coordinate Systems on P(d, N) 218 8.5.3 Evaluation of Codimension of the F(N) 218 8.5.4 End of the Proof of Theorem 25 .................... 222 Genericity of Codimension One Singularity 222 Proof of Theorem 26 222 8.7.1 The "Bad" Set for Theorem 26 223 8.7.2 Evaluation of the Codimension of BC(N, q) 224 Singularities of the Singular Flow of Minimal Order 226 8.8.1 Preliminaries (see Sect. 4.1.3) ...................... 226 8.8.2 Local Classification near C ........................ 227 8.8.3 Local Classification near S\C ...................... 228 8.8.4 The Quadratic Case

.................................... 213

.................... ............. ................ .................................... ..................... .......... .........

.............................. 229

Exercises .................................................. 231

9

Singular Trajectories in Sub-Riemannian Geometry....... 233 9.1

9.2

Introduction ........................................... 233

......................... 234 ....................................... 234

Generalities About SR Geometry Definition Optimal Control Theory Formulation

9.2.1 9.2.2

............... 234

XIV

Table of Contents

9.2.3 Computations of the Extremals and Exponential mapping

............................................ 235 ....................... 239 .................................... 239 .......................... 239

Research Program in SR-Geometry 9.3.1 Classification 9.3.2 Singularity Theory of the Exponential Mapping and of the Distance Function 9.4 Privileged Coordinates and Graded Normal Forms .......... 240 9.4.1 Regular and Singular Points 9.4.2 Adapted and Privileged Coordinates 241 9.4.3 Nilpotent Approximation 9.4.4 Graded Approximation 9.3

....................... 240 ................ .......................... 242 ............................ 242

9.5 The Contact Case of Order -1 or the Heisenberg-Brockett Example .............................................. 242

........................... 242 ............ 243 ...... 243

9.5.1 9.5.2 9.5.3

The Contact Case in R3 Symmetry Group in the Heisenberg Case Heisenberg SR-Geometry and the Dido Problem

9.5.4

Geodesics

....................................... 244

................................. 245 ........................... 245 ...... 246 The Generic Contact Case ............................... 247 9.5.5 Conjugate Points 9.5.6 Sphere and Wave Front 9.5.7 Conclusion About the SR-Heisenberg Geometry

9.6

................. 247 .......................... 248 ..................................... 249 .................................... 249 .................................... 250 .............................. 253 ............................. 253

9.6.1 Normal Forms in the Contact Case 9.6.2 Generic Conjugate Locus 9.7 The Martinet Case 9.7.1 Preliminaries 9.7.2 Normal Form 9.7.3 Orthonormal Frame 9.7.4 Graded Normal Form 9.7.5

Geodesics ....................................... 254

.............. 255

9.7.6 Riemannian Metric on the Plane (x, y) 9.7.7 Asymptotic Foliation Associated to the Normal Form

at Order 0 ....................................... 256

............. 257

9.7.8 Properties of the Asymptotic Foliations 9.7.9 Integrable Case of Order 0 and Elliptic Integrals

..... 258 9.7.10 The Martinet Flat Case ........................... 262

Estimates of the Sphere and of the Wave Front near the Abnormal Direction in the Flat Case and exp -In Category .... 266 9.8.1 Intersection of S(0, r) with the Cut Locus ........... 266 9.9 Conclusion Deduced from the Martinet SR Flat Geometry Concerning the role of Abnormal Geodesics in SR-Geometry . 268 9.9.1 Behaviors of the Normal Geodesics near the Abnormal Direction - Geodesic C°-Rigidity 268 9.9.2 Nonproperness of the First Return Mapping R1 and Geometric Consequence 9.8

...................

........................... 269

Table of Contents

XV

9.10 Cut Locus in Martinet SR-Geometry ...................... 271 Exercises .................................................. 273

10 Micro-Local Resolution of the Singularity near a Singular Trajectory - Lagrangian Manifolds and Symplectic Stratifi-

cations ................................................... 281 10.1 Introduction ........................................... 281

10.2 Lagrangian Manifolds ................................... 281 10.3 Application to Classical Calculus of Variations ............. 282 10.4 Singularity Theory of the Generating Function - Generating

Family ................................................ 283

10.5 Application to Optimal Control Theory and to SR-Geometry 285

10.5.1 The SR-Normal Case ............................. 285

10.5.2 Jacobi Fields .................................... 287

10.5.3 Reduction Algorithm in the SR-Case ................ 288

10.6 The Singular Case ...................................... 289

10.6.1 A Geometric Remark ............................. 289 10.6.2 Heisenberg Case .................................. 289 10.6.3 Martinet Flat Case ............................... 289

10.7 Resolution of the Singularity in the Hyperbolic Case ........ 292

10.7.1 Normal Form .................................... 292

10.7.2 Intrinsic Second-Order Derivative ................... 293

10.7.3 Goh Transformation .............................. 294 10.8 Lagrangian Manifolds in the Hyperbolic Case .............. 294

10.9 Examples .............................................. 296 10.10 Resolution of the Singularity in the Exceptional Case ....... 302

10.10.1 Normal Form - Intrinsic Derivative ................. 302

10.10.2 The Operators .................................. 303 10.10.3 Isotropic Manifolds in the Exceptional Case ......... 303 10.11 Micro-Local Analysis of SR Martinet Ball of Small Radius -

Lagrangian Stratification of the Martinet Sector ............ 306

10.11.1 Preliminaries .................................... 306 10.11.2 The Smooth Abnormal Sector ..................... 307 10.11.3 The Smoothness of the Sphere in the Abnormal Di-

rection .......................................... 307

10.11.4 The Martinet Sector ............................. 309 10.11.5 Symplectic Stratifications ......................... 312 10.11.6 Transcendence of the Sector ....................... 312

10.11.7 The Micro-Local Analysis of the Whole Martinet Sphere313 Exercises .................................................. 314

11 Numerical Computations ................................. 321 11.1 Introduction ........................................... 321 11.2 Numerical Algorithm ................................... 321

11.3 Applications ........................................... 324

XVI

Table of Contents

11.3.1 Contact Case in Dimension 3 ...................... 324

11.3.2 The Martinet Flat Case ........................... 324

11.4 The Tangential Case .................................... 328 11.4.1 The Elliptic Case ................................. 329 11.4.2 The Hyperbolic Case ............................. 330 12 Conclusion and Perspectives .............................. 335 12.1 The Category of the Distance Function in SR Geometry ..... 335 12.2 SR-Distances and Control of the Oscillations of Solutions of

Ordinary Differential Equations .......................... 335 12.3 Optimal Control Problems with Bounded State Variables .... 338

12.3.1 Maximum Principle with State Constraints .......... 338 12.4 Singular Trajectories and Regularity of Stabilizing Feedback . 339

12.5 Computations of Singular Trajectories .................... 340 12.6 Computations of Conjugate Points ........................ 340

Exercises .................................................. 341 References .................................................... 343 Index ......................................................... 349

1 ' Linear Systems and the Time Optimal Control Problem

1.1 Linear Systems In this section we consider the class of linear systems described by an equation of the form:

x(t) = A(t)x(t) + B(t)u(t)

(1.1)

where x(t) E RI and: - A(t) is a real n x n matrix, B(t) is a real n x m matrix and each components of A(t) and B(t) are continuous on the line 0 < t < +oo; - an admissible control is a function u(.) E U, where U is the set of bounded measurable mapping defined on some interval [to, ti 1, 0 < to < ti and with values in a fixed subset 1? C Rm. The set (1 is called the control domain. A trajectory is defined as an absolutely continuous solution x(.) of the system (1.1). In particular we shall denote by x(., to, xo, u) the trajectory issued from x(to) = xo and associated to a given control u(.) E U. If A and B are constant

matrices the system is said to be autonomous. Lemma 1. The solution of the system (1.1) corresponding to u(.) E U and issued from x(to) = xo is given by x(t, to, xo, u) = 0(t) (so +

t

it,

-1(s)B(s)u(s)ds)

where 0(.) is the fundamental matrix solution of the differential equation fi(t) = A(t)45(t) with 0(to) = Identity. Proof. See [34] for instance.

1.2 Accessibility Set and Controllability Definition 1. If we fix the initial state to be xo at to, the accessibility set at time t1 > to denoted by A(xo, ti) is the set of all points x(ti, xo, to, u) to which xo can be steered by an admissible control. The accessibility set is A(xo) = Ut,>t0A(xo,t1)

2

1 Linear Systems

Definition 2. The system is said to be controllable at time t1 if for each xo E 1Rn, the accessibility set is the whole space: A(xo, t1) =1Rn. The system is said to be controllable if for each xo E 1R", we have A(xo) = R"

1.3 Controllability and Feedback Classification in the Autonomous Case See [77], [1091.

In this section, we consider linear autonomous systems:

i(t) = Ax(t) + Bu(t)

(1.2)

where A, B are real constant matrices. We assume the control domain I? to be Rm and U = L°°([0, +oo[).

1.3.1 Controllability Proposition 1. The system (1.2) is controllable if and only if the n x nm matrix R = [B, AB, . , An-1 B1 is of rank n. Proof. Let T > 0 and consider the mapping ET : L°O((0, 71) - IRn u(.) fo exp(-sA)Bu(s)ds

-

where exp(tA) is the exponential matrix E o The set L°°([0,T]) is a linear space and ET is a linear mapping. Hence the image FT = { fo exp(-sA)Bu(s)ds; u(.) E LOO([0,T])} is a linear space and when T increases, FT increases. We conclude that F = UT>OFT is a linear space. Assume first that the system (1.2) is controllable and that we have rank[B, , An-IB] < n. We will reach a contradiction. As rank(R) < n, there exists a nonzero (row) vector p such that pv = 0 for each column v E R. Using Cayley-Hamilton theorem (any square matrix satisfies its own characteristic equation), there exist some constants ai E hR such that we have n-1

An = > aiA'. i=o

Therefore, since exp(tA) = Ek o

"A' we obtain

pexp(tA)B = 0

Vt > 0.

Let x(., u) be the trajectory associated to u(.) and issued from the origin: x(0) = 0. Hence

1.3 Controllability and Feedback Classification in the Autonomous Case

3

x(t, u) = / t exp((t - s)A)Bu(s)ds. p

This implies

p x(t, u) =0

`dt > 0

and therefore x(t, u) belongs to the hyperplane p- L. This is a contradiction with the controllability assumption. Conversely assume now that rank(R) = n. We must show that the system (1.2) is controllable. Let T > 0. As previously, the image FT of L°° ([O, T)) by the map ET is a linear space. If dimFT < n, there exists a row vector p E R"\{0} such that: T

1

pexp(-sA)Bu(s)ds = 0

Vu(.) E LO°([O,T]).

As a consequence, the following equation holds for almost every s E [0, TI:

pexp(-sA)B = 0. Since the mapping s H pexp(-sA)B is continuous, we can deduce that the previous equation is satisfied for all s E [0, T[. By successive differentiation

and using that the mapping s '--+ pexp(-sA)B is in fact analytic, this last equation is shown to be equivalent to pAkB = 0 for all k > 0. Hence

pv=0

Vv ER.

This contradicts the assumption rank(R) = n. Therefore for each T > 0, we have FT = R". Since x(T, 0, xo, u) = exp(tA) (0 + fTexp(_sA)Bu(s)ds)

U

we deduce that

x(T, 0, xo, u) = R", for all T > 0 and then the

u(.)ELOO((O,T])

system is controllable.

Corollary 1. The following assertions are equivalent:

1. rank(R) = n. 2. VT > 0, the system is controllable at time T. S. The system is controllable.

1.3.2 Linear Classification Let n, m E N be fixed and let Ln,,,,, = {(A, B)} the set of linear autonomous

systems k(t) = Ax(t) + Bu(t) where x(t) E R", u(t) E R'". We denote by ,Cn,m the subset of £n,," of controllable systems.

Definition 3. Two systems (A, B) and (A', B') are said to be linearly equivalent if there exists P E GL(n, R) such that

A' = P-'AP,

B' = P-1B.

4

1 Linear Systems

Geometric interpretation. Consider the linear autonomous system i(t) = Ax(t) + Bu(t) and introduce the following change of coordinates: x = Py where P E GL(n, R). Then y(.) = P-'x(.) is solution of the following linear autonomous system:

y(t) = (P-'AP)y(t) + (P-'B)u(t). Remark 1. For linearly equivalent systems, the controllability is an invariant

property. Indeed, we have R' = [B', . , A'"-'B') = P-' [B, , A"-'BI = P-'R. As P is an invertible matrix, it implies rank(R) =rank(R'). Definition 4. A linear autonomous single-input system is a system of the form (1.3) i(t) = Ax(t) + u(t)b, b E R", U(t) E R. Proposition 2. The linear autonomous single-input system defined by the xi")(t)+alxln-1i(t)+.. +afzxl(t) = u(t), where xlkl (t) = dk, (t), is equation: controllable. Moreover each linear autonomous single-input controllable system i(t) = Ax(t) + u(t)b is linearly equivalent to such a system.

Proof. The equation x(in) (t) + ala(I"-1)(t) + as a system (A1ib1) of 4, 1 by setting

+ anx1(t) = u(t) can written dxl

(t) = x2(t)

(1.4)

dx dt 1(t) = xn(t)

(1.6)

(t) = x i"1 (t) = -a"x1(t) - ... - a1xn(t) + u(t).

(1.7)

dt

dX" dt

We have 0

1

0

0

0

0

1

0

A1=

0 0

b1= 0

-an -an-1 ...

0

0

1

-a2 -al

0 1

A straightforward computation shows that (A1, b1) is controllable. Conversely, let (A, b) be a linear autonomous single-input controllable system. It , vn-1 = follows that b is a cyclic vector for A and if we denote vi = A"-' b, , vn} forms a basis of 1R". By conAb, vn = b, hence the set of vectors {v1i struction, in this basis the linear mapping defined by A is represented as

1.3 Controllability and Feedback Classification in the Autonomous Case

5

al

a2

A' _ an-1 an

1

00...0

where the coefficients as are given by

A n b = aiAn-lb +...+ anb and the vector b' is represented as 0

0

V

Applying to the matrix Al the Cayley-Hamilton theorem, we get

Al =

-a1Ai-1

- ... - anl.

Hence using the previous change of coordinates, we can give to (Al, bl) the canonical form (A', b') in which a; = -at. The proposition is then proved. Remark 2. In the previous normal form, the coefficients al, an are a complete set of invariants, i.e. two single-input controllable control systems are linearly equivalent if their sequence al, , an is similar.

1.3.3 Feedback Classification Definition 5. A linear feedback is a transformation of the form u = Fx+Gv where F and G are respectively a m x n matrix and a m x m invertible matrix. It acts on Gn,,n as follows: (A, B) - (A+BF, BG). The linear feedback group is the group induced on the set (P, F, G) E GL(n, R) x M(n, R) x GL(m, R) by the transformations: 1. Linear change of coordinates: y = Px. 2. Feedback transformations: u = Fx + Gv. Proposition S. Let any (A, b) E 4n,, single-input linear controllable system. Then (A, b) is feedback equivalent to the equation xln)(t) = u(t). Proof. From Proposition 2, a linear single-input controllable system (A, b) is linearly equivalent to an equation x(n)(t)+alx(n-i)(t)+ +anxi(t) = u(t).

Hence, using the feedback u = a,xli-1)+ +anxl +v, the previous equation is shown to be feedback equivalent to the equation x(n)(t) = v(t). 0

1 Linear Systems

6

1.3.4 Stabilization Definition 6. Let i(t) = Ax(t) be a differential equation where A is a constant n x n matrix. We denote by a(A) = (v1 i , on) the spectrum of A, i.e. the set of eigenvalues of A over C. We say that A is stable if o(A) C {z E C; 9e(z) < 0}. Let us now consider the following linear autonomous system: i(t) = Ax(t) + Bu(t), x(t) E R", u(t) E R"`. The System (A, B) is said to be stabilizable if there exists a m x n matrix D such that (A + BD) is stable. Theorem 1. Let (A, B) be a controllable linear autonomous system. Then (A, B) is stabilizable.

We are going to prove that the system (A, B) E Gn,,n is feedback equivalent to a stable control system. But first let us prove the following two lemmas.

Lemma 2. Let (A, b) be a single-input controllable linear system. Then it is stabilizable.

Proof. By Proposition 3, the system (A, b) can be assumed to be of the form 0 1

0

00

1

0 0

0

...

0

A=

b=

0...

.

0 01

0

0 0

1

0 0

, d,) be any vector of R". Hence the characteristic polynomial Let D = (d1i of A + bD is by definition:

P(A) = An - d1 Hence by choosing (d1i

,

A"-1

- ... - dn.

dn) in a suitable way, we can assign any prescribed

roots ( A I ,--', An) to P(A) and the lemma is proved. 0 Lemma 3. Let (A, B) E Gn,,n be a controllable system and b E ImB, b 34 0. Then there exists a n x m matrix F such that (A + BF, b) is controllable. Proof. Since (A, B) is controllable, the linear space generated by the columns , v" with the following , An-1B] is Rn. Let us construct a basis v1, properties: of [B,

1. v1 = b E ImB;

2. v;-Av;_1 EImB

fori=2,

n.

We proceed as follows. We set b1 = b and let El = Span{b1, dimE1 = n1. We define v1 = b and

v? =Avv_1+b1,

j=2,...,n1.

,A"-lbl},

1.4 Controllability in the Nonautonomous Case

7

If n1 = n, the result is proved. Otherwise, there exists b2 E ImB such that b2 0 El. Let E2 =Span{b2, - , An-'b2} + El, dimE2 = n1 + n2. We set vn,+i = Avn,+i-i + b2,

.7 = 1,..-,n2-

If n1 + n2 = n, the construction is ends. If it is not the case, we proceed by reccurence. Using this construction, we produce a basis {v1, , vn } satisfying the properties 1 and 2 above. Therefore we can choose u; E 1R"' such that

vi = Avi-1 + Bui-1,

i = 2, ... , n.

We then define a linear mapping F : Itn - lR' by setting

Fvi=ui

for

and Fvn is chosen arbitrarily. By construction, we have v1 = b and (A + , n - 1 and the lemma is proved. BF)vi = vi+1 f o r i = 2, Proof (Theorem 1). Let b be any nonzero vector that belongs to ImB and let F be chosen as in Lemma 3. Then (A, B) is feedback equivalent to (A + BF, B). Let v E 1R'" such that b = By and restrict the controls to the singleinput controls of the form u(t) = A(t)v, where A(.) is any measurable bounded function. For this class of controls, the system (A + BF, B) takes the form

i(t) = (A + BF)x(t) + a(t)b. By Lemma 3, this single-input system is controllable and using Lemma 2 it is stabilizable. The theorem is then proved.

1.4 Controllability in the Nonautonomous Case See [34].

Lemma 4. Let T > 0, G(t) be a n x m matrix with t E [0, T] and t - G(t) be a continuous map. If we set L(u) = fo G(t)u(t)dt, u E L°0([0,T]) and W = fT G(t)tG(t)dt, then ImL = ImW. Proof. Let x E ImW, then 3p E lRn such that Wp = x, i.e T

J0

G(t)'G(t)p dt = x.

Let u(.) be the control defined by u(t) = tG(t)p. Then L(u) = x. Hence ImW C ImL.

1 Linear Systems

8

To show that ImL C ImW we observe that W is symmetric and hence

KerW = ImW1,

R" = ImW ® KerW.

Hence if x1 f ImW, 1x2 E KerW such that (x1, x2) # 0. Assume x1 E ImL. Then there exists u(.) such that fT

G(t)u(t)dt = x1.

T

Therefore

`x2G(t)u(t)dt = (x1i x2) 76 0.

0

Since Wx2 = 0, we have tx2Wx2 = 0 and tx2Wx2 =

r

T

tx2G(t)tG(t)x2dt =

rT

J

( tG(t)x2, tG(t)x2)dt = 0.

Since t s-' G(t) is continuous we get

tG(t)x2 = 0,

Vt > 0.

Hence we have a contradiction since it implies

fT tx2G(t)u(t)dt = 0. 0

0

Corollary 2. Consider the linear system x(t) = G(t)u(t) where G(t) is as above. There exists a control steering x(0) = xo to x(T) = x1 if and only if x1 - xo E ImW where W = fT G(t)tG(t)dt. Such a control is given by u(t) = tG(t)p where p is solution of Wp = x1 - xo. Proof. We have x(T) = xo + fa G(t)u(t)dt and x(T) = x1. Hence x1 - xo E Im W = ImL, and we apply the previous construction.

1.4.1 Application Consider the system i(t) = A(t)x(t) + B(t)u(t). Let fi(t) be the fundamental

matrix solution of fi(t) = A(t)fi(t), 0(0) = Identity. By setting x(t) _ 0(t)z(t) we get

z(t) = 0-1(t)B(t)u(t). Moreover, we know by Lemma 1 that

x(t) = 415(t) (xo +t f1(s)B(s)u(s)ds).

1.5 Time Optimal Control for Linear Systems

9

The steering control is determined by corollary 2 as follows. If x(T) = x1, in the z-state the problem of steering ±o =:zo to x1 can be written

z(t) = G(t)u(t),

G(t) _ --1(t)B(t)

z(0) = xo,

z(T) V 1(T)xl.

Proposition 4. The linear system x(t) = A(t)x(t) + B(t)u(t) is controllable on [0,T] if and only if for G(t) = --1(t)B(t) one of the previous conditions is satisfied:

1. ImL = lR";

2. ImW = R"; 3. there exists no (row) nonzero vector p such that p -0-1(t)B(t) = 0,

Vt E [0, T].

1.5 Time Optimal Control for Linear Systems 1.5.1 Notations In this section we consider a linear system

±(t) = A(t)x(t) + B(t)u(t),

x(t) E R",

(1.8)

where t .- A(t), B(t) are assumed to be continuous on an interval 0 < t < T. The control domain 12 is taken as a compact subset of R"' and the set of admissible control U is the set of bounded measurable functions defined on

[0, T] with values in 0. Let to E 10, T] and let us fix the initial state by x(to) = xo. From Lemma 1 the solution of (1.8) is given by

x(t, to, xo, u) = 0(t) (xo + Jt -1(s)B(s)u(s)ds) o

where fi(t) = A(t)46(t), #(to) = Identity. The accessibility set at time t is denoted by An(t). The set of compact subsets is endowed with the Hausdorff metric defined by d(A, B) = max d(a, B) + max d(b, A). aEA

bEB

If 12 is a subset of R1, con(17) denotes the convex hull of D.

Proposition 5 ([77]). Assume that the control domain 12 is a compact, convex subset of Rm. Then the accessibility set An(t) for to < t < T is convex, compact and t i-- An(t) is a continuous mapping.

10

1 Linear Systems

Proof. By definition, we have An (t) = U x(t, to, xo, u) where u(.)En

rt

x(t, to, xo, u) = fi(t) (xo +

-1(s)B(s)u(s)ds)

.

to

Since f? is convex and the mapping u(.) i-+ x(t, to, xo, u) is linear, the accessibility set An(t) is convex. To prove that An(t) is compact we use the following standard argument. Let I . I be a norm on R. The set of admissible controls u(.) : [to, t] -+ 17 is a subset of the Banach space L°°([to, t]) endowed with the norm

hull = sup I u(s) 1, sE )to,t)

where L°°([to,t]) is the dual of the separable set E = L1([to,t]). Since 12 C Rm is bounded, there exists M > 0 such that the set of admissible controls is a subset of the ball BM = {u(.) E L°°([to, t]); Dull < M}. From the BanachAlaoglu-Bourbaki theorem (see [31] for instance), the set BM is compact for the weak'-topology o(E', E) where E' = L°°([to, t]). Moreover BM is metrizable for this topology. Therefore for any sequence un(.) E U, there exists a subsequence u,,,+(.) converging *-weakly to u(.) E E', that is rc

k

lim +O°tJto

rt

dt = j Hudt o

for any integrable p x m matrix H, on [to, t]. We know that u(.) E BM and we must prove that u(.) E 1? almost everywhere. Since ( is a convex compact subset of R"', it is the intersection of a countable number of closed half-spaces. Let 17 be such a space defined by: H = {v E Rm; av + b < 0).

Let J be the subset of r E [to, t] such that au(r) + b > 0. Using the weak convergence we have urn

b)dr =

Jtot X,

Ito

X, (r)(au(r) + b)dr

where X, is +1 on J and 0 on [to, t]\J. By construction, the following inequality holds: L

J to

x, (t)(aun, (t) + b)dt < 0.

If J has positive measure, we would have that L

JLp

X, (t)(au(t) + b)dt > 0.

It contradicts the previous inequality. Hence J has zero measure and u(r) lies on the appropriate side of 17 a.e. Since fl = nnENI7n, u(r) E 92 a.e. Hence

1.5 Time Optimal Control for Linear Systems

11

the accessibility set A(xo) is closed. It is clearly bounded. Therefore it is a compact subset of R'. To prove the continuity of the mapping t'-- A(xo, t) when the set of compact subsets is endowed with the Hausdorff topology is an easy exercise left to the reader.

Proposition 6 (Bang-bang principle [55]). Let .Ro be a subset of Rm such that eon(Qo) = can(Q). Then Ago(t1) = An(ti). Proof. This result is a consequence of a theorem due to Liapunov that we state as a lemma. The proof indicates the way of proving the general theorem. 0

Lemma 5. Let y(.) be an element of L' ([tc, t]), 'P be the set: {u(.) E LO0([to, t]); 0 < u(s) < 1) and 'Po be the subset {XE; E is a measurable subset of [to, t] }. If we denote

M = {J c y(s)u(s)ds; u E W}, to

/t

Mo = { J y(s)uo(s)ds; uo E'Po}

c

then M0=M. Proof. Let T : L°° ([to, ti]) -' R be the linear mapping defined by: u(.) fto y(s)u(s)ds. As T is a continuous map for the norm topology then T is continuous for the weak'-topology. Clearly we have that Mo C M, we must prove that M C Mo. Let x E M, then T-1(x) is a closed convex subset of the weak'-compact unit ball of L00 ([to, t i ]) and hence T -I(x) is compact. Using the Krein-Milman theorem, T-i(x) has extreme points, and we shall show than an extreme point u(.) of T-i(x) is of the form XE for some measurable subset E of [to, t]. Let u(.) be an extreme point of T - i (x) and suppose that there is an e > 0 and a subset E of [to, ti] with positive measure such that

e < u(t) < 1 - e on E. There exist El and E2, both with positive measure such that E = El UE2 and El nE2 = 0. Clearly there exist a,,8 E R satisfying ]< z and such that if we define h = aXE, + 0XE2 the (o,#) # (0, 0), 1 a following equality holds: y(s)ds +,3

1E y(s)h(s)d,9 = a

JE1

y(a)ds = 0. fZ

Hence h(.) is non zero, I h(t) IS a and we have t y(s)u(s)ds = x.

y(s)(u(s) ± h(s))ds

to

t0t

Therefore both functions u(.) ± h(.) have values between 0 and 1 and u(.) is the middle point of u(.) ± h(.) E T-i(x) proving that u(.) is not an extreme

12

1 Linear Systems

point. We have shown that u(.) has the only values 0 and 1 a.e. in [to, t] and that u(.) is the characteristic function of a set E. This proves the assertion.

0 Corollary 3. An(t) = Aen(t). Corollary 4. The accessibility set An(t) is compact, convex and varies continuously with t. Definition 7. Let x(.) be the corresponding trajectory to an admissible control u(.), both defined on the time interval [to, ti]. If x(t1) E 8An(ti), then u(.) is called an extremal control and x(.) an extremal trajectory.

Theorem 2 (Maximum principle (PMP) for linear time optimal problems [771). Assume the domain of control .0 to be a compact, convex subset of Rm. An admissible control u(.) and its corresponding trajectory x(.) both defined on [to, t1] are extremal if and only if there exists a nonzero absolutely continuous (row) vector p(.) solution of the adjoint equation

p(t) = -p(t)A(t)

a.e. on [to, t1]

(1.9)

such that

p(t)B(t)u(t) = m p(t)B(t)u.

n

(1.10)

Such vector is called an adjoint vector.

Proof. Assume x(.) to be the extremal trajectory corresponding to the extremal control u(.), both defined on [to,t1]. By definition, we have x(ti) E 8An(tl), where x(t) is given by

x(t) = 0(t) (xo +

P- 1(s)B(s)u(s)ds)

Jt

.

co

From Corollary 4, the accessibility set An(ti) is compact and convex and since x(t1) E 8An(t1), there exists a support hyperplane If to An(tl) at x(ti). Let p be a nonzero normal row vector to fI at x(ti), outward with respect to An(ti). Let p(t) be defined for to < t < ti by p(t) = pofi-i(t), p(ti) = p. We have p(t)x(t) = poxo +

J

t p(s)B(s)u(s)ds.

(1.11)

to

Let us assume that there exists a subset of [to, ti] of nonzero measure such that for all t in this subset we have p(t)B(t)u(t) < manxp(t)B(t)u.

1.5 Time Optimal Control for Linear Systems

13

Using Filippov selection Theorem [77] we can define a measurable control u(.) satisfying a.e. on to < t < t1: p(t)B(t)u(t) = manx p(t)B(t)u. uc-

Let i(.) be the trajectory associated to u(.). We have c,

p(ti)x(ti) = poxo + J p(s)B(s)u(s)ds. to

Moreover, by construction of u(.) and from (1.11) the following inequality holds:

r t p(s)B(s)u(s)ds < Jto

fit p(s)B(s)u(s)ds. to

Hence, we deduce that

p(ti)x(ti) < p(ti)x(ti) This contradicts the fact that x(ti) E 0Aa(tl) and that p(ti) is the outward normal to 17 at x(tl ). Therefore we must have a.e.

p(t)B(t)u(t) = maxp(t)B(t)u. p(t)B(t)u. Conversely, if u(.) satisfies a.e. the equality p(t)B(t)u(t) = mEanxp(t)B(t)u(t), u

we show that x(t1) E OAn(tl). Indeed assume that x(ti) E IntAn(tl). Therefore there exists xl E An(tl) such that

p(t1)x(t1) < p(tl)i1.

(1.12)

Let u(.) be a control defined on [to, t1] steering xo to i1 and I(.) the corresponding trajectory. It follows that

p(t)B(t)u(t) < p(t)B(t)u(t) a.e. Hence, by computing we get

p(tl)x(tl) = p(tl)xl <_ p(t1)x(t1) which contradicts the inequality (1.12). 0

Proposition 7. Consider the problem of reaching in minimum time a continuously varying compact target C(t), 0 < t < +oo. If there exists an admissible control steering xo to C(ti), then there exists a minimal time optimal control u*(.), to < t < t` < ti steering xo to C(t*).

14

1 Linear Systems

Proof. We are proving the result when the target is a continuous curve t '-+ c(t), the generalization to a continuously compact varying target is straightforward. By assymption, there exists a time ti such that to < ti

and An(tl) n c(tl) 0 0. Let t' = inf{t; c(t) E An(t)}. Then there exist a decreasing sequence tn, limn.+ooto = t' and a sequence un(.) of controls defined on (to, tn] such that the corresponding trajectories xn(.) satisfy x(tn, un) E An(tn) n c(tn). Moreover,

I C(t') - x(t*,un) I<-I C(t*) -C(tn) I + I x(tn,un) -x(t*aun) I

and by continuity of c(.) and xn(.) the right-hand side - 0 when n -- +oo. Hence c(t') = limn+o x(t*, un). Since x(t*, un) E An(t') for each n and since from Proposition 5 the accessibility set An(t*) is closed, then c(t*) E AQ(t') and the assertion is proved. 0

Theorem 3 ((77]). We consider the autonomous linear system i(t) _ Ax(t) + Bu(t), x(t) E Rn, U(t) E (1 where:

1. A, B are constant matrices; 2. the control domain 12 is a convex polyhedron and for all nonzero vector

w parallel to an edge of Q, the vectors Bw,

,

AnBw are linearly

independant.

Then if x1 is accessible to xo, there exists an unique optimal control (up to a set of zero measure) steering xo to x1. Moreover such a control is extremal and is equal almost everywhere to a control taking its values on the vertices of fi and with a finite number of discontinuities.

Proof. According to Proposition 7, there exists an optimal control u'(.) defined on 10, t'J steering xo to x1. Let x'(.) be the corresponding trajectory. First observe that P = x* (t*) E 8An (t' ). Indeed, let us assume P E IntAn(t*). Since t ' - An(t) varies continuously, there exist a neighborhood V of P and 6 > 0 such that V C IntAn(t) for I t - t' I< 6. Hence there exists a control such that its corresponding trajectory steers the system from xo to P in time t < t*. This contradicts the optimality of u'(.). Since u*(t), 0 < t < t* is extremal, from Theorem 2 there exists a nonzero vector p'(.) where p' = p' (t') is the outward normal to An(t*) at x*(t*) and such that the maximum principle holds:

P* (t) = -p'(t)A p*(t)Bu*(t)

n p'(t)Bu a.e.

m

We must show that u'(.) is uniquely defined by the maximum principle up to a set of measure zero. Indeed assume there exist u1(), u2(.) such that

p'(t)Bu1(t) = p'(t)Bu2(t)

(1.13)

1.6 Feedback Equivalence to a Linear System

15

and ul(t) # u2(t) on a subset S C [0, t'] of nonzero measure. Consider the linear mapping Ft(u) = p*(t)Bu. Since 1? is a polyhedron Ft takes its maximum on the sides of 1? which are:

if dim(1= 1: vertices, if dims? = 2: edges, etc ...

.

Hence for t E S, Ft takes its maximum at two different points of .R and therefore on an edge. Since .fl has only a finite number of edges, there exists a subset S1 C S of nonzero measure such that Ft on S1 reaches its maximum on a given edge denoted w1. Let w be a nonzero vector along w1. From (1.13), we have p'(t)Bw(t) = 0. Moreover, we have P* (t) = -p'(t)A and since A is constant it implies: p* (t) = po exp(-tA). Therefore

poexp(-tA)Bw = 0

for t E S1.

Since t i--' po exp(-tA)Bw is analytic and vanishes on S1 and since Si is of nonzero measure, we must have

po exp(-tA)Bw = 0,

Vt.

Differentiating and evaluating at t = 0 we get rank[Bw, ABw,

,

A"`-'Bw] < n.

This is a contradiction with the assumption that the vectors Bw, , An-1 Bw are linearly independent. Similarly we can show that Ft reaches its maximum

on an unique vertice of 0, except for a finite number of t E [0, t'] proving that u'(.) is (up to a set of zero measure) piecewise constant with values on the vertices of 51.

0

1.6 Feedback Equivalence to a Linear System See [33].

1.8.1 Problem Statement In this section we consider a single-input a line control system

i(t) = X(x(t)) + u(t)Y(x(t))

(1.14)

where X, Y are two germs at 0 of analytic vector fields of R". We assume that X(0) = 0. In the sequel, the set of such systems (X, Y) will be denoted by A. For (X, Y) E A we allow two types of transformations: 1. local change of coordinates in the state space: y = cp(x), p(0) = 0 where W is a germ at 0 of an analytic diffeomorphism;

16

1 Linear Systems

2. local feedback of the form: u = a(x) + v, with a(O) = 0 where a is a germ at 0 of an analytic function. The respective actions are the following: Z o cp-1 is the image of Z by V

1. (X, Y) '-+ (cp * X,

2. (X, Y) " (X +Yo,Y) We denote by G the group induced on the pairs (gyp, a) by the previous transformations. G is called the feedback group. The aim of this section is to characterize the set of systems in A which are feedback equivalent to a single-input linear system: i(t) = A(t)x(t)+u(t)b(t).

Proposition 8. Let (X, Y) E A. Then (X, Y) is feedback equivalent to a linear controllable system if and only if the following conditions are satisfied

1. Span {adkX (Y)(0); k = 0, , n - 1} =Rn; 2. Vk, m E N, k, m E [0, n - 1) there exist germs at 0 of analytic functions d; such that: max(k,rn)

di (x)ad-1 X (Y)(x)

[adkX (Y), admX (Y)J(x) _ i=1

where adX is the endomorphism defined by adX (Y) = [X, Y), [X, Y](x) Y(x) - TX- X (x). being the Lie bracket Proof. Using for X, Y : Rn

R' and f, g : Rn -i R the identity

[fX, 9Y] = f9[X,Y] - f (Lxg)Y + g(Lyf)X where Lzh = DhZ is the Lie derivative, we check easily that the conditions 1 and 2 of Proposition 8 are feedback invariants. Moreover, if X (x) = Ax and Y(x) = b, the following two equalities hold: adkX (Y) = Akb, [adkX (Y), ad-X (Y)] = 0.

Hence conditions 1 and 2 of the proposition are necessary. Conversely, if these two conditions are satisfied we show that the system is feedback equivalent to the equation: x(n) (t) = u(t). We proceed as follows. Since from condition

1 we have Y(0) # 0, there exists a coordinate system such that Y = 8 and using a proper feedback, the system can be written as:

il(t) = Xi(x(t)) (1.15)

xn-1(t) = Xn-1(x(t))

in(t) = u(t) If n = 1, the reduction is achieved. Otherwise, using the condition

1.7 Time Minimal Synthesis

17

[adX (Y), Y] = d1Y with Y =TX-; 0 ,

we get that 88 = 0 for i = 1, , n - 1. Hence by analyticity X can be written as Xo + xnX where X0 and X depend only on (xl, x2, , xn_l ). From condition 1, X(0) 96 0. Hence there exists a coordinate system on the space (x1, , xn_ 1) such that X = a and the system can be assumed to be under the following form: Xl(xl,'-',xn-1)

i1 =

(1.16) Xn-1(x1,...,xn-1)+xn

in-1 =

xn=u Replacing xn by xn = xn + Xn_1, it can be written as

xl =

X1(xl,...txn-1) (1.17)

xn-1 = xn n-1

r` 8Xn-1

8x-lx;

i=1

where in = u. Hence, using a feedback and the notation xn for xn we obtain a system of the form

xl =

X1(xle...,xn-1)

xn-2 = Xn-2(x1+"',xn-1)

(1.18)

xn-1 = xn

xn=U If n = 2, the result is proved. Otherwise we repeat the previous algorithm and the proposition is proved by recurrence.

1.7 Time Minimal Synthesis 1.7.1 Notion of Synthesis In this section we consider a time minimal problem for a linear autonomous system:

i(t) = Ax(t) + Bu(t),

X(t) E 1R", U(t) E .f1

18

1 Linear Systems

where Q is a convex polyhedron and assume the target to be the point 0. When dealing with applications, we are concerned with stability properties, hence the optimal control is not computed as an open loop function u' : [0, t'J --' .R but as a closed loop function u` : x -+ Q, that is we compute the optimal control corresponding to each initial condition. Such a feedback u'(.) is called a synthesis. To compute a synthesis we proceed as follows. We integrate the system

P(t) = -p(t)A

i(t) = Ax(t) + Bu(t),

backwards in time from the terminal condition x(0) = 0 and for each p(O) E S"-1 (sphere in R" with radius 1), where at each time the optimal control u'(.) is computed according to the maximum principle using p(t)Bu'(t) = manxp(t)Bu. If the problem satisfies the assumption of Theorem 3, for each x E R" that can be steered to the target point 0 this algorithm computes backwards in time the optimal trajectory x' : [0, t'] --+ R" corresponding to x*(t*) = x. We shall explain this computation using two examples before we present the general algorithm. Example 1. We consider the single-input systems on R2 given by i(t) = u(t), where u(t) E R and we assume the control domain !l to be defined by I u(t) I< 1. We introduce the system:

i(t) = y(t) (1.19)

W) = u(t),

I u(t) 1< 1

10

and the problem is to steer any point (x(0), y(0)) = (xo, yo) E R2 to the target (0, 0) in minimum time. Here A = ( 0

) and B =

()

.

The adjoint

0

vector is written as p = (pl,p2). According to the maximum principle we must integrate the following differential equations:

i(t) = y(t), W) = u(t),

P1(t) = 0

(1.20)

P2 (t) = -pi(t)

(1.21)

where an optimal control u'(.) must satisfy p2(t)u'(t) = maax p2(t)u.

Therefore u'(t) = sign(p2(t)) almost everywhere and p2(t) = -p°t+p2 where (p°, p2) = p(0). It implies that the optimal control u'(.) is almost everywhere a piecewise constant mapping, taking its values into the set {-1,+1} and

1.7 Time Minimal Synthesis

19

with at most one switching since p2(t) has at most one root. This means that to compute the optimal synthesis we have to consider concatenations of at most two trajectories corresponding to u =- I or u -1. There are respectively given by

x=

2y2+c1

foru=1 (1.22)

x=-2y2+c2

for u=-1

where cl, c2 are two constants. Both foliations are represented on Fig. 1.1 and we represent on Fig. 1.2 a solution of the system th(t) = y(t) (1.23) y(t) = sign(p2(t)), p2(t) = -pit + P20

integrated backwards in time from (x(0), y(0)) = (0, 0) and corresponding

to p2 > 0,p2 < 0. The switching point (x y,) corresponds to the time t.

Fig. 1.1. solution of p2(t,) = 0. Making p(0) varies in Si we get the optimal synthesis, see Fig. 1.3. Example 2. We consider on R3 the system:

x(t) = y(t) + ui (t), I ui (t) I< -1, (1.24)

v(t) = -x(t) + u2(t), I U2(t) I< 1.

Here the control domain Il is the unit square defined by:

ul(t) I<- 1, I 1. Hence the system satisfies the assumption: rank[Bw, ABw) = 2 u2(t) 15

-

I

20

1 Linear Systems y

x

Fig. 1.2.

Fig. 1.3.

where A = I

°l 0

1)

,

B= (b?) for each vector tw = (1, 0) or tw = (0,1)

parallel to an edge of the square Q. Therefore, from Theorem 3 each optimal control is (up to a set with zero measure) a piecewise constant mapping taking only its values on the vertices of Q. The adjoint system is given by: 01 (t) = p2(t),

P2(t) = -Pi(t)

and an optimal control u'(.) satisfies pi (t)ui (t) + P2(t)u2(t) =

max plus + p2u2. Iu, I<<1 IU2I<<-1

It follows that

ui(t) = sign(pi(t)),

u2(t) = sign(p2(t))

1.7 Time Minimal Synthesis

21

where (pl,p2) are computed by integrating the adjoint system. We have:

pi (t) = A sin(t + a),

p2(t) = A cost + a)

where A is a nonzero constant. We observe that the vector p(.) is rotating clockwise in the plane and that the time between two switchings is at most '. On Fig. 1.4, we represent the switching rules when we integrate the adjoint equation backwards in time. The trajectories of the system corresponding to

Fig. 1.4.

a constant control u° = ±1, u2 = ±1 are family of circles in the plane (x, y) whose center is denoted by uo _ (uz, u°). We represent on Fig. 1.5 the construction of an optimal trajectory and on Fig. 1.6 the optimal synthesis. The set of switching points splits into two types: the bold curves which are optimal terminating arcs and are arcs of circles; the dotted curves which are not admissible optimal arcs. They are easily geometrically constructed and are isometric to arc of circles.

1.7.2 The General Algorithm in the Plane Consider a general linear system in the plane: x(t) = Ax(t) + Bu(t), u(t) E £2 C Rm. To compute the time minimal synthesis we proceed as follows. First of all we choose a coordinate system in the state space 1R2 such that the matrix A is under the real Jordan canonical form:

y(t) = Jy(t) + Cu(t),

J = P-' AP, C = P-' B, x = Py.

Secondly let us introduce V, the image of 17 by C. As 11 is a convex polyhedron, V is also a convex polyhedron. Thus we obtain two linear systems:

22

1 Linear Systems

Fig. 1.5.

Fig. 1.6.

y(t) = Jy(t) + Cu(t), y(t) = Jy(t) + v(t),

U(t) E ,fl, v(t) E V.

Clearly, since the optimal control is unique, both synthesis are equivalent and we shall consider the simplest system:

W) = Jy(t) + v(t),

v(t) E V.

1.7 Time Minimal Synthesis

23

We note ty = (yl, y2) and let el, -

, e9 be the vertices of V. The adjoint system is written in the dual coordinates p = (pl, p2) as

p(t) = -p(t)A and the solution is p(t) = poexp(-tA). If 1v = (VI, V2), then the optimal control v`(.) satisfies pi (t)vi (t) + p2(t)v2* (t) = maVx(pl (t)VI + p2 (t)v2)

and the maximum of the linear mapping v F-+ pv on V is easily computed. Indeed, we draw q rays having the directions of outward normals to the sides

of the polyhedron V, the sector containing ei being denoted ai. Clearly if p is contained in the sector ai, the maximum is attained for v = ei, see Fig. 1.7. Having solved the maximization problem, the optimal synthesis is

Fig. 1.7.

easily computed. Indeed, given po E S' we have p(t) = po exp(-tA) and the optimal control v' (.) is computed at each time t < 0 as above. Then the optimal trajectory is computed by integrating the system: y(t) = Jy(t)+v' (t) where v(.) is a piecewise constant function.

Notes and Sources The results presented in this chapter are standard. A more complete theory concerning the feedback classification of autonomous linear systems can be

1 Linear Systems

24

found in Wonham [109] where the Brunovsky canonical form is constructed. For the problem of feedback equivalence to a linear system, we have followed the heuristic original article of Brockett [33], see [61] for the multivariable

case. The presentation of the time optimal control is based on the presentation of the Maximum principle due to Lee-Markus [77], completed by the concise book of Lasalle-Hermes [55). See also Cesari [38] which contains all the material of functional analysis needed to make a neat description of the

properties of the accesibility set in order to get the existence theorem and the bang-bang principle.

Exercises 1.1 We consider the linear autonomous system in R": i(t) = Ax(t) + Bu(t), u(t) E .R = R' and let F be the accessibility set of 0.

1. Prove that F is the smallest linear subspace of R" containing ImB and invariant by A. 2. Show that there exists a linear coordinate system (XI, X2) of R' such that F is the set x2 = 0 and in which the system can be written xl (t) = Aiixi(t) + A12x2(t) + Biu(t) th2(t) = A22x2(t)

and the restriction to F is a controllable linear system.

1.2 Consider a linear control system i(t) = Ax(t) + Bu(t), x(t) E R", u(t) E R' linearly observed by a mapping y(t) = Cx(t), y(t) E RP, A, B, C being real constant matrices. Denote by y(t, to, xo, u) = Cx(t, to, xo, u) the observed

trajectory. We say that the system is observable if Vxo,xl E R", xo # xl, 3u(.) such that y(., xo, u) 76 y(., xi, u). Prove that the system is observable if and only if the linear system £(t) = tAx(t)+ tCv(t), v(t) E RP is controllable.

1.3 (Luenberger observer) Consider the observable control system i(t) = Ax(t)+Bu(t), y = Cx(t). Show that there exists a differential equation 1(t) = Jz(t) + Ky(t) + Bu(t), z(0) = zo such that the error e(t) = x(t) - z(t) -+ 0 when t -- +oo for each x(0) = xo. Hint: Find K such that J = A - KC is a stable matrix. 1.4 (Brunovsky canonical form) Let x(t) = Ax(t) + Bu(t), x(t) E R", u(t) E R'", A, B real constant matrices, be a linear controllable system. Show that (A, B) is feedback equivalent to a system

ii(t) = Aixi(t) + u(t)bi, where

i = 1,...,p

1.7 Time Minimal Synthesis

00 00 1 ... 0 01

Ai=

0

0 bi

, .

.

.

25

0

1

000 00

1

1.5 Consider the controlled harmonic oscillator: x(t)+x(t) = u(t), I u(t) (< 1.

Compute the time minimal synthesis to steer the initial state x(0) = xo, z(0) = yo to the origin. 1.6 Consider the following second order linear system: i(t)+2bx(t)+cx(t) _ u(t), u(t) I< 1 where a, b are real constants. Compute the time minimal synthesis to steer x(O) = xo, z(0) = yo to the origin. I

1.7 Let A1, , \,, be n real distinct numbers and P1(t), nomials with real coefficients and respective degree k 1, function n

f (t) _

Pi (t) exp(Ait)

,

, P,(t) be n polykn. Prove that the

(1.25)

i=1

has at most k1 +

+ kn + (n - 1) real roots.

1.8 Consider the single input linear autonomous controllable system: i(t) _ Ax(t) + u(t)b, x(t) E 1R", I U(t) I< 1.

1. Assume that the eigenvalues of A are real. Show that a time optimal control has at most (n - 1) switchings. 2. Assume that every eigenvalue of A has a nonzero imaginary part. Prove that for each N > 0 there exists an initial state xo for which the time optimal control steering xo to the origin has more than N switchings. 1.9 Consider the autonomous linear system in R": i(t) = Ax(t) + Bu(t) with u(t) E 12 where £2 is a compact set. Assume:

i) u(t) = 0 belongs to the interior of fl; ii) the system (A, B) is controllable; iii) A is a stable matrix. Prove that for each xo E 1Rn, there exists a time minimal control steering xo to the origin. 1.10 Consider the following system in 1R2:

y(t) = -x(t) + u(t),

x(t) = y(t), u(t) < 1.

Let r > 0. Compute the time minimal synthesis to steer (xo, yo), xp + y02 > r2 to the circle xp + yo = r2. Hint. Use the transversality condition of Theorem 9, Chap. 2.

26

1 Linear Systems

1.11 Consider the following system in R2:

x(t) = y(t) + u(t) y(t) _ -y(t) + u(t), I u(t) 1< 1.

(1.26)

Compute the time minimal synthesis to steer (xo, yo) to the target x = 0, Iyl<_1.

2 Optimal Control for Nonlinear Systems

2.1 A Short Visit into the Classical Calculus of Variations See [47].

In this section we briefly recall the technics and results which are the core of the classical calculus of variations. The problem we consider is the following. Let C be a family of parametrized curves t i- x(t) and one wants to minimize on C a functional of the form

C(x) =

to to

L(t, x(t), b(t))dt,

where the function L is assumed to be C. The space C of the admissible curves represent the constraints of the system. For instance, in holonomic mechanics C is the set of curves on a manifold M satisfying fixed boundary conditions x(to) = xo,x(tl) = x1. In the case of nonholonomic mechanics the constraints are of the form f (t, x,.+) = 0 and are generally nonintegrable. Hence in general the space C is complicated and the optimal problem requires a precise analysis of the set C which in this book means to make a singularity theory of C.

2.1.1 Statement of the Problem in the Holonomic Case We consider the set C of all curves x : [to, t1] --+ 1R of class C2, the initial and

final times to, t1 being not fixed and the problem of minimizing a functional on C:

C(x) = rte L(t, x(t), i(t))dt to

where L is C2. Moreover, we impose extremities conditions: x(to) E Mo, x(tj) E Ml where Mo, M1 are C1-submanifolds of lRn. The set C is endowed with the Cl-topology defined as follows. A curve x(.) defined on the interval [to, t1] is considered as curve (t, x(t)) E Rn+' with extremities Po = (to, xo), Pi = (ti, xl ). Let x(.), x* (.) be two such curves. The distance between x(.) and x'(.) is given by

2 Optimal Control for Nonlinear Systems

28

p(x, x') = max II x(t) - x*(t)II + max 11i(t) - i' (t) II + d(Po, P) + d(Pi, Pi), where 11.11 is any norm on R" and d is the usual distance mapping on R"+i The two curves x(.), x`(.) being not defined on the same interval they are by convention C2-extended on the union of both intervals.

Proposition 9 (Fundamental formula of the classical calculus of variations). We adopt the standard notation of classical calculus of variation, see [47]. Let y(.) be a reference curve with extremities (to, xo), (ti, xl ) and let y(.) be any curve with extremities (to+bto,xo+bxo), (tl+6tl,x1+6xl)

We denote by h(.) the variation: h(t) = -y(t) - -y(t). Then, if we set AC = C(y) - C(ry), we have

AC - It

o

d 8L

8L

t1

C

I x - dt ax) + [(L

where

.

8L

h(t)dt +

ai,' '

I

- BL.i)l,ybt] 8i

tl

bx

to

(2.1)

t1

to

+ o(p(7, y))

denotes the scalar product in R".

Proof. We write

QC =

rt1+6t1

L(t, y(t) + h(t), y(t) + h(t))dt - f t1 L(t, y(t), ry(t))dt to

+6to

J

=f'

L(t, y(t) + h(t), y(t) + h(t))dt -

to

+ rtl+atl

J

1

L(t, y(t),7(t))dt

to

L(t, y(t)+h(t),y(t)+h(t))dt.

L(t, to

t1

We develop this expression using the Taylor expansions keeping only the linear terms in h, h, bx, bt. We get

aC =

(0L J

to 1

8x

"A(t) +

OL 8x I., ,h(t)) I7

+ [L(t, y, y)bt to + o(h, h, bt).

The derivative of the variation h is depending on h, integrating by parts we obtain

'ac ,.,

" (8L - d 8L1 .h(t)dt + [8L h(t), t1 + [L1ot] tt 82 I,, to to Jo 8x dt 8i J ly

We observe that h, bx, bt are not independent at the extremities and we have

for t = to or t = tl the relation

h(t + bt) N h(t) ' bx - ibt.

2.1 A Short Visit into the Classical Calculus of Variations

29

Hence, we obtain the following approximation:

dC

ft t' c9L o

d aL

.h(t)dt +

ax - dt

OL

t,

bx

+ (L -

aLi)17bt i ax

where all the quantities are evaluated along the reference trajectory -y(.). In this formula h, bx, bt can be taken independent because in the integral the values h(to),h(tl) do not play any role. From (2.1), we deduce the standard first-order necessary conditions of the calculus of variations.

Corollary 5. Let us consider the minimization problem where the extremities (to, xo), (tl, xl) are fixed. Then a minimizer y(.) satisfies the EulerLagrange equation

aL d 8L _ 0 TX - at ax I

(2.2)

Proof. Since the extremities are fixed we set in (2.1) bx = 0 and bt = 0 at t = to and t = ti. Hence d aL °C = Lot (OL ax - at ax) `ry.h(t)dt + o(h, h)

for each variation h(.) defined on [to,t1J such that h(to) = h(ti) = 0. If y(.) is a minimizer, we must have AC > 0 for each h(.) and clearly by linearity, we have

ftl (OL

/.h(t)dt

=0 I siiy 8L),, is continuous, it must be for each h(.). Since the mapping t '- (at - dtd S1identically zero along y(.) and the Euler-Lagrange equation (2.2) is satisfied. o

ax

TX_

0

2.1.2 Hamiltonian Equations The Hamiltonian formalism, which is the natural formalism to deal with the maximum principle, appears in the classical calculus of variations via the

Legendre transformation. Definition 8. The Legendre transformation is defined by

p=

si (t, x, i)

(2.3)

and if the mapping cp : (x, i) - (x, p) is a diffeomorphism we can introduce the Hamiltonian:

H : (t,x,p)'-' p.x - L(t,x,P).

(2.4)

30

2 Optimal Control for Nonlinear Systems

Proposition 10. The formula (2.1) takes the form t, 8L (_. 8x o

d 8L l

QC t

dt 8X I

t, 1.,,'h(t)dt + [ & - Hot1 to

(2.5)

and if -y(.) is a minimizer it satisfies the Euler-Lagrange equation in the Hamiltonian form

±(t) =

Xt)

(t, x(t), p(t)),

e (t, x(t), p(t))

(2.6)

2.1.3 Hamilton-Jacobi-Bellman Equation Definition 9. A solution of the Euler-Lagrange equation is called an extremal. Let Po = (to, xo) and P1 = (tl, xl ). The Hamilton-Jacobi-Bellman (HJB) function is the multivalued function defined by t,

S(Po, Pi) = f L(t, 7(t),1'(t))dt to

where 'y(.) is any extremal with fixed extremities xo, x1. If -y(.) is a minimizer, it is called the value function.

Proposition 11. Assume that for each Po, P1 there exists an unique extremal joining Po to Pl and suppose that the HJB function is C1. Let P0 be fixed and let S : P - S(Po, P). Then S is a solution of the Hamilton-JacobiBellman equation 69S

8t

49S

(Po,

P) + H(t, x, 8x)

(2.7)

0.

Proof. Let P = (t, x) and P + bP = (t + dt, x + bx). Denote by y(.) the extremal joining Po = (to,xo) to P and by ry(.) the extremal joining Po to P+BP. We have

aS = S(t + dt, x + dx) - S(t, x) = C(y) - C(y) and from (2.5) it follows that:

A3=AC,

t(8L-

10 \ 8x

d8LI ,ti'h(t)dt + dt x I

[pbx -

L

H8t ] to,

where h(.) = ry(.)-y(.). Since y(.) is solution of the Euler-Lagrange equation, the integral is zero and

AS = 'dc , In other words we have

[P,6x -

HUI

to

2.1 A Short Visit into the Classical Calculus of Variations

31

dS = pdx - Hdt. Identifying, we obtain 63-

at

= - H,

Hence we get the HJB equation. Moreover p is the gradient to the level sets {x E R°; S(t, x) = c}. 0

2.1.4 Euler-Lagrange Equations and Characteristics of the HJB Equation Under some extra regularity conditions, the extremals are the characteristics of HJB equation. Indeed, let u(.) be a solution of the HJB equation. Hence we can write (2.7) as

F(t'x'

as as as at' ax) = at

as

+H(t'x' ax) = 0

and let us assume the map F to be C2. Introduce p = as, T = as and z = S(t, x). Then, according to (63] the characteristic curves parametrized by s are solutions of. dx ds

OF

OH

ap

ap

dpOF_OF _ _ax p Ts

dz

OH

az

ax

' aH

OF

T =pap + T =pap - H

dtOF1 _

dT Ts

OF at

ds OF az

OT

aH p

at

In particular since 7, = 1, we deduce that dx

OH

dt - ap'

dp dt

OH

ax

which is the Hamiltonian form of the Euler-Lagrange equation.

2.1.5 Second Order Conditions The Euler-Lagrange equation has been derived using the linear terms in the Taylor expansion of W. Using the quadratic terms we can get a necessary and sufficient second order condition. For the sake of simplicity, from now

2 Optimal Control for Nonlinear Systems

32

on we assume that the curves t - x(t) belongs to R, and we consider the problem with fixed extremities: x(to) = xo, x(tl) = xl. If the map L is taken C3, the second derivative is computed as follows:

LC = f", L(t, y(t) + h(t),7(t) + ih(t)) - L(t, y(t),7(t)))dt ti 8L d8L hZ (t) +28'L J to Jtp e2L 8x Wt 8i I7' h(t)dt+12 (t' ( azft h(t)h(t) &2

+ o(h, h)2.

If y(.) is an extremal, the first integral is zero and the second integral corre-

sponds to the intrinsic second-order derivative 62C, that is: 62C

1

-2

f1,

o

a2 L

\ axe I, h2+

2a.92 L

xax IY

h(t)h(t)

a2 L

+ ax

h2dt.

(2.10)

Using h(to) = h(ti) = 0, it can be written after an integration by parts as 62C

where

=f'

P1822

2ax 6

(th2ct) + Q(t)h2(t))dt

(2.11)

d 82L Q = 2 ( axe - dt axax 1

82L

Using the fact that in the integral (2.11) the term Ph2 is dominating (see (471), we get the following proposition.

Proposition 12. If y(.) is a minimizing curve for the fixed extremities problem then it must satisfy the Legendre condition: 82L

(2.12)

2.1.6 The Accessory Problem and the Jacobi Equation The intrinsic second-order derivative is given by b2C =

f ' (P(t)h2(t) + Q(t)h2 (t)) dt,

h(to) = h(ti) = 0,

where P, Q are as above. We write 62C

= /o ((P(t)h(t))h(t) + (Q(t)h(t))h(t))dt

and integrating by parts using h(to) = h(ti) = 0, we obtain

2.1 A Short Visit into the Classical Calculus of Variations

b2C = Jol (Q(t)h(t)

-

33

dt(P(t)h(t)))h(t)dt.

Let us introduce the linear operator D : h

Qh - dt (Ph). Hence, we can

write

52C = (Dh, h)

(2.13)

where (,) is the usual scalar product on L2([to, t1]). The linear operator D is

called the Euler-Lagrange operator. Definition 10. From (2.13), 62C is a quadratic operator on the set Co of C2-

curves h : [to,tl] - R satisfying h(to) = h(tl) = 0, h # 0. Rather to study 62C > 0 for each h(.) E Co we can study the so-called accessory problem: min b2C. hECo

Definition 11. The Euler-Lagrange equation corresponding to the accessory problem is called the Jacobi equation and is given by

Dh = 0

(2.14)

where D is the Euler-Lagrange operator: Dh = Qh order linear differential operator.

d(Ph).

It is a second-

Definition 12. The strong Legendre condition is: P > 0 where P = z elry. If this condition is satisfied, the operator D is said to be nonsingular.

2.1.7 Conjugate Point and Local Morse Theory See [52, 85].

Definition 13. Let y(.) be an extremal. A solution J(.) E Co of DJ = 0 on [to, tll is called a Jacobi curve. If there exists a Jacobi curve along y(.) on [to,tll the point -y(tl) is said to be conjugate to -y(to).

Theorem 4 (Local Morse theory [851). Let to be fixed and let us consider the Euler-Lagrange operator (indexed by t > to) Dt defined on the set Co of curves on [to, t] satisfying h(to) = h(t) = 0. By definition, a Jacobi curve on [to, t] corresponds to an eigenvector Jt associated to an eigenvalue At = 0 of Dt. If the strong Legendre condition is satisfied along an extremal y : [to, T] - R", we have a precise description of the spectrum of Dt as follows.

There exists to < ti < . . . < to < T such that each y(ti) is conjugate to -y(to). If ni corresponds to the dimension of the set of the Jacobi curves Jt{

associated to the conjugate point y(ti), then for any T such that to < tl < < tk < T < tk+I < . < T we have the idendity -

k k

n , =

ni

(2.15)

34

2 Optimal Control for Nonlinear Systems

where nT = dim{linear space of eigenvectors of DT corresponding to strictly negative eigenvalues}. In particular if t > tj we have min

hECo

ft. (Q(t)h2(t) + P(t)h2(t))dt = -oo.

(2.16)

2.1.8 Scalar Riccati Equation Definition 14. The quadratic differential equation

P(t)(Q(t) + tb(t)) = w2(t)

2.17)

is called the scalar Riccati equation.

Its connection with the problem is the following. Assume P > 0 on [to, ti] and assume that there exists a solution u(.) of the Jacobi equation such that this solution does not vanish on the interval [to, t,]. Let h(.) be any C2-function such that h(to) = h(ti) = 0. Then t,

d(w(t)h2(t))dt = 0

fto and

b2C =

t ((Ph2 + Qh2) + d(wh2))dt

Jto

= f (Ph+ 2whh + (Q + w)h2)dt.

If w(.) is a solution of the Riccati equation, the previous expression can be written as 62C h(t)) 2dt. P(t) (h(t) + r_

to

1

Hence 62C

=

tl

P(t)cp2(t)dt

to

where W(t) = h(t) + "'tit . Now observe that if we set w(t) = -u t P(t) where u(.) is nonvanishing on [to,ti], then u(.) is a solution of the Jacobi equation:

Q(t)u(t) and

h(t) +

d

(P(t)u(t)) = 0

w(t)h(t)

h(t)u(t) - h(t)u(t)

P(t)

U(t)

Hence cp =- 0 is equivalent to

h(t)u(t) - h(t)u(t) = 0.

2.1 A Short Visit into the Classical Calculus of Variations

35

This is possible if and only if h(.) = Cu(.) where C is a constant. It contradicts

the fact that u(.) does not vanish on [to, t1] and that h(to) = h(ti) = 0 if h 0 0. Hence ca 0 unless h- 0 and 62C

t,

=

J to

is nonzero for each h(.) E Co and 62C > 0.

2.1.9 Local CO Minimizer - Extremal Field - Hilbert Invariant Integral Definition 15. Consider the time-minimal problem with fixed extremities: Let 7(.) be a reference trajectory. It is called a C°(to, xo), (ti, xi) E minimizer if it is a local minimum for the C°-topology: Rn+i.

d(x, x*) = ma`el IIx(t) - x (t)II tE

To obtain C°-sufficient optimality conditions we use the concept of ex-

tremal (or Mayer field). Definition 16. Let 7 : [to, t1] - W1 be a reference extremal issued from xo at t = to: 7(to) = xo. An extremal field is a mapping 0 : (a, t) -- Rn+1, a E D = parameter space C R" such that: 1. 4)(ao, t) = (t, 7(t)) is the reference extremal and 10(a,.); a E D} is a family .T of extremals;

2. the image of 0 denoted T is a tubular neighborhood of 7(.) and through each point (t, x) of T there passes an unique extremal of F whose derivative is denoted by u(t, x). We assume that t '- u(t, x) is Cl and we use the following notations: P

aL

L=Ll,(

(2.18) =),

H=HIa=p

where H = p.x - L is the Hamiltonian. We can easily prove the following lemma.

Lemma 6. The following relations hold: Oj3

8Pk

8xk = 8xi '

OH

Opi

8xi

at

In particular the one form w = -Ifdt + Pdx, called Hilbert-Cartan form, is closed.

36

2 Optimal Control for Nonlinear Systems

Theorem 5 (Hilbert invariant integral theorem). The integral

Jr

-Hdt + pdx

is independent of the curve r(.) on T. Moreover if n) is an extremal of F it is given by fr Ldt. Proof. The first assertion is a consequence of the fact that the form w is = u(t, x) and -Hdt+pdx =

closed. Moreover if T(.) is an extremal we have (L - 3.u(t, x))dt + pdx. 0

Remark 3. 1. The resolution of c:i = dS on the domain where S : (t, x) F-+ R is a smooth function, is equivalent to solve the Hamilton-Jacobi equation. 2. In practice we use a central Mayer field, i.e. all the extremals are starting at to - E from a single point xo, for e > 0 small enough.

Corollary 6. Let -y(.) be the reference extremal with extremities (to,xo), (t1,x1) and let rd be any curve of T with same extremities. We define by E the excess Weierstrass mapping:

E(t, x, x, w) = L(t, x, w) - L(t, x, z) - (w - z) 8x (t, x, z)

z = u(t, x), (t, x) E T. Then, if E > 0 we have that y(.) is a Co-minimizer on T. Proof. We have

L(t, x, i)dt =

J(L - p.u(t, x))dt +

J7

zC = fr L(t, x, i)dt - fy L(t, x, i)dt = fr ((L - L) - (i - u(t, x).p) dt = fr E(t, x, u(t, x), i)dt. This proves the assertion.

2.2 Optimal Control and the Calculus of Variations 2.2.1 Problem Statement We consider the autonomous control system

2.2 Optimal Control and the Calculus of Variations

i(t) = f (x(t), u(t)),

x(t) E R", U(t) E .R

37

(2.19)

where f is a C'-mapping. Let the initial and target sets Mo, M1 be given. We assume M0, M, to be C'-submanifolds of R. The control domain is a given

subset Sl C Rm. The class of admissible controls U is the set of bounded measurable mappings u : [0,T(u)] -i 1. Let u(.) E U and xo E R" be fixed. Then, by the Caratheodory theorem, see [77], there exists a unique trajectory of (2.19) denoted by x(., x0, u) (in short x(.)) such that x(0) = xo. This trajectory is defined on a nonempty sub-interval J of [0, T(u)) on which t F- x(t, xo, u) is an absolutely continuous function and is solution of (2.19) almost everywhere. To each u(.) E U defined on [0, T] with response x(., xo, u) issued from x(0) = x° E Mo defined on [0, T], we assign a cost T

C(u) = J f°(x(t),u(t))dt

(2.20)

0

where f ° is a C'-mapping. An admissible control u*(.) with corresponding trajectory z ' (., x' (0), u') defined on 10,T*] such that x' (0) E M0 and x'(T') E M1 is optimal if for each admissible control u(.) with response x(., x(0), u) on [0, T], X(0) E M0 and x(T) E M1, then

C(u*) < C(u).

2.2.2 The Augmented System The following remark is straightforward but is geometrically very important

to understand the maximum principle. Let us consider f = (f, fo) and the corresponding system on R"+' defined by the equations I = f (±(t), u(t)), i. e:

±(t) = f (x(t), u(t)), x°(t) = f°(x(t), u(t)).

(2.21)

(2.22)

This system is called the augmented system. Since f is C1, according to the Caratheodory theorem, to each admissible control u(.) E U there exists an admissible trajectory &(t, $o, u) such that io = (x0i x°(0)), x°(0) = 0 where the added coordinate x°(.) satisfies x°(T) = fu f°(x(t),u(t))dt. Let us denote by AMo the accesibility set Uu(.)Eu (T, xo, u) from Mo =

(Mo, 0) and let if, = M1 x R. Then, we observe that an optimal control u'(.) corresponds to a trajectory !'(.) such that :to* E Mo and intersecting M1 at a point &* (T*) where x° is minimal. In particular &* (T) belongs to the boundary of the accessibility set AMo , see Fig. 2.1.

38

2 Optimal Control for Nonlinear Systems

Fig. 2.1.

2.2.3 Related Problems Our framework is a general setting to deal with a large class of problems. Examples are the following:

1. Nonautonomous systems:

i(t) = f (t, x(t), u(t)). We add the variable t to the state space by setting as = 1, t(so) = so. 2. Fixed time problem. If the time domain [0, T(u)J is fixed (T(u) = T for all u(.)) we add the variable t to the state space by setting d8 = 1, t(so) = so

and we impose the following state constraints on t: t = 0 at s = 0 and t = T at the free terminal time s. Some specific optimal problems important for applications are the following.

1. If f ° __ 1, then min fo f °(x(t), u(t))dt = min T and we minimize the time of global transfer. 2. If the system is of the form: ±(t) = f (t, x(t), u(t)), f (t, x, u) = A(t)x(t) +

B(t)u(t), where A(t), B(t) are matrices and C(u) = fo L(t, x(t), u(t))dt where L(x, u,.) is a quadratic form for each t, T being fixed, the problem is called a linear quadratic problem (LQ-problem).

2.2.4 Optimal Control and the Classical Calculus of Variations Classical problems of calculus of variations can be easily stated as optimal control problems as follows.

2.2 Optimal Control and the Calculus of Variations

39

1. Holonomic problems: min f L(t, x(t), x(t)dt. We introduce the control

system by setting x(t) = u(t) and we must minimize a cost C(u) _ f L(t, x(t), u(t))dt. In particular the accessory problem min h(.)

J

(P(t)h2(t) + Q(t)h2(t))dt

is transformed into the LQ-problem h(t) = u(t),

min u(.)

J

(P(t)h2(t) + Q(t)u2(t))dt.

2. Nonholonomic problems: more generally we consider the problem min x(.)

J

L(t, x(t), x(t)dt)

among a set of curves satisfying the constraints x(t) E D(x(t)) can be reformulated into an optimal control problem when the differential inclusion can be restated as a system x(t) = f (t, x(t), u(t)). An important example in our study is the sub-Riemannian problem: T

min fa (z(t), x(t))g dt with x(t) E D(x(t)) and

D(x) = Span{F1(x), ... , F,(x)}

where the distribution D generated by the vector fields F's is of constant rank and (,)g is the scalar product associated to a Riemannian metric g.

2.2.5 Singular Trajectories and the Weak Maximum Principle Definition 17. Consider a system of R": ±(t) = f (x(t), u(t)) where f is a Coo -mapping from R" x R"' into R" . Fix xo E R" and T > 0. The end-point mapping (for fixed xo, T) is the mapping Exo.T : u(.) E U -. x(T, xo, u). If u(.) is a control defined on [0, T] such that the corresponding trajectory x(., x0, u) is defined on [0, T), then E--°,T is defined on a neighborhood V of u(.) for the L°°([0,TJ) norm.

2.2.6 First and Second Variations of E,O,T It is a standard result, see for instance [99], that the end-point mapping is a C°°-mapping defined on a domain of the Banach space L°D ([0, T] ). The formal computation of the successive derivatives uses the concept of Gateaux

derivative. Let us explain in details the process to compute the first and second variations.

40

2 Optimal Control for Nonlinear Systems

Let v(.) E L°°([O,T]) be a variation of the reference control u(.) and let us denote by x(.) + f (.) the response corresponding to u(.) + v(.) issued from x0. Since f is C°O, it admits a Taylor expansion for each fixed t:

(x, u)f + L (x, u)v + lku (x, u)(6, v)

f (x + l;, u + v) = f (x, u) +

e8

2

+2

2

(x,u)(C1C)+28 (x,u)(v,v)+

Using the differential equation we get

k(t) + fi(t) = f (x(t) + fi(t), u(t) + v(t)) Hence we can write 1; on the form: 61x + 52x + where 51x is linear in v, Sex is quadratic, etc and are solutions of the following differential equations: 61x

a2x

+ 8f

8x

(x, u)51 x + au (x, u)v

(x, u) a2x +

02

(2.23)

(x, u ) (a lx, v)

-

+2 a (x'u)(6 1x, 61x)+ 2au

(x,u)(v,v).

( 2 . 24 )

Using 1;(O) = 0, these differential equations have to be integrated with the initial conditions 61x(0) = 52x(0) = 0. (2.25) Let us introduce the following notations:

A(t) = Lf (x(t), u(t)), ax Definition 18. The system

B(t) =

(x(t), u(t)) 49U

6x(t) = A(t)6x(t) + B(t)5u(t) is called the linearized system along (x(.), u(.)).

Let M(t) be the fundamental matrix on [0, T] solution a.e. of

Af(t) = A(t)M(t),

M(O) = identity.

Integrating (2.23) with 61x(0) = 0 (see Lemma 1) we get the following expression for 61x: T

81x(T) = M(T) fo M -1(t)B(t)v(t)dt This implies the following lemma.

(2.26)

2.2 Optimal Control and the Calculus of Variations

41

Lemma 7. The Frechet derivative of Ex0,T at u(.) is given by rT Eu O.T(v) = &,x(T) = M(T) J M-1(t)B(t)v(t)dt. 0

Definition 19. The admissible control u(.) and its corresponding trajectory x(., x0i u) defined both on [0, T] are said to be regular if the Frechet derivative E1x0,T is surjective. Otherwise they are called singular.

Proposition 13. Let A(xo, T) = Uu(.)EUx(T, x0, u) be the accessibillity set at time T from x0. If u(.) is a regular control on [0, T] then there exists a neighborhood U of the end-point x(T, x0i u) contained in A(xo, T).

Proof. Since E'xO,T is surjective at u(.), we have using the open mapping Theorem [31] that ExO,T is an open map. Theorem 6. Assume that the admissible control u(.) and its corresponding trajectory x(.) are singular on [0, T]. Then there exists a vector p(.) E R"\{0} absolutely continuous on [0, T) such that (x, p, u) are solutions a.e. on [0, T] of the following equations: (x (t),

dt (t) = an

p(t), u(t)),

dt (t)

_ - eH (x(t), p(t), u(t)) (2.27) a (x(t), p(t), u(t)) = 0 (2.28)

where H(x, p, u) = (p, f (x, u)) is the pseudo-Hamiltonian, (,) being the standard inner product. Proof. We observe that the Frechet derivative is solution of the linear system

81x(t) = A(t)6,x(t) + B(t)v(t).

Hence, if the pair (x(.), u(.)) is singular this system is not controllable on [0,T]. We use the proof of Proposition 1 to get a geometric characterization of this property. The proof which is the heuristic basis of the maximum principle is given in details. By definition, since u(.) is a singular control on [0, T] the dimension of the linear space

{jT M (T)M1(t)B(t)v(t)dt; v(.) E L°°([O,T])} is less than n. Therefore there exists a row vector p E R"\{0} such that

EM(T)M-1(t)B(t) = 0 for a.e. t E [0, T]. We set

42

2 Optimal Control for Nonlinear Systems

p(t) = pM(T)M-1(t) By construction p(.) is solution of the adjoint system

P(t) = -p(t) Lf (x(t),u(t)) Moreover, it satisfies almost everywhere the following equality:

p(t) Lf (x(t), u(t)) = 0. Hence we get the equations (2.27) and (2.28) if H(x, p, u) denotes the scalar product (p, f (x, u)). 0

2.2.7 Geometric Interpretation of the Adjoint Vector In the proof of Theorem 6 we introduced a vector p(.). This vector is called an adjoint vector. We observe that if u(.) is singular on [0, T], then for each 0 < t < T, ul10,, is singular and p(t) is orthogonal to the image denoted K(t) of E'x0,T evaluated at u11011. If for each t, K(t) is a linear space of codimension

one, then p(t) is unique up to a factor.

2.2.8 The Weak Maximum Principle Theorem 7. Let u(.) be a control and x(., xo, u) the corresponding trajectory, both defined on [0, T]. If x(T, xo, u) belongs to the boundary of the accessibility set A(xo,T), then the control u(.) and the trajectory x(.,xo, u) are singular.

Proof. According to Proposition 13, if u(.) is a regular control on [0, T] then x(T) belongs to the interior of the accessibility set.

Corollary 7. Consider the problem of maximizing the transfer time for system i(t) = f (x(t), u(t)), u(.) E U = L°°, with fixed extremities xo, xl. If u`(.) and the corresponding trajectory are optimal on 10,,r*), then u`(.) is singular. Proof. If u* (.) is maximizing then x' (T) must belongs to the boundary of the accessibility set A(xo, T) otherwise there exists E > 0 such that x` (T - e) E A(xo, T) and hence can be reached by a solution x(.) in time T: x* (T - E) =

I(T). It follows that the point X*(T) can be joined in a time t > T. This contradicts the maximality assumption.

Corollary 8. Consider the system +(t) = f (x(t), u(t)) where u(.) E U = L01([0, T)) and the minimization problem: min

rT

J

u(.)EL/ p

L(x(t),u(t))dt, where

the extremities xo, xI are fixed as well as the transfer time T. If u'(.) and

2.2 Optimal Control and the Calculus of Variations

43

its corresponding trajectory are optimal on [0, T[ then u' (.) is singular on [0,T] for the augmented system: x(t) = f (x(t), u(t)), to (t) = L(x(t), u(t)). Therefore there exists P* (t) = (p(t),po) E Rn+1\{0} such that (i*j5*,u*) satisfies

X(t) _ 4V (1(t),P(t),u(t)),

p(t)

a (x(t),*t),u(t)), (2.29)

aH u

WO, p(t), u(t)) = 0

where 1 = (z, x°) and H(x, P, u) = (p, f (x, u)) + poL(x, u).

Proof. We have that x' (T) belongs to the boundary of the accessibility set A(:io,T). Applying (2.27),(2.28) we get the equations (2.29) where Po = = 0 since ft is independent of x°. Hence po is a constant. 0

-

2.2.9 Abnormality In the previous corollary, P*(.) is defined up to a factor. Hence we can normalize po to 0 or -1 and we have two cases:

1. Case 1: u(.) is regular for the system x(t) = f (x(t), u(t)). Then po 0 0 and can be normalized to -1. This case is called the normal case (in calculus of variations), see [18].

2. Case 2: u(.) is singular for the system x(t) = f (x(t), u(t)). Then we can choose po = 0 and the Hamiltonian H evaluated along (x(.), p(.), u(.)) doesn't depend on the cost L(x, u). This case is called the abnormal case.

2.2.10 The Weak Maximum Principle and Euler-Lagrange Equation We can deduce from Corollary 8 the standard Euler-Lagrange equation. Indeed consider the problem of minimizing fo L(t, x(t), x(t))dt, where L R2n+1 -- IR is a smooth map, among the set of absolutely continuous curves

t t-- z(t) in R", with bounded derivative and satisfying the boundary conditions: x(0) = xo, x(T) = xl. We introduce for almost every t, the linear system +(t) = u(t), u(t) E R", with u(.) a measurable bounded function. We have H(i,1P, u) = p.u + poL(x, u)

where p is a row vector in R". Since the linear system is controllable, any = 0, we get optimal control is normal and we can set po = -1. Using

44

2 Optimal Control for Nonlinear Systems

Pi (t) = -Po au W* u(t)) = 8u; (x(t), u(t)),

i = 1, ... , n.

Moreover

AM

eH (x(t), P(t), u(t)) = -P0 8x (x(t), u(t)) = gxi (x(t), u(t)).

Integrating this last equation with respect of t, we get Pi (t) = Pi (to) + f

tax (x(s), u(s))ds

and we write the Euler-Lagrange equation in the integral form (satisfied a.e.):

rt

8xi W* u(t)) = Pi (to) +

8L (x(s), u(s))ds,

(2.30)

Moreover, if the curve t'- x(t) is C2 we obtain by differentiating at a (x(t), u(t)) = ax WO, u(t))

everywhere on [0, TI.

2.2.11 Comparison with the Calculus of Variations Although we can recover the Euler-Lagrange equation from the weak maximum principle, the two viewpoints are radically different and the point of view of the maximum principle is much superior to the one of the calculus of variations in several ways: 1. we impose minimal regularity assumptions on the set of curves; 2. we use the concept of the augmented system where the derivative of the

cost is a state variable and the adjoint vector have a clear geometric explanation; 3. we obtain a set of equations in the Hamiltonian form without using the Legendre transformation which is not in general well-defined.

2.2.12 LQ-Control and the Weak Maximum Principle We can apply the maximum principle to get optimality necessary conditions in the LQ-problem. For the sake of simplicity we analyze only the autonomous case. We consider the problem of minimizing the cost

IT( C(u) =

tx(t)Rx(t) +t u(t)Uu(t)dt

2.3 Pontryagin's Maximum Principle (PMP)

45

among the set of curves satisfying

zb(t) = Ax(t) + Bu(t), X(t) E Rn, U(t) E Rm where A, B, R, U are constant matrices, and R, U are symmetric. We assume to have fixed boundary conditions: x(O) = xo, x(T) = x1. Moreover we impose the (strong Legendre) regularity condition: U > 0

and we assume that the linear system i(t) = Ax(t) + Bu(t) is controllable, that is the rank(R) = n where of 7Z = [B, , An-1B]. From this last assumption, any minimizer is normal and according to Corollary 8, a minimizer is solution of the following constraint Hamiltonian system:

x(t) =

8p

(x(t), p(t), u(t)),

p(t)

aH (x(t), p(t), u(t) ),

a WO, p(t), u(t)) = 0 where FI (x, p, u) = (p, Ax + Bu) - 1( txRx + tuUu). Solving the linear = 0, we get with U > 0, that an optimal control is defined by equation the (dynamic) feedback

u(p) = U-' tB tp

where p is written as a row vector. Introducing the Hamiltonian function

H(x,p) = H(x,p,u(p))

.,

-

= 0, we get _ . Hence and using the constraint an optimal trajectory is the projection on the x-space of a solution of the following Hamilton system:

i(t) =

8

(x(t),p(t)),

P(t)

aft

a (x(t),p(t))

2.3 Pontryagin's Maximum Principle (PMP) In this section we state the Pontryagin maximum principle and we outline the proof. We adopt the presentation from Lee and Markus [77] where the result is presented into two theorems. The complete proof is complicated but rather standard, see the original book from the authors [91].

Theorem 8. We consider a system of R': i(t) = f (x(t), u(t)), where f Rn++n -- R' is a Cl-mapping. The family U of admissible controls is the set of bounded measurable mappings u(.), defined on [0, T] with values in a control domain Il C Rm such that the response x(., xo, u) is defined on [0, T].

Let U(.) E U be a control and let 7(.) be the associated trajectory such that

46

2 Optimal Control for Nonlinear Systems

T(T) belongs to the boundary of the accesibility set A(xo, T). Then there exists p(.) E 1111\{0), an absolutely continuous function defined on [0,21 solution almost everywhere of the adjoint system:

At) = -p(t) of (T (t), u(t))

(2.31)

such that for almost every t E (0, T] we have

H(T(t), p(t), u(t)) = M(Y(t), p(t))

(2.32)

where

H(x, p, u) = (p, f (x, u)) and

M(x, p) = max H(x, p, u). UEf1

Moreover t F-- M(T(t), p(t)) is constant on [0,T].

Proof. The accessibility set is not in general convex and it must be approximated along the reference trajectory T(.) by a convex cone. The approximation is obtained by using needle type variations of the control u(.) which are closed for the L1-topology. (we do not use L°° perturbations and the Frechet-derivative of the end-point mapping computed in this Banach-space) 0

Needle type approximation. We say that 0 < t1 < T is a regular time for the reference trajectory if d

dt a1

J

f (7(7-), u(r))dr = f (Y(tl), u(tl))

and from measure theory we have that almost every point of [0, T] is regular. At a regular time t1, we define the following L'-perturbation U,(.) of the reference control: we fix 1, e > 0 small enough and we set

ue(t) =

u1 E .R constant on [tI - le,tl] -u(t) otherwise on [0, T]

We denote by 7,(.) the associated trajectory starting at TE(0) = x0, see Fig. 2.2. We denote by a -+ at(e) the curve defined by at(e) = Te(t) for t > ti. We have

xf(tl) = (t1 - lE)+

t1

t1 -IE

f (Ye(t),uf(t))dt

where uE = ul on [t1 - lE, t1]. Moreover

Y(ti) = 7(t1 - lE) +

1

t1

!e

f (T (t), u(t))dt

2.3 Pontryagin's Maximum Principle (PMP)

47

xo

Fig. 2.2.

and since ti is a regular time for a(.) we have

Ye(ti) -Y(ti) = 1E(f (x(ti),ui) - f (x(tl), u(ti)) + o(E). In particular if we consider the curve E '- at, (E), it is a curve with origin T(ti) and whose tangent vector is given by

v = 1(f(T(ti),u1) -f(x(ti),'u(ti)).

(2.33)

For t > ti, consider the local diffeomorphism: ¢t(y) = x(t,ti,y,u) where x(., ti, y, u) is the solution corresponding to u(.) and starting at t = ti from y. By construction we have at(e) = cbt(at, (e)) for e small enough and moreover for t > ti, vt = W16-0 t t(E) is the image of v by the Jacobian V.L. In other words vt is the solution at time t of the variational equation dV

Of

dt = ax

(x (t), u(t))V

(2.34)

with condition vt = v for t = ti. We can extend vt on the whole interval [0,T). The construction can be done for an arbitrary choice of ti, I and ul. Let 17 = {t1,1, ui } be fixed, we denote by v[j (t) the corresponding vector vt.

Additivity property. Let t1, t2 be two regular points of u(.) with ti < t2 and 11, 12 > 0 small enough. We define the following perturbation ii (t) =

I

ui on [ti - lie,ti] t2] u2 on [t2 u(t); otherwise on [0, T]

48

2 Optimal Control for Nonlinear Systems

where u1, u2 are constant values of (1 and let 7E(.) be the corresponding trajectory. Using the composition of the two elementary perturbations III = {t1i l1, u1 } and 172 = {t2, 12, u2} we define a new perturbation

H : {t1i t2, ll)l2, ui, u2}. If we denote by vn, (t), vn2 (t) and v,7 (t) the respective tangent vectors, a computation similar to the previous one gives us:

vn (t) = vn, (t) + vn, (t),

fort > t2.

We can deduce the following lemma.

Lemma 8. Let 1 7 = It,,

, t87 A111 i , Asia, ul, , us} be a perturbation at regular times ti, ti < < t8i li > 0, Ai > 0, Fi=1 Ai = 1 and corresponding to elementary perturbations Hi = {ti, li, ui} with tangent vectors vn: (t). Let 7E(.) be the associated response with perturbation H. Then we have

xE(t) = x(t) +

EAivrfi (t) + o(e)

(2.35)

i=1

where

0, uniformly for 0 < t < T and 0 < Al < 1.

Definition 20. Let u(.) be an admissible control and a(.) its associated trajectory defined for 0 < t < T. The first Pontryagin's cone K(t), 0 < t < T is the smallest convex cone at x(t) containing all elementary perturbation vectors for all regular times ti.

Definition 21. Let vi,

, v,, be linearly independent vectors of K(t), each vi being formed as convex combination of elementary perturbation vectors at distinct times. An elementary simplex cone C is the convex hull of the vectors vi.

Lemma 9. Let v be a vector interior to K(t). Then there exists an elementary simplex cone C containing v in its interior. Proof. In the construction of the interior of K(t), we use convex combination of elementary perturbation vectors at regular time not necessary distincts. Clearly by continuity, we can replace such a combination by a cone C in the interior with n distinct times.

Approximation lemma. An important technical lemma is the following topological result whose proof uses the Brouwer fixed point theorem.

Lemma 10. Let v be a nonzero vector interior to K(T), then there exists A > 0 and a conic neighborhood N of Av such that N is contained in the accessibility set A(xo,T). Proof. See [77].

2.3 Pontryagin's Maximum Principle (PMP)

49

The meaning of this lemma is the following. Since v is interior to K(T), there exists an elementary simplex cone C such that v is interior to C. Hence for each w E C there exists L() a perturbation of u(.) such that its corresponding trajectory 7e(.) satisfies

7,(T) = Y(T) + ew + o(w). In particular there exists a control U,(.) such that we have

Y,(T) = x(T) + ev + o(e). This geometric situation is illustrated on Fig. 2.3, by construction 7e(T) E K(T). In other words K(T) is a closed convex approximation of A(xo, T).

Ev

Fig. 2.3.

Separation step. To finish the proof, we use the geometric Hahn-Banach theorem. Indeed if x(T) E OA(xo, T) there exists a sequence x V A(xo, T) such that x,, -, Y(T) when n - +oo and the unit vectors x^-x T have

a limit w when n - oo. The vector w is not interior to K(t) otherwise from Lemma 10 there would exist A > 0 and a conic neighborhood of Aw in

A(xo, T) and this contradicts the fact that x,, 0 A(xo, T) for any n. Let a be an hyperplane at a(T) separating K(T) from w and let p' be the exterior unit normal to 7r at !(T), see Fig. 2.4. Let us define p(.) as the solution of the adjoint equation P(t) = -p(t) 8x (x(t), u(t)) satisfying p(T) = p. By construction, we have

p(T)v(T) < 0 for each elementary perturbation vector v(T) E K(T) and since for t E [0, T] the following equations hold:

50

2 Optimal Control for Nonlinear Systems

A(x0,T)

Fig. 2.4.

P(t) = -p(t) ax (x, u),

v(t)

ax

(Y, u)v

we have

p(t)v(t) = 0.

Hence p(t)v(t) = p(T)v(T) < 0, Vt. Assume that the maximization condition (2.32) is not satisfied on some subset S of 0 < t < T with positive measure. Let t1 E S be a regular time, then there exists ul E S7 such that p(t1)f(7(t1), 9(t1)) < p(t1)f(T(t1), ul) Let us consider the elementary perturbation 171 = {t1,1, ul } and its tangent vector

VIII (ti) = 1[f(x(tl),ul) -f(x(t1),u(t1)}. Then, using the above inequality we have that P(tl )vn, (t1) > 0

which contradicts p(t)vn, (t) < 0, for all t. Therefore the equality

H(T(t),p(t),u(t)) = M(Y(t),p(t)) is satisfied a.e. on 0 < t < T. Using a standard reasoning we can prove that t - + M(x(t), p(t)) is absolutely continuous and has zero derivative almost everywhere on 0 < t < T, see [77].

2.3 Pontryagin's Maximum Principle (PMP)

51

Theorem 9 (Maximum Principle). Let us consider a general control system: i(t) = f (x(t), u(t)) where f is a continuously differentiable function and let Mo, M1 be two Cl submanifolds of Rn. We assume the set U of admissible controls to be the set of bounded measurable mappings u : [0, T(u)] .0 E Rm, where !l is a given subset of Rm. Consider the following minimization problem: minUEU C(u), C(u) = fo f °(x(t), u(t))dt where f ° E C', x(O) E Mo, x(T) E Ml and T is not fixed. We introduce the augmented system:

i°(t) = f°(x(t),u(t)),

x°(0) = 0

(2.36)

i(t) = f (x(t), u(t)),

(2.37)

x(t) = (x°(t),x(t)) E R"+1, f = (f°, f). If (x*(.),u*(.)) is optimal on [0,T'1, then there exists P'(.) = (p°, p(.)) : [0, T*] -+ R"+1\{0} absolutely continuous, such that (i' (.), P' (.), u' (.)) satisfies the following equations almost everywhere on 0 < t < T*: x (t) = 8p WO , P(t) , u (t)) , p- (t) = - aH (x (t) , P(t) , u (t)) fI(x(t), P(t), u(t)) = )cf(x(t), P(t))

(2 . 38)

(2.39)

where

H(x,0,u) _

Ax, u)), M(x,P) = En H(x,P,u)

Moreover, we have

M(x(t),P(t)) = 0,dt, p° < 0

2.40)

and the boundary conditions (transversality conditions):

x'(0) E Mo, x*(T*) E Ml, P (0)1T .(o)Mo, p'(T')1T=.(T)Ml.

(2.41)

(2.42)

Proof (for the complete proof, see [771 or [91]). Since (x'(.),U*(.)) is optimal

on [0,T*] the augmented trajectory t -4 x' (t) is such that x' (T) belongs to the boundary of the accessibility set A(x' (0), T*). Hence by applying Theorem 8 to the augmented system, one gets the conditions (2.38),(2.39) and k constant. To show that M = 0, we construct an approximated cone K'(T) containing K(T) but also the two vectors ± f (x' (T ), u' (T)) using time variations (the transfer time is not fixed). To prove the transversality conditions, we use a standard separation lemma as in the proof of Theorem 8. Definition 22. A triplet (x(.), p(.), u(.)) solution of the maximum principle is called an extremal.

2 Optimal Control for Nonlinear Systems

52

2.4 Filippov Existence Theorem In order to solve optimal control problems, we need an existence theorem about optimal trajectories. The following existence theorem can be found in [77] with a complete proof (in a more general setting than the one stated here).

Theorem 10. Consider a control system in 1R": ±(t) = f (x(t), u(t)), where f is C1, with the following data: - The initial and target sets M0, Ml are nonempty compact sets of R. - The control domain 42 is a nonempty compact set in Rm. - The state constraints are of the form hi (x) > 0, i = 1, , p where the hi are Co functions on R". - The set U of admissible controls is the set of measurable mappings u(.) [0, T] -> Si, such that each u(.) has a response x(.), 0 _< t < T steering

x° E Mo to x(T) = x1 E Ml, and t i-- x(t) is entirely contains in the restraint set: h; (x) > 0. - The cost to be minimized is for each u(.) E U of the form:

C(u) =

o 0

f°(x(t),u(t))dt

where f ° is a C1 function. We assume the following.

1. The family U is not empty, that is there exists u(.) steering xo E M° to

x1EM1. 2. For each response x(.) defined on [0,T] corresponding to u(.) E U, there exists an uniform bound:

Ix(t)I
0
3. The extended velocity set:

V(x) = {(f°(x,u), f(x,u)); u E (1) is a convex subset of R"+1 for each x in the state space. Then, there exists an optimal control u' (.) E U minimizing C(u).

2.4.1 Comments about the Existence Theorem The proof is essentially the same than the one about the existence of optimal controls for the time minimum problem for linear systems we presented in Chap. 1, Proposition 7. The convexity assumption is necessary as shown by the following example:

2.5 Dynamic Programming and the Maximum Principle

C(x) = min u(.)

53

r1

J0

(1 - u2(t) + x2(t))dt,

Wi(t) = u(t),

u(t) 1< 1, x(0) = xo, x(1) = 0.

We can construct a sequence {x"(.)} converging uniformly to 0 such that

in(t) = 1 a.e., and such that C(x") -, 0 when n - +oo. But if fo (1 u2(t) + x2(t))dt = 0, then 1

- u2(t) + x2(t) = 0 a.e. and x(t) = 0 on [0, 1).

This is a contradiction because the cost corresponding to the trajectory identically zero is 1 and hence not minimum. If the convexity assumption is not satisfied, we can convexity the problem in order to get a well-posed optimal control problem, see [13]. Some classical optimal control problems like the time minimum problem for affine systems ±(t) = Fo(x(t)) + F(x(t))u(t), u(t) E .f2 with fl a convex set are convex. Hence it is only required to check the uniform bound of assumption 2.

2.5 Dynamic Programming and the Maximum Principle We give in this section a brief account of the relationship between the dynamic programming of Bellman and the maximum principle. It is also an extension

of the Hamilton-Jacobi theory. It is based on a very simple principle: each

piece of an optimal trajectory is optimal. For the sake of simplicity it is presented for the time minimum problem. We consider a system: d(t) = f (x(t), u(t)) in R", where f is a Cl-mapping.

We fix the terminal point x1 and let u(.) defined on an interval [to, t1] be an optimal control transfering xo to x1. We denote by T(xo) the optimal transfer time t1 - to. We make the following assumption:

Assumption (H1) Each optimal trajectory is C' and t i- T(x(t)) is C'.

Let w(x) = -T(x). If x(.), to < t < tl is the corresponding optimal trajectory and since every piece of x(.) is optimal we have T(x(t)) = ti - t, T(xo) = ti - to Hence

w(x(t)) = -T(xo) + t - to. Differentiating this last relation with respect to t, one gets:

a

(x(t))fa(x(t),u(t)) = 1.

Consider now the response with initial condition x(t) at t and apply a constant control v E .R on the small time interval dt. At time t + dt the position

2 Optimal Control for Nonlinear Systems

54

is x(t) + dx, where dx = f (x(t), v)dt. Moreover the time needed to transfer x(t) + dx to x1 is T(x(t) + dx). We have

T(x(t)) < dt + T(x(t) + dx), which can be written as

w(x(t) + dx) - w(x(t)) < dt. Therefore

n

E a a (x(t)) fQ (x(t), v)dt < dt ct=1

which is equivalent to the relation

a=1

ax

(x(t))f0(x(t),v) < 1, dv E n.

Moreover the left-hand side is 1 for v = u(t). Therefore we get the fundamental relation of the dynamic programming generalizing Hamilton-JacobiBellman equation: manxQ>

as (a)

fo.(x,u) = 1

(2.43)

and the maximum is reached for the optimal control. Let us now make the following additional assumption:

Assumption (H2) The function x " T(x) is C2. We can derive the equations of the maximum principle as follows. Consider the function n 9(x,u) _

YX-

Let u(.) be an optimal control and x(.) be the corresponding trajectory. Fix t and consider the mapping

x " g(x, u(t)) Then for all x, we have g(x, u(t)) < 1 and for x = x(t) the maximum is reached. Hence differentiating with respect to x we get 09(x, u(t)) = 0 at x = x(t). T

Simple computations lead to the following relation at x = x(t): n

n (a

a2 axa

(X)f. (x, u) +

a (x) axa (x, U)) = 0

for i = 1, ... , n.

2.5 Dynamic Programming and the Maximum Principle

55

We can write n

2

E as

ax;(x(t))fa(x(t),u(t)) =

E a- (ax) (x(t)) ac (t) a=1

a=1

d (0,

d ax (x(t))). Therefore along an optimal pair (x(.),u(.)) we have d

dt

(

n

(t))

E 8x. (x(t)) a (x(t),u(t))

aax;

, pa (t))

Introducing the adjoint vector p(t) = (pl (t), level set w(x) = constant: P=(t)

as the gradient to the

= ax (x(t)),

we get the equation a

8i (x(t), u(t))P.(t) and the maximization relation n

pa(t)fa (x(t), u(t)) = m

n

P. (t)fe(x(t), v).

Moreover the maximum is constant and equal to 1. These are the relations of the maximum principle.

Notes and Sources The results about the calculus of variations are standard. See the book by Bolza for his historical interest and the books by Caratheodory [37), GelfandFomin [47] for a modern presentation. The reference [30] makes a concise and

nice introduction of the subject and contains interesting examples and exercises. The book by Bliss [18) is a classic and rather complete reference of the discipline. For the Pontryagin maximum principle standard references are Pontryagin and al. [91], Lee-Markus [77] and Fleming-Rishel [44]. A maximum principle with minimum regularity assumptions was obtained recently by Sussmann [101] but is rather technical; see also Clarke [41]. For existence theorems, see [77) or [38].

56

2 Optimal Control for Nonlinear Systems

Exercises 2.1 We consider the following functional in R2: rep

J(x, y) =

f (t, x, y)

1 + ±2(t) + 0(t)dt

Jto

defined on the set of smooth curves whose extremities belong respectively to surfaces parametrized by y = w(t, x), y = ii(t, x).

Use the fundamental formula of the classical calculus of variations to writ necessary optimality conditions (transversality conditions). Give a geometric interpretation.

2.2 We consider the minimization problem in R, with fixed extremities: min fto' L(t, x(t), i (t))dt. Let t'--, x(t) be a piecewise smooth minimizer, with an angular point at c E)to, tl [. Use the fundamental formula to get necessary

optimality conditions at c (Erdmann and Weierstrass conditions (1877)). Present those conditions using Hamiltonian formalism. Compare with the maximum principle.

2.3 Let J(f) = fo (f'2(x) - b f 2(x))dx) where a, b > 0, evaluated on the set of smooth functions f : [0, a) R, such that f (0) = f (a) = 0. Find a relation between a, b for which

1. J(f) > 0 for f 0 0. 2. J(f) > 0. Deduce the following inequality: f07r

f 2(x)dx > f 2.4 Consider the linear system: i(t) = A(t)x(t) + B(t)u(t), x(t) E 1R", u(t) E 1R' and the quadratic cost T

C(u) = tx(T)Qx(T) + f (tS(t)W(t)x(t) +t u(t)Imu(t))dt o

with x(to) fixed and x(T) free. We assume t --, A(t), B(t), W (t), U(t) be continuous matrices and Q, W (t), U(t) being symmetric matrices with Q, W(t) > 0. The control u(.) is taken in L2([to,TJ). We call Riccati equation associated to the LQ- problem the equation

k(t) = -tA(t)K(t) - K(t)A(t) + K(t)B(t)tB(t)K(t) - W(t)

(2.44)

and we denote by K(t, K1, t1) the maximal solution defined on a subinterval J of IR", with K(tl) = K1.

2.5 Dynamic Programming and the Maximum Principle

57

1. Consider the scalar equation: k(t) = k2(t) + 1 and k(O) = 0. Show that the solution is not defined on R. 2. Assume there exists a symmetric matrix K(.) solution of the equation (2.44) on [to, TJ and let x(.) be a curve solution of ±(t) = A(t)x(t) + B(t)u(t). Show the relation T (t Jeo

t BK

r 0

tAK+KA][ux])dt-[

l [uxJlKB K+

T xKxJto

-- 0.

3. Assume that the solution 17 = K(t, Q, to) is defined on the whole interval to < t < T. Show that there exists a control u' (.) minimizing C(u) among

all the solutions satisfying i(t) = Ax(t) + Bu(t), x(to) = xo fixed, x(T) free, given by the feedback

u'(t,x) = - LB(t)17(t)x*(t) and that the minimal cost is

C(u*) = tx'(to)17(to)x'(to). 4. Consider a general Riccati equation

W(t) = A(t) + W(t)B(t) + C(t)i4'(t) + W (t)D(t)W(t)

(2.45)

where W, A, B, C, D- are respectively n x k, n x k, k x k, n x n, k x n matrices

and t i-- A(t), B(t), C(t), D(t) are continuous. Show that (2.44) is a particular case of (2.45) with: k = n, A = -W, B = -A, C = -'A, ,b = BtB. 5. Consider the matrix equation

.k(t) X (t) C(t) A(t) (1'(t)) _- ( -D(t) -B(t)) ( Y(t) )

(2,46)

where X (t) is a n x k matrix and Y(t) an invertible k x k matrix. For to < t < T, set W(t) = X(t)Y-1(t) and show that W(.) is solution of the Riccati equation. 6. Show that the PMP equations associated to the LQ-problem with A, B, W constants are of the form

(i) -H(x)'

H- (BtB A)

Lifting this equation to

CY) =H(Y)

X, Y: n x n matrices

write the corresponding Riccati equation. Compare with equation (2.44). 7. Show that for each t, H(t) satisfies JH +t HJ = 0 where J is the 2n x 2n

matrix:

In

I0 ).

58

2 Optimal Control for Nonlinear Systems

2.5 Consider the following system on Rn: b(t) = X (x(t)) + u(t)Y(x(t)), where X, Y are analytic vector fields.

1. Show that a singular trajectory x(.) corresponding to u(.) satisfies the following equations for some p(.), p(t) E R'\{0}:

= X(x(t)) + u(t)Y(x(t)),

x(t)

(2.47)

At) _ -At)(g x(t) + u(t)

x(t)) (p(t), Y(x(t))) = 0, Vt.

(2.48)

2. Let [Z1, Z21 = s Z2 - s Zi be the Lie bracket for Z,, Z2 : Rri - R" and adZi (Z2) = [Zl, Z2]. Show that if (p(t), ad2Y(X)(x(t))) t then the singular control is given by (p, ad2X (Y)(x))

u (x, p) _ + (p, ad2Y(X)(x)

0 for each

(2 . 49)

and the trajectory t ' x(t) is analytic and solution of

i(t) =

aH(x (t) ,p (t))

(2 . 50)

(p(t), Y(x(t))) = (p(t), [X, Y](x(t))) = 0

(2.51)

ap

( x (t) ,p (t))

, P(t)

with H(x, p) = (p, X (x) + uY(x)). 3. Assume that t i- x(t) is a trajectory on [0, T] corresponding to the (analytic) control u(t) = 0. Show that t i- x(t) is singular if and only if there exists p(t) E R"\{0} such that (p(t), adkX (Y)(x(t))) = 0

(2.52)

for each k E N, and each t E [0, T]. Deduce that for 0 < t < T, the image of L- ([0, t]) by Frechet derivative of the end-point mapping evaluated at u =_ 0, x(0) = xo is K(t) = Span{adkX (Y)(x(t)); k E N).

4. Consider now the system +(t) = X(x(t)) + u(t)Y(x(t)) defined on the plane. Show that the singular trajectories are contained in the set S : {x E R2; det(Y(x), [X,Y](x)) = 0}. Assume X and Y linearly independent on

a nonempty open set U C R2. Define the one-form on U. a = aldxl + a2dx2 by the relations (a, x) = 1 and (a, X) = 0, where (a, Z) = a(Z) for a given vector field Z on R2. Show that S n U is the set of points where dot is not a volume form i.e. {x E R2; da(x) = 0}. 2.6 The objective of this exercise is to solve the Dido problem: among all closed curves in the plane whose length is fixed to L find the one enclosing a domain D with maximal area. 1. Show that D is convex and that we may divide D into two domains D+ and D_ with same area and equal length 2 .

2.5 Dynamic Programming and the Maximum Principle

59

2. Deduce that the problem is equivalent to maximize A = J

1 - y'2(s)ds

y(s)

0

where s represents the length of the curve, y(O) = y(Z) = 0 and y(s) > 0 for s E [0, ]. a 3. Apply the Euler-Lagrange equation to prove that the solution is given by the half-circle with radius r = 2 . 4. Consider the following optimal control problem: th(t) = u(t)

W) = v(t) z(t) = 2 WOO) - i(t)y(t)) T

min

J0

22(t) + y2(t)dt.

a) Show that z(T) - z(0) represents the area swept by the curve t '(x(t), y(t)), t E [0, T]. b) Apply the maximum principle and compute the extremals. What is conclusion do you reach? 2.7 [Bernouilli problem] We consider the following problem: determine for an heavy particle the curve of quickest descent in a vertical plane between two

given points. We take the positive y-axis vertically downward and denote by g the gravitational constant, by yo the initial y-coordinate and by vo the initial velocity, which we suppose to be nonzero.

1. Show that the problem can be set as:

t, min

to

x2(t) + y2(t) dt

y(t --yo + k

9

where k = 2g 2g

2. Assume that we restrict the problem to curves given by a graph of the form (x, y(x)). Show that then the problem can be restated as: 1 -TO (x)

min Z'01'

--yo y(x)

+ k

dx.

Write the corresponding Euler-Lagrange equation.

3. Let L :

(y, y)

R be a smooth mapping. Prove that H =

y-L

is constant along the solutions of the Euler-Lagrange equation. LYse this property to deduce the smooth graphs (x, y(x)) solutions of the Bernouilli problem.

60

2 Optimal Control for Nonlinear Systems

4. Let a E R and consider the following minimization problem: min

xl y°`

Jxo

1 + y2(x)dx

among the set of smooth curves y e -+ y(x) with y(xo) = yo, y(xl) = yi. Discuss the solutions according to a.

2.8 [Weierstrass 1879] Let F : (x, y, i, y) -+ R be a smooth mapping and assume that for all k > 0 the following is satisfied: F(x, y, ki, ky) = kF(x, y, i, y).

Consider the minimization problem given by: min ft. t F(x(t), y(t), i(t), y(t))dt.

(2.53)

to

1. Prove the following relations:

F. = ±F±1 + iFyx

FU =iFiy+yFyv iFyz + yFyy = 0

iFxy+yFii,=0 and deduce that for (i, that:

# 0, there exists a smooth mapping F1 such F00 = i2F1

Fyi-y2F1 F:jo _ -iyF1. Compute F1 for F = y\,/±2 + y2. (0, 0). Prove 2. Let T = Fxy - F11i + F1 (1+9 - #) and assume that (i, that the Euler-Lagrange equation is equivalent to the Weierstrass equation: T = 0. 3. Consider the problem: min f F, among the set of piecewise smooth curves with fixed extremities and contained in a domain D with a boundary. Let

(i(t), y(t)) be a curve contained in the boundary and joining 2 to 3 as represented in Fig. 2.5. By taking variations (7, y) with fixed extremities

i = i + , y = y + n, prove that if (a, y) is a minimizer, then

T<0

(2.54)

where T denotes T evaluated along (1, y). Assume F1 > 0 in the domain and let ,1-, be the curvature

2.5 Dynamic Programming and the Maximum Principle

61

(2.55)

Take P a point in the boundary and let be the curvature of (1, y) at P and let .1 be the curvature at the same point P of the extremal tangent to the boundary. Prove that (2.55) can be written: 1

1

- > r

Assume F =

(2.56)

r

x2 -+0, find a geometric interpretation of the above

condition.

4. Let 023 be a minimizing curve meeting the boundary at the point 2. By taking variations indicated on Fig. 2.5, prove the following necessary condition: E(x2, y2) ±2, /2, x2, y2) = 0

(2.57)

where E is the excess Weierstrass function: E(x, y, x, 1/l, X-, Y-) = F(x, y, x, y) -

(F(x,y,±,) + yF'F (x, y, x, y)) (2.58)

and (d2, y2), (x2, y2) are the respective tangents at point 2 of the arcs 02 and 23.

Assume Fl to be nonzero in the domain. Prove that a minimizer must hit the boundary tangentially. 5. Let 2 = 3 in Fig 2.5 and let 021 be a minimizing curve with a single point 2 in common in the boundary. Obtain the necessar o timality conditions and find a geometric interpretation if F = i2 + y2. 2.9 Consider the following quadratic functional: 6

J(h) = j ( P(x)h 2(x) + Q(x)h2(x))dx

(2.59)

2 Optimal Control for Nonlinear Systems

62

where P(x) > 0 for a < x < b and h(a) = h(b) = 0. We divide the interval [a, b] into (n + 1) equal parts of length dx:

dx = xi+1 - xi = (n + 1)'

2 = 0,1,...,n

(2.60)

and let Pi, Qi, hi be the value of the functions P(x), Q(x), h(x) at the point

x=xi.

1. Show that the functional J can be approximated by a quadratic form R in n variables h1, , hn defined by the symmetric n x n matrix: 0

0

0

0

0

0

0 b2 a3 0 0

0

f a1 bl 0

...

b1 a2 b2

0 0 0 ... bn-2 an-1 bn-1 000 0 bn-1 an where

az

ns

1.

2. If Di are the descending principal minors, prove the following recurrence relation:

Di=a1Di_1_b?_1Di,

i=3,...,n

(2.61)

and set be convention Do = 1, D_1 = 0. Deduce the relation:

Di = (Qidx +

Pi-1 + Pi

dx

)D,-1

_ P? Di-2, i = 1, ... , n. (1x)2

(2.62)

3. Making the following change of variables: __

D`

P1...Pi Zi+11 i = 1,

,n

(AX)i+1

D-1=Zo=0, prove for i = 1,

, n the following relation:

1 (PZi+1-Zi

QiZi - dx

AX

`

-Pi-1Zi+1-A

Ax

= 0.

(2.63)

Passing formaly to the limit Ax - 0, obtain the following differential equation:

-

(PZ') + QZ = 0.

Compare with the Jacobi equation.

(2.64)

2.5 Dynamic Programming and the Maximum Principle

63

4. assume R > 0 and use the Sylvester criterion to prove that Z; > 0 for i = 1, , n + 1 and deduce formally that the solution of (2.64) with initial condition Z(O) = Za = 0, Z'(0) = limax.o Z1-Z0 = 1 does not vanishe for a < x < b.

3 Geometric Optimal Control

Geometric optimal control is the computation of the optimal control using geometric methods. It is mainly based on the geometric analysis of the equar tions of the maximum principle. The objective of this chapter is to give an introduction to this analysis. We use three articles for this presentation. One is the article by Ekeland [42], which is historically the first one using singularity theory to locally classify the extremals under generic assumptions. The second one is from Kupka [72] and is a basic article to deal with the time optimal control problem for single-input affine systems. Also [711 gives a model for the Fuller phenomenon. Finally, we present the results from Sussmann [102] for planar systems. It is an introduction to finite closed loop parametrization to compute the synthesis for time minimum problem of affine control systems.

First of all we give an introduction to symplectic geometry which plays an important role in geometric optimal control.

3.1 Introduction to Symplectic Geometry In the sequel, all objects are assumed to be COO.

Definition 23. Let V be a It-linear space of dimension 2n. This space is said to be symplectic if there exists an application w : V x V -> R which is bilinear, skew-symmetric and nondegenerate, that is if w(x, y) = 0 for all x E V then y = 0. Let W be a linear subspace of V. We denote by W- the set

Wl = {x E V; w(x, y) = 0 dy E W}.

The space W is isotropic if W C W -L. An isotropic space is said to be Lagrangian if dimW =dim a . Let (Vi, wi ), (V2, w2) be two symplectic lin-

ear spaces. A linear mapping f

:

V1 - V2 is symplectic if wl (x, y) _

W2 (f (x), f (y)) for each x, y E V1.

Proposition 14. Let (V, w) be a linear symplectic space. Then there exists a basis fell , en, fi, ',fn} called canonical defined by w(ei, ej) = w(fi, fj) = 0 for/1 for( 1 < i,, j < n and w(ei, fJ) = 8ij (Kronecker symbol). If

J is the matrix

U

)

where I is the identity matrix of order n, then

66

3 Geometric Optimal Control

we can write w(x, y) _ (Jx, y) where (,) is the scalar product (in the basis (e,, f;)). In the canonical basis, the set of all linear symplectic transformations is represented as the symplectic group defined by Sp(n, R) = IS E

GL(2n, R); tSJS = J}. Definition 24. Let M be a C°°-manifold with dimension 2n. A symplectic structure on M is defined by a 2-form w such that dw = 0 and w is regular, that is Vx E M, wz is nondegenerate.

Proposition 15. For any C°°-manifold M with dimension n, the cotangent bundle T*M admits a canonical symplectic structure defined by w = da where

a is the Liouville form. If x = (x1, .. , x") is a coordinate system on M and (x, p) with (pi, , pn) the associated coordinates on T'M, the Liouville form is written locally as a = p;dx' and w = doe = Enj=1 dp; A dx'. 1

Proposition 16 (Darboux). Let (M, w) be a symplectic manifold. Then given any point of M, there exists a local system of coordinates called Darboux-coordinates, (xl, , xn, p1i , pn) such that w is given locally by dpi A dx'. (Hence the symplectic geometry is a geometry with no local F n1 invariant)

Proof (sketch of the proof). A standard proof uses a recurrence argument on the dimension of the space, but we use here the homotopy method which

is a basic technique to compute normal forms in geometric problems. Since the result is local we work in a neighborhood of U C R2n of the origin. We consider the 2-form wo = dp A dq and we want to construct a germ of diffeomorphism cp such that yo*wo = w1, where w1 = w (notation). We proceed as follows:

- Step 1. Using a linear change of coordinates, we may assume w(0) _ (dp n dq)I0.

- Step 2. Let us consider the following deformation: wt = (1 - t)wo + tw, where t E [0,1]. The idea of the proof is to construct a family of diffeomorphisms Pt, 0 < t < 1 such that Vt * wt = wo. This family is obtained by integrating a time depending vector field: n

x(t, x) _

+ E x;(t, x) 8xi i=1

Since 'Pt * wt = wo for all t, we have 0

=

d wt-(Wt * Wt)-

- Step S. To solve the previous equation, we use the Cartan-Baker-CampbellHausdorff formula: Lx 12 = d(ix 12) + ixdfl

3.1 Introduction to Symplectic Geometry

67

exp tx.11-tt where 0 is a smooth form, X a vector field, Lx fl = (Lie derivative) and ix is the interior product (see Definition 25). Hence

we get dwt

0 = Sot * (dt + ixdwt + d(ixwt)) and since wt is closed, we deduce that: dwt = 0. Therefore, we must solve the equation: d(ixwt) att.

Hence, we must find a 1-form at = ixwt such that dat = - dt = WO - w1. Since d(wl - wo) = 0 we can apply the Poincare lemma i.e., there exists a one form ,Q = at such that dQ = wo - wl. To conclude the proof, one needs to compute the vector field X such that at = ixwt. This equation is locally solvable since wt is of full rank at 0: wtlo = (1 - t)wolo + twilo = wolo

Definition 25. Let (M, w) be a symplectic manifold and let X be a vector field on M. We note ixw the interior product defined by ixw(Y) = w(X, Y) -, for any vector field Y on M. Let H : M -+ R. The vector field denoted by H and defined by ig(w) = -dH is the Hamiltonian vector field associated to H. If (x, p) is a Darboux-coordinates system, then the Hamiltonian vector field is expressed in these coordinates by:

H=

"8H8 -aHa i=1

api ax=

axi

a i

Definition 26. Let F, G : M - R be two mappings. We denote by {F, G} H the Poisson-bracket of F,G defined by IF, G} = dF(G).

Proposition 17 (Properties of the Poisson-bracket). 1. The mapping (F, G) " IF, G} is bilinear and skew-symmetric. 2. The Leibnitz identity holds: {FG, H} = G{F, G} + F{G, H} S. In a Darbouz-coordinates system, we have

IF, G}

n 8G OF =

api axi

8G 8F axi api

4. If the Lie bracket is defined by [F, G] =G o F - F o G, then its relation with the Poisson bracket is given by: [F, G] ={F,G}. 5. The Jacobi identity is satisfied: {{F, G}, H} + {{G, H}, F} + {{H, F}, G} = 0

68

3 Geometric Optimal Control

Definition 27. Let H be a Hamiltonian vector field on (M, w) and F : M R. We say that F is a first integral for H if F is constant along any trajectory of H, that is dF(H) _ IF, H} = 0. Definition 28. Let (x, p) be a Darboux-coordinates system and H : M - R. The coordinate x1 is said to be cyclic if = 0. Hence F : (x, p) H p1 is a first integral. Definition 29. Let M be a n-dimensional manifold and let (x, p) be Darbouxcoordinates on T *M. For any vector field X on M we can define a Hamiltonian vector field Hx onT'M by H(x, p) = (p, X (X)); Hx is called the Hamiltonian lift and Hx= X 8 - t ox p 8 . Each difeomorphism V on M can be lifted into a symplectic diffeomorphism co on T'M defined in a local system of coordinates as follows. I f x = co(y), then cp: (y, q) --, (x, p) _ (cp(y), t !q) .

Theorem 11 (Noether). Let (x, p) be Darboux-coordinates on T*M, X a vector field on M and Hx its Hamiltonian lift. We assume Hx to be a complete vector field and we denote by Wt the associated one parameter group.

Let F : T*M - R and let us assume that F o cot = F for all t E R. Then Hx is a first integral for F.

Definition 30. Let M be a manifold of dimension 2n + 1 and let w be a 2 -form on M. Then for all x E M, wz is bilinear, skew-symmetric and its rank is <- 2n. If for each x, the rank is 2n, we say that w is regular. In this case ker w is of rank one and is generated by an unique vector field X up to a scalar. If a is a 1 -form such that do is of rank 2n, the vector field associated to da is called the characteristic vector field of a and the trajectories of X are called the characteristics.

Proposition 18. On the space T"M x R with coordinates (x, p, t) the characteristics of the 1 -form E 1(p{dx` - Hdt) project onto the solutions of the Hamilton equations:

x(t) = ep (x(t), p(t), t),

p(t) = -

a (x(t), p(t), t).

Definition 31. Let co : (x, p, t) '- (X, P, T) be a change of coordinates on T*M x R. If there exist two functions K(X, P, T) and S(X, P, T) such that

pdx - Hdt = PdX - KdT + dS then the mapping cp is a canonical transformation and S is called the generating function of W.

3.2 One Dimensional Problems of Calculus of Variations

69

Proposition 19. For a canonical transformation the equations dx (t) = 8H (x(t), p(t), t), 8p Iii

dp (t)

8H (x(t),

dt

8x

p(t), t)

transform onto

dX (T) 8K (X (T), P(T), T), = 8P dT

dP (T) = - 8K (X(T), P(T), T). 8X dT

If T = t and (x, X) forms a coordinate system, then we have

dt (t) = with

WP-(X(t)1P(t)1t)1

(X (t), P(t), t)

dt (t)

as

ax 19S

p(t) = ax (X (t), P(t), t), P(t) = - x (X (t)' P(t)' t)' H(X(t), P(t), t) = K(X(t), P(t), t) - 5 (X (t), P(t), t). Remark 4 (Integrability). Assume that the generating function S is not de-

pending on t. If there exists coordinate such that K(X, P) = H(x, p) is not depending on P, we have k (t) = 0, X (t) = X (O); hence P(t) _ P(O) + t3Xlx=x(0) . The equations are integrable. With H(x,p) = K(X) we get

H(x, T) = K(X). Since X (t) = (X1(0), , X,, (0)) is fixed, if we can integrate the previous equation we get solutions of Hamilton equations. A standard method is by

separating the variables. This is called the Jacobi method to integrate the Hamilton equations. In particular this leads to a classification of integrable mechanical systems in small dimension, see [74].

3.2 Geometric Classification of Extremals in one Dimensional Problems of Calculus of Variations In this section we present the historical starting point for the classification of extremals using the singularity theory. It is due to Ekeland [42].

3.2.1 Problem Statement and Preliminaries Consider the following minimization problem min

J

0

f (x(t), x(t))dt; x(0) = xo, x(T) = xl

70

3 Geometric Optimal Control

where the transfer time T is fixed and f : R2 - R is a smooth mapping, among the set t ti x(t) of curves in R, defined and absolutely continuous with bounded derivative. The control system is

±(t) = u(t).

(3.1)

According to the maximum principle, an optimal solution can be computed as follows. The Hamiltonian is

H(x,p,u) = pu +pof(x,u) where po is a constant normalized by po = -1. The adjoint equation is P(t) = ax

(x(t),

u(t)).

(3.2)

We know that an optimal control must satisfy the constraint K = 0, that is p(t)

- au (x(t), U(0) = 0.

(3.3)

H(x, p, u) is constant along a solution. We asMoreover M(x, p) = sume that for each (x, p) E R x R, u h-' H(x, p, u) has at least one maximum over R.

Let .11 be the subset of (x, p) E R x R such that H(x, p, u) has an unique maximum denoted u(x, p) and moreover assume that for (x, p) E 12, the quadratic form s?1 is nondegenerate. According to the implicit func-

tion theorem, on 12 the control u(x, p) is smooth and from (3.3) we have p(t) = k(x(t), u(x(t), p(t)) everywhere. If we define a smooth Hamiltonian function by: ft (x, p) = H(x, p, u(x, p)),

along an extremal the following equations hold:

aH _ ax

au afafau_af

- p ex

ax

au ax ax' au a f au

aH 8p =u+pap -8p =u.

Hence on 12 the extremal solutions of the maximum principle are the C°Osolutions of the Hamiltonian equations:

i(t) =

OP

(x(t),p(t)),

P(t)

a(x(t),p(t)) x

(3.4)

3.2 One Dimensional Problems of Calculus of Variations

71

3.2.2 Singularity Analysis On Q, the map u H H(x, p, u) reaches its maximum at an unique regular point. Outside we must analyze the singularities. They are classified as follows:

- the mapping H(x, p, .) reaches its maximum at K distinct points uk, 1 < k < K where 1 < K < +oo; - each point Uk is a critical point with a singularity of codimension ck. (In this classification, we consider only singularities corresponding to local maxima)

Definition 32. Let (x, p) E R2. 1. The codimension of the singularity of H(x, p, .) is given by K

codH(x,p,.)=(K-1)+1: ck. k=1

2. We define a partition of R2 into subsets E/ 0 < i < +oo by

Ei = {(x,p) E R2; cod H(x,p, ) = i}. Let us describe the sets Eo, E1, E2. Description of E0 If (x, p) E E0 then K = 1 and Cl = 0. Hence we have

Eo=.R. Description of El. We have two cases:

1. K = 1, c1 = 1. This situation corresponds to an inflexional point. Hence we do not consider this case as it does not correspond to a maximum.

2. K = 2, c1 = c2 = 0. The maximum is reached for two distinct points u1i u2 both regular. Hence El corresponds to a competition between two regular local maxima. If we introduce locally Qj = { (x, p); H is maximal at u; } we obtain Fig. 3.1.

Description of E2. The case K = 2, cl = 1 or c2 = 1 does not correspond to a situation of maximum. We must consider the two following cases: 1. K = 1, cl = 2. The maximum is reached at a point u1 with codimension 2. The corresponding subset of E2 is denoted E. 2. K = 3, c1 = c2 = c3 = 0. The maximum is reached at the three regular points u1, u2, u3 and the corresponding subset is denoted E. We represent on Fig. 3.2, the corresponding stratification of R2 for a generic f, where we have only strata of codimension < 2.

72

3 Geometric Optimal Control

U2

(x.P)E a1

U1

U1

U2

U2

(x.P) E n2

E1

Fig. 3.1.

'.a

2

Fig. 3.2.

3.2.3 Generic Classification near El Near El we have a competition between two local maxima UI, u2. Take (xo, po) E El and solve locally using the implicit function theorem the equation = 0 to get two (local) COO-functions u;(x, p). Let H8(x, p) _ H(x, p, u;(x, p)) be the corresponding C°°-local Hamiltonian functions. We define H = max(Hi, H2). By construction, we have near (xo, po) a partition of 12 into

El = {(x, p); Hi = H2}, 12i = {(x,p); max(Hi,H2) = H;}. By definition, the Hamiltonian lift are given by

Hi= ui(x, p)

a of ax + 8x

(x, ui

a (x, p)) aP

and let us introduce the speed vector vi defined by

3.2 One Dimensional Problems of Calculus of Variations

73

Vi = (ui(x,P), Lf The set E1 is given by

El = {(x,P); H1 - H2 = 0} and a vector normal n to Ei is defined as

n = grad H1 - grad H2. Using .,(t-)

p(t)

=

aps

(x(t),P(t)) = ui(x(t),P(t)),

aH' (x(t), P(t)) = ax

(x(t),

ui(x(t), P(t)))

we get

n=

(af

(x,

u2) - of (x, ul ), ui - u2

where by definition ul # u2. Hence Ei is a C°°-curve whose tangent space is generated by b, b 56 0, where (n, b) = 0. We can take Far

6=

u2 - U1,

(x, u2) -

of

(x,

U1)

Moreover we check the following refraction law on El.

Lemma 11. V2 = v1 +6

The following important corollary is expressing the conservation of the

component of the normal speed. Lemma 12. We have (n, vi) = (n, v2) = { H1, H2 } (Poisson bracket)

This relation is the basis to make the generic classification of extremals new the set El. We have to consider two cases.

1. {HI, H2 } # 0 at zo = (xo, po) E El. Recall that El is locally a smooth curve and if {H1, H2} L 0, both vector fields Hl, H2 are transverse to E1 and a broken extremal crosses Ei at zo, see Fig. 3.3(a). Moreover the normal component of the speed is conserved. 2. Case {H1i H2} = 0 at zo = (xo, po) E El. Since at zo, (n, vi) _ {Hi, H2} = 0, both vectors H1, H2 are tangent to Ei at zo. We make the following

generic additional assumption.

74

3 Geometric Optimal Control

Assumption 1 At zo, the contact of the integral curves of H1, H2 is of

order 2, that is if El is c = 0, c : R2 - R, CO*, then 6(z;(t))1t=0 = 0 but 6(zi(t))1t=o 54 0 where t .-* z;(t) is the integral curve of H; passing through zo at t = 0. With this assumption, the geometric picture is clear. Indeed, near zo there

exists two coordinate systems Vi such that E, is given by Y = 0 and H1= A(X,Y)+ X . To compute the extremals we must connect pieces of trajectories of Hi (parabolas) near zo. For this it is sufficient to use the conservation of the normal speed and we have three situations, see Figs. 3.3(b) and 3.4(c)(d). To have the phase portrait in case (d) we

EI

02

Fig. 3.3.

must use the fact that from the maximum principle along an extremal t '- maxue$7 H(x(t), p(t), u(t)) is constant. Hence the curves are broken closed curves. Observe that the period for one revolution goes to 0 when the amplitude tends to 0.

Definition 33. A point zo E E1 corresponding to the case (a) is called a regular switching point. A point zo E El corresponding to the case (b) is called a parabolic point, to the case (c) an hyperbolic point and to the case (d) an elliptic point.

3.2 One Dimensional Problems of Calculus of Variations

75

Fig. 3.4.

3.2.4 Classification and Existence of Optimal Solutions There is a connection between the existence of optimal solutions and the classification of extremals. This is illustrated by the standard example: mint( .) f° f (x(t), i(t))dt, where f (x, i) = X2 + (22 - 1)2. Here the mapping V : i - (±2 - 1)2 is not convex and its convexification is given by

V (x)

_

if jil<1 (i2-1)2ifIxI> 1 0

See the respective graphs of V, V' on Fig. 3.5. By choosing T = 1, x(O) =

x(1) = 0, we know that the problem has no optimal solution because we can construct a sequence of curves on [0, 11, with xn(0) = xn(1) = 0 converging to the function identically zero for the C°-topology and such that 1= 1 a.e., the cost tending to 0 when n - +oo. The cost of I

x' - 0 being 1 for f and 0 for the relaxed problem. If we consider the previous classification, the Hamiltonian function is given

by: H(x, p, u) = pu - (x2 + (1 - u2)2), and the equation Off 67 = 0 is then

p + 4u(1 - u 2) = 0. We observe that p = 0 is in the stratum El, the maximum being reached at the two regular points ul = 1 and u2 = -1. The corresponding Hamiltonian is

H: (x, 0, uj) = -x2. The point (x, p) = (0, 0) is an hyperbolic point. The role of the relaxed prob-

lem is to add the curve t 1-- (x(t),p(t)) - 0 as an extremal. For the details,

76

3 Geometric Optimal Control

V i

J

+1

x graph V*

graph V

Fig. 3.5.

see [42].

We shall now consider the problem of classification of extremals for time minimum problem for single-input affine control systems.

3.3 Time Minimum Control Problem Definition 34. Consider the time minimium control problem for a singleinput affine control system ±(t) = X(x(t)) +u(t)Y(x(t)), x(t) E R", [ u(t) [<1, where X, Y are COO -vector fields. Let H(x, p, u) = (p, X (x) + uY(x)), p # 0, be the Hamiltonian. An extremal is a solution of the equations from the maximum principle: p (t)

8 (x (t

), p( t ), mo( t ))

( 3 .5 )

H(x(t), p(t), u(t)) = max H(x(t), p(t), v)

(3.6)

Wi(t) = aH (x(t) , p (t) , u (t)) ,

An extremal defined on [0,T] is called regular if for a.e. t E [0,T], we have u(t) = sign(p(t), Y(x(t))). It is called bang-bang if u(t) is piecewise constant with values in {-1,+1}. It is called singular if for every t E [0,T], we have

(p(t),Y(x(t))) = 0.

3.4 Determination of the Singular Extremals

77

3.4 Determination of the Singular Extremals Let (z, u), z = (x, p) be a singular extremal defined on [0, TJ. By definition it is a solution a.e. on [0, T] of the following equations:

±(t) = X(x(t)) + u(t)Y(x(t)),

fi(t) _ -p(t) ((x(t)) + u(t)

a (x(t)))

and it is contained for each t in the set El : {(x, p); (p, Y(x)) = 0}.

Since t ,--, z(t) is an absolutely continuous curve in El, differentiating t '-+ (p(t), Y(x(t))) = 0, one gets (p(t), [Y,X](x(t))) = 0 a.e. on [0, TJ where the Lie bracket is computed with the convention

[Zi, Z2](x) = a (x)Z2(x)

-

Since t -' (x(t), p(t)) is continuous, the curve t t E [0,T] in the set

z(t) is contained for each

EZ : {(x,p) E E1; (p, [Y, X](x)) =0}.

Hence differentiating t -. (p(t), [Y, X] (x(t))) = 0, we get the relation

(p(t),([Y,X],X](x(t)))+u(t)(p(t),[[Y,X],Y](x(t))) = for almost every t E [0, T].

This last relation allows us to compute u(.) in many cases and justifies the following definition.

Definition 35. For any singular extremal (z, u) defined on (0,T], R(z, u) will denote the set {0 < t < T; (p(t), [Y, X], Y](x(t))) # 0. The set R(z, u) possibly empty is always an open subset of [0, T].

Definition 36. A singular extremal (z, u) defined on [0, T] is called of minimal order if 7(z, u) is dense in [0,T].

Proposition 20. Let (z, u) be a singular extremal defined on [0, T] and assume that R(z, u) not empty. Then: 1. for a.e. t E 1(z, u),

u(t) _ '(z(t)) _ -

(p(t), [[Y, X1, X](x(t))) (p(t), ([Y, X],YJ(x(t)))

(3.7)

3 Geometric Optimal Control

78

2. z restricted to R(z, u) is smooth and is solution for every t of the equations:

x(t) = X(x(t)) + u(z(t))Y(x(t)) p( t)

= -p( t ) (

a

WO) + u(--( t ))

8

(x( t ))).

(3.8) ( 3. 9)

The minimal order singular extremals are the easiest to compute. Moreover, from the following proposition we can see that the existence of such extremals is a very common phenomenon.

Proposition 21. Let (X, Y) be a system such that the open subset .(l = {z; (p,([Y,X],Y)(x)) # 0} is not empty. On n we define the mapping X (X) + u(x, p)Y(x)), u(x, p) = v. d'x x H(x, p) = (p, . Then any tmjectory of the Hamiltonian vector field H, starting at t = 0 from the set E2 : {(x,p); (p,Y) = (p, [Y,X](x)) = 0} is a singular extremal with minimal order.

3.4.1 Hamiltonian Formalism and Singular Extremals Using the notations from Definition 29, we have that Hx (x, p) = (p, X (x)) and Hy(x, p) = (p, Y(x)). We observe the following:

(p, [X, Y] (x)) _ {Hx, Hy}(x,p) Hence

El : {(x,p); Hy(x,p) = 0},E2 : {(x,p); Hy(x,p) = {Hx,Hy}(x,p) = 0}, and

{{Hy,Hx},Hx}(x,p) {{Hy,Hx},Hy}(x,p)

Let us take a point zo E 12 such that locally E2 is a manifold of codimension 2. Then using { {Hy, Hx}, Hy } # 0 at zo, w = dpAdx is nondegenerate at zo and locally E2 is a symplectic manifold of codimension 2. The trajectories of

H for this symplectic structure are the singular extremals of minimal order described in Proposition 21.

Proposition 22. Let (z, u) be a singular extremal defined on [0, T], T > 0, such that: 1. z is contained in .(l = {z; (p, [Y, [X,Y]J(x)) 0 01; 2. x is contained in the set

12' = {x; X (x) and Y(x) are linearly independent}.

3.5 Geometric Classification of Extremals near E;

79

Then t ,-> x(t) is a C°°-curve, not reduced to a single point and moreover the mapping t h-' x(t) is an immersion.

Proof. According to Proposition 21, t '-- z(t) = (x(t),p(t)) is a COO-curve and the control t N u(t) is COO. Since i(t) = X(x(t)) + u(t)Y(x(t)), if X, Y are linearly independent then t 1--+ x(t) cannot be reduced to a point and this mapping is of rank 1.

3.5 Geometric Classification of Extremals near Ei In this section we describe the geometry of regular extremals obtained by Kupka [71].

Definition 37. Let (z, u), with z = (x, p) be an extremal defined on [0,T]. A time s E [0, T] is called a switching time if s belongs to the closure

of the set of t E [0, T] where z(.) is not C'. The set z(s), z(.) being any extremal is called the switching set. Observe that this set is a subset

of E, = {(x, p); (p,Y(x)) = 01. Let z = (x, p) be any C°°-solution of i(t) = W (x(t), p(t), u(t)), p(t) = - aH (x(t), p(t), u(t)) defined on [0,T], H(x, p, u) = (p, X (x) + uY(x)), corresponding to a C°°-control u(.). The switching function fi is the mapping t - (p(t), Y(x(t))) evaluated along z(.).

If u = +1 (reap. -1), we set z = z+ and tA = P+ (resp. z = z- and 5 = 0-). Lemma 13. The mapping t - 0(t) is COO and its first two derivatives are given by:

fi(t) = (p(t), [Y, X] (x(t)))

(3.10)

gj(t) = (p(t),[[Y,X],X](x(t)))+u(t)(p(t),[[Y,X],Y](x(t)))

(3.11)

3.5.1 Normal Switching Points Let zo = (xo,po) E El and let us assume that Y(xo) # 0 and zo E E1\E2i where E2 = {(x, p); (p,Y(x)) = (p, [Y, X](x)) = 0}. The point zo is then called normal. The behavior of regular extremals near zo is given by one of the two situations of Fig. 3.6 where E+ (reap. E-)= {(x, p); (p,Y(x)) > 01 (resp. < 0). F1som relations (3.10),(3.11) we have the following lemma.

Lemma 14. Let to be the switching time given by z+(to) = z-(to) = zo. Then, the following equalities hold:

0+ (to) =

(to) = (po,Y(xo)).

Moreover, let z = (x, p) be an extremal passing through zo. We have:

1. if (po, [Y,X](xo)) < 0, then x = -y+7-; 2. if (po, [Y,X](xo)) > 0, then x = 7-7+

(3.12)

3 Geometric Optimal Control

80

Fig. 3.6.

where 'y+ (resp. y_) is an arc solution of the system and corresponding to u = +1 (resp. u = -1) and -y+-t- represents the trajectory corresponding to the concatenation of u = +1 with u = -1, i.e. an arc ry+ is followed by an

arc -f-

3.5.2 The Fold Case Let zo = (xo,po) E E2. We assume that Y(xo) 96 0 and we suppose that E2 is a smooth surface of codimension 2. If we denote H+ (x, p) = (p, X (x) + Y(x)) and H_ (x, p) = (p, X (x) - Y(x)), then El is locally a C°°-manifold of codimension one and is defined by: {z; H+ (x, p) - H_ (x, p) = 0}. Moreover E2 : {z; H+ (x, p) - H_ (x, p) = {H+, H_ } (x, p) = 0}. According to (3.10) -.

-4

and (3.11), at zo E E2 both vector fields H+ and H- are tangent to E1 at zo.

If we set

At = (po, [[Y,X],Xj(xo) ± (po, [[Y,X],Y](xo))

(3.13)

then if both A* 0 0, the contact of the trajectories of H+ and H- with El is of order 2. Such a point is called a fold. According to [73] we have the three distinc cases:

a) A+A_ > 0: parabolic case b) A+ > 0 and A_ < 0: hyperbolic case c) A+ > 0 and A_ < 0: elliptic case The respective behaviors of regular extremals are represented on Fig. 3.7 and we have the following. In the parabolic there exists a neighborhood V

of zo such that each regular extremal in V has at most two switchings and in the hyperbolic case at most one switching. In the elliptic case the situation is more complex: there exists a neighborhood V of zo such that each

regular extremal in V has a finite number of switchings, although there

3.5 Geometric Classification of Extremals near E,

81

exists no uniform bound for the number of switching for all the extremals.

(a)

(b)

(C)

Fig. 3.7.

Another important geometric remark is the following. The surface E2 is the set containing all the singular extremals. In the hyperbolic and the elliptic

case, then (po, [[Y,X],Y](xo)) 0 0 and through zo there passes locally an unique singular control satisfying I u 1< 1 and hence is admissible for the bound I u I< 1. Moreover according to Fig. 3.7, in case (b) we can connect at zo a regular extremal with a singular arc to form an extremal but not in case (c). In the parabolic situation, if zo E fl there passes through zo a singular extremal, but the singular control does not satisfied the bound I u 1:5 1 and hence the singular arc is not an admissible one. Hence from our analysis we deduce the classification of all extremals

near a fold point in fl. Proposition 23. Let zo be a fold point in fl. Then there exists a neighborhood V of zo such that: 1. In the hyperbolic case, each extremal trajectory has at most two switchings

and is of the form ryfry,yf (by convention each arc of the sequence can be empty).

2. In the parabolic case, each arc is bang-bang with at most two switchings and has one of the two form: ry+7-7+ or 8. In the elliptic case, each arc is bang-bang but with no uniform bound on the number of switchings.

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3 Geometric Optimal Control

3.6 Comparaison Between the Calculus of Variations and the Time Minimum Problem for Affine Systems. The Role of Singular Extremals It is striking to notice the similarity between the classification of extremals in the calculus of variations and the one of regular extremals for the time minimum problems for affine systems. Ideas and techniques involved for the proofs are even the same, but the structure of optimal trajectories is not the same. In the calculus of variations we are dealing with non convex problem and we must convexify the problem to get optimal trajectories. Convexifying the problem means exactly to add admissible optimal trajectories in the set El. For the time minimum problem for affine systems, the problem is already convex and if we convexify the problem with only regular extremals where u E { -1, +1 }, we get the original problem with J u J.-5 1. Here the role of

the extremals in E1 for the convexifled problem is precisely played by the singular trajectories.

3.7 The Fuller Phenomenon In the elliptic case, the main problem when analysing the extremals is to prove that each extremal defined on [0, T] has a finite number of switchings. One of the main contribution of geometric optimal control was to prove that it is not a generic situation.

Definition 38. An extremal (z, u) defined on [0, T] is called a Falter extremal if the switching times form a sequence: 0 < t1 < t2 < . . . < T such

that t - T when n - +oo and if there exists k > 1 with the property: tn+1 - tn - k^ as n - +00. 3.7.1 Fuller Example In 1963 was published an article [45] given an example in which optimal control problem has an optimal trajectory which is a Fuller extremal. This example is the following:

x(t) = y(t), y(t) = u(t), min u(.)

r

I U(t) JS 1

+00

x2(t)dt.

This problem is a LQ-problem where u(.) is not penalized in the cost. Such a LQ-problem is called a cheap LQ-control problem. The system has no singular trajectory and the Hamiltonian is given by:

H(x, p, u) = -x2 + Ply + P2u,

P = (Pi, P2)

3.7 The Fuller Phenomenon

83

Since there is no boundary condition at t = +oo, it can be shown, see (77], that the adjoint vector must satisfy the transversality conditions

pi(+oo) = P2(+00) = 0. Introducing the augmented system

i(t) = y(t), y(t) = u(t), z(t) = x2(t) the adjoint system is then

Pi(t) = 2x(t),

P2(t) = -pi(t)

and an extremal control is defined by

u(t) = sign p2(t). It turns out that the optimal synthesis is characterized by a switching locus S in the (x, y) space given by the equation

x+hyIyI=0 where h 0, 4446, and each optimal non trivial solution exhibits a Fuller phenomenon as represented on Fig. 3.8. Here k = 11-24 > 1, where is the

positive root of x4 + i2

- is =

0. Such optimal trajectories provide Fuller

Y

Fig. 3.8. extremals for the time minimum problem where the system is the augmented system. Hence we have Fuller extremals for systems in R3. This example is not

stable, for instance observe that the switching locus is a curve symmetric with respect to 0. Hence the following theorem is important. Theorem 12. If the dimension of the state space is large enough there exists a stable model (X, Y) exhibiting Fuller extremals.

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3 Geometric Optimal Control

The (complicated) proof of this result can be found in [71] and semi-algebraic

conditions F are given for describing the model. They involve the Poisson brackets of H± = X ± Y at zo up to order 5 and all the Poisson brackets up to order 4 must be 0. The Fuller example satisfies those conditions at the point x0 = (0, 0, 0) and po = (0, 0, -1) (see the transversality conditions). In the Fuller example, there exist optimal trajectories x"(t),0 < t < T with an infinite number of switchings. Such phenomenon is not admissible to construct optimal synthesis using the tools of the analytic geometry like sub-analytic sets. This makes the problem very difficult. Nevertheless, the Fuller phenomenon is not a stable phenomenon for systems in dimension 2 or 3. This leads to the research program initialized by Sussmann and completed by Schattler to describe the stucture of optimal synthesis for systems in small dimension. We shall here present some results initially obtained by Sussmann, see [102] for all the details.

3.8 Geometry of the Time Optimal Control in the Plane 3.8.1 Preliminaries If Z is a C°°-vector field we denote by {exp tZ} the local one parameter group of Z, i.e. (exptZ)(zo) = z(t,zo) is the unique solution to the Cauchy problem z(t) = Z(z(t)), z(0) = zo. Hence we consider a COO-single affine control system in the plane: z(t) = X(z(t)) + u(t)Y(z(t)), I u(t) I< 1 with z = (x, y). We take zo E R2 and the program is to investigate the local structure of the time optimal control law in a small neighborhood U of zo. Using Filippov existence theorem such an optimal control exists if U is taken small enough. Indeed, the problem is convex and moreover since the controls are uniformly bounded, there exists T > 0 such that every solution z(t, xo) is defined on [0, TJ and stays in a compact set K. Hence using a cut-off we can restrict the system to K and apply Theorem 10. According to the maximum principle an optimal trajectory is solution of the equations:

z(t) = M (z(t), p(t), u(t)), p(t)

aH(z(t), p(t), u(t))

(3.14)

H(z(t), p(t), u(t)) = max(p(t), X (z(t)) + vY(z(t)))

(3.15)

X(z(t)) + vY(z(t))) > 0

(3.16)

w1<1

From the second condition, the trajectories candidate to optimality are arcs y+ and y_ corresponding to the vector fields Z+ = X + Y and Z_ = X - Y and singular arcs y, where the singular control is generically given by us(z,p) _

(p,[[Y+X],X](z)) (p, [[1', X],Y](z))

(3.17)

3.8 Geometry of the Time Optimal Control in the Plane

85

Moreover (p, Y(z)) = (p, [X, Y](z)) = 0 along a singular are.

The program is to find the admissible optimal synthesis near zo using concatenation of arcs 'y+, ry_ and %. This is a classification program

based on the structure of the Lie brackets of the system (X, Y) at zo and the general program is to analyze all the situations of codimension < 2. To investigate the problem we use two basic tools: - Classification of extremals in the situations of small codimensions, as given in Sect. 3.5; - Optimality criteria to exclude some extremal trajectories.

The classification of extremals is based on the analysis of the switching function:

0(t) = (p(t),Y(z(t))).

(3.18)

Differentiating the switching function with respect to t, one gets q(t) = (p(t), [Y, X)(z(t))) j'(t) = (P(t), ([}', X ], X)(z(t)) + u(t)(p(t), [[1', X], Y[(z(t))) In particular, since from the maximum principle we must have along an extremal p(t) 0 0, the singular trajectories are located in the set

S = (z; det(Y, [Y,X))(z) = 0}. We denote by SO its complementary.

Another important singularity set is the set of points such that X, Y are collinear:

C = {z; det(X, Y)(z) = 0}.

On its complementary C`, we can define the clock-form a by

a(X) = 1,

a(Y) = 0

(3.19)

and if z(t), 0 < t < T is a trajectory in C° then we have r.)

Jz(

rT

a=J dt=T. 0

3.8.2 Generic points Take a point zo E S` n CO, then the classification is straightforward.

Proposition 24. Let zo E SO n CO, then there exists a neighborhood u of zo in which each optimal trajectory is bang-bang and has at most one switching. Moreover each optimal trajectory is of the form 7+y_ or -t--r+.

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3 Geometric Optimal Control

Proof. If we lift zo = (xo, yo) E R2 into mo = (zo, po), then (po, Y(zo)) 34 0 or (po, Y(zo)) = 0. Since zo E Sc fl C° in the second case, we have according

to Sect. 3.5.1, that mo is a normal switching point. Hence each extremal is of the form -j+-f- or 'y 'y+ where an are of the sequence can be empty. Moreover consider two points zl, z2 near zo and 7i = 7+'Y-, -y2 = 'Y-'y+ be two curves defined respectively on 10, T,] and [0, T2] and connecting zl to z2, see Fig. 3.9. An easy computation left in exercise shows that if a is the clock

7+

ZI

1+

Fig. 3.9. form, then da = 0 is exactly S. Hence only one of the two curves 7i or rye is optimal since using Stoke's theorem we have

Tl-T2=J

ry

a-Ja=Jdo 7a D

where D is the domain limited by -fl U -y2.

3.8.3 Singular arc We shall now consider the situation when zo E S fl C. In this case, if ad2Y(X)(zo) is not collinear to Y(zo), through zo there passes a C°°-singular arc ry, : t H [0, T] which is locally an immersion. The singular control is given by

(p, ad2XY(z))

u,(z,p) _ (p, ad2YX (z)) and y,(.) is contained in the set S : (p,Y(z)) = (p, [Y, X] (z)) = 0. In particular using (p, Y(z)) = 0 and the homogeneity with respect to p, we see that u, is depending only on z and is a feedback control. Moreover according to the maximum principle, along the singular arc we have H1,., = (p, XI.,.) and this value is a positive constant. Since X, Y are

3.8 Geometry of the Time Optimal Control in the Plane

87

linearly independent this constant is non zero and we have: (p, Xly,) > 0 and po is unique up to a positive scalar. Hence zo admits (up to a positive scalar for po) an unique lifting mo = (zo, po) E E2 and the corresponding point mo is a fold point according to Sect. 3.5.2. Introducing

of

= (po, [[Y,XJ,X](zo)) ± (po, [[Y,XJ,Y](zo))

we have under the generic assumptions \+,.\- # 0 three cases: parabolic case, hyperbolic case, elliptic case. The two first cases are the easiest to analyze since each extremal law is bang-bang with at most two switchings, see Sect. 3.5.2. The elliptic case is more complex because we do not have the existence of a uniform bound on the number of switchings for the extremals. We describe the two first cases in the following proposition.

Proposition 25. Let zo E S n C` and mo = (zo, po) be its canonical lift (unique up to a positive scalar on po). Then 1. If mo is a parabolic point, every time-optimal trajectory near zo is of the

form: ^y+7-'+ or i'-'+ i-

2. If mo is an hyperbolic point, then every time-optimal trajectory near zo is of the form: ry+7s'Y+, 7+ Ys y-, y-- -Y+ or'_1's yIn the hyperbolic case, the singular arc is indeed optimal as it can be shown

using the clock form: a, and da = 0 on S. Also a stronger result can be obtained even without any uniform bound on u.

Proposition 26. Consider the time optimal control problem ,z(t) = X (z(t)) + u(t)Y(z(t)), z(t) E R2, U(t) E R.

Assume that',(.) is a simple singular arc on [0,TJ such that: i) X, Y are linearly independent along -y. (.). ii) Y and ad2YX are linearly independent along 73 ( ) Then we have two situations:

1. If X1,y, and ad2YXI.y, point with opposite directions with respect to the plane RY y,, then the singular arc is time minimizing with respect to all trajectories contained in a C°-neighborhood of the image of',(.). 2. If X1.y, and ad2YXIy, point on the same side of RY y, then 'ys(.) is time maximizing with respect to all trajectories contained in a C°-neighborhood

of Im',. Proof. We use condition i) to construct a coordinate system alongsuch that:

1. rys:t'-'(t,0) 2. Y is the vector field ' .

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3 Geometric Optimal Control

Hence the system can be written as: z(t) = 1 + X, (X(t), y(t)) y(t) = X2(x(t), y(t)) + u(t)

(3.20) (3.21)

(x, 0) = 0, since t '-+ (t, 0) is singular. Therefore the where X1 (x, 0) _ system is feedback equivalent to the system: ±(t) = 1 + a(x(t))y2(t) + R(x(t), y(t)) y(t) = u(t)

(3.22) (3.23)

where R(x, y) = o(y2). The conditions 1 and 2 of the proposition are feedback

invariant. The case 1 corresponds to a(x) < 0 and case 2 to a(x) > 0. If we

consider the truncated system: x(t) = 1 + a(x(t))y2(t), y(t) = u(t), the assertion of the proposition is trivial. Indeed consider the case a(x) < 0, then we decrease the speed x(.) at each time where y 54 0. Clearly the singular arc y,(.) identified to t - (t,0) is optimal. Now observe that a C°-neighborhood of Imy, is given by I y 1< E. In [261 it is rigorously proved that the assertions are true also for the non truncated system. (We shall give later in Chap. 6 a complete proof of a more general result)

3.8.4 Elliptic Case - The Concept of Conjugate Point The second case analyzed in the previous proposition is the situation where a singular arc is time maximizing if u(t) E R. If mo = (zo, po) is the canonical

lifting and if mo is a fold point we are precisely in the situation where a singular are is slow and satisfies the constraint u,(t) El - 1,+1[. Let us examine the time optimal synthesis. Since the point m0 is elliptic, then there exists no uniform bound on the number of switchings for the extremals near zo. More precisely, there exists of bang-bang extremals whose projection in the state space a sequence tends in the C°-topology towards the image of the singular arc, see Fig. 3.10. Nevertheless the arc y,(.) is not time minimizing. It follows that a natural

guess is that an optimal trajectory has to stay as far as possible from the singular arc. Indeed we have the following proposition from (102].

Proposition 27. Let z° E S fl Cc and m0 = (z0, p0) its canonical lifting. If mo is an elliptic point, then there exists a neighborhood of z0 such that every time optimal trajectory is bang-bang with at most two arcs, i.e., is of the form

y+y- or y- y+ Proof. Since mo is an elliptic point, there exists a coordinate system such that z0 = (0, 0) and such that the system can be written as: z(t) = 1 + Ey2(t) + Ri (x(t), y(t)),

e>0

(3.24)

y(t) = -u,(0) + u(t) + R2(x(t), y(t))

(3.25)

3.8 Geometry of the Time Optimal Control in the Plane

89

Fig. 3.10. where u8(0) is the value of the singular control at zo and R1 = o(II(x,y)II2), R2 = o( II (x, y) II) Direct evaluation of the time transfer exclude the bangbang extremals with more than two arcs and a time optimal law is represented on Fig. 3.11.

Fig. 3.11.

Unicity. We are in the situation where the optimal control is not in general unique. Indeed consider for instance the system i(t) = 1 + y2(t), y(t) = u(t). Take two points on the singular line: y = 0, zo = (xo, 0), zl = (XI, 0), Si > xo. Then there exists two optimal controls steering zo to zl and the singular line is a cut locus, see Fig. 3.12. The original proof of Sussmann use the concept of conjugate point that we present briefly. Definition 39. Let (z (t), p(t), u(t)) be an extremal defined on [0, T]. We say

that the two points z(ti) and z(t2) are conjugate if the switching function 0(t) = (p(t), Y(z(t))) vanishes at both times tl and t2.

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3 Geometric Optimal Control

Fig. 3.12. From this definition, we have

(p(tl), Y(z(tl))) = (p(t2), Y(z(t2))) = 0. Now observe that if v(t) is solution of the variational equation

v(t) = ( _ wt)) + u(t) az (z(t)))v(t)

(3.26)

then, the scalar product (p(t), v(t)) is constant. Hence if we integrate this equation with v(t2) = Y(z(t2)) and if we denote by w(.) the solution, we get

(P(tl),w(tl)) = 0. Therefore, this implies the constraint:

det(Y(z(tl)),w(tl)) = 0. This concept can be used to obtain bounds on the number of switchings as shown by the following analysis. Consider an arc y+ (resp. 'y_) corresponding

to Z = X + Y (resp. X - Y) and denoted byz(t),0
(z(t))w(t), w(T) =Y(z(T)) g The vector w(.) can be computed as follows. Consider the curve a : [0,e] w(t) =

R2 given by a(e) = expeY(z(T)). It is a curve starting at e = 0 from z(T) and whose tangent vector is a(0) = Y(z(t)). At each point of z(t), we can consider the curve a '- a(t, e) = exp -tZ(a(e)) = exp -tZ exp eY exp TZ. In particular for t = T one gets a curve at zo. Using the ad-formula, its Taylor expansion is given by

3.8 Geometry of the Time Optimal Control in the Plane

91

n

a(T,e) =

(-1)n ni adnZ(eY)(zo).

(3.27)

n>O

In particular

a(T,e) _ (expe E(-1)nTadnZ(Y))(zo) _ (expe(Y +

))(zo). (3.28)

n>O

Since Z = X ± Y, we observe that [Z, YJ = [X, Y. And by definition

w(T) =

8a 8e i£=o

= (Y - T[X, Y] + o(T))(zo)

(3.29)

where T is non zero. Hence we observe that if Y and [X, Y] are linearly independent at z(O) we cannot have for T small enough the relation

det(Y(z(0)),w(T)) = 0. Hence near zo we cannot have for an extremal more than one switching. This result was already proved by the classification of normal switching points, see

Sect. 3.5.1. but it is the starting point to obtain bounds on the number of switchings for optimal trajectories. The idea is illustrated by the following. In the fold case, the two vector fields Y and [X, Y] are linearly dependent at zo. In this case let us consider near zo an arc z(t), t E [0, T] of the form -y- -y+ -y-,

where 0 < tl < t2 < T are the switching times. Denoting Z+ = X + Y, Z_ = X - Y, we have: z(t) = exp t3Z_ exp t2Z+ exp tl Z_ (z(0))

where ti, t2, t3 are positive times such that t1 + t2 + t3 = T. We denote by zl and z2 the switchings points on z(.). We observe from Fig. 3.13 that if we introduce the variations (--1, 62) on the switching times we can construct a curve

a(E1, E2) = exp t3Z- exp (t2 - E2)Z+ exp (t1 - el )Z- (z(0))

with t3 = T - (ti + t2) + e1 + e2 such that Ima is in general a neighborhood of z(T) contained in the set of points accessible from z(O) in time T. Hence z(T) is not optimal, see Fig. 3.13. The structure of extremals near zo is as follows. Take an extremal (z(t), p(t), u(t)), t E [0,T] with two switchings at z1, z2. First by assumption we are near an elliptic point and at zl, Z2, the vectors Y, [X, Y] are linearly independent but at zo, Y and [X, Y] are linearly dependent and zl, z2 are lifted into two points in the switching surface near the lifting of zo. Since zl and z2 are switching points, then Y(zi) and w = Y - T [X, YJ + 2 [X +Y, [Y, X]]+o(TZ)(zi) are linearly dependent where w is given by (3.29)

92

3 Geometric Optimal Control

z2

Fig. 3.13.

and T is the transfer time from zl to z2, and is approximated by solving a polynomial equation of order 2. This can be done uniformly near zo using the

model: i(t) = 1 +ey2(t), y(t) = -u,(t)+u(t), where zo = (0, 0). Then we can construct near (z(0),p(0)) a one parameter family of extremals (ze, pE) such that z, (0) = z(0), pE(0) = p(0)+e, see Fig. 3.14 where the switching sequence is well defined. In the state space we get a one parameter family of extremals

z(0),p(0)

Fig. 3.14.

represented on Fig. 3.15. We observe that z2 is a limit point of end-points of extremals defined on the interval [0, tl + t2]. This corresponds as noticed by [102] to a concept similar to the geometric concept of conjugate point in the calculus of variations, see [47]. If we represent the piece of the switching curve S from z(0), we observe that the reference extremal has a reflexion point at z2, see [11] and Fig. 3.16. This concept of conjugate points is at the origin of

3.8 Geometry of the Time Optimal Control in the Plane

93

Fig. 3.15.

the bounds on the number of switchings obtained by Sussmann in dimension 2, see [102] and later by Schattler in dimension 3, see [97].

Fig. 3.16.

Notes and Sources The article from Ekeland [42] is historically important. Based on the idea of the singularity theory, the classification of extremals is viewed as a compe-

94

3 Geometric Optimal Control

tition between Hamiltonian vector fields, this idea being already in [36). It also introduces the use of normal forms. This classification for problems of calculus of variations in dimension 2 was pursued by Klok, [66]. Both articles from Kupka [71],[73) about the Fuller phenomenon and the generic classification of extremals use the singularity theory. They are basis to deal with time minimal problem for single-input control systems. Finally the results from Sussmann [102], and Schattler [97] are important because they give local bounds for the number of switchings for optimal trajectories of systems in dimension 2 and 3. These results also provide algorithms to solve the time minimal problem for systems by gluing the local results using topology. See also the book by Hermes-Lasalle [55] using the clock form to analyze time optimal control problems in the plane.

Exercises 3.1 Consider the planar system z(t) = Q(z(t)) + u(t)b, u(t) E R, z = (x, y) where Q = (Q1, Q2), Q: being a quadratic form and b is a constant vector in R2. We assume that b and [[b, Q], b] are linearly independent. 1. Compute the singular trajectories and discuss their time optimality status using the one clock-form. 2. Show that there exists a linear coordinate system and a feedback of the

form: u = a(z) + Qv where a(z) is a quadratic form and A 4 0 is a constant such that the system is transformed into one of the following systems:

1(t) = x2(t) + y2(t)

W) = u(t)

t(t) = x2(t) - y2(t) y(t) = u(t)

Wi(t) = y2(t)

v(t) = u(t)

3.2 Consider the planar system

i(t) = 1 + y2(t), y(t) = -u9(t) + u(t) where I u(t) 1< 1 and u3 (t) E] -1, +1 [. Let zo = (xo, yo), I zo I small enough. Find the time optimal feedback law to steer zo to (0, 0) in minimal time.

3.3 Consider the planar system z(t) = X(z(t)) + u(t)b, I u(t) I< 1, z = (x, y),

where X is CO* and b a constant vector in R2. Assume X (O) = 0 and set Vl (0) = A. Assume b, Ab linearly independent. Compute the time optimal feedback law to steer zo to (0, 0) in minimal time, with I zo I small enough.

3.8 Geometry of the Time Optimal Control in the Plane

95

3.4 Consider the following planar system: th(t) = u(t) with I u(t) 1< 1.

y(t) = 1 + y(t)x3(t)u(t)

Show that u s 0 yields a singular trajectory to join (0, 0) to (0, 1). Use the clock form to prove that this solution is not optimal. 3.5 Consider the time optimal control problem to steer (xo, yo) to the origin for the system:

X0

±(t) = y(t) f (x(t), y(t)) + u(t)

where f is smooth and I u(t) I< 1. 1. Prove that an optimal trajectory is solution of the following equations:

x(t) = y(t) y(t) = -f (x(t), y(t)) + u(t) Pi (t) = py(t) of (x(t), y(t))

Py(t) = -px(t)

PV

with u(t) =signpy(t) almost everywhere. 2. Let u(.) be an optimal control defined on the time interval [0, T]. Prove

that u(.) can be taken piecewise constant and is given by: u(t) =sign py(t) = -1 if py(t) < 0, 0 if py(t) = 0 and +1 if py(t) > 0. Show that the adjoint vector p(t) = (px(t), py(t)) is C' and that py(t) has a finite number of 0 which are simple. 3. Let u(.) be an optimal control and (x(.), y(.)) be the corresponding trajectory with adjoint vector p(.) = (p.(.), p,(.)), all defined on [0,T]. If

t1, t2 are times such that 0 < tl < t2 < T, then prove the following assertions:

a) If py(tl) = py(t2) = 0 and if y(ti) = 0 then y(t2) = 0b) If py(tl) = py(t2) = 0 and if y(ti) 0 0, then y(t2) # 0 but there is a zero of y(.) on the interval tl < t < t2. c) If Y(t1) = y(t2) = 0, y(t) 0 0 on t1 < t < t2 and if py(tl) = 0 then py(t2) = 0-

d) If y(tl) = y(t2) = 0, y(t) 0 0 on tl < t < t2 and if py(t1) 34 0 then P1, (t2) 76 0 but p2(.) vanishes on the open interval ti < t < t2.

3.6 Consider a smooth control system in R^ of the form:

i(t) = X (x(t)) + u(t)Y(x(t)),

I u(t) (< 1.

(3.30)

We denote by A(xo, T) the set of points that are accessible from xo in time T and A(xo) the accessibility set.

96

3 Geometric Optimal Control

1. Let u(.) be an admissible control and x(.) the corresponding trajectory starting at xo and defined on [0,T]. Assume x(T) E 9A(xo) for all t E (0, T]. Show that there exists an absolutely continuous curve p : [0, T] R°\{0} such that we have almost everywhere:

ax

ey

At) = -P(t)(a (x(t)) + u(t) 8x (x(t))) (p(t), Y(x(t)))u(t) = max(p(t),Y(x(t)))v(t) H(q(t), p(t), u(t)) = (p(t), X(x(t)) + u(t)Y(x(t))) = 0

2. Let (X, Y) be a 2-dimensional system and suppose that X, Y are ind pendent at xo. Represents the small time accessibility set from xo. 3. Assume the system (X, Y) to be a 3-dimensional system and suppose that X, Y and [X, Y] are independent at xo. Show that all possible trajectories that can lie in the boundary of the small time accessibility set from xo are of the form -y+-y- or ^i--y+.

4. Assume the system (X, Y) to be a 4-dimensional system and set f = X + Y, g = X - Y. Suppose that the following conditions hold: (HI) f, g, If, g], If, If, g]] are independent at xo. (H2) f, g, If, g], [g, If, g]] are independent at xo. Write near xo: If, If, g]] = «f + f3g + 7[f, g] + 5(g, If, g]].

(3.31)

a) Compute the singular trajectories for the system (X, Y) contained in the level set H = 0. b) Assume b > 0. Prove that each trajectory that lies in the boundary of the small time accessibility set from xo is bang-bang with at most two switchings.

3.7 Consider the following system in R4:

zo(t) = 1

i1(t) = u(t) ±2(t) = x1(t)

x3(t) = 2xi(t) with [ u(t) 1< 1 and x(0) = 0. 1. Prove that all Lie brackets with length > 4 are 0. 2. Prove that if u(.) is a control defined on [0, T] and x(.) is the corresponding trajectory, then the accessibility set at time 1 can be obtained from

the accessibility set at time T by letting: u(t) = u(+), &i(t) = Txi(

fori=1,2,3.

3. Compute the accessibility set at time 1.

)

4 Singular Trajectories and Feedback Classification

The objective of this chapter is to enlight the relationship between feedback classification of systems and classification of singular trajectories. This relation is somehow obvious. Indeed a singular trajectory corresponds to a

singularity of the end-point mapping and this singular point is clearly feedback invariant. It follows that the classification of singular trajectories parametrized by the maximum principle is related to the feedback classification. It is important to remark that the classification of singular trajectories is easiest because we consider only change of coordinates on the state and we are faced with a well studied problem of classifying differential equations up to a set of diffeomorphisms. Less trivial is the assertion that for generic

control systems it will allow to compute a complete set of invariants, that is both classification problems are equivalent. Hence we can use singular extremals to compute feedback invariants. Conversely the feedback group is a basic tool to investigate the properties of singular extremals, like time optimality as it will be shown later. Hence both problems are deeply interconnected and have to be simultaneously studied when dealing with the control of nonlinear systems. The analysis presented in this book is mainly based on the article [20]

which deals with affine control systems and is completed by results from [58] on general systems. Both problems are not independent and the rela-

tion is given by the Goh transformation: the system i(t) = f (x(t), u(t)) can be seen as a very specific affine system in the extended space a(t) = f (x(t), u(t)), ii(t) = v(t). Hence the classification of general control systems is a subcase of the classification of affine systems i(t) = Fo(z(t)) + ,=I v;(t)F;(z(t)), where the distribution Span{Fi(z), tive and with constant rank. Definition 40. Consider control systems of the form

a(t) = f (x(t), u(t)),

,

F,,,(z)} is involu-

x(t) E R", u(t) E IIt""

(4.1)

where f is C°° or C' (real analytic) and we denote by S = {f (x, u)} the set of such systems. Two systems f (x, u) and f'(y, v) are said to be feedback equivalent if there exists a C°° or CW -diffeomorphism of the form q : (x, u) -' (y, v), y = cp(x),v = z,1i(x, u) which transforms f into f i.e. d.iof (x, u) = f'(y, v)

98

4 Singular 'trajectories and Feedback Classification

and f is called the image of f by W. We use the notation: f = O * f. We give a global definition of feedback equivalence, but they are various useful local concepts which have to be used in various situations:

- local feedback equivalence at a point xo E R" of the state space; - local feedback equivalence at a point (xo, uo) E R" x Rm; - local feedback equivalence near a given trajectory: x(t) or (x(t), u(t)) of

the system (4.1).

Also we can restrict the feedback group G = {0;

to a given

subgroup. In particular an important subclass of systems is the class of affine systems. This leads to the two following definitions.

Definition 41. Consider the class of affine control systems

i(t) = Fo(x(t)) + F(x(t))u(t), where F = (Fl,

X(t) E R",u(t) E Rm

(4.2)

F,,,.). We identify such systems to the set A of (m + 1)uplet of vector fields: {Fo, F}. The vector field Fo is called the drift and let D be the distribution defined by D(x) = Span{Fj(x), , Fm(x)}. The system ,

is called regular if for each x E R', rank D(x) = m. For affine systems we restrict the feedback transformation to the subgroup G' = {(cp, a,#) } (the product law is well defined by the action on A) of the form -O(x, u) = (sc(x), O(x, u)),

where O(x, u) = a(x) + p(x)u. We observe the following: if we take (Fo, F) E

A and 0 = (sp, a, p) E G', then the image of (Fo, F) is the affine system (FO', F') given by

i) FO' = cp * (Fo + Fa)

ii) F =cp*Ff3 In particular the second action corresponds to the equivalence of the two dis-

tributions D and D' associated to F and P.

Definition 42. Consider a general control system: x(t) = f (x(t), u(t)), x(t) E R", u(t) E Rm. The control can be viewed as a state variable: y = u and let us introduce a new control v(.) by setting u(t) = v(t). Such a transformation is called a Goh or an I - D transformation and we can canonically associate to the original system the following affine control system: Wi(t) = f (x(t), u(t))

u(t) = v(t) where (x, u) E R"i"m is the state space and v E Rm the control.

(4.3)

4.1 Classification of Affine Systems

99

Such a transformation is not only a trick but has many applications in control engineering. A practical application will be given in Chap 7 where we deal with the study of the time minimal control problem for chemical batch reactors. For the feedback classification problem we have the following.

Proposition 28 (Global statement). Two general systems (4.1) are feedback equivalent if and only if the corresponding affine systems (4.3) are feedback equivalent (up to feedback affine transformation)

Proof. The proof is straightforward and is given in [59]. Hence solving the feedback classification problem for general control sys-

tems is a subcase of solving the feedback classification problem for affine systems. In fact it is much simpler because the distribution associated to (4.3) is the flat distribution and in this subclass we avoid the problem of classifying the distribution. We shall now present the results concerning affine systems.

4.1 Classification of Affine Systems 4.1.1 Computations of Singular Controls In this section, we will consider affine control systems:

i(t) = Fo(x(t)) + F(x(t))u(t),

x(t) E Rn, u(t) E R"'

, F,,, are assumed to , F,,,) and the vector fields FO, F1i be C°'. According to Chap. 2, the singular extremals are the solutions of the

where F = (Fl,

following equations:

x(t) = a WO, p(t), u(t)),

OH

At) OH

WO, p(t), u(t)) = 0

where H(x, p, u) = (p, f (x, u)) and f (x, u) = Fo(x) + Zcn l u;F;(x). The constraintH = 0 has to be satisfied almost everywhere, it is equivalent to (p(t), Fo(x(t))) = 0, Vi = 1,

m

(4.4)

Since (x(.), p(.)) is continuous it is satisfied for all t. The constraint (4.4) define a subset El of the space Rn x R"\{0}. We shall now generalize the algorithm of Sect. 3.4 to compute generic singular trajectories. Differentiating relation (4.4) with respect to t, we get the equation

L(x(t), p(t)) + O(x(t), p(t))u(t) = 0

(4.5)

100

4 Singular Trajectories and Feedback Classification

where L(x, p) is them x 1 matrix (Li) with Li = (p, [Fi, Fo] (x)) and O(x, p) is the m x m matrix Oij with Oi, = (p, [Fi, FJ}(x)). Let s be the max(,,,p) rank O(x, p) and observe that O(x, p) is a skewsymmetric matrix. Let Al be the set of affine systems (FO, F) such that: if m is even then s = m and if m is odd then s = m - 1. To compute the control we must distinguish two cases: Case 1: The number of inputs is even. Take (Fo, F) E Al and let u be defined on an open dense subset of 1R" x R's by

u(x,p) = -0-1(x,p)L(x,p)

(4.6)

Now let us choose (Fo, F) such that det 0 is not identically zero on El Define the singular control is defined a.e. on El by u(t) = u(x(t), p(t)).

Case 2: The number of inputs is odd. The computation is similar to the previous one, the only complication being the existence of a kernel for O(x, p).

Take (Fo, F) E A1, then for a.e. z = (x, p) the dimension of kernel O(z) is one. Let zo be such a point. Then from equation (4.5) we have L(zo) = 0 for each u E ker O(zo). Near zo (our computations are local) there exists an orthogonal matrix P(z) such that P-1(z)O(z)P(z) = (01, 0)

where 01 is a skew-symmetric matrix of dimension m - 1. Moreover P(z) can be chosen analytic since the eigendirection corresponding to the zero eigenvalue of O(z) depends analytically on the coefficients of O(z). P-1L can be written in two blocks (L1, L2) where L1 is a (m-1) column vector and L2 is a scalar. Let us write :9 = P-1u as 1i +u2, where ul E and U2 E R. Equation (4.6) is then equivalent to 1Rm-1

11 (Z) + 01(z)ul = 0 L2(z) = 0.

(4.7) (4.8)

Let ul(.) be the map defined a.e. by

ul (z) = -01 1(z)L1(z). To compute the component of u(.) in the kernel of 0 we differentiate (4.8) along a solution. We must observe that L2 (z) is near zo analytic in the coefficients of O(z) and L(z). These coefficients are of the form (p, Z(x)) where Z belongs to the Lie algebra generated by { FO, F1, , Fm }. Therefore by differentiating we get a relation of the form

f(z(t)) + gl(z(t))ul(t) + g2(z(t))u2(t) = 0.

(4.10)

Let (FO, F) be chosen such that 92 is not identically zero and let u2(.) be the map defined a.e. on 1R2n by

4.1 Classification of Affine Systems u2(z) = -(f (z) - 9i(z)ui(z))g2 1(z).

101 (4.11)

A control u(t) = Pi(t) defined by the analytic restriction of ul(z(t)),'2(z(t)) to El fl L2(z) = 0 is a singular control.

Definition 43. Let V be the analytic set defined by: if m is even, V = El and if m is odd V = El n {L2 = 0}. Observe that in each case the number of equations defining V is even. Let A. be the subset of good systems in Al given by:

(i) if m is even, det O is not identically 0 on El; (ii) if m is odd, 0 is a.e. of rank (m - 1) on El and 92 is a.e. invertible on V.

Take (Fo, F) E Ag and let u be the map defined a.e. on 1W1 x R'1 by:

(i) if m is even, u is given by (4.6); (ii) if m is odd, ii = Pu with t- given by (4.9) and (4.10). By construction u is meromorphic and let S be the set of points of V where u is not analytic. From our computation, on V\S the singular extremal is said of minimal order. Let us denote by H the meromorphic function defined a.e. on R2s by (4.12) H(x, p) = (p, Fo(x) + F(x)u(x, p)).

ti

We note H the associated constrained Hamiltonian vector field where a solution is by definition an analytic curve: t i--+ z(t) which stays in El\S and

solution a.e. of the vector field H. Observe that z(t) is contained in V\S and at the regular points of V it is solution of an analytic vector field denoted Hreg.

We have the following proposition.

Proposition 29. Let (Fo, F) be a system of Ag. Then the singular extremals of minimal order are solutions of the constrained meromorphic differential equation :k(t)

=ap8H

(x(t), P(t)),

P(t)

8H

a x (x(t), P(t)),

(x, P) E El.

(4.13)

Conversely through each point zo = (xo, po) of V\S there passes an unique singular extremal of minimal order solution of the above equation.

Remark 5. The previous algorithm to compute singular extremals is valid for generic systems (FO, F) for the Whitney topology. Nevertheless in the nongeneric situation similar computations hold. For instance, if the distribution D(x) = SpanF(x) is involutive, see [20]. In each case we have to

102

4 Singular Trajectories and Feedback Classification

solve only linear equations. Even for a system (Fo, F) in A. we have computed only singular extremals of minimal order. See Chap. 8 for a discussion on the existence of higher-order singular extremals in the generic situation.

Also observe that linear controllable systems (A, B) are in the bad set and indeed they have no singular trajectories. In the single-input case: ±(t) = Fo(x(t)) + u(t)F(x(t)), u(t) E R we have the following: V = E2 = {(x, p); (p, F(x)) _ (p, [F, Fo](x)) = 0}

and S is the set of points of V where (p, ad2F(Fo)(x)) = 0. At the regular points of V\S, the symplectic form da = dp A dx is nondegenerate. It follows that singular extremals of minimal order are locally solutions of an Hamiltonian vector field on (V\S,da). This result can be extended to the general case. Observe also that by making the change of parametrization da = dt((p,ad2F(Fo)(x))), we can extend the differential equation to R2n.

4.1.2 Singular Trajectories and Feedback Classification Definition 44. Let E, F be R-linear spaces and let G be a group acting linearly on E and F. A homomorphism X : G -+ IR\{0} is called a character. Let X be a character. A semi-invariant of weight X is a map A : E --+ R such that for all g E G, we have A(gx) = X(g)X(x) for every x E E. If X = 1 it is called an invariant. A map A : E F is a semi-covariant (resp. covariant) of weight X if for all g E G, we have A(gx) = X(g)gA(x) dx E E (resp. X = 1). Given a constrained meromorphic differential equation, we must give a precise definition of equivalence up to diffeomorphisms.

Definition 45. Let (Zi, ci, Si) and (Z2, C2, S2) be two constrained meromorphic differential equations on IIY9 where ci are the constrained sets. Let G be a subgroup of the group of diffeomorphisms over II89. The two constrained differential equations are called G-equivalent if there exists a diffeomorphism i E G such that

i) maps cl onto C2, S1 onto S2; ii) ip maps trajectories of (Zl, cl, Sl) onto trajectories of (Z2i c2, S2). (By convention a trajectory is an analytic solution of Zi which stays in ci\Si)

Definition 46. Let (Fo, F) be a system in A9 and let (H, El, S) be the associated Hamiltonian constrained differential equation. Let A be the map A : (FO, F) H (H, El, S). If (gyp, a, (3) is an element of the affine feedback

group G', we define the action of (cp, a, (3) on (H, El, S) as follows. We can lift cp into a symplectomorphism W defined by

a -1 x = jo(y),

p = q 8x

4.1 Classification of Affine Systems

(Here p, q are row vectors)

103

y

The action of (cp, a, p) on (H, E, cp) is by definition reduced to the action induced by the change of coordinates defined by gyp. In other words the feedback

acts trivially. The following result is straightforward, for the proof see [20].

Theorem 13. The diagram of Fig. 4.1 is commutative: In other words A is

A Ag

G'

A(Ag)

I

Ag

G'

A

A(Ag)

Fig. 4.1.

a covariant.

Less obvious is the assertion that two systems are feedback equivalent if their singular flow are diffeomorphic. Of course, this result is not true in general, for instance both planar systems x(t) = x2(t) + y2(t), y(t) = u(t)

and i(t) = x2(t) - y2(t), y(t) = u(t) have the same singular trajectories ±(t) = x2(t) but they are not feedback equivalent. Hence we must assume that we restrict our analysis to systems with "enough singular trajectories".

Definition 47. Let D(x) = Span F(x) be a distribution and denote by HD the stabilizer of D i.e. HD

C°'-diffeom.with W *Fj

Two vector fields Fo, Fo are called equivalent modulo D if there exists p E HD such that cp * F0 = Fo (mod D), i.e. (cp * Fo - FF)(x) E D(x), Vx E IR".

The following lemma is obvious.

104

4 Singular Trajectories and Feedback Classification

Lemma 15. The following assertions are equivalent: i) (Fo, F) and (Fo, F) are feedback equivalent; ii) a) There exists a C"-difeomorphism tp such that 'p * D = D' b) cp * F0 and Fo are equivalent modulo D' Assumption Let .Ar be the set of affine systems (Fo, F) such that

i) (Fo, F) E A9 ii) ir(V\S) contains a non empty open subset U of R", where 7r is the projection on the state space (x, p) '-+ x. Theorem 14. Let (Fo, F) and (Fo, F) be systems in ,A,.. Then the following assertions are equivalent: i) (Fo, F) and (Fo, F') are feedback equivalent; ii) The constrained Hamiltonian vector fields A(Fo, F) and A(FQ, F') are G' equivalent.

Proof. Let q = maxx rank D(x) and q' = maxz rank D'(x). Hence q, q':5 m where m is the dimension of the control space and we may assume q = q' = m.

Assume A(Fo, F) = (H, El, S) and A(FF, F') = (H', Ei, S') be equivalent. Then there exists a diffeomorphism cp with symplectic lifting cp which maps

El onto E'. Observe that by definition

E, = {(x,p); (p,Y(x)) = 0,`dY E D}

and El is D. Similarly E; = (D')-L. Take a non empty open set W on which the rank of D is maximal. Since the restriction of 0 to W maps the restriction of El onto the restriction of Ei, then P maps locally D onto D' and then globally by analycity. Since D and D' are diffeomorphic we may assume D = D'. Since A(Fo, F) and A(FF, F) are equivalent, there exists a diffeomorphism 'P of R" such that 'P preserves D and satisfying V * X =1Y'

on V'\S'. In particular we have on V'\S'

V * Fo+'p*Fiio V= Fo+F'u' where cp E HD. Therefore

cp*Fo=cp*Fo (modD=D') for x E 7r(V'\S'). By assumption, the equation holds a non empty subset of R'. By analycity it is true everywhere. Thus using Lemma 15 we have proved ii) - i). Hence the theorem is proved. Corollary 9. If n is the dimension of the state space and m is the dimension

of the control space, let us assume m < n - 1 if m is even and m < n - 2 if m is odd. Then, for an open dense set of systems (Fo, F) for the Whitney topology, the feedback classification is equivalent to the G'-classification.

4.1 Classification of Affine Systems

105

Proof. The set V almost everywhere foliated by singular extremals is defined by m equations in the even case and (m + 1) equations in the odd case. They

are linear in p. Hence 7r(V\S) contains generically an non empty subset if m < n - 1 in the even case and m < n - 2 in the odd case.

4.1.3 Time Optimality and Feedback Classification The time optimality status of singular trajectories is feedback invariant and can be used to compute feedback invariants. We shall use this remark to

compute feedback invariants for single input systems in 1R2 and 1R3. Both situations are different: for a planar system, there exists no enough singular trajectories to compute all the feedback invariants; in dimension 3 the singular flow is rich enough to generate all the invariants. In both cases it is important to relate feedback invariants to the geometric classification of optimal control problems.

The planar case. We consider a system in 1R2: 1(t) = X(z(t))+u(t)Y(z(t)), u(t) E R, z = (x, y). Along a singular extremal, the following constraints are satisfied: (p,Y(z)) = (p, [X,Y](z)) = 0. Hence the singular extremals are contained in the set Sl defined by {z; det(Y(z), [X,YJ(z)) = 0}. The singular control of minimal order is given by u(z) = - p.nd2JC Y z and is defined outside the set S2 : det(Y, ad2Y(X )) = 0. Another feedback invariant is the set of points C such that det(X, Y) = 0 where X, Y are collinear. The geometry of the time optimal control in the plane was investigated in Sect.

3.8. Outside the set C it can be investigated using the clock one form. We have two situations: - hyperbolic case: a singular extremal of order 2 is time minimizing. - elliptic case: a singular extremal of order 2 is time maximizing. Example 3. Consider both systems

x(t) = x2(t) + y2(t) y(t) = u(t)

i(t) = x2(t) - y2(t) y(t) = u(t)

The singular trajectories are contained in Sl : y = 0 and are solutions of the one dimensional differential equation: i(t) = x2(t). To distinguish between

the two cases we use the optimality status of the singualr arc in y = 0, x > 0. In the first case it is time maximizing and in the second case it is time minimizing.

The local feedback classification when Y is non zero is analyzed in details in [92] in the generic case. We work around 0 and Y can be taken as . It is based on the classification of the clock form. A basic invariant is the first integer k such that adkXY is not collinear to Y at 0. We state the theorem.

106

4 Singular Trajectories and Feedback Classification

Theorem 15. Any generic C°° pair (X, Y) such that Y does not vanish is locally feedback equivalent at 0 E 1R2 to one of the following canonical forms:

Y = y and for X : (M2) yex,

(M1) (y + War; (M3) (y2 + 1)

(1V14) (y2 - 1) 0

,

(NF5) (y2 + Ax) (NF6) (y3 + xy + a(x)) go; where A E R is an invariant and a(x) is a C°°-function such that a(0) # 0 with the following invariance property: two systems (NF6) with a(x) and a(x) respectively are equivalent if and only if a(x) = a(x) for x < 0. The proof is left in exercise to the reader or see [92].

a

Systems in 1183. Consider the following system

i,(t) = X (v(t)) + u(t)Y(v(t)), where v = (x, y, z) E R3 and X, Y are C°°-vector fields. Introduce D1 = det(Y, [Y, X), [f Y, X], Y]), D2 = det(Y, [Y, X], [[Y, X], X]), D3 = det(Y, [Y, X], X).

(The Des are depending on X, Y and we use the notation DX'Y when necessary)

Convention. In the sequel of this section we consider only systems (X, Y) such that D1 is not identically zero. By singular trajectories we mean a singular trajectory contained in 1R3\{D1 = 01. Such a trajectory is of minimal order.

Proposition 30. The singular trajectories are the solutions of

i(t) = X(v(t)) - D2(v(t))Y(u(t)) D1(v(t))

(4.14)

in 1183\{D1 = 0}.

Proof. Computing the singular control we get (p, Y(v)) = (p, [Y, X1(v)) _ (p, [[Y, X1, X](v) + u[[Y, X],Y](v)) = 0

which implies D2 (v) + uD1(v) = 0. Hence the singular control is the feedback u(v) Di °

Definition 48. Let Z be the vector field X - aY on II83\{D1 = 0}. We observe that Z is the restriction of ir* H to R3\{D1 = 0} where zr is the canonical projection (v, p) ,- v. We denote by A, the mapping (X, Y) +-o Z. The feedback group G' = {(cp, 8)} induces the following action on the set of such vector fields in the state space: (,p, a. O)Z = cp * Z.

4.1 Classification of Affine Systems

107

Clearly we have the following proposition.

Proposition 31. The map A,r is a covariant. Moreover in the analytic case the feedback classification of systems (X, Y) with DI non identically zero is equivalent to the G' classification of pairs (Z, RY). In particular the time optimality status of singular trajectories is encoded in (Z, RY). We shall now describe how the small time optimality can be determined. In Chap. 6 we shall give a definition of curvature associated to the concept of conjugate points to deal with optimality on fixed intervals.

Lemma 16. Let (cp, a,13) E G', then: i) ii)

(v) = det (?X) (cp-1(v)) for i = 1, 2,3 i 8V DX'Y Dx+Y«,Y" =13 Dx'Y

iii) D3 +Y°'YQ iv) D3

= f33 (D2 'Y + aDX'Y)

+Ya'Y(3 = 132D3 'Y

Proof. Computations.

Corollary 10. Let f : R3 - 118 be a C°°-mapping. Define the action of (
1. Al : (X, Y) - Dl ,Y 2. A3 : (X, Y)

D3 'Y

The geometric interpretation is given by the following.

Proposition 32. The set defined by D3 = 0, D1D3 > 0 and D1D3 < 0 are, invariant sets for the solutions of (4.14). Proof. On 1R3\{D1 = 0}, the two vector fields Y and [X,YJ are linearly independent and D3 = 0 is the set of points where X belongs to the linear span of Y and [X, Y]. The Hamiltonian H(x,p, u) = (p, X (x) + uY(x)) is constant along a singular extremal and since (p, Y(x)) = (p, [X, YJ(x)) = 0 we have (p, X (x)) = 0 on D3 = 0. Hence D3 represents the projection of singular extremals with H = 0. Now D1D3 > 0 and D1D3 < 0 are invariant since along a singular trajectory, the Lie bracket ad2Y(X) cannot cross Span{Y, [X, Y]}.

Definition 49. A singular trajectory y(.) is called i) exceptional if y(.) belongs to D3 = 0 ii) hyperbolic if y(.) belongs to D1D3 > 0 iii) elliptic if y(.) belongs to D1 D3 < 0

108

4 Singular Trajectories and Feedback Classification

Small time optimality. Consider a singular trajectory ('y(.), u(.)) defined on [0, TJ and let (ry, p, u) be the associated singular extremal where p is unique

up to a scalar and is oriented with the convention H > 0. Take any M > 0 such that the singular control satisfies I u(t) I < M on [0, T]. Then observe that

each point m(t) = (ry(t), p(t)) is an hyperbolic fold point in the hyperbolic situation and an elliptic fold point in the elliptic case, where fold points are defined in Chap. 3. It can be shown that if T is small enough, as for planar systems, the singular arc is time minimizing in the hyperbolic case and time maximizing in the elliptic case. This result is true in a small C°-neighborhood of -y(.) even if u(t) E R. This can be easily seen using a normal form where y(.),Y and [X,Y] form a frame.

Singular trajectories are characteristics. On R3\{D1 = 0}, the vector fields Y, [X, Y] are independent and we can define the one form w = pdv using

the relation (p, X (x)) = 1 and (p, Y(x)) = (p, (X, Y](x)) = 0. Using Cartan's formula it can be checked the following.

Proposition 33. The vector field Z defined by (4.14) is the characteristic field associated to the one form w.

4.2 Singular Trajectories and the Problem of Classification of Distributions 4.2.1 Preliminaries Consider an affine driftless system (F0 = 0): m

±(t) = E ui(t)Fi(x(t))

(4.15)

i=1

where the vector fields Fi are smooth and defined on an open subset U of itt". In this case the feedback classification is equivalent to the classification of

the distribution: D(x) = Span{Fl(x), , Fm (x)). We shall assume that D is of constant rank m on U. One of the main geometric property of a driftless system (4.15) is that any smooth immersion t - x(t) solution of (4.15) can be reparametrized in order to satisfy the constraints ( u(t) 1= 1 where u = (ul, , u,,,) and I . is any norm on the control domain u E R"'. In particular we can locally use the charts of the projective space P(R'") obtained by fixing the parametrization with ui = 1 for some i between 1 and m. Hence to the smooth driftless affine system (4.15) we can associate m I

affine control systems defined by

i(t) = F,(x(t)) + E ui(t)Fi(x(t)), j = 1, ... m. i#i

(4.16)

4.2 Singular Trajectories and the Problem of Classification of Distributions

109

To compute feedback invariants for the distribution D we can use the singular trajectories of (4.15). They are related in an obvious way to the singular trajectories of systems (4.16). Let (x, p, u) be a singular extremal of (4.15) defined on [0, T], then the following relation holds:

(p(t), Fi(x(t))) = 0 f o r i = 1,

(4.17)

, m and observe that the Hamiltonian M

H(x,p,u) = (p,uiFF(x)) i=1

is zero along a singular extremal. We call such singular extremals exceptional. To compute the singular extremals, we differentiate (4.17) and we obtain for i = 1, , m the constraint m

(p, E u, [Fi, Fj] (x)) = 0.

(4.18)

j=1

Hence to compute the singular extremals of minimal order we can proceed like for affine systems. We distinguish between two cases: - if the number of inputs is odd the singular control is computed by solving the linear equation (4.7); - if the number of inputs is even, we must differentiate the additional relation (4.9) with u E ker O(x, p) where O(x, p) = (p, [Fi, Ft] (x)) to compute the singular control. In any case, we observe that since the Hamiltonian is constant along a singular

arc, then any piece of singular extremals such that the control u(.) belongs to the chart uj = 1 for j = 1, , m can be computed using the algorithm for the affine systems. Hence we have the following proposition.

Proposition 34. The singular extremal arcs of minimal order of driftless affine systems (4.15) are singular extremals of one of the affine system (4.16).

They are exceptional, that is H = 0 and they form a subset of codimension one in the set of all extremals of minimal order of (4.16). Having computed such trajectories we shall use the singular trajectories to compute feedback invariants. Of course we have clearly the following.

Proposition 35. The singular extremals of minimal order are feedback invariants.

4.2.2 Local Classification in Dimension 3, with rank D = 2 See [110] for details. Let us consider a driftless affine system:

110

4 Singular Trajectories and Feedback Classification

NO = u1(t)Fi (v(t)) + u2(t)F2(v(t)),

v = (x, y, z) E R3

and with rank D = Span{FI, F2} = 2. Our classification is localized near vo and we consider only the generic cases, which are all the cases with codimension_< 3. We have three situations which are classified according the nature of the singular flow. The singular extremals must satisfy (p(t), F1(v(t))) = 0,

i = 1,2

(p(t), [F1, F2](v(t))) = 0

and hence they are contained in the set M : {v E R3, det(Fl, F2, [FI, F2J) _ 0). The singular controls of minimal order satisfy (p, u1 [[FI, F2], Fi](x) + u2[[Fi, F2], F2](x)) = 0.

(4.19)

We define the singular set S by S = Si n S2 where Si = {v; det(Fi, F2, [[Fi, F2], F;J) = 0}. We have three generic situations.

Case 1: Take a point vo 0 M, such a point is called a contact point and through xo there passes no singular are. In this case, D is C°° (or C") isomorphic to ker a where a is the contact normal form: a = ydx + dz. For this normalization da = dyAdx and 8 is the characteristic direction. Case 2: (Codimension one) We take a point vo E M\S. Such a point is called

a Martinet point. Since one of the Si is non zero, we observe that M is near vo a C°° (or C'') surface. This surface is foliated by the singular trajectories. A normal form is given by D = ker a, where a = dz - Zdx and if the vector fields Fi are analytic then the isomorphism is analytic. In this normal form we have the following identification:

-M:{y=0}

- The singular trajectories are the solutions of Z = j restricted to y = 0

Case 3: (Codimension 3) We take a point vo e M n S and we assume that the point vo is a regular point of M. The situation has been analyzed in [111]. We can proceed geometrically as follows. If Z is the vec-

tor field direction defined locally near vo on M\{vo} whose solutions are singular trajectories, it can be smoothly extended at vo and has a singularity at vo. The extension is obtained as follows. If we introduce Di = det(Fi, F2, [[F1, F2], F1]), the singular control is defined by the re-

lation D1u1 + D2u2 = 0, then u2 = -"D a.e. and the vector field Z is given by Z = D2F1 - DIF21,, and is tangent to M. Introduce A the linearization of Z at vo and let al, a2 be the eigenvalues of A. Then we have al + a2 = 0 and we have two generic situations: - hyperbolic case: v1, o2 E 1R\{0} - elliptic case: a1, a2 E ilR\{0}

4.3 Feedback Classification and Analytic Geometry

111

The computations in [1101 show that there exists in the C°°-category an unique feedback invariant b in the two normal forms corresponding to the two cases.

Hyperbolic case: D = ker a, a = dy + (xy + x2z + bx3z2)dz Elliptic case: D = ker a, a = (dy + xy + 3 + xz2 + bx3z2)dz --1, the The invariant b has the following interpretation. Since singularity is resonant and b is an obstruction to the formal linearization.

Of course in the analytic category we have numerous additional invariants.

4.3 Feedback Classification and Analytic Geometry 4.3.1 Preliminaries Although the classification of general control systems ±(t) = f (x(t), u(t)) is equivalent to the classification of the sub-class of affine systems of the form x(t) = f (x(t), u(t)), fl(t) = v(t), we can make a direct research of feedback invariants for nonlinear systems using mainly Weierstrass preparation theorem. This is the point of view adopted by Jakubczyk in [581. In essence the starting point is the same. Introducing the Hamiltonian H(x, p, u) = (p, f (x, u)), the singular extremals are the solutions of H contained in the set TU- = 0 and in general they are several solutions Hi. As shown by the discussion of the article of Ekeland in Chap. 3, the time optimal synthesis is related to a competition between the Hamiltonian vector fields Hi. Hence the idea is to recover all the

feedback invariants from the Hi s. Of course it has an interpretation with the time minimal synthesis for the associated affine systems.

Here we shall present several points of the article [581, restricting our study for the sake of simplicity to single input control systems. Definition 50. We consider a single input control system ±(t) = f (x(t), u(t)) and our study is localized near a point (xo, uo) E X x U. Assume that f is real analytic and extend locally f to a complex analytic mapping using the Taylor expansion of f at (x0, uo). We assume the following regularity assumption: Assumption 1

©

(xo, uo) # 0.

Assumption 2 Let F(x) = f (x, U) and F = U=EXF(x). Then assume F C TX is a regular submanifold of TX. The Hamiltonian associated to the system is H(x, p, u) = (p, f (x, u)) where

p 0 0 and it is defined on T'X x U, and linear with respect to p. Hence

4 Singular Trajectories and Feedback Classification

112

its admits a complex extension on en X W where W is a complex neighborhood of (xo, uo). The constraint OH = 0 defines locally an analytic set E : {(x, p, u); g(x, p, u) = 0) where g(x, p, u) = OH. Let po 76 0 be a real direction such that (zo, uo) E E where za = (xo, po). We assume the following:

Assumption 3 E is irreductible near (zo, uo) and is of order p, 1 < p < +oo with respect to u.

4.3.2 Critical Hamiltonians and Symbols Let us consider a point (zo, uo) of multiplicity p with respect to u(.). Then, from the Weierstrass preparation theorem the equation g(z, u) = 0 has the same roots than a polynomial equation

Let V be the discriminant of the above polynomial and S be the set {z; D(z) = 0). If we take a point zl S the equation g = 0 has p distinct roots near zl, in fact the solutions form a p-sheeted covering above zj, denoted fil (z), , u, (z). We define the critical Hamiltonians by

Hk(z) = H(z,u:(z)) These Hamiltonians are not in general holomorphic because the singularity set S. To define holomorphic functions near the branching point (zo, uo) we use the symmetrization procedure from Galois. We introduce for z 0 S, the following functions:

Sk = FIi +

+ Hp

called symbols by Jakubczyk and they are extended by continuity on S. We have the following proposition, see [58].

Proposition 36. a) The functions Sk, k > 1, are holomorphic near (zo, uo) and are also real analytic; b) for k > p, the function Sk are polynomials of S1i , SP; c) the functions Sk are homogeneous of order k with respect to p.

Now we can use the symbols to analyze the feedback classification problem. Indeed if we take two Hamiltonian lifts H(x, p, u) = (p, f (x, u)) and H(x, p, u) = (p, j (x, u)) such that assumptions in Definition 50 hold at (zo, uo) and (zo, uo) respectively we have the following theorem, see [58].

4.3 Feedback Classification and Analytic Geometry

113

Theorem 16. a) If the systems t(t) = f (x(t), u(t)) and i(t) = f (x(t), u(t)) are locally feedback equivalent near (xo, uo) and (xo, uo) respectively, then the corresponding local symbols are related up to a symplectic lifting

Sko'P= Sk, for k > 1.

b) Conversely, if there exists a symplectic lifting 0 such that

SkoS=Sk, fotl
Hence we can study feedback equivalence using the symbols. Of course this approach is not different of the approach based on the singular flow of the affine extension of the system. This will be clarified in Chap. 7 when analyzing the time optimal synthesis for chemical batch reactors.

Notes and Sources We have introduced the idea of using the Hamiltonian singular flow to analyze the feedback classification under the heuristic influence of the article from Ekeland [421, and also the one from Sussmann [1041 where a normal form

is computed for the switching function under a suitable hypothesis implying the nonexistence of singular trajectories. More recent works from Agrachev and Gamkelidze (11] are in the same directions. This idea and pre and postHilbert invariant theory concerning classification of tensors and the numerous algorithms are then available to compute feddback invariants. The classification of distributions is a preliminary problem to be studied when dealing with affine systems. Also it is especially important in sub-Riemannian geometry, see Chap. 9. It is an old problem studied by Darboux, Frobenius, Cartan, Coursat. More recent works are due to Martinet, Zhitomirskii. Also see Mormul [40] for classifying distributions associated to chained systems. See the recent book by Jakubczyk and Respondek (60) containing survey articles about the geometry of feedback and optimal control.

Exercises 4.1 Consider a local contact distribution in R3: a = ydx + dz. Compute the infinitesimal stabilizer group, that is the set of local vector fields X such that X (O) = 0 and (exptX) * a = a.

114

4 Singular 'I'Yajectories and Feedback Classification

4.2 (see [93]) Compute the singular trajectories for the following systems in R3: v(t) = f (V(0) + u1(t)91(v(t)) + u2(t)g2(v(t)) when g1 =A and: .f

(MI) (M2)

M) M)

92

-

8z

(I +Y) f

(18

(M>)

ya y&

(Ms)

Y

'Fz

+ x20

Y)

x28 + X 28 a7z

8 T

e - x20

(NF7) p(z + 2 - 9xy) (NF8) p(z 2 - 9(x2 + y2)) 37Z

x37Z

Ty- - x Z 7X

where p, 0 E R.

4.3 Consider the controlled Euler equation: W(t) = Q(w(t)) + u(t)b,

u(t) E R

(4.20)

W = (WI, W2, W3), 'Q(W) _ (w2w3, -WIw3, wiw2), tb = (bi, b2, b3) where bi are

constant. 1. Compute the singular trajectories. 2. Represent the projections on the sphere S2 of the singular trajectories when: a) tb = (b1, b2, 0), b1b2 # 0,

b) tb = (b1, 0, b3), b1b3 # 0, bi # b3, and deduce that both systems are not feedback equivalent.

c) Assume tb = (b1,b2,e), btb2 # 0 ands $ 0 small enough. Prove that the system is not feedback equivalent to the case tb = (b1, b2, 0).

4.4 Let Dn be the set of germs of analytic distributions of rank m in RI. 1. Give an example of two elements D1, D2 of D without nontrivial singular trajectories and which are not locally diffeomorphic. 2. Give conditions on (n, m) for which there exists an open dense set of such distribution which are locally diffeomorphic if and only if their singular trajectories are locally diffeomorphic.

4.5 Consider the following control system in RI: (4.21) ±(t) = F0(x(t)) + u(t)F1(x(t)). 1. Assume that the system is feedback equivalent near xo to a linear controllable system. Prove that there does not exist a singular trajectory passing through xo. 2. Assume n = 2, 3. Compare the set of pairs (Fo, F1) such that: a) the system is feedback equivalent near xo to a linear system; b) there does not exist a singular trajectory passing through xo.

4.3 Feedback Classification and Analytic Geometry

115

4.6 We consider the following class of control system in the plane:

z(t) = Fo(z(t)) + u(t)Fj (z(t)),

(4.22)

z = (x, y) E R2, u(t) E IR, where the vector fields are germs of analytic mappings near 0.

1. Assume F1(0) # 0 and prove that the system is locally feedback equivalent around the origin to a system of the form: x(t)

= ax"(y'

ar-2(x)yr-2 +

+

... + a,(x)y) + ao(x)

y(t) = u(t) where a = 0, ±1, p > 0, r > 1 and a°(.), , ar_2(.) are germs of analytic functions such that a,(0) = 0 for i = 1, , r - 2. 2. If two systems in the above normal form are locally feedback equivalent near 0, then either a = 0 for both systems or a = ± 1 for both of them and then they both have the same numbers p and r.

4.7 let M be a smooth manifold and let D be a distribution of rank k. We set D1 = D, D2 = D + [D, DJ, , D,,,+1 = D. + [D, D + [D, DI, , D(-+1) = D(-) + [D(-), D(-)]. The small growth vector of D at q E M is the sequence {nl, n2i n3, } of dimensions at q of and the the ascending flag of modules of vector fields D1 C D2 C D3 C big growth vector of D at q the sequences {k1, k2, k3, ...} of dimensions at q of the flag D(O) C Dill C D(3) C . 1. Prove that if two distributions are equivalent up to smooth local diffeomorphisms they have the same small/big growth vector. 2. Let D be a 2-dimensional distribution on a (n+2)-dimensional manifold. We say that D is a Goursat form if D has at every point the big growth vector a) Classify Goursat distributions in dimension 3 and 4.

b) Let D be a Goursat distribution in R5. Prove that at every point q, D is locally equivalent to one of the two distributions around 0: 8 8 8 a s G1: Span{ -,+ql-+q3-+q4-} aql aq2 aqs aq4 aqs

a aql

s

a

ft

a

a

aq4

aqs

G2 : Span{ - , - +q, - + q, q3 - + q, q4 - } 49q3

Show that the small growth vector is (2,3,4,5) in the first case and (2,3,4,4,5) in the second case.

4.8 A quadratic control system is a system in R" of the form: 4(t) = Q(q(t)) + u(t)b

(4.23)

116

4 Singular Trajectories and Feedback Classification

where Q is a quadratic mapping and b is a constant n x 1 matrix. Two systems (Q, b) and (Q', b) are called isomorphic if there exists a linear diffeomorphism such that:

Q' = P-1Q o P,

b' = P"1b

(4.24)

and they are called feedback equivalent if there exists such a P as well as a feedback u = a(x) + /3u' where a is quadratic and ,a is an invertible scalar such that: Q' = P-1(Q + ab) o P, b' = P-1(fb). (4.25) Let {e1,

,

en} be a Rn-canonical basis and let n

xi(t) _

ajkxi(t)xk(t) i,k=1

be a quadratic differential equation with ajk = ak,.

ajkei. Prove 1. Endow 1R" with the multiplication defined by: ej.ek = 1 that the associated algebra is commutative but nonassociative in general. 2. Let vi, v2 E R', prove: a) [Q,vlj(v2) = 2v1.v2, b) [[Q, v1], v2] = 2v1.v2.

3. Assume n > 3. Prove that for an open dense set of pairs (Q, b) the system is feedback equivalent to an unique system with the following normalization: i) b = en

ii) E lank+lei=2ek iii) ant =0 forp=

k=1, ,n-1 1

iv) when n is odd all = a or when n is even all = z

v) ask=0 for 4.9 Let 4(t) = Q(q(t)) + u(t)b be a quadratic control system in 1R3. Compute all the singular trajectories when {Q, b}AL(q) =1R3, for all q E 1R3.

5 Controllability - Higher Order Maximum Principle - Legendre-Clebsch and Goh Necessary Optimality Conditions

In the time optimal control problem, one of the main question is to determine the optimality status of the singular trajectories. They are two distinct problems: small time optimality and time optimality over a fixed interval. We

shall discuss the first problem in this chapter, the second one is related to the concept of conjugate point along a singular arc and it will be discussed in the Chap. 6. To solve the small time optimal problem we use an extension of the maximum principle. Indeed, in the singular case the first order Pontryagin cone K introduced in the proof of the maximum principle is a linear space and hence is not a good approximation of the accessibility set. Hence the idea is to extend this cone using higher order directions. This idea formalized by Krener [67] and Hermes [54) provides higher order necessary optimality conditions. The most useful are the Legendre-Clebsch condition and the Goh condition. These are conditions obtained using the intrinsic second order derivative of the end-point mapping but here we present these conditions using the Baker-Campbell-Hausdorff formula. Before explaining these conditions we shall present two geometric results about local and global controllability that we shall need later.

5.1 Some Notations and Formulas from Differential Geometry In this chapter we shall denote by M a smooth (C°° or C") manifold of dimension n, connected and second countable. We denote by TM the fiber bundle and by T'M the cotangent bundle. Let V(M) be the set of smooth vector fields on M and Dif f (M) be the set of smooth diffeomorphisms.

Definition 51. Let X E V (M) and let f be a smooth function on M. The Lie derivative is defined as: LX f = df (X). If X, Y E V(M), the Lie bracket is given by

adX(Y)=[X,Y]=LyoLX-LXoLy.

If x = (x',

,

xn) are a local system of coordinates we have: X (X)

n

_

Xi(x) axi

118

5 Controllability and Higher Order Maximum Principle

Lx f (x) = axf X (x)

[X, Y] (x) = 8 (x)Y(x) - 8 (x)X (x) The mapping (X, Y) H [X, Y] is IR-linear and skew-symmetric. Moreover, the Jacobi identity holds:

[X, [Y,ZII+[Y,[Z,XII +[Z,[X,Y1] =0. Definition 52. Let X E V(M). We denote by x(t,xo) the maximal solution of the Cauchy problem: x(t) = X (x(t)), x(0) = xo. This solution is defined on a maximal open interval J containing 0. We denote by exp tX the local one parameter group associated to X, that is: exp tX (xo) = x(t, xo). The vector field X is said to be complete if the trajectories can be extended over R. Definition 53. Let X E V (M) and cp E Di f f (M). The image of X by cp is V * X = dcp(X o cp-1).

We recall the following results.

Proposition 37. If X, Y E V (M) and

E Di f f (M), we have:

1. The one parameter local group of Z = p * X is given by: exp tZ = p o exp tX o

(p-1

2. W * [X, Yl = [gyp * X,

3. The Baker- Campbell-Hausdorf (BCH) formula is:

expsXexptY = exp((X,Y) where ((X, Y) belongs to the Lie algebra generated by [X, Y] with: 2

((X, Y) = sX + tY + [X, Y] + 12 [[X, Y], YI 2[X,[Y,[X,Y111+..., z _=4

-i 2

[[X, YI, X]

the series being converging for s, t small enough in the analytic case.

4 We have

exptXexpEYexp-tX = exprl(X,Y)

with rl(X, Y) = e

tk

E k. adkX (Y) and the series is converging for e, t k>O

small enough in the analytic case. 5. The ad-formula is: k tadkX(Y)

exptX*Y= k>o

k!.

where the series is converging for t small enough.

5.1 Some Notations and Formulas from Differential Geometry

119

Definition 54. A polysystem D is a family {V; i E I} of vector fields. We denote by the same letter the associated distribution, that is the mapping

x '- Span{V(x); V E D}. The distribution D is said to be involutive if

[V,V]CD,VV,VJED. Definition 55. Let D be a polysystem. We design by DAL the Lie algebra generated by D. By construction the associated distribution to DAL is involutive. The Lie algebra DAL is constructed recursively as follows:

Dl = Span{D}, D2 = Span{D1 + [DI, D1]}, , Dk = Span{Dk_I + [Di,Dk_I]} and DAL = Uk>1Dk. If x E M, we associate the following sequence of integers: nk(x) = dim Dk(x).

5.1.1 Controllability with Piecewise Constant Controls Definition 56. Consider a smooth system on M, given in local coordinates by

±(t) = f (x(t), u(t)), x(t) E M, U(t) E U C l[2'n

(5.1)

The set of admissible controls u(.) is the setU of piecewise constant mappings.

If x(t, xo, u) is the solution of (5.1) associated to u(.) starting at x(O) = xo, we denote by A(xo, T) the accessibility set xo, u) in time T and A(xo) the accessibility set UT>oA(xo,T). The system is controllable at time T if for each xo we have A(xo, T) = M and is controllable if for each xo we have A(xo) = M.

Definition 57. Consider a control system (5.1) on M. We can associate to this system the polysystem D = { f (., u); u constant, u E U}. We denote by ST(D) the set k

kEN,tj>Oand

t,=T,Vi E D}

and by S(D) the local semi-group: UT>OST(D). We denote by G(D) the local group generated by S(D), that is G(D) = {exp t1 V1

exp tk Vk; k E N, t; E 1R, V E D}.

Properties. 1. The accessibility set from xo in time T is:

A(xo,T) = ST(D)(xo) 2. The accessibility set from xo is the orbit of the local semi-group:

A(xo) = S(D)(xo). Definition 58. We call orbit of xo the set O(xo) = C(D)(xo). The system is said weakly controllable if for every xo E M, O(xo) = M.

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5 Controllability and Higher Order Maximum Principle

5.2 Integrating Distributions 5.2.1 Preliminaries Let D be a polysystem and DAL the Lie algebra generated by D. We consider

the distribution d : x t-, DAL(X). It is an involutive distribution and the problem of integrating d at a point xo is to find a submanifold N, containing

xo such that for each y E N, TyN = A(y). It is a generalization of the Cauchy problem for integrating a single vector field. Here, we are presenting two results:

- if near xo the rank of d is constant, then we have the Frobenius theorem which is a generalization of theorem of linearization of a smooth vector field

X near a regular point; - if the rank is not constant but if the vector fields of the polysystem are real analytic, then the result is still true. It was proved by Nagano-Sussmann. The proofs of both results are radically different.

5.2.2 Frobenius Theorem Theorem 17. Assume that the rank of the distribution d is constant near the point xo: rankd = p. Then there exists a local coordinate system x = 1. In particular near (x' , . , xn) such that d is generated by { ., , xo the integral manifolds are given by x' = constant, i = p + 1, , n. Proof. The proof is standard and is a reccurence on p.

- If p = 1, then locally d = RX where X E V(M), X(xo) 0 0. We use the linearization theorem for ordinary differential equations. - If p > 2, locally d = Span {Y1, , Yp}. We choose a coordinate system y = (y1, . . . , yn) centered at xo such that Y1 = -. Consider the following p vector fields Z1, , Zp of d defined by Z1 =Y1,

Zk=Yk-(LYky1)Y1, 2
By construction LZk y' = 0 for 2 < k < p. Hence locally Zk is tangent for

2 < k < p to the submanifold S : yl = 0. Therefore there exist (p - 1) vector fields Vk on S which are the restriction of the vector fields Zk to

S. They define on S an involutive distribution a of rank (p - 1). We use the recurrence assumption which asserts that there exists on S a local coordinate system z = (z2, , zn) such that

d = Span{V2i

,

Vk) = Span{

a az2

a

"&p)

We can define a local coordinate system centered at xo by

5.2 Integrating Distributions

121

xi=yl, x'=z` 2
tribution Span{r,

,

0}.

, Z,} coincides locally with the flat disTo prove it, it is sufficient to check that

LzkxP+r = 0 for r > 1 and 1 < k < p. 1.

Fork=1: since Zi=

r, we

have Lz,x`=0fori>2.

2. For k > 2: first of all we observe that axl (LzkxP+r) = Lz (LzkxP+r) Since Lz, xP+r = 0, we can write

axi (LzkxP+r) = L(zk,Zil (XP+r) and we know by construction that

[Zi, Zk] E Span{Zj; j > 2}. Hence we can write

,(LzkxP+r)Aj(Lz,p+r) j=2

where the Aj are scalar. It is a linear differential equation with respect to x1. For x1 = 0, we have Zk = Vk and by construction LzkxP+r = 0. Since the solution of a linear system with values 0 at x1 = 0 is the identically zero solution, we have Lzk Xp+r = 0. The theorem is proved.

0 5.2.3 Nagano-Sussmann Theorem See [105]. When the rank condition is satisfied (rankL =constant) we get from the Robenius theorem a description of all the integral manifolds near xo. If we only need to construct the leaf passing through xo the rank condition is clearly too strong. Indeed, if D = {X } is generated by a single vector field

X, there exists an integral curve through xo if X is locally Lipschitz. For a family of vector fields this result is still true if the vector fields are analytic.

Theorem 18. Let D be a family of analytic vector fields near xo E M and let p be the rank of A : x H DAL(x) at xO. Then through xo there exists locally an integral manifold of dimension p.

122

5 Controllability and Higher Order Maximum Principle

Proof. Let p be the rank of A at xo. Then there exists p vector fields of DAL : X1, , XP such that Span{X l (xo), . the mapping

,

XP(xo) } = A(xo). Consider

a : (t1,...,tp) i--+ exptjX1 ...exptpXp(xo)

It is an immersion for (t1, , t p) = (0, , 0). Hence the image denoted by N is locally a submanifold of dimension p. To prove that N is an integral manifold we must check that for each y E N near xo, we have TAN = A(y). It is a direct consequence of the equalities DAL(exptX;(x)) = dexptXi(DAL (x)),i = 1,

,p

and x near x0, t small enough. To show that the previous equalities hold, let V(x) E DAL(x). Then there exists Y E DAL such that V(x) = Y(x). By analycity and the ad-formula for t small enough we have = 1: k (dexptXi)(Y(x)) k>O

Hence for t small enough, we have

(dexptXi)(DAL(x)) C DAL(exptXi(x)) Changing t into -t we show the second inclusion. O

5.2.4 C°°-Counter Example To prove the previous theorem we use the following geometric property. Let

X, Y be two analytic vector fields and assume X(xo) # 0. From the adformula, if all the vector fields adkX(Y), k > 0 are collinear to X at x0, then for t small the vector field Y is tangent to the integral curve exp tX (xo). Hence it is easy to construct a C°°-counter example using fiat CO°-

mappings. Indeed take f : R

R a smooth mapping such that f (x) = 0

for x < 0 and f (x) # 0 for x > 0. Consider the two vectors fields on 1R2: X = a and Y = f(x)s. At 0, DAL is of rank 1. Indeed, we have [X, Y](x) = -f'(x) = 0 at 0 and hence [X, YJ(0) = 0. The same is true for all high order Lie brackets. In this example the rank of DAL is not constant along exp tX (0), indeed for x > 0, the vector field Y is transverse to this vector field.

5.3 Nonlinear Controllability and Chow Theorem Theorem 19 (Chow). Let D be a C°°-polysystem on M. We assume that for each x E M, DAL(x) = TIM. Then we have G(D)(x) = G(DAL(x)) = M,

for each x E M.

5.3 Nonlinear Controllability and Chow Theorem

123

Proof. Since M is connected it is sufficient to prove the result locally. The proof is based on the BCH-formula. We assume M = R3 and D = {X, Y} with rank{X, Y, [X, Y]} = 3 at xo; the generalization is straightforward. Let A be a real number and consider the mapping 'Pa : (t1, t2, t3) r--' exp AX exp t3Y exp -AX exp t2Y exp t1 X (xo).

We prove that for A small be non zero, spa is an immersion. Indeed using BCH formula we have

WA(tl, t2, t3) = exp (t1X + (t2 + WY +

At3

2

[X, Y] + - )(xo),

hence

acl (0, 0, 0) = X (xo).

(0, 0, 0) = Y(xo),

09

8t3 (0) = Y(xo) + 2 [X, Y] (xo) + o(ff).

Since X, Y, [X, Y] are linearly independent at xo, the rank of (pa at 0 is 3 for

,\:A 0 small enough. 0 Definition 59. The polysystem D is called weakly controllable if the orbit O(x) of G(D) is M for every x E M. The polysystem D is called controllable if the orbit A(x) of S(D) is M for every x E M. The polysystem D is said symmetric if for every X E D, we have -X E D.

Example 4. Let D = I -L,, aL } on R2. Hence 0(0) = JR2 and A(O) = {x > 0, y > 0}. In general we have that A(x) is a strict subset of O(x). Corollary 11. Let D be a symmetric polysystem. Assume that rank DAL(x) = n (dim M) for every x. Then D is controllable. In the analytic case this rank condition is also necessary.

Proof. Since D is symmetric, we have: orbit S(D) = orbit G(D). Then we apply the Chow theorem. In the analytic case, we apply the Nagano-Sussmann theorem. 0 Remark 6. The symmetric case is the only case where we can conclude trivially that S(D) = G(D). In general the problem to decide if a semi-group is transitive is difficult. Nevertheless the following weaker result is true, see [106].

Proposition 38. Let D a polysystem. If dim DAL(X) = it (dim M) for every x E M then for each neighborhood V of x there exists a non empty open set U contained in V n A(x).

5 Controllability and Higher Order Maximum Principle

124

Proof. Let x E M. If dim M > 1, then there exists X 1 E D such that Xl (x) # 0, otherwise we would have dim DAL(X) = 0. Consider the integral curve a 1 t i--+ exp tX 1(x). If dim M > 2, then there exists in every neighborhood V of x a point y E M such that y = exp t1 X 1(x) and vector field X2 E D such that X2 and X 1 are not collinear at y, otherwise we would have dim DAL = 1. Consider the mapping a2 : (tl,t2) -. expt2X2expt1X1(x). If dim M _> 3, then there exists in every neighborhood of y a vector field X3 transverse to the image of a2 With this method we construct in every neighborhood V of x a mapping a - exp t 1 X 1(x) such that in a point z = a,,(ti, , t',) of V, a is an immersion. This construction provides a nonempty open set U contained in V fl O(x). :

5.4 Poisson Stability and Controllability Definition 60. Let X be a Coo -vector field on M. The point xo E M is said Poisson stable if for every T > 0 and every neighborhood V of x0 then there exists tl, t2 > T such that exp t1X (xo) and exp -t2X (xo) E V. The vector field X is called Poisson stable if the set of Poisson stable points is dense in M.

Theorem 20 (Poincar6 [49]). If M is a compact manifold with a volume form w, each conservative vector field X is Poisson stable.

Proposition 39. Let D be a polysystem. Assume the following: i) for every x E M, rank DAL (x) = n (dim M); ii) every vector field X E D is Poisson stable. Then the system is controllable.

Proof. Here we outline the proof, see [80] for details. Let x, y E M, we must show that there exists X1i , Xk E D and t1, tk > 0 such that -

y = expt1X1 ...exptkXk(x).

Since D satisfies the rank condition, we can apply Proposition 38 to D and

-D to find the existence of two points x', y' and two open set U and V such that x' E U, y' E V such that x' can be steered using Xi E D, -D to each point of U and each point of V can be steered to y'. To prove the proposition it is sufficient to show that there exists two points x" E U and y" E V such that x" can be steered to y". Since the polysystem satisfies the rank condition, there exists p vector fields Y1i

,

Yp and p non zero (positive

or negative) real numbers si such that y' = exps1Y1 ... expspYP(x"). In the previous sequence each element exp skYk corresponding to a negative time sk can be nearby replaced by an arc exp skYk using the Poisson-stability of Yk. The result follows.

5.5 Controllability and Enlargement Technique

125

5.4.1 Application A first application of Proposition 39 is to construct many controllable polysystems on compact manifolds using Poincard theorem.

Example 5. Take M = G a compact Lie group and D a polysystem whose each vector field is right invariant. Then the polysystem is controllable if and only if the rank condition is satisfied. In this case, observe that DAL is a Lie sub-algebra of g ^_- TG and hence is finite dimensional. There exist algorithms to compute DAL.

5.5 Controllability and Enlargement Technique See [641.

5.5.1 Problem Statement Let D be a polysystem satisfying the rank condition: rank DAL(x) = dim M

for all x E M. To study the controllability of such a polysystem, a powerful technique is the enlargement technique which was codified by Jurdjevic-Kupka, see for instance 165]. The principle is simple, we enlarge D using operations which are not modifying the controllability of D. We shall briefly explain these operations.

Lemma 17. Let D be a polysystem such that rank DAL(x) = dim M for all x. Then the polysystem D is controllable if and only if the adherence of S(D)(x) is M for every x E M. Proof. Use Proposition 38.

Definition 61. Let D, D' be two polysystems satisfying the rank condition.

We say that D and D' are equivalent if for every x E M: S(D)(x) = S(D')(x). The union of all polysystem D' equivalent to D is called the saturated of D and is denoted by sat D.

Clearly a polysystem D is controllable if and only if sat D is controllable. Now, we define the codified operations.

Proposition 40. Let D be a polysystem, then the convex cone generated by sat D is equivalent to D.

Proof. Clearly if X E D then AX E sat D for every A > 0 (reparametrisation). Let X, Y E D, using BCH formula we have

H exp!Xexp!Y =exp(t(X+Y)+o()).

n times

Taking the limit when n -. +oo, we have X + Y E sat D.

126

5 Controllability and Higher Order Maximum Principle

Proposition 41. Let X E D and assume X Poisson stable, then -X E D. Proof. It is a consequence of the proof of Proposition 39.

Proposition 42. If ±X, ±Y E D, then ±[X, Y] E sat D. Proof. Apply the BCH-formula.

Definition 62. Let D be a polysystem on M. The normalizer N(D) of D is the set of diffeomorphisms cp on M such that for every x E M,
Proposition 43. Let D be a polysystem, X be an element of D and 9 E N(D). Then cp * X belongs to sat D. Proof. Let Y = cp * X, then we have y = (exp tY) (x) = cp o exp tX o cp- 1(x).

Hence y belongs to the adherence of S(D)(x). The proposition follows.

Proposition 44. If D is a polysystem, then the closure of D for the topology of uniform convergence on the compact sets belongs to sat D.

Proof. If X X when n -- +oo for the above topology, then exp tX,a Fexp tX when n

+oo on each compact set.

Application [Controllability of linear systems.] Consider the linear system of R": _+ (t) = Ax(t) + Bu(t), u(t) E IRP, A, B constant matrices. Then the linear system is controllable if and only if the rank of R = [B, , An-1B] is n. Indeed, the system can be written i(t) = Ax(t) + EP I ui(t)bi and we introduce the polysystem D = {Ax + p 1 uibi; ui E 1RP}. Computing we have

DAL =Ax ®ImR and the rank is minimal at 0 and equals to the rank of R. Since the system is analytic, we must have rank?Z = n. Let us prove the converse. Let i E {1, , p} then ±bi E sat D. Indeed for every n E N n (Ax + Bu) E sat D

and setting ui = ne, e = ±1 we have lim 1(A + nebi) = ebi E sat D.

"-.+oo n

Hence for every A E R, we have exp Abi * Ax E sat D.

5.6 Evaluation of the Accessibility Set

127

Computing we obtain k

exp.lb; * Ax =

k! adkb;(Ax) k>O

and adkbi(Ax) = 0 for k > 2. Hence for A # 0 we have

expAbi*Ax = --1(Ax-AAb;)esat D. Taking the limit when A oo we obtain ±Ab; E sat D. Then we repeat the same operation, replacing bi by Abi. At the end we have

=7ZE sat D

i= The result is proved.

Proposition 45. Consider the following affine system on M: P

x(t) = Fo(x(t)) +Eui(t)F,(x(t)),

ui(t) E R.

i=1

Let D be the distribution x -+ Span{Fi (x),

FP(x)}.

If rank DAL(X) = dim M, for all x E M, then the system is controllable.

Proof. As before for every n E N, ui = ne, e = f1, uj = 0 if j 0- i 1(Fo + nEF;) E sat D n

and by making n - +oo we have ±F; E sat D for i = 1,

p. Applying Proposition 42 we have DAL E sat D. If DAL is of rank n, the system is controllable.

5.6 Evaluation of the Accessibility Set The Baker-Campbell-Hausdorff formula can be used to make evaluation of the accessibility set by constructing an approximating cone. In spirit it is similar to the idea of Pontryagin maximum principle where we construct the first order Pontryagin cone. This was extensively used by Hermes, see for instance [54J, to get higher order necessary optimality conditions along a singular arc.

128

5 Controllability and Higher Order Maximum Principle

Definition 63. A rational polynomial is an expression of the form Fa_1 G tq, where l E N, t>_0 small enough, 4: E Q and c; E R. It is called positive if ci > 0 for all i = 1, 1. Let X, Y E V (M), D be the polysystem {X,Y}, DAL the Lie algebra generated by D. We denote by £ the set of 1

germs of vetor fields W such that there exists k E N and rational polynomials [0,e] IIt such that

exp r1(t)X exp s1(t)Y = exp (tW + 0(t°))

exp rk(t)X exp sk(t)Y

with a > 1. From the BCH formula, the set 6 is contained in DAL. We shall prove the following result.

Theorem 21. The set £ is DAL In order to prove this result we need several lemmas and the following two formulae n

k

expt1Xexpt2Wexp-t1X =exp(t2(E

(5.2)

k=0

exp t1Xexp t2 W = exp(t1X + t2W +

t 22

t12 [[X, W[, X]

[X, W]+ 924

t

2

22 [[X, WJ, WJ

[X, [W, [X, w))) + ...) (5.3)

Lemma 18. The set .6 is convex.

Proof. Let A E [0, 1] and W1, W2 E 6. Then there exists k1, k2 E N and rational polynomials such that

exp(tW1 +0(t°)) exprk2(t)X . expsi(t)Y = exp(tW2 + 0(t°)) exprk,(t)X

.

Hence we have

exp (rk1(At)X) ... exp (si (At)Y) exp (rk, (1 - A)t)X ... exp (si (1 - A)t)Y) = exp (AtW1 + (1 - A)tW2 + 0(t°)) where a > 1. Hence AW1 + (1 - A) W2 EE.

Lemma 19. The vextor fields -X and -Y belong to 6. Proof. For a, 0 E lit we have exp (atX) exp (/3tY) = exp (t(aX +,QY) + 0(t2)).

If we seta=-1,Q=0, we have -XE£,and if we set a=0, 3=-1, we have -YEE.

5.6 Evaluation of the Accessibility Set

129

Proof (Theorem 21). We must show that if ±W E E, then ±[X, W] and ±[Y, W] E E. Since ±W E E, there exists rational polynomials such that

exprk(t)X . exp si (t)Y = exp (tW + 0(t°)) exprk(t)X . . .expsi(t)Y = exp(-tW +O(t°))

witha> 1. Let33EQwith 0< 6< 1 and0 > 1. Then exp (t'-"X) exp (OW + 0(t°A)) exp (-tl-'9X) exp (-taW + 0(t°')) = exp (tpW + t[X, W] + ) exp (-tOW + 0(t°p)) = exp (t[X, W] + ). This proves that [X, W] E E. Hence by reccurence we show that ±[adk^Y, [adk^-' X, [... , [adki X, y] ...]

belongs to E. Since these Lie brackets are generating DAL the theorem is proved.

5.6.1 Legendre-Clebsch Condition We can use the same approach to get necessary higher order optimality conditions along a singular arc. We proceed as follows. Consider a single-input afiine control system

x(t) = X (x(t)) + u(t)Y(x(t))

(5.4)

where X, Y are smooth vector fields of R'. Then we have the following lemma.

Lemma 20. Let y : [0, T] - 1R be a singular arc. Assume that y(.) is reduced to a point or y(.) is a smooth one-to-one immersion and that the vector field Y is nonvanishing along y(.). Then there exists a CO -neighborhood

U of y(.) and a feedback u = a(x) + v, a(.) smooth on U, such that on U the singular are y(.) is a trajectory of the system

±(t) = X (x(t)) + v(t)Y(x(t)),

(5.5)

where f ((x) = X (x) + a(x)Y(x), corresponding to the control v =_ 0.

Proof. If y(t) is reduced to a point xo, the singular control can be taken as u(t) = uo where uo is such that X(xo) + uoY(xo) = 0 and we can take a(x) = a(xo) = uo. If y(.) is a smooth one-to-one immersion, then there exists

a smooth coordinate system along y(.) such that y(.) is reparametrized as t -- (t, 0, , 0) for t E [0, T]. Since Y is non vanishing along y(.), then the singular control u(.) is uniquely determined by y(.) using (5.4). In particular it is a smooth mapping. Since y(.) is t ' - (t, 0, , 0) it can be written u(xi ) for xi E [0,T). We set a(x) = a(x' ). In both cases -y(.) is the trajectory corresponding to v = 0. The lemma is proved.

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5 Controllability and Higher Order Maximum Principle

Proposition 46. Consider the follouring C"-system on liY":

i(t) = X(x(t)) + u(t)Y(x(t)), I U(t) I< 1. Let y(.) be a singular trajectory defined on [0, T] corresponding to the singular control u =_ 0 on [0, T). Then for every t > 0, the first order Pontryagin cone

K(t) is the set K(t) = Span{adkX(Y)(y(t)); k E N}.

Proof. (In the C°° case we have K(t) C Span{adkX (Y)(y(t)); k E N}; the proof is straightforward) Since y(.) is singular on [0, TJ then there exists

p(t) E Rn\{0}, p(t) _ -", H(x, p, u) = (p, X (x) + uY(x)) and (p(t),Y(y(t))) = 0, Vt.

(5.6)

Differentiating recursively this relation and using u - 0 along y(.), one gets (p(t), adkX (Y)(ry(t))) = 0, Vk.

(5.7)

From Chap. 2, we know that p(t)1K(t). In the C' case both relations (5.6) and (5.7) ar equivalent, hence we have the identity K(t) = Span {adkX(Y)(y(t)); k E H}. 0

From the above proposition the first order Pontryagin cone is with empty interior in the singular case. We shall construct an extension of this cone to get better necessary conditions. For this we use the procedure of Hermes [54) which is similar to the one of the begining of this section to show the identity DAL{X,Y} = S. Definition 64. Consider the Coo-control system

.i(t) = X(x(t)) + u(t)Y(x(t)), I u(t) 1< 1 and let y(.) be the reference singular trajectory associated to u - 0 and starting from xo: y(t) = exp (tX)(xo). A vector W belongs to the higher order Pontryagin cone K'(t) at y = exp tX (xo) if there exist positive rational polynomials o2k, r2k, . ' , of , ri such that a(e) = exp o2k(E)X exp r2k(E)(X + Y) exp a2k-l (E)X exp r2k-1(E)(X - Y) 2k

exp a1(E)X exp r1(E)(X - Y) exp (- J(o;(E) + r (e)X )) expTX (xo) i=1

=exp(EW+O(e))(y) with Q > 1. By construction the left hand side belongs fore > 0 small enough to A(xo, T). Hence a(E) is a curve in the accessibility set and W is the tangent vector at E = 0.

5.7 The Multi-Inputs Case: Goh Condition

131

Proposition 47. The set K(t) is a convex cone containing the first order Pontryagin cone. Moreover the vector [Y, [Y, X]](ry(t) belongs to K'(t).

Proof. By construction K'(t) contains Span{adkX(Y)(ry(t))} = K(t). The convexity is proved in [54], see also the proof of lemma 18. To prove the last part of the proposition we have from the BCH formula exp e i (X - Y) exp 2c (X + Y) exp l (X - Y) exp - 4e X

= exp (2ead2Y(X)

- 2Ead2X(Y) + o(c)).

Hence 3ad2Y(X) - 2ad2X (Y) E K'(t) and since K'(t) is a convex cone containing ±ad2XY, we deduce that ad2Y(X) E K'(t).

0

As in the PMP, K'(t) is an approximating cone of the PMP and we deduce the so-called higher-order maximum principle.

Proposition 48. [67] If the trajectory x(t) associated to u - 0 is a time optimal trajectory then there exists p(t) E IR"\{0} such that the following equations are satisfied for every t E [0, T]:

dt (t) = X WO) d. (t) = -p(t) 8X WO)

(p(t), Y(x(t))) = 0 (p(t), X (x(t))) > 0 (Legendre Clebsch condition),

(p(t), [Y, [Y, X]](x(t))) < 0

(5.8)

Remark 7. We observe that if K(t) is of codimension one then in general K'(t) is a cone with non empty interior and is then a good approximating cone. We shall discuss later the genericity of the codimension one assumption and the sufficiency of the above optimality conditions.

5.7 The Multi-Inputs Case: Goh Condition Definition 65. Consider the multi-inputs affine control system m

x(t) = Fo(x(t)) +

>ui(t)Fi(x(t)),

x(t) E R°,ui(t) E R.

i=1

Let (x(t), p(t), u(t)) be a reference extremal defined on [0, T]. We say that it satisfies Goh condition if

132

5 Controllability and Higher Order Maximum Principle

(p(t), [Fv, Fw](x(t))) = 0

for all v, w E lRm, Vt E [0, T], where F = (Fl,

,

(5.9)

F,,,).

Remark 8. A slight modification of the enlargement technique allows to prove the following: if F denotes the polysystem {F1i - , F.} then if FAL (X) is of rank n for every x E R", for every T > 0, the accessibility set A(xo, T) from -

x0, in time T is all R" because the polysystem D = {Fo + Fu; u E Rm} is equivalent to {Fo,FAL}. In this case the time minimal control problem, where the domain of control is lR'" has no solution. Observe that [F, F] is contained in FAL. Therefore the Goh condition is somehow connected with necessary optimality conditions for the time optimal control problem and we shall precise this connection in the next section.

5.8 The Concept of Rigidity - Strong Legendre Clebsch and Goh Conditions as Necessary Conditions for Rigidity Definition 66. Consider the affine control system

i(t) = Fo(x(t)) + > ui(t)Fi(x(t)) i=1

where ui(t) E R, X(t) E M. Let x0 be a fixed point and (x(t),u(t)) be a reference trajectory defined on [0, T] and starting at t = 0 from xo. Assume that the control u(.) is left continuous at T and it is extended on IT, +oo[ by the constant u(T). The control u(.) and the corresponding trajectory are called C' -rigid on [0, T] if there exists e > 0 such that there exists no control v(.) contained in a e-neighborhood of u(.) in LOD([0,T]), v(t) 76 u(t) (a.e.) which can steer xo to x(T) in a time T' E [T - e, T + e].

Definition 67. The (timex input) end-point mapping E is the mapping R x L°°([0,T]) -, M which maps T x u(.), u(.) E LOD([0,T]) onto the end-point x(T, xo, u) of the trajectory of the system. Observe that this mapping is a slight extension of the end-point mapping defined in Chap. 3, where the time T is fixed.

As previously we can compute the Frechet derivative of E and we get the following first order necessary condition for rigidity which is a slight extension of the weak maximum principle.

Proposition 49. If (x(.), u(.)) is C'-rigid on [0,T], then there exists an absolutely continuous vector p(.) on (0, T], nonzero such that the triplet (x, p, u) satisfies the following equation a. e.:

5.8 Rigidity-Strong Legendre Clebsch and Goh Condition

i(t) = a (x(t), p(t), u(t)), At)

aH (x(t), p(t), u(t))

133

(5.10)

where H(x, p, u) = (p, Fo(x) + F(x)u) is the Hamiltonian, together with the conditions for every t: (x (t), p(t), u(t)) = 0 8u H(x(t), p(t), u(t)) = 0.

(5.11) (5.12)

Proof. The proof is similar to the proof of the weak maximum principle and required the computation of the Frechet derivative of E. Conditions (5.10) and (5.11) are the conditions of the weak maximum principle where T is fixed. The additional conditions (5.12): H = 0 uses variation of the time T, see also the conditions H = 0 in the PMP where the time is not fixed. 0

Remark 9. According to the previous proposition, a necessary condition for

C'-rigidity is that (x, p, u) is an exceptional singular extremal.

5.8.1 Intrinsic Second Variation - Morse Index Definition 68. By construction pr = p(T) is orthogonal to the image of Eu. The intrinsic second variation is Q. = pTE"(S, v) where E" is the second variation and (b, v) belongs to the kernel of the first variation (Here b is a time direction and v is a control direction for the variations of the form (T + b, u + ev)). The Morse index of the extremal (x, p, u) is the maximal dimension of the subspaces where Q. is negative definite. The Morse index of the singular trajectory (x, u) is the minimum of indices of quadratic form Qp1 for all possible pT orthogonal to Im Eu. One of the main contribution of [11] is the following.

Theorem 22. A necessary condition for C' -rigidity is that the Morse index of the reference trajectory (x, u) is finite. Both Goh-condition and Legendre Clebsch condition are necessary conditions for the finitness of the Morse index. Hence we get, see [11).

Proposition 50. Let (x, p, u) be a singular exceptional trajectory defined on [0, T]. Then the following conditions are necessary for finitness of the Morse index:

- Goh condition: (p(t), [Fv, Fwj(x(t))) = 0, Vt E [0, T[ - Legendre-Clebsch condition:

8 d2 8H au ate au

>0

134

5 Controllability and Higher Order Maximum Principle

Remark 10. The Legendre-Clebsch condition is the multi-inputs generalisation of the necessary optimality condition established in Sec. 5.6.1. Remark 11. Consider a single-input control system

x(t) = Fo(x(t)) + u(t)F1(x(t)). In this case, if (x, p, u) is a singular extremal one has (p(t), [Fo, Fi](x(t))) = 0. Hence Goh condition is a consequence of the extremality condition. Of course

it is not true in the multi-input case. We shall clarify in the next chapter the concept of C°-rigidity for single-input control systems.

Notes and Sources Higher order necessary optimality conditions for singular extremals are due to Goh, Kelley and others; see the survey article ([46]). The higher order maximum principle is due to Krener [67] and Hermes [54]. The enlargement technique exists in an heuristic form in Hirschorn [56], but was conceptualized by Jurdjevic-Kupka [65] and fully exploited to get controllability conditions for right-invariant systems on Lie groups. The problem of rigidity is everywhere present in the litterature about the abnormal problem in calculus of variations, see for instance Bliss [18]. It also appears in the article [26] in control theory and the neat analysis concerning this problem is due to [11].

Exercises 5.1 Let G be a Lie group with identity e, TeG its tangent space at e, L(G) and R(G) be the Lie algebra of left invariant and right invariant vector fields. We define as usual on TG Lie algebra structures as follows. If A, B E TeG, we define their Lie product by [A, B] = [X, Y] (e) where X and Y are right (or left) invariant vector fields such that X(e) = A, Y(e) = B. 1. Prove that the diffeomorphism on G, .0 : g -+ g-1 induces an isomorphism between L(G) and R(G).

2. Prove that the Lie product induced on TeG by the left invariant vector fields is equal to the negative of the Lie product induced by the right invariant vector fields.

5.2 Let G be the Lie group GL(n,R) of invertible n x n matrices with TeG = g1(nlR) the set of n x n matrices. 1. Prove that a right (resp. left) invariant vector field is of the form X (g) _ Ag (resp. gA) with A E gl(n, lR). 2. If X (g) = Ag, Y(g) = Bg are right invariant vector fields prove that the Lie bracket is [X, Y) (9) = (AB - BA)g and therefore gl(n, R) is a Lie algebra with the matrix commutator as Lie bracket.

5.8 Rigidity-Strong Legendre Clebsch and Goh Condition

135

5.3 Let G be a Lie group acting on a manifold M. To each X in g = TeG identify to the Lie algebra of right invariant vector fields, we can associate a vector field X+ on M by the formula:

(X+f)(m) = lim

f (exp tX m) - f (m)

t-o

(5.13)

t

for f smooth function on M. 1. Prove that [X+, Y+J = [X, Yj+. 2. Let G = GL(n,1R) and M =1R". a) If X (g) = Ag where A is a n x n matrix, compute X+. b) Let D be a polysystem of right invariant vector fields on GL(n,1R) and D+ be the induced system on 1R". Prove that D++L is a finite dimensional Lie algebra. Assume D weakly controllable (resp. controllable) on GL(n, R). Prove that D+ is weakly controllable (resp. controllable) on 1R"\{0).

5.4 Let G be a Lie subgroup of GL(n, R) and consider the subgroup E of GL(n + 1, R) of all matrices of the form: I 1 R

in 1R" andREG.

)

where v is a vector column

1. Prove that the Lie algebra of E is the set of matrices M = I 0

0), with

a being a column vector in 1Rn and A is a matrix in the Lie algebra of G, the Lie bracket being given by: [(a, A), (b, B)] = (Ab - Ba, [A, B]).

(5.14)

2. Prove that E acts on the affine space 1R" =I (1,x) where x is a column vector in R" with the law:

(v R) (x)

- (Rx+v)

(5.15)

Compute the vector field induced by (a, A) on 1R".

5.5 Consider the controlled Euler equation: c:w1(t) = W2(t)W3(t) + u(t)b1

L,2(0 = -W1(t)w3(t) + u(t)b2 W3 (t) = w1(t)W2(t) +u(t)b3

where b = (b1ib2,b3) is constant and IIu(t)II < 1.

1. Assume u = 0 (free motion). Show that the differential equation is Poisson stable. 2. Give necessary and sufficient condition on b to get a controllable system.

5 Controllability and Higher Order Maximum Principle

136

5.6 We consider the following right-invariant control system on SO(3)

dX

(t) = (A+ u(t) B)X (t), X (t) E SO(3), U(t) E IR

where

A=

-100 01010

,

0 00

B=

0 O1 0 001

.

-100

1. Show that the Lie algebra generated by {AX, BX } is the Lie algebra of SO(3). Deduce that the system is controllable on SO(3). 2. Consider the same system but with X = x E S2. Show that the system

is controllable on S2. Given any points xo and xl on S2 construct a piecewise constant control to reach xl from xo.

5.7 Consider a bilinear system on 1R2: x(t) = u(t)Ax(t) + (1 - u(t))(Bx(t) +

b) where A, B are matrices, b is a vector and u(t) E 10, 1). Classify the controllable systems.

5.8 Consider the following system in 1R3: x(t) = X (x(t)) + u(t)Y(x(t)), u(t) 1< 1. We denote by -y+,-y- the arcs corresponding respectively to u T +1, u = -1. Let A(xo, t) be the accessibility set in time t from xo and I

A' (xo) = Ut>o,small enough A(xo, t) the small time accessibility set. Assume

that X(xo),Y(xo) and [X,Y](xo) are linearly independent. Show that the boundary of A'(xo) is formed by the following two surfaces: Sl = U-Y+ -y- (X0)'

S2 = Ury_ry+(xo) and that each point in the interior can be reached by an arc y+'y-'y+ and t-'y+'Y-. 5.9 Consider a rigid spacecraft controlled by gas jets and described by the following system:

R(t) = S(w(t))R(t) P

w(t) = Q(w(t)) + E Uk(t)bk k=1

where R E SO(3) =group of rotations in 1R3 with determinant 1, S(w) is the 0

antisymmetric matrix

w3 -w2

-W3 0

wl

,

Q is given by

W2 -WI 0 a a a a3wlw2 , Q = alw2w3 aw + a2wlw3 aw + aw3 2 1

al =

a2

a3= f,I,>I2>I3>0,thebksarecon-

stant vector fields anJ uk(.) are piecewise constant mappings with values in

{-1,0,1).

5.8 Rigidity-Strong Legendre Clebsch and Goh Condition

137

1. Show that the free motion corresponding to a control identically zero is Poisson stable. 2. Prove that the system is controllable if and only if {Q, b1, , b,} is of rank 3 at each point w E 1R3.

3. Assume p = 1. Prove that the system is controllable if and only if the constant vector field b1 does not belong to one of the axis Owi, i = 1, 2,3 or one of the two planes given by the equation: a3w2 - a1w3 = 0.

5.10 Let Hk be the set of polynomial homogeneous vector fields in R' with degree k and consider a quadratic control system: P

2(t) = Q(x(t)) + E ui(t)bi

(5.16)

i=1

where Q E N2 and bi E H°. Let L be the Lie algebra generated by {Q, b1,

,

b,,} and let us define:

i) L° is the smallest subspace of 7.{° such that:

a) bi EL°, for all b) if v1, v2 E L°, then [[Q, VI 1, v2] E L°.

ii) Ll is the Lie subalgebra of W1 generated by the vector fields: {[Q, v]; v E L°}. iii) L2 is the Lie subalegbra of ek>2Hk generated by the vector fields {adL, o

..oadLm(Q); mEN;L1EL1} 1. Prove that L = L° ®L1 ® L2. 2. Let 0(0) be the orbit of 0, prove that 0(0) = L°. 3. Define the following sequence of subspaces: i) So is the linear span of {b1, , by}; ii) for k > 1, Sk is the linear span of Sk_1 U {[[Q, V11, v2]; vi, v2 E Sk_1}. Prove that L° = Uk>°Sk and that there exists k < n such that L° = Sk_1.

4. Consider the planar single-input system: i(t) = Q(a(t)) + u(t)b, u(t) E R. Prove that the system is weakly controllable if and only if we have Span{b, [[Q, b], b]} =1R2 and show that the system is not controllable.

5.11 Consider a control system in R" of the form: P

x(t) = H(x(t)) + > ui(t)bi

(5.17)

i=1

where H is an homogeneous polynomial vector field of degree m, bi are constant vector fields and the controls ui (.) are piecewise constant vector fields with values in R. Let L be the Lie algebra generated by {H, b1, , b,} and L° be the Lie subalgebra of all constant vector fields.

1. Prove that L° is equal to the smallest vector space which contains the , b,} and all element of the form adv1 o ... o advk(H) for vectors {b1, an arbitrary element v1i -, vk in the vector space.

5 Controllability and Higher Order Maximum Principle

138

2. Prove that L° is the smallest vector subspace of R" which contains the vectors bl, , by and which is invariant under the mapping H. 3. Prove that the system is weakly controllable if and only if L° =1R" 4. Assume the order of H to be an odd integer m. Use the enlargement technique to prove that the system is controllable if and only if L° = R'. 5.12 We note SL(n, IR) the Lie group of n x n matrices with determinant 1 and sl(n, R) its Lie algebra of n x n matrices with trace equal to 0. Consider the right invariant control system on SL(n, IR):

R(t) = (A + u(t)B)R(t)

(5.18)

where R(t) E SL(n, IR), A, B E sl(n, IR) and u(.) is a piecewise constant mapping with values in R.

1. Prove that the system is controllable if and only if the accessibility set from the identity contains an open neighborhood V. 2. Assume that the system is controllable on SL(n, R). Prove the controllability of the following bilinear system in IR"\{0}: ±(t) = Ax(t)+u(t)Bx(t). 3. Assume that A is diagonalizable over the set of complex numbers with imaginary spectrum:

a(A) = {±iak; ak E lR, k = 1,

, n}.

Prove that the system (5.18) is controllable on SL(n, IR) if the Lie algebra generated by {A, B} is sl(n,R).

4. Assume that the spectrum of B is real and disctinct: Al < . . . < Ai E Ill and let e 1,

, e be the corresponding eigenvectors. Let (aid)

be the matrix representing A in the basis e1. We make the following assumptions:

i) the numbers Ai - A, are all distinct; ii) a22

0forI i-j1=1;

iii) 0. Use the enlargement technique to prove that the system is controllable on SL(rt, R).

5.13 Let A, B1, , B. be n x n matrices and assume that the system ±(t) = Ax(t) + Ep, ui(t)B1(x(t)), I ui(t) I< M is controllable on )R"\101. Prove that for all b1, , br,, bi E lR" there exists R > 0 such that for each xo, xl E {x; I x 1> R} there exists a control steering x0 to x1 for the system: Wi(t) = Ax(t) + F,?1 ui(t)(B,x(t) +bi). 5.14

1. Let A1, A2 and A3 be respectively the matrices:

000 10 0 1

0-10

00-1 ,

00 0

100

0 10

-101 0 00

5.8 Rigidity-Strong Legendre Clebsch and Coh Condition

139

and denote by exp A the exponential matrix. Let R E SO(3) =group of direct rotations in 113. Prove that there exists angles Vi, 0, 95 E [0, 2ir[ such

that: R = (expi/.'A3)(exp9A1)(expOA3)

(5.19)

2. Consider a rigid spacecraft controlled by two gas jets: 3

R(t) _ (Ew;(t)A;)R(t) =1 2

w(t) = Q(w(t)) +

z-

u=(t)b=

where R E SO(3) and Q(w) =

a a1w2w3C7W1

a

a

+a2wlw38W2 +a3WIW2C7W3,

with al, a3 > 0 and a2 < 0. The controls u;(.) being piecewise constant mappings with values in [-1,+11. a) Assume b1 =' (1, 0, 0) and b2 =' (0, 0, 1). Using part 1, construct a control law to steer (Ro, 0) to (R1, 0). b) Prove that each (R, w) can be steered to some (Ro, 0) and is accessible from some (R1i 0).

c) Construct a control law to steer the spacecraft from (Ro,wo) to (RI,wl). 5.15

1. Consider the smooth differential equation on R': do (t) = X(x(t)). Let V : 1R" -+ IR be a smooth function such that:

i) V(x)>0and V(x)-* oo when Jlxii-++oo. ii) LXV < 0 for all x E lR". Let E = {x E IR ; LXV = 0} and M the largest subspace of E invariant for the solutions. Prove that all the solutions are bounded when t -+ +oo and tend towards M. 2. Consider the smooth control system on Rn:

d (t) = X(x(t)) + u(t)Y(x(t)).

(5.20)

a) Assume X (x) = Ax where A is a n x n matrix whose spectrum consists of n disctinct imaginary eigenvalues and suppose that for each x 0 0, the system (X, Y) satisfies the ad-condition:

Span{adkXY(x); k = 1, , +oo} =1R". Prove that all the solutions of th(t) = Z(x(t)), with Z = X - (x, Y)Y where (,) is the inner product are tending towards 0 when t -- +oo.

140

5 Controllability and Higher Order Maximum Principle

b) Assume that there exists a smooth function V : 1R -' R such that V >_ 0, V(x) -' +oo when DDxli - +oo, LXV(x) = 0 for all x E R". Prove that all the solutions oft(t) = Z(x(t)), with Z = X - (LXY)Y are when t oo tending towards

W = {x E R'; LXV(x) = Lad'-X(y)V(x) = 0, k = 1,

,+oo}.

3. Consider the planar differential equation:

x(t) = x2(t) - y2(t), y(t) = u2(t) with u2(t) = alx2 (t) + a2y2(t) + a3x(t)y(t), a1, a2i a3 being constant. a) Prove that this differential equation admits a positive definite first

integral V = Axe + 2Cxy + By2 if and only if: a3 > 0, a2 = f 1, a1 + a2 + a2a3 = 0b) In this case, prove that all the solutions of x(t) = x2(t) - y2 (t), y(t) _ u1(t)+u2(t), ul(t) = -LyV are tending to the origin when t +oo and draw the phase portait.

5.16 Let G be a connected Lie group, g its Lie algebra of right-invariant vector fields on G. Let A, B1, system:

,

B,, in g and consider the following control m

9(t) _ (A+>ui(t)B2)g(t)

(5.21)

awl

where u . (t) E R. We denote by go the Lie algebra: {B1,. , Bm } AL and by Go the corresponding connected subgroup of G. Suppose that go has codimension one in L. Prove the following:

1. If Go is closed in G, then the system is controllable if and only if A 0 go and G/Go is isomorphic to S'.

2. If Go is not closed in G, then the system is controllable if and only if

Ago. 5.17 Let G be a connected Lie group, with identity e and let D be a family of left invariant vector fields on G. 1. Prove that the polysystem D is controllable if and only if the accessibility set A(e) contains a neighborhood of e. 2. Assume G compact. Prove that the polysystem D is controllable if and only if D is weakly controllable.

5.18 Let G be the Heisenberg group represented by 3 x 3 upper diagonal matrices:

lxy G=

01z

001 with the usual matrix multiplication.

;

(x, y, z) E R3

(5.22)

5.8 Rigidity-Strong Legendre Clebsch and Goh Condition

141

lxy 1. Identify G to R3 via the mapping

01z

" (x, y, z) and compute the

001 induced multiplication in R3. Prove that the vector fields:

Yl=a

Y2=ay +xaz' Y3=az

,

(5.23)

form a basis of the Lie algebra of left invariant vector fields on R3. 2. Let the canonical coordinates of the first kind (x, y, z) F-+ exp(xY1 + yY2 + z[Y1iY2))

(5.24)

defined by the exponential mapping. Prove that in these coordinates, Yl and Y2 are represented by:

Fl

a ax

xa 2 az,

a F2

X49

a y + 2 az

(5.25)

Use the Baker-Campbell-Hausdorff formula to prove that the multiplication rule is:

(x,y,z).(x',y',z')=(x+i,y+y',z+z'+2(xy'-yx)

(5.26)

5.19 let Sp(1, Ill;) = SL(2, R) be the Lie group of 2 x 2 matrices with determinant 1 and its Lie algebra of right invariant vector fields identified to sp(1, R)

the set of 2 x 2 matrices with zero trace. Let P = {z E C; Im(z) > 0} be the Poincare plane; Sp(1,R) acts on P with the law: S.z = (az + b)(cz + d)-1

if S E Sp(1, R), S

=Cad

I .

Let A E sp(1, R), denotes by Ap the induced

vector field on P. 1. Draw the integral curves of Ap if: a) the eigenvalues of A are imaginary; b) the eigenvalues of A are real and nonzero. c) the eigenvalues of A are zero. 2. Let D = {A, B} be a pair of right invariant vector fields on Sp(1, IR). Prove that D is controllable on Sp(1,1R) if and only if Dp = {Ap,Bp} is controllable on P.

5.20 Let SO(n) be the group of n x n matrices of rotations with determinant 1. Construct on SO(n) a controllable pair {A, B} of right invariant vector fields.

5.21 Let K be a compact connected Lie group which acts linearly on a finite dimensional vector space V and let G = V x 8 K be the semi-direct product. If k is the Lie algebra of right invariant vector fields on K, the Lie algebra

5 Controllability and Higher Order Maximum Principle

142

g of G consists of V ® k with the Lie bracket defined by [(x, A), (y, B)] = (Bx - Ay, (A, B]) with x, y E V, A, B E k and [A, B] is the Lie bracket on k. Prove the following: suppose that V and K are such that V admits no fixed nonzero point. Then a necessary and sufficient condition of controllability of a polysystem D of right invariant vector fields on G is that the Lie algebra generated by D is g = V ® k.

5.22 Let x(s) be a (generic) curve in Rn parametrized by arc-length, v1(s), , v (s) its Frenet frame and V (s) the matrix whose columns are the vts. 1. Show that the Frenet formula: dx V1

TS dv1

ds = dv;

s=

k1v2

-kt_1vi_1 + k;vt+1, 1 < i < n dvn

ds

-

-kn-lvn-i

can be written ds

where e1 =c (1, 0,

(s) ,

V(s)A(s),

ds

(s)

V(s)e1

(5.27)

0) and A(s) is then x n matrix

0

... kn_1

0

and interpreted as a differential equation on the semi-direct product SO(n) x, Rn. 2. Let n = 3, k1(s) = k(s) be the curvature and k2(s) = r(s) be the torsion.

0-k 0

a) Assume that k, r are constants and that A =

k 0 -r Ur

. Prove

0

that exp sA is a rotation with axis tw = (r, 0, k) and angle sr + k2 and x(s) is an helix along w. b) Assume k(s) =(nonzero) constant and r(s) = r1 or 7-2, r1 0 r2. Prove that each helix is a concatenation of helices along: wl =t (r1, 0, k), W2 =1' (r2, 0, k).

6 The Concept of Conjugate Points in the Time Minimal Control Problem for Singular Trajectories, C°-Optimality

6.1 Single-Input Case 6.1.1 Preliminaries In this section we consider a real analytic (or C°°) single-input affine control system:

x(t) = X(x(t)) + u(t)Y(x(t)), u(t) E R. Let ry

:

(6.1)

[0, T] -+ lR" be a reference singular trajectory corresponding to a

control u(.) E L°°([0,T]) and starting at t = 0 from y(0) = x°. We shall give necessary conditions for time-optimality under some assumptions. The topologies we consider are the following: C°-topology and strong C°-topology.

Definition 69. The C°-topology is defined as follows. Let y(.), 7(.) be two C°-curves defined respectively on [0,T], [0,T] and with extremities Po =

(0,7(0)), Pi = (T,y(T)), Po = (0.-;Y(0)). P1 = (T,=y(T)) in the time extended state space. The distance between y and

is given by:

P(y,y) = max 11y(t) -7(t)II + d(Po, Po) + d(P1, Pt)

is any norm in IRn. The curve y(.) is said to be C°-time minimal (resp. maximal) if y(.) is a time minimum (resp. maximum) trajectory with respect to all solutions of (6.1) with extremities (0, y(0) = x°), (T, %T) = y(T) = xl) and contained in a C°- neigborhood of -y(.). A stronger optimality concept called strong C°-optimality is defined by considering all piecewise C°-trajectories ry(.) with extremities ro and xl and contained in a tubular neigborhood of the image of y(.). where

Jump direction and generalized control. Since the system is affine, the set of singular trajectories solutions of the maximum principle with u(t) E R is not sufficient to provide optimal solutions for every boundary points. This leads to the following definition. Definition 70. The control direction JRY is called the jump direction. A control jump of height a at time t is the control impulse defined as the limit when

144

6 The Concept of Conjugate Points

n - +oo of the sequence un(s) = na for s E [t, t + ,,] and 0 otherwise. We extend the class of admissible control U = L°° to the set of generalized controls U. defined as follows. The control u(.) defined on [0, T] belongs to U9 if there exists 0 < t1 < < tk < T such that the restriction of u(.) to ]ti, ti+1 [ belongs to L°° and at every ti, u(.) has a control jump of height ai. This leads to well defined generalized trajectory solution of (6.1).

Basic assumptions. Let (z(), u(.)), z = (x, p) be a singular extremal defined on [0, T]. We shall assume the following.

Assumption (Ho). z(.) is contained in the set .R = {z; (p,ad2Y(X)(x)) 34 0}

and x(.) is contained in the set if = {x; X (x) and Y(x) are linearly independant} and is one-to-one. Then according to Proposition 22, t i-, x(t) is a C°'-curve not reduced to a single point and the mapping t i-, x(t) is an one-to-one immersion. According to (Ho), z(.) is a singular extremal of minimal order and is a solution of the

Hamiltonian vector field H where H(x, p) = (p, X (x) + u(x, p)Y(x)) and (p. [ Y,X ,X x u(x , P) Y (x p. We shall consider the following assumptions:

Assumption (HI). For each t E [0, T] the linear space 71

-

K(t) = d7r (Span{adk H (HY)(z(t)); k = 0,

,

+oo})

is of codimension one and is generated by the vectors corresponding to k = 0,. , n-2 where Hy (x, p) = (p, Y(x)) and ir is the projection (x, p) '- x. Assumption (H2). If n > 3, for every t E [0, T], we have

X(z(t)) 0 dlr(Span{adk H (HY)(z(t)); k = 0,

n - 3}).

Geometric interpretation of the condition (HI). The condition (H1) means that for t > 0 the first order Pontryagin cone denoted K(t) is of codimension one and is generated by the first (n - 1) vectors.

6.1 Single-Input Case

145

Non-intrinsic formulation. If the singular control is identified to u = 0 the previous assumptions take the following simple form:

(H1) Vt E [0, T], Span{adkX (Y)(x(t)); k = 0, , +oo} is of codimension one and generated by Span{adkX (Y)(x(t)); k = 0, , n - 2};

(H2) If n > 3, Vt E [0, T], X(x(t)) V adkX (Y)(x(t)) for k = 0, - n - 3.

Definition 71. Let (z(.), u(.)), z = (y, p) be the reference extremal defined on [0, T] and assume that the previous assumptions (H°), (H1) and (H2) are satisfied. According to (H1) the vector p is unique up to a scalar and it can be oriented such that H(x, p) = (p, X 0. We say that y() is exceptional if H = 0 along y. Let D be the quantityBua -7 7V = (p(t), [[Y, X], Y](x(t))).

The trajectory y(.) is said to be hyperbolic if D is strictly positive along y and said to be elliptic if D is strictly negative. Remark 12. Recall that from Proposition 48, the condition

e d2OH>0 Ou dt2 8u called the Legendre-Clebsch condition is a necessary time-optimality condition. We can now formulate our main result.

Theorem 23. Let y(.) be a reference singular trajectory defined on [0, T] and satisfying assumptions (Ho),(H1),(H2). Then 1. For T small enough, y(.) is a strong C°-time minimum (resp. maximum) for the generalized trajectories of (6.1) if -y(.) is exceptional or hyperbolic (resp. elliptic). 2. If -y(.) is an exceptional or a hyperbolic (resp. elliptic) trajectory, it is a strong C°-time minimum (resp. maximum) if and only if T < t1c where t1c is called the first conjugate time along -y.

Remark 13. The previous theorem gives a necessary and sufficient strong C°optimality condition for a singular trajectory satisfying (Ho), (Hl) and (H2)

and we shall give an algorithm to compute the first conjugate time. An important point which must be emphasized is that we have two different algorithms: one in the hyperbolic-elliptic cases and one in the exceptional case.

Sketch of the proof: The complete proof is given in [26] and we shall only outline the different steps and indicate the geometric intepretation. The main idea of the proof is to make a direct computation of the accessibility set A(x°i T) along the reference singular trajectory. This requires the computation of a feedback semi-normal form and the theory of cheap LQ-control using differential operators.

146

6 The Concept of Conjugate Points

6.1.2 Feedback Semi-Normal Forms in the Hyperbolic and Elliptic Cases Proposition 51. Assume that -y(.) is a hyperbolic or an elliptic singular trajectory. Then the system is feedback equivalent in a C°-neighborhood of -y(.) to a system (X1,Y1) of the form: X1 = a

8x1

+n-I E x'+'

a

:=2

n

E a;j(x')x2)

1

+R

:,3==2

a Y1

axn

where ann is strictly positive (resp. negative) on [0, T] if 'y(.) is elliptic (reap. i R, i is a vector field such that the weight of Ri hyperbolic) and R =

is of order greater or equal to 2 (resp. 3) for i = 2, , n - 1 (reap. i = 1), the weights of the variables x' being 0 for i = 1 and 1 for i = 2,- , n.

Geometric interpretation. - The reference singular trajectory y(.) is identified to t '- (t, 0, corresponds to u = 0. - The first order Pontryagin cone along y is given by KI, = Span{

... , a--

xe

IY

,

0) and

}

17,

and the linearized system is autonomous and in the Brunovsky canonical form l2(t) = t;3(t),. .. , £n(t) = u(t).

- If py(.) is the adjoint vector associated to y(.), then we can set p.1 = (e, 0, , 0) where e = +1 in the elliptic case and c = -1 in the hyperbolic case, the Hamiltonian being e. The second order variation at T is then e

with t2(t) =

f

aij(t)e'(t)(t)dt

(6.3)

i,j=2

en(t),tn(t) = v(t) and the intrinsic

derivative is obtained by taking the boundary conditions 2(s) = . . . = l;n(s) = 0 for s = 0 and s = T. Moreover (py,ad2Y1(Xl),,1) = wan(t)-

Exceptional case. Proposition 52. Let -y(.) be an exceptional trajectory. Then n > 3 and there exists a C°-neigborhood of y(.) in which the system (6.1) is feedback equivalent to a system (X2, Y2) of the form:

6.1 Single-Input Case

X2 = s1 +

n-2

E

147

n-1

xt+1

ax' +

i,j=2

i=1

an + R

aij (x

Y4 2 =

where an-1,n-1 > 0 on [0,T], R = E 1 RI

a axn-1

;, Rn-1 = 0 is a vector field

such that the weight of Ri has order greater or equal to 2 (resp. 3) for i

, n - 2 (resp. i = n), the weight of the variables xi being zero for i = 1, one for i = 2, , n - 1 and two for xn. 1,

Geometric interpretation. - The reference trajectory y(.) is identified to t H (t, 0, sponds to u 0.

,

0) and corre-

- The first Pontryagin cone along y is given by: KI.y = Span{

a ax117'

... ,

a I,y}.

axn-1

- The adjoint vector p.y associated to -y can be normalized to (0, moreover (p.y, ad2Y2(X2)17) = -an-1,n-1(t) - The intrinsic second-order derivative at time T is identified to

-

,

0, -1)

rTn-1

J° Ei,j=2aii (t)t`

(t)dt

(6.4)

/ t Sn-1(t) = u(t), n(t) = v(t) and with 41(t) = t 2(t), ... , n-2 (t) = en-1(t), the boundary conditions at s = 0: c1(s) = . . . = to (s) = 0.

6.1.3 LQ-Model Having computed the semi-normal form we can identified in both cases the LQ-model which will be used to evaluate the end-point mapping. It is simply obtained by taking Ri = 0 in the semi-normal forms and replacing x1 by t. We get the following. Hyperbolic-Elliptic case:

21(t) = 1 +

aij (t)xt (t)xi (t) i,j=2

x2(t) = x3(t)

xn(t) = u(t) Exceptional case:

148

6 The Concept of Conjugate Points

xl (t) = 1 + x2(t) x2(t) = x3(t)

in-1(t) = u(t) n-1

in (t) = > a:; (t)xi (t)xi (t) i,j=2

This leads to a concept called accessory problem similar to the one in the calculus of variations introduced in Sect. 2.1.6. Definition 72. Let 0 < t f < T be fixed. The accessory problem is - Hyperbolic-elliptic case:

e = f 1, subject to . x2(t) = x3(t), . . , xn(t) = u(t)

min ex' (t f ),

and the boundary conditions

x(0) = A, 1(tj) = B where i = (x2, ... , xn-l - Exceptional case: minxn(t1), subject to

xl(t) = 1 + x2(t), x2(t) =

x3(t)...... n-1(t) = u(t)

and the boundary conditions

x(0) = A, 1(tt) = B where x =

(x1,...,xn-2)_

We observe that the previous equations are integrable and they will be used to approximate the end-point mapping of the original problem in a C°neighborhood of the reference singular trajectory y(.). We proceed as follows.

6.1.4 Approximation of the End-Point Mapping Using the LQ-Model in the Elliptic Case The model is written

i1(t) = 1 + q(t, x2, ... , xn) ±2(t) = x3(t)

in-1(t) = xn(t)

6.1 Single-Input Case

149

where q is the quadratic form n

q(t,x2,...,x^) = r aij(t)x.(t)x'(t) i,j=2

We set x1(t) = t + 1(t) and we get n

bl(t) = E

=Sn(t)

i,j=2

and the accessory problem is, using the notation l; = f 2, equivalent to min f (, -2)

ref

J0

q(t, c

,

&-2) )dt

where q() = rii 20 bij(t)F,(i) (t)e(j) (t) and the bij are symmetric and defined by

bi-2,j-2 =

aij + aji 2

if i 0 j,

bi-2,i-2 = aii.

Observe that the control is taken as &-2) and this corresponds to make the Goh transformation for the system where xn taken as control. We have the following standard concept and results concerning the above problem. Definition 73. Let 0 < t < T and let us denote by Ct the set of C2(n-2)curves defined on [0, t] and satisfying the following boundary conditions: C(0) =

= &-3) (t) = 0.

_ C(n-3)(0) = C(t) =

Let D be the differential operator of order 2(n - 2) defined by n-E=1 2(-1)iai8

D(C)-2 S

()(S) M(C)

This operator is called the Euler-Lagrange operator associated to the accessory

problem. We denote by b the restriction of D to V. The time t, is said to be conjugate to 0 along the elliptic trajectory ry(.) for D if there exists a non trivial solution (.) E V such that D(1:) = 0. The differential operator b is self-adjoint and we have the following results.

Proposition 53 (Hilbert-Schmidt theorem [89]). For every t E]0,T] there exists a sequence (ea, Aa), a = 1,

, +oo such that:

1. ea(.) E Ct, (ea, ep)La((o,tJ) =6! (Kronecker symbol) and Dea = Aaea.

2. AI
150

6 The Concept of Conjugate Points

3. Every element l;(.) E C can be written as an uniformly converging Fourier series: +00

where &« E R.

Moreover if there exists no conjugate time on ]0, t] then a« > 0, otherwise a1 < 0. Proof. See [89] and Morse theory from [52].

Proposition 54. Assume that there exists no conjugate time on ]0, T]. Then for every 0 < t < T there exist C2(n-2)-curves Ji, J., i = 1, , n - 2 defined on [0, T], unique solutions of the equations:

i) DJ, = DJi = 0

ii) j(k) (0) = Jik) (t) = 0, k = 0, ,n-3 iii) J(k) (t) = -(k) (O) = 6(k i , k = 0, ... , n - 3

Every C2(i-2) -curve l; (.) can be written as an uniformly converging series too 6

n-2

n-2

i=1

i=1

(6.8)

,,,,

where l a E 1R, vi = t; (s-1) (0), wi = C('-') (t) and ) > 0, a = 1, -

. + oo.

Proof. The existence of the curves Ji, Ji is proved in [18], p. 233. The relation (6.7) is then a consequence of the previous proposition.

Using this representation, we can evaluate the end-point mapping.

Proposition 55. Let tic be the first conjugate time for the operator D. Assume 0 < t < t1 and let l;(.) be a C2(n-2) -curve defined on [0,t] represented as:

+oo

n-2

n-2

i=1

tol

1: «e«+viJi+wiJi, vi = e(i_ l) (0), wi = e(i-1) (t), i = 1, ... , n - 2. Then

Qw =

f

q(t, l;, ...

(n-2))dt ,

+00

=

«a« + f (v, w)

0

where v = (VI,' ',V,,-2), w = (WI,'

,

wn_2) and f is a quadratic form.

6.1 Single-Input Case

Ei Proof. We have io product defined by q and B(e, n) = fo

bij = bji. Let

151

r!) be scalar

i (t))dt. We have

c

q(6(t))dt =

J0 Integrating by parts we get

q)

rl) = for every obtain

E

C2(n-2) and every q(.) E V. In particular, computing we n-2

+00

Q(t) _

vivjB(Ji, Jj) +

titjB(ei, ej) + n-2

+oo n-2

wiwjB(Ji,J,)+EEtivjB(ei,Ji) +oo n-2

+E E tiwjB(ei, Jj)

i=1 j=i n-2 viwjB(J1, Ji). + t,j=1

i=1 j=1

We use the following relations:

- B(ei,ej) _ (Dei,ej)La = Aiai - B(Jj, ei) _ (DJj, ei)L2 = 0, since DJ, = 0 - B(Jj, ei) _ (DJj, ei)L2 = 0, since DJj = 0 The proposition follows.

Corollary 12. Let 0 < t f < tjc and () be a curve on [0, t1] given by n-2

+00

_

wiJi,

&ei + i=1

i=1

then the end-point mapping for the system .+I (t) = 1 + Q(t, x2(t), ... , xn(t)),

2(t) = x3(t), ... , xn-1(t) = xn(t)

with x(0) = 0 is given for xn = &-2) at time t f by

,l

r

M+(xn) = (tf + w1, W2, ... , wn-2, L.r

a=1

where Bij = B(Ji,Jj).

n-2

+oo

`C

\,,,,t.2

+ E Bijwiwj) i,j=1

152

6 The Concept of Conjugate Points

Corollary 13. Let is a curve with 0 boundary conditions: 0 = .(') (0) = e(') (t) for i = O, ... n - 3 represented as

+oo

a=1

The end-point mapping for the system 21(t) = 1 + q(t, x2(t), ... , xn(t)), x2(t) = x3(t),...

,

2n-1 (t)

= xn(t)

with x(O) = 0 is given for xn = &-2) at time t f by .O(xn)

= (tf + L>'g,2

0'..

.

0).

If b is without conjugate point on [0, T] the minimum of x1(t f) for 0 < t f < T on the set of controls t'(n-2) is reached for = 0 and is x1(tf) = tf. If t f > t1c where t1, is the first conjugate time, then the minimum is -oo.

We represent on Fig. 6.1 the end-point x1(t f) for t f < tic and t f > t1,.

Fig. 6.1. Elliptic case

Remark 14. In the hyperbolic situation, the analysis is similar replacing the minimization of x1 (tf) by the maximization of x1(t f ), see Fig. 6.2.

6.1 Single-Input Case

153

Fig. 6.2. Hyperbolic case

6.1.5 Conclusion By Inspecting Fig. 6.1,6.2. we can deduce the following. For the LQ-model, we have:

- in the hyperbolic case, the reference singular trajectory y(.) is C°-time minimal up to the first conjugate time tl, of the operator D. - in the elliptic case, the reference singular trajectory is C°-time maximal up to the first conjugate time ti of the operator D. Clearly, those results are still true for the strong C°-topology. Also they are valid for the original system when we consider the generalized trajectories, see [26] for the details. Moreover, our analysis proves the following.

Proposition 56 (C°-one side rigidity). Assume T < tl,,. Then the reference singular trajectory y(.) defined on [0, T] is in the hyperbolic (resp. elliptic) case the only trajectory ry(.) contained in a C°-neighborhood of y(.) and satisfying the boundary conditions ^y(0) = y(0), y(T) = y(T) in a time

T < T (resp. T > T).

6.1.6 The Exceptional Case We shall use a similar technique to evaluate the end-point mapping in the exceptional case. The model is then i1(t) = 1 + x2 (t)

x2(t) = x3(t) thn-2(t) = xn-1(t)

in (t)

= 9(t,

x2 (t), .

. .

,

xn-1(t))

154

6 The Concept of Conjugate Points

where q is the quadratic form n-1

q(t, x2 ... , xn) _

aij (t)x`xj i, j =2

which can be written as n-1 q(t,x2...... n) =

bij(t)xiy

i,j`=2

where bij(t) = bji(t). We write x1(t) = t + fi(t)

and the quadratic form can be written with x2(t) = e(t) as n-1 i,j=2

t(n-2)) not depending on S

It is a quadratic form on the space n = t;. If 0 < t < T we introduce

Q(t) =

(n-2)(T))dT

J

and B(C, r7) the associated scalar product. As previously if C(.),77(.) are C2(n-2)_curves and n(.) E V, i.e. 77(0) = ... = 77 (n-3) (O)

= n(t) _ ... =

7(n-3) (t) = 0

we can write

B(t, n) = 77)Lwhere D is as in Definition 73. Its restriction to G` is denoted f) and is self-adjoint. We denote by t1c the first conjugate time for the operator D. Assume that t < t1c. Using the Proposition 55, we can expand each curve l;(.) E

C2(n-2) such that

e(O) = ... = as

+00

&-3)(O) = O

n-2

_ E C. e. + ?.wiJi 1

where (O) = 'w1, ... ,

&-3)(0) _ Wn-2

6.1 Single-Input Case such that

= +0 AaSa CC

Q(tt)

155

n-2

+ E Bijwiwj. i,j=1

C.=1

Hence the end-point mapping for the model, where xn-1(t) = (n-2) (t) is the control, is then given a t time t f < t1c by:

n-2 (tf + w1, w2, ... , wn_2, E Aac2a + > bijwiwj) too

C

O(S(n-2))

a=1

(6.10)

i,j=1

where Aa > 0, Va = +1, , +oo. Let x(.) be a trajectory of (6.8) defined on [0, t1] and satisfying the boundary conditions: i(0) = 0, x1(t f) = T, x2(tf) = xi-2(t f) = 0. Then we have using (6.10): x(tf) = (T, w2 = 0, ... , wn-2 = 0, xn(tf )) where tf+w1 = T, xn(tf)

_ EQ° Ab ara2 +wiB11 and B11 is given by Q(J1).

In particular, we have

min in(t f) = wiB11 = (T - tf)2B11

=n-1

and this is consistent with Hamilton-Jacobi-Bellman equation

8minin(tf)

f

Itf=T

Definition 74. The time tcc is said to be conjugate to 0 along the exceptional Jin-2) )dt = 0. trajectory -Y if Q(J1) = focc q(t, J1: Ji,... On can prove the following, see [26].

Proposition 57. We have: 1. The first conjugate time satisfies the inequality: 0 < tlcc < tic. 2. Assume tlcc < t1ci then Q(J1) = fog q(t, J1(t), .. , Jin-2)(t))dt > 0. 3. If n = 3, Q(J1) > 0 for every t f and if n > 4, tcc are the conjugate points of the differential operator D of order 2(n - 3) defined by D = - A(D A )

Geometric representation of the accessibility set. On Fig. 6.3 we represent xn(t f) for the model with the boundary conditions w2 = = wn-2 = 0 for t f < tlcc and t1cc < t f < tic, the control being xn_1.

6.1.7 Conclusion Inspection of the previous figures shows the following for the model:

156

6 The Concept of Conjugate Points

I1\1

tf=T

tf
If

=T

tf

tf=lcc

If

tlcc < t f
Fig. 6.3.

1. the reference singular trajectory y(.) is strongly C°-time minimal up to the first conjugate time tl,; 2. the reference singular trajectory y(.) is C°-isolated (i.e. C°-rigid) up to the first conjugate time. Using approximation lemmas proved in [26], this result still holds for the original system and we have the following proposition.

Proposition 58. Up to the first conjugate time tlcc, the reference singular trajectory is strongly C°-time minimal and moreover it is C°-rigid.

6.1.8 Comparison of the Elliptic-Hyperbolic Case and the Exceptional Case The two main differences between both cases are:

- Rigidity status. In the exceptional case, the reference singular trajectory is C°-rigid but in the hyperbolic case it is only one-side C°-rigid. This is consistent with the results of Sect. 5.8: under the assumptions (H1), (H2), (H3) the reference hyperbolic or elliptic trajectory is not satisfying the necessary C1-rigidity conditions of Proposition 49. - Conjugate points. Unlike to the classical situations of the calculus of variations, the concept of conjugate points depends on the hyperbolic-elliptic or exceptional status. In the first case it is related to a differential operator of order 2(n - 2) and in the second case of order 2(n - 3).

6.2 Applications

157

6.2 Applications Our analysis will be one of the main tool to analyze the time minimal control of chemical batch reactors in Chap. 7. They are also other applications.

6.2.1 Time Optimal Synthesis for Planar System Consider a planar system i(t) = X(x(t))+u(t)Y(x(t)) and let ry : [0,T] -+ Rn be a reference singular trajectory satisfying the assumptions (Hl) - (H2) (H3). Then, in order to analyze the time optimal problem we can use the model: dxt

(t) = 1 + a(t)y2(t),

j(t) = u(t)

where the reference singular trajectory is identified to -y : t i- (t, 0) and the

reference singular control to u =_ 0. In the elliptic case, we have a(t) > 0 on 10, T] and in the hyperbolic case, we have a(t) < 0. In the planar case, there exists no conjugate time and for the set of generalized controls we can describe the C°-time minimal synthesis in the hyperbolic case, see Fig. 6.4. In the elliptic case, the time minimal control is not well defined locally and

y vo

I

Fig. 6.4.

depends clearly on the C°-neighborhood. But the elliptic trajectory is time maximizing and the time maximal synthesis is given by Fig. 6.4.

6.2.2 Example in Dimension 3 and Connection with the Hamilton-Jacobi-Bellman Equation We consider a system v(t) = X(v(t)) + u(t)Y(v(t)), where v = (x, y, z) E ]R3. Let us introduce D = det(Y, [Y, X], [[Y, X], Y]),

D' = det(Y, [Y, X], [[Y, X], X])

158

6 The Concept of Conjugate Points

and

D" = det(Y, [Y, X], X]).

The singular trajectories satisfying (HI), (H2), (H3) are contained in D 0 0, v'('3. They are: and the singular control are given by the feedback (1(v) ITT

- hyperbolic if D"D > 0 - elliptic if D"D < 0 - exceptional if D" = 0 and X, Y are linearly independent along -y. Consider the following example:

X=

0

yax

0

1

8

2

+Exay + 2(ex +y2) az,

a

Y = ay

where e = +1 or -1. Here D = 0 is empty and D' = 0 on 1R3. The singular control is then given by u(v) - 0. The exceptional trajectories are contained in D" = 0 and hence must satisfy the equation: Ex2 - y2 = 0. The system is not in the semi-normal form but from our study, the time optimal problem can be analyzed using the following problem: min z(T) V

under the constraints:

1(t) = v(t), z(t) = 2(ex2(t) +v2(t)), x(0) = A,x(T) = B, where A, B and T > 0 are fixed. We denote by S(A, B, T) the value function. We must distinguish two cases:

1. Case e = -1: The system is without exceptional trajectory. The EulerLagrange equation associated to the concept of conjugate point is x(t) + x(t) = 0.

The time tx, = zr is then the first conjugate time. The optimal cost is given by

S(A, B, T) =

2sinT[(A2+B2)cosT-2ABJ

and if A 0 B and T = n, its values is -oo. Let A, B be fixed, A $ B. The graph of T --' S(A, B, T) is represented on Fig. 6.5. The inspection of this graph shows that all the singular trajectories are time minimal on [0, 7r[. This is consistent with our theory because they are hyperbolic. 2. Case e = 1. The Euler-Lagrange operator is without conjugate point. The exceptional trajectories are contained in y = ±x. The optimal cost is given by

6.2 Applications

0

n

159

T

Fig. 6.5. S(A, B, T) =

. 2sT

!_[(A2

+ B2)chT - 2ABJ

and for A, B fixed we have:

(A, B, T) =

2sh2T

[2ABchT - (A2 + B2 )]

5T 2 . Representing the graph T '--+ S(A, B, T) which is 0 for chT = A2 whith A, B fixed, we have two cases, see Fig. 6.6. The time optimality status is deduced by inspecting the graph. 2,

hyperbolic

exceptional

AB>O

T

Fig. 6.6.

AB
T

160

6 The Concept of Conjugate Points

6.3 The Case in ][fig. Intrinsic Computations of Conjugate Points - Connection with the Time Minimal Synthesis Problem. The Concept of Curvature In this section, we consider a system in 1R3: v(t) = X(v(t)) + u(t)Y(v(t)), v = (x, y, z). If 'y(.) is a reference singular trajectory defined on [0, T] and satisfying (Hl) - (H2) - (H3), we have the following: - If y(.) is exceptional, then since we are in dimension 3 the system is without conjugate point and y(.) is strongly C°-time optimal. - If -y(.) is elliptic, y(.) is not satisfying the necessary Legendre-Clebsch optimality condition and is indeed strongly C°-time maximizing if T is small enough.

- If y(.) is hyperbolic, y(.) is strongly C°-time minimizing up to the first conjugate point ti, Hence in the remaining of this section we shall assume that y(.) is hyperbolic. Our aim is the following:

- give an intrinsic algorithm to compute the conjugate points, that is without integrating the reference singular trajectory and the vector field Y;

- give a geometric interpretation of conjugate points; - make a connection with the time minimal synthesis problem and the concept of conjugate point introduced by Moyer, see [88]. All these different points can be generalized for systems in R". More specific to the 1R3 case is the following:

- Introduction of a concept of curvature. This concept is very important for the feedback classification problem localized in a C°-neighborhood of y(.) and will provide a non trivial new feedback invariant, see [20, 19].

6.3.1 Euler-Lagrange Equation In the hyperbolic case the model is:

i(t) = 1 + L(t, y(t), z(t)) y(t) = z(t) z(t) = u(t) where L(t, y, z) is the quadratic form a(t)z2+2b(t)yz+c(t)y2 with a(t) < 0 on [0, T] and the reference singular trajectory is y : t '- (t, 0, 0) and is associated to the control u = 0. By setting v(t) = (t, 0, 0) + V(t) with cp = (API, P2, <'3), then gyp(.) is solution of

6.3 The Case in R3, Conjugate Points, the Concept of Curvature

161

01 (t) = L(t, 402, 403), 02 = 403, 403 = u.

Let 0 < t < T be fixed, the accessory problem is

max 41(t), 02 ='P3, 03 = u with '2(0) = 403(0) = 402(t) = 403(t) = 0. By making Goh-transformation we can take 4p3(.) as the control and we consider the problem t

Max J(t), J(t) _ f (a(s)co (s)3+ 2b(S)402(s)4P3(s) + c(s)4P2(s))ds 2 0

with the boundary condition: 402(0) = 4P2(t) = 0.

By definition, the first conjugate time tic is the first t such that J vanishes on a non trivial curve 402(.) satisfying the boundary conditions. This is equivalent to the following.

Proposition 59. The first conjugate time tic is the first t such that there exists a non trivial solution 402(.) of the Euler-Lagrange equation

d 8L

8L

dS 8W2 _ 8402 with 40i(0) = 4P2(t) = 0 and L(s, 402, 402) = a(s)02 + 2b(s)40202 + c(8)4p2.

6.3.2 Geometric Interpretation We consider the reduced system:

i(t) = 1 + L(t, y, z), y(t) = z(t)

where z(.) is the control variable. By definition of tic there exists a curve

y(.) such that y(0) = p(t1c) = 0, y(t) = 1, y(t) # 0 on ]0, t1c[ and fo'` L(t, y(t), y(t))dt = 0. Hence the corresponding solution of the reduced system starting at t = 0 from (0, 0) satisfies 2(tic) = tic and (T, p) intersects (]0, tic[, 0) only at (tic, 0). Let e E R and (ie, ye) be the solution of the reduced system issued from

Te(0) = %(0) = 0 and associated to the control z(t) = e (t). We have ye(t) = y(t) and from our analysis the family of curves (tee, cy') intersects (]0, tic[, 0) only at (tic, 0), see Fig. 6.7, (i). As in the classical theory, one can shows that (tic, 0) is the limiting points of the intersections of the projections

of the extremals on the (x, y)-space, when -y(.) is identified to t " (t, 0, 0) and Y = 8 is the jump direction. This gives us a geometric interpretation of conjugate points.

162

6 The Concept of Conjugate Points

y

(i) Model

(ii) System

Fig. 6.7.

6.3.3 Intrinsic Computation To compute the conjugate points we can use the following algorithm that does not require any integration: the hyperbolic trajectory y(.) is solution of the constrained Hamiltonian equations: a (t) dt

=

OH

8p

(v(t), p(t))

dp dt

off

= - 8v (v(t), p(t))

(p(t),Y(v(t))) = (p(t), [X,Y](v(t))) with A(v, p) = (p, X (v)+u(v)Y(v)) and fi(v) is the singular feedback control. If (y(.),p(.)) is the reference singular extremal we can consider the linearized Hamiltonian system along (y(.), p(.)) with the linearized constraints: aw(t) = Z(y(t), p(t))aw(t) (6.11)

l(bw) = 0

with aw = (av, ap) and l is the linear mapping defined by linearizing (p,Y(v)) _ (p,[X,Y](v)) = 0. The time t. is conjugate if there exists a non trivial solution t -- 6w(t) of equation (6.11) such that 6v(0) = 6v(t,) _ 0 (mod RY).

6.3 The Case in R3, Conjugate Points, the Concept of Curvature

163

6.3.4 The Concept of Curvature Assume the system to be in the semi-normal form:

x(t) = 1 + L(x(t), y(t), z(t)) + Rj(x(t), y(t), z(t)) y(t) = z(t) + R2 (x(t), y(t), z(t)) .z(t) = u(t) + R3(x(t), y(t), z(t))

The variational equation along ry : t

ax(t) = 0,

by(t) = bz(t),

(t, 0, 0) takes the form

bz(t) =

c(t)

(t)

(t) by(t) - a(t) bz(t) (6.12)

and the two last equations can be written as the following second-order differential equation: ay(t) + a(_t) by(t) + b(t a(t) (t) by(t) = 0.

(6.13)

The existence of conjugate points means that there exists a non trivial solution of (6.13) satisfying 8y(0) = by(t,) = 0. If we set A(t) = a ' B(t) = e eac = and by(t) = C(t)Y(t) where C(t) = exp fo -A2 ds, the previous equation can be written in the normal form as:

Y(t) + K(t)Y(t) = 0 where

K(t) = C(t) + A(t)C(t) + B(t)C(t), C(t) = -

(6.14)

and K(.) is defined on the set DD" > 0 containing the hyperbolic trajectory satisfying the assumptions (Hl) - (H2) - (H3).

Definition 75. The function K(.) is called the curvature in the hyperbolic case.

6.3.5 Conjugate Points and Time Minimal Synthesis Let 0 < M < +oo and consider the CO-control system v(t) = X(v(t)) + u(t)Y(v(t)), where the set of admissible controls is the set UM of bounded

measurable mappings taking their values in [-M, +M). Let S be given by S(X, Y) = X - aY where D = det(Y, [Y, X], [[Y, X], Y]) and D' _ det(Y, [Y, X], [Y, X], X]). Let -y(.) be the reference hyperbolic singular trajectory defined on [0, T] and satisfying (Hl) - (H2) - (H3); we assume the following:

164

6 The Concept of Conjugate Points

Assumption (H4) t1, E) - M, +M[. Let V(t), 0 < t < T be the solution of the variational equation

6v(t) = 7(.y(t))ov(t)

(6.15)

with initial condition v(0) = Y(y(0)). Let E, E' = ±1 and let f be the mapping f : (ti, t2) t3, E, E') 1- exp t3(X + E'MY) exp t2S exp t1(X + EMY)(y(0)).

from the analysis of Chap. 3, such a trajectory is an extremal for t1, t3 _> 0, small and M < +oo. Let F be the image of f for t2 E [0, T] and t1, t3 sufficiently small. According to [88] if det(V(y(t)), Y(y(t)), S(y(t))) never vanishes on ]0, T], then F is an extremal field about the arc y(.) in the following sense. There exists a C°-tubular neighborhood U of y(.) such that each point of U is the image of an unique (t1, t2, t3, E, E'). Moreover it is the time minimal synthesis in a neighborhood of the reference singular trajectory for the fixed end-point problem. Our previous analysis shows that this result is still valid for M = +oo if we consider the generalized trajectories.

Definition 76. Let us denote by t1 the first time in 10, TJ such that det(V(t), Y(y(t)), S(y(t))) vanishes.

The following lemma is proved in [24].

Lemma 21. We have the following:

1. V(t) E Span{Y(y(t)),[X,Y)(y(t))} 2. det(V(t), Y(y(t)), S(y((t))) = 0 for 0 < t < T if and only if V (t) and Y(y(t)) are colinear. 3. tic = tic Hence we get an equivalent definition of the concept of conjugate point. Also it shows the relation with the time minimal synthesis when I u(t) I< M.

6.4 Connection with the Liu-Sussmann Example 6.4.1 Preliminaries In 179J Liu and Sussmann consider the following system:

x(t) = ui(t) y(t) _ (1 - x(t))u2(t) z(t) = x2(t)ul(t)

(6.16)

6.4 Connection with the Liu-Sussmann Example

165

and the time minimal control problem. The system is written as

v(t) = ul (t)F1(v(t)) + u2(t)F2(v(t))

(6.17)

where v = (x, y, z), F1 = YX- , F2 = (1- x) + x2 . The system is symmetric and the singular trajectories satisfy the following equations: (P, Fi (v)) = (p, F2 M) = 0

=0 u1(, [[F1, F21, F1J(v)) + u2 VI, [[Fl, F21, F21) = 0

Hence they are exceptional and are contained in det(Fi, F2, [F1, F21) = x(x - 2). Moreover

D1 = det(Fi, F2, [[Fl, F2J, F1]) = 2(x - 1), D2 = det(Fi, F2, [[F1, F2J, F1J) = 0.

Hence they are contained in x = 0 or x = 2 and the singular control satisfies ul = 0 almost everywhere. If they are parametrized by the length ui (t) +

u2(t) = 1, we get u2(t) = ±1. We observe that the system is left invariant by:

S : (x, y) z, ul, u2) '-' (x, -y, -x, u1, -u2) and we can assume u2(t) = +1. We consider the reference singular trajectory ry : t F-4 (0, t, 0) starting from 0 and corresponding to u2 = +1. The connection with the Sect. 6.1 is the following.

Remark 15. The system (6.17) with v = 1 is the model corresponding to z = a(y)x2, a(y) = a(0) (small time model) used in Sect. 6.1 to study the C°-time optimality of the reference singular trajectory. In dimension 3 the following result is obvious but it can be generalized to R".

Corollary 14. The reference singular trajectory ry t 14 (0, t, 0) is strongly C°-time minimal for the system (6.16) when ul =_ 1 and u2(t) E R. :

We deduce the following.

Corollary 15. The reference singular trajectory y : t ' - (0, t, 0) is strongly C°-time minimal for system (6.16) when 0 < u1(t) < 1 and u2(t) E R. Proof. The system (6.16) is symmetric, hence the set of trajectories is invariant for the following control homothety: (ul, u2) '-+ (Aul, Au2). In particular each trajectory such that 0 < ul (t) < 1 at a Lebesgue point can be reparametrized on a smaller interval as a trajectory with ul (t) = 1. Remark 16. The above proof shows that we cannot extend the control domain beyond the point u1(t) = 1 and keeping the time optimality. But we leave as an exercise to the reader the possible extension to ul (t) < 0, using integrability of the equations.

166

6 The Concept of Conjugate Points

6.4.2 Conclusion The Liu-Sussmann example fits in our framework and is really equivalent to

the C°-rigidity of the exceptional trajectories satisfying (Hl) - (H2) (H3). In the remaining of this section we shall present the original proof of both authors which is different of our approach in which we parametrize the boundary of the accessibility set using extremals. Also our approach proved that the example is a generic model, compare with the extension of Sussmann in [79].

6.4.3 Statement and Proof of Liu-Sussmann Result Lemma 22. Let T = 3 and 0 < t < T. Consider two measurable mappings ul, u2 : R -+ [-1, +1] such that:

i) x(t) = fo ul (s)ds = 0; ii) z(t) = fo x2(s)u2(s)ds = 0.

Then fo(1 - x(s))u2(s)ds < t and the equality is true only if ul - 0 and u2 =I on [0, t]. Proof. We set A = fo(1 - x(t))u2(t)dt and h(t) = fo u2(s)ds. Then -t < h(t) < +t. Let a = t - h(t) and /3 = sup,Elo,tl I x(s) [. Since I u2(t) 1< 1, we have It

J0

x(s)u2(s)ds I<- jit.

Hence

A = fot u2(s)ds -

t x(s)u2(s)ds

< h(t) + /3t = t - a + Qt.

J0 Asa>0,wehave A<0iff3t
3'

We shall prove ,13 <

2

a. We show the following: 233

<

f

x2 (s)ds < ,32a. t

- fo x2(s)ds < 0 a: We have f t x2(s)u2(s)ds = 0. then //

t

0

< 02

J0

x2(s)ds =

J0

t x2(3)(1

- u2(s))ds

(1 - u2 (s))ds = ,132 (t - h(t)) < ,32ee.

6.4 Connection with the Liu-Sussmann Example

167

Fig. 6.8. 9

-

<

foCx2(s)ds:

Let a such that I x(a) 1= supSEfo,tl I x(s) I. Since x(O) = x(t) = 0 and

Ix(t)I/3,t-a<0,seeFig. 6.8.

Hence if Il = [a - 0, a] and I2 = [a, a + Q], then 11,12 E [0, t] and on each interval I X I>_ V1, X02 where lpj are the affine functions with values A in a and

O at a -,0 or a +,8. We have 3

r x2(s)ds

ft

>J,=3 33

x2(s)ds >

11

x2(s)ds + f x2(s)ds >

J 12 This proves the inequality. Clearly the equality is true only if x =- 0. 0

6.4.4 Conclusion We proved that if T < 3, then each trajectory -77(.) distinct of ry(.) and satisfying x(O) = x(T) = 0 and z(O) = z(T) = 0 satisfies y(T) < T. Hence 7 : t i-, (0, t, 0) is for T < 3 the only trajectory joining -y(O) = 0 to y(T) _ (0, T, 0) in a time < T. In particular we have the following proposition.

Proposition 60. The reference singular trajectory ry

:

t ' - (0, t, 0) is time

minimal on [0, T] if T < 23'

Remark 17. The main property used in Sect. 6.1 and in the previous proof is the integrability of the equations (the Lie algebra generated by F1 and F2 is nilpotent).

168

6 The Concept of Conjugate Points

Also an important remark is the following.

Remark 18. The trajectory y(.) is Cl-isolated. Indeed let y(.) be another trajectory on [0,T] with y(0) = 0 and z(T) = 0. Since z(t) = u2(t)x2(t), the control u2(.) must have a change of sign but it is impossible in a neighborhood

of u2 .= 1. Since y(t) = (1 - x(t))u2(t), then near x - 0 to satisfy z(T) = 0 the variable y(.) must change of sign. This will cause loops in SR-geometry studied in Chap. 9 for the projections of geodesics on the space (x, y), see Fig. 6.9.

Fig. 6.9.

Remark 19. If T is large enough we can construct faster trajectories satisfying

the boundary conditions. Indeed if (1 - x) < 0, if u2 < 0 then ! > 0. Hence

we can have z(T) = 0 without loop. Moreover, we can have y > 1 and be increasing the speed along the axis y. Hence we get shortest trajectories satisfying the boundary conditions x(T) = z(T) = 0, see Fig. 6.10.

6.4.5 Connection with the Sub-Riemannian Geometry The previous problem is connected with the sub-Riemannian problem: rT min

J0

(ui (t) + u2(t)) i dt.

Indeed parametrizing by arc-length: u2 + u2 = 1 is equivalent to time minimization. By homogeneity it is equivalent to the time optimal problem with the convex constraints u2+u2 < 1. Replacing the sphere Sl at ul = 1, u2 = 0 by its tangent space we get the affine problem: (t) = F,(v(t))+u2(t)F2(v(t)).

6.4 Connection with the Liu-Sussmann Example

169

y

Fig. 6.10.

Notes and Sources One earliest reference concerning the concept of conjugate point along a reference singular extremal for the time optimal control problem is due to Sarychev [96]. It covers roughly the hyperbolic situation. It was an inspiration to make the contribution of [26] introducing the tools (normal form, differential operators) to analyze the exceptional case. The connection with sub-Riemannian geometry is briefly analyzed in this section which unifies all the approaches. This connection has a neat treatment in [12]. The key remark is the C°-rigidity concept in the exceptional case. This will be analyzed in great details in Chap. 9 on sub-Riemannian geometry introducing the concept of C°-rigidity for geodesics. This concept is presented in Kupka [73], Montgomery [87] but also in our earliest article [26]. The concept of curvature in the hyperbolic situation is due to [24], see also [2].

7 Time Minimal Control of Chemical Batch Reactors and Singular Trajectories

7.1 Introduction The choice of best temperature schedule in a batch reactor to maximize the yield per year is one of the main problems in chemical engineering. Until now, in practice the reaction temperature is held constant over the duration of the batch and hence an optimal law is computed among all the constant temperatures. Clearly, by varying the temperature of the reaction we may improve the yield. In this chapter we shall analyze this optimization problem using the concepts and results introduced in the previous chapters. In particular, we shall explain the role played by singular trajectories. We restrict our study to a batch reactor in which three species X, Y, Z are reacting according to the

scheme X - Y - Z and every reaction is irreversible and of first order, but our analysis can be used to analyze more complicated problems and to provide suboptimal control laws. Our study is based on the analysis of the equations of the maximum principle. The optimization problem is posed as a time optimal control problem,

where the terminal state has to belong to an hypersurface, which corresponds to a desired repartition of the concentrations of the end of the batch. The control is the temperature T(t) or the derivative T(t) of the temperature of the reactions. The relation between both cases being precisely the Goh transformation. The system is a single-input control system and we make an interesting application of the results of [26] explained in Chap. 6. In particular we compute the conjugate points along a reference singular trajectory as well as the curvature of the system. Also in this optimal problem, the terminal manifold is an hypersurface and we have to develop in this chapter a generic classification of optimal feedback law near the terminal manifold. This classification problem is related to the generic classification of extremals presented in Chap. 3 and uses also the local bounds obtained by SussmannSchattler for time optimal controls problems with fixed extremities, which is discussed in Chap. 3. Finally the optimal synthesis is computed using geometric properties of the control problem and global bounds of the number of switchings. Our presentations is based on the articles [24],[27],[28].

172

7 Chemical Batch Reactors

7.2 Mathematical Model of Chemical Batch Reactors and Description of the Control Problem 7.2.1 Chemical Kinetics We model our systems using the standard laws of chemical kinetics, see for instance [43]. They can be resumed as follows. Consider first a single chemical reaction, e.g. 2N0 + 2H2 - N2 + 2H20 written as nt .2

i=1

i=1

where the R;s are the reactants, the Ps the products and a;, Q; are the stoichiometric coefficients of the reactions. Let X be a species R; of P; with

stoichiometry yx and let nx be the number of moles of X at time t. The quantity I ((t) I defined by <(t) =

nx(t)

- nx(O) 7x

depends only on the reaction and is called the molar extent of the reaction. Taking the limit, we get {(t) = and vx =fax is the rate of evolution of x the species X. We denote by V the volume of the batch which is constant and introduce the specific rate

rx

vx

dnx _dcx

V

Vdt

dt

where cx = "VL is the molar concentration of species X;. Let r = Z&, this rate is given at constant temperature by an empirical law due to Gullerg-Waage nl

rl=k(T)flca' where the c;s are the concentrations of the reactants. The reaction is said of b;-th order with respect to Rs and the overall order is m = F,!` 16,. The numbers 61 are not generally related to the stoichiometric coefficients ai and have to be determined experimentally. If the reactions are simple we have b; = a1. The coefficient k depends on the temperature of the reaction and is given by Arrhenius law:

k(T) = Aexp (-i)

(7.1)

where the parameters A, E are respectively the frequency factor and the activation energy and R is the gas constant. We can now consider a set 7Z of p reactions which are assumed to be simple. We denote by Xk, k = 1, , n the set of all species occuring in the

7.2 Mathematical Model of Chemical Batch Reactors

173

reaction scheme and ck their respective concentrations. Every reaction can be written as yi - yj where yi, yj are linear combinations of the X's with coefficients in N and the reaction network can be identified to an oriented graph in R2 whose vertices are the yi, yj. Since the reactions are simple, if yi is given by F_k 1 ry{kXik we can associate to the reaction y; -+ yj: m

Ki-; = ki3 (T) [ C,14 k k=1

where ki3 is given by Arrhenius law. Hence the equation describing the evolution of the system can be written (7.2)

c = > Ki-.i (yi - yi) R

where c = t(cl,

,

c,,) is the vector whose components are the concentra-

tions. For instance consider a network X1 first order reactions. Since we have n+1

Xn+1 of consecutive

X2 n+1 mo(o)

if we introduce x'(t) =

"I t

and v = k1 the reaction scheme is modelled

(0)

by

dx

at

= K(v)x

dv

= h(v)u dt

where x =t (x1, . .

.

,

xn), u = AT,, h(v) = *vln2(A_ ), and K is the matrix

-k1

u-

0

...

k1 -k20 0

0

0

0 0

kn_1 _k,,

Moreover x, v satisfy the inequalities P

0<Ex'<1, 1
and xn+1 can be computed using the law of mass conservation.

174

7 Chemical Batch Reactors

In particular if we consider a network of two reactions X k + Y k + Z we get the system

z(t) = -v(t)x(t) y(t) = v(t)x(t) - Qv°(t)y(t) v(t) = h(v(t))u(t)

(7.4)

where a = jj is the ratio of the activation energies and ,Q = A We must have x(O) > 0 and the physical space is: x > 0, y > 0, x + y < 1, 0 < v < A1. The control u(.) is the derivative of the temperature in the batch and in practice we have constraints on T and hence on v. This problem will be shortly discuss at the end of this chapter.

7.2.2 Control Device In practice, the control T in the batch is tuned using an heat exchanger, see Fig. 7.1.

Te

Ts

T

Fig. 7.1. T =temperature in the reactor, Te =temperature of the fluid at the entrance of the exchanger, T. =temperature of the fluid at the exit, T =temperature of the inner shell.

We need a regulator to track the computed optimal law. This regulator must take into account: 1. The dynamic of the heat exchanges; 2. The heat transfert of the reactions which can be exothermic or endothermic.

7.2 Mathematical Model of Chemical Batch Reactors

175

In practice our control law is implemented for small reactors (about 5 liters) and a PID-regular allows to regulate T which can be taken as a control.

7.2.3 The Optimal Control Problem The optimal control problem is the following: fix a desired product quantity for a batch and minimize the batch duration. If we consider a network of the Z, we have different choices for the desired product. The form X --+ Y

most obvious is to fix the quantity d of the intermediate species Y at the end of the batch. The problem is then: reach the terminal manifold y = d in minimum time for system (7.4). It is a problem in R3 whose analysis, presented in details in [27), is extremely complicated. In order to simplify the

analysis we shall consider that the desired product is the ratio It = d. For this terminal constraint we shall see later using the symmetry of the problem that it can be reduced to a time optimal control problem for a planar system.

Definition 77. We consider the following time optimal control problem (P): Minimize the time transfert for the system:

i(t) _ -v(t)x(t) y(t) = v(t)x(t) - 0i0°` (t)y(t) i,(t) = h(v(t))u(t)

(7.5)

with the terminal constraint: z = d and the control being u = T and satisfying bounds: u_ < u < u+ with u_ < 0 < u+. The state (x, y, v) of the system must belongs to the physical space x > 0, y > 0, 0 < x+y <- 1 and 0 < v < A1. The reduced problem (P) is the previous optimal control problem where v is taken as a control and satisfies the inequalities: vm < v < vM. This corresponds to make Goh transformation and physically since v = Al exp (-4} this means that the control is the temperature.

7.2.4 Projected Problem For a batch reactor in which every reaction is a first order, the system is invariant for the following symmetry: (x, y) -+ (Ax, Ay). This will imply a nice projection property.

Definition 78 (every object is smooth (C°D or Cw)). Let M, M' be two manifolds, let 1r be a submersion from M into M' and let N (resp. N') be a regular submanifold of M (resp. M') with N = i-1(N'). Consider now the system on M given by T, (t) = f(x(t),u(t)), u(t) E Q and the associated time optimal control problem with terminal manifold N, denoted (P). Let us assume that for every fixed u E .fl, the vector field x +-+ f (x, u) can be 7r-projected and is complete for each admissible control u(.). Hence one may define the projected system on M': do (t) = f'(x'(t),u(t)), u(t) E .R where

176

7 Chemical Batch Reactors

x' = ir(x), f' = d7r o f and the associated time optimal control problem (P') with terminal manifold N'. It is called the projection of (P). Our aim is to compare both problems. We denote respectively by x(t, xo, u), x'(t, x'0, u) the trajectories starting at t = 0 from xo and xo = zr(xo). We have the following.

Lemma 23. The trajectory x* (t, xo, u') defined on [0, t') is a solution of (0) if and only if x'(t, xo, u') is a solution of (?P'). Proof. By completeness we have for every t: x'(t, xo, u) = 7r (x (t, xo, u*)) and

by definition N = 7r-1(N'). Lemma 24. Every extremal (x', p', u) defined on [0, T] of the projected problem (2') can be lifted onto an extremal (x, p, u) of the original problem (1').

Moreover if (x', p') satisfies the boundary conditions x'(T') E N' and p(T) orthogonal to T ,(T)N' imposed by (P'), then (x, p) can be selected to satisfy the boundary conditions x(T) E N, p(T) orthogonal to Tx(T)N imposed by (7')

Proof. The system 4 (t) = f (x (t), u(t)) can be written locally as dx'

it

(t) = f'(x'(t), u(t)), d, (t) = f"(x (t) x"(t), u(t))

Hence every extremal (x', p', u) can be lifted into (x, p, u) = ((x', x"), (p', 0), u)

where x" is any solution of the second equation. Clearly (x, p, u) is an ex-

tremal of (P) and we have x'(T) E N', p(T) orthogonal to ,,'(T)N' x(T) E N and p(T) orthogonal to TX(T) N.

Conclusion. Not every extremal of (75) can be projected onto an extremal of (2'). Hence although (P') is equivalent to (P), its analysis using the maximum principle is simplier because we have less extremals.

7.3 Singular Extremals - Curvature - Conjugate Points First we compute the singular extremals for the general control system (7.3) and then we deduce the singular extremals for the system (7.4) in dimension 3.

Lemma 25. The singular extremals of minimal order for system (7.3) with ((x, v), (p, A), U) E 1Rn x R x IIYn x R x R are the solutions of

d = K(v)x,

p

= -pK(v),

is (x, p) =

dv

= h(v)u(x,p)

p[K'(v),K(v)](x) pK"(v)xh(v)

A

=0

7.3 Singular Extremals - Curvature - Conjugate Points

177

contained in E' = {(x,p); pK'(v)x = 0}, where p is taken as a row vector, K' and K" are the first and second derivative of K with respect to v and [K', K) represents the bracket KK' - K'K.

Proposition 61. Consider system (7.4) in dimension 3. If a # 1, all the singular extremals in the physical space are hyperbolic singular extremals sat-

isfying assumptions (Hi) - (H2) - (H3) of Chap. 6 and are solutions of dx

=-vx' dt

dy

dt

= vx - ,0vay,

dv = h(v)u

dt

(7.7)

restricted to the physical space and the singular control feedback is u = _

V2I

hvy'

Proof. For the system (7.4) in 1R3 written as (X, Y) with IRY = 0 we observe first that X, Y are linearly independent in the physical space since i = -vx is strictly negative. In particular there exists no singular point and no periodic

trajectory. The control is given by u = -o, and the hyperbolic trajectories are contained in DD" > 0 where D, D', D" are defined in Chap. 6. Computing

we get D = h4a(a-1)$v's-2xy, D' = h3i(a-1)v°x2 and D" = h2,3v''xy(a1). This proves the proposition.

Basic assumption. The value a = 1 is a bifurcation value for which the system is not controllable since d is not depending on v. In the remaining of this chapter we shall make the following assumption: a > 1. It is the case physically interesting where the optimal problem is nontrivial.

7.3.1 Projected System We use the invariance of the system by the transformation (x, y, v) p--' (Ax, ay, v), A E IR\{0}. Hence it can be projected onto 1P' x R where IP' = projective space. More precisely, since x never vanishes in the physical space, it becomes in the coordinates (x, z = s, v) dx

dv

dz

= -vx' dt = v - /3v°z + vz dt = h(v)u at and the system

dt = v - /3v°z + vz, dt = h(v)u

(7.9)

is the projected system. Now we can project the differential equation (7.7) onto dz

dt

_

- v - /3vax + vz,

dv

dt

_

v2

az

(7.10)

whose solutions are the projections of the singular trajectories. As observed in Sect. 7.2.4, not every solution of this equation corresponds to a singular

178

7 Chemical Batch Reactors

trajectory of the projected system. Indeed if (X, Y) is a planar system the singular trajectories are contained in the set S : det(Y, [X, Y]) = 0

the singular control being given by a feedback on S. By computing for the projected system we get the following lemma.

Lemma 26. The singular trajectories for the projected system (7.10) are contained in S = {(z,v); z(a,3v°-1 - 1) = 1} and the singular control is given by u(z, v) = - h v2QZ . The differential equation describing the evolution on the singular arc is di (t) _ ! ! (1 - a,Ov°-1(t)).

Conjugate points and curvature. Since the singular trajectories corresponding to the system (7.4) of J3 are satisfying assumptions (H1) - (H2) (H3) of Chap. 6, we can apply our algorithm to compute the first conjugate time t1c where a reference singular trajectory ry(.) ceases to be optimal. The variational equation associated to (7.7) is the equation ax = -vbx - xbv az = v(1 - ,Ov°-1)6z + T/i(z, v)6v

by =

v2

az2

(7.11)

6z - 2v 6v az

where t'(z, v) = 1 + z(1 - a,6v°`1) and 0 = 0 is the set S. Let Q = S x 1R and G = {(x, z, v); z = 0}. To compute the conjugate points we must compute the Jacobi field V defined by integrating the previous equation with 6x(0) = bz(O) = 0 and bv(0) = 1. The Jacobi field V is contained in Span[Y, [X, Y]] along y(.) and ti, is the first conjugate time t such that V(t) is collinear to Y = ]R& To simplify our computations we can use the projected system. We have to distinguish two cases.

Lemma 27. A singular trajectory in P n Q, where P is the physical space, is without conjugate point.

Proof. Integrating (7.11) with the initial condition v(0) E IRS and with = 0 we get:

6z=0

6v=exp

2v(s))ds (- az(s) /

I0 \

and clearly 6x(t) < 0 for every t > 0. Therefore we cannot have 6x(tic) = 0. Now we consider the singular trajectories not contained in PnQ. The system (7.7) is written as th(t) = X(w(t)) +u(t)Y(w(t)) where w = (x, z, v). Let us denote by it the projection (x, z, v) H (z, v) and let X'", Y' being respectively the ir-projection of X, Y. Let y(.) be a singular trajectory contained

7.3 Singular Extremals - Curvature - Conjugate Points

179

in P\Q. The vector V(t) can be written A1(t)Y(7Y(t))+.2(t)[X,Y](ry(t)), its projection on the space (z, v) is d7r(V(t)) = )1(t)Y"(ir(7(t))) + A2(t)(Xx,Y"](ir('Y(t)))

and t1, is the first t such that .1z(t) = 0. Since on P\Q, V(t) is collinear to Y(ry(t)) if and only if d7r(V (t)) is collinear to Y'"(ir(ry(t))), we have proved the following lemma.

Lemma 28. The time t1c is the first time such that the solution of the projected variational equation bz = v(1

- /3v°-1)oz + O(z, v)8v (7.12)

by =

vz

azz

oz -

2v

az

by

passing through (0,1) at t = 0 is such that bz(tlc) = 0. 6 Computations. By setting J = 70T v-

previous equation can be written in the canonical form: J + (K o -y) J = 0 where K is the curvature for z<' 34 0 of the system and is given by:

K

= (a - 1)/3v°+1 (7.13) az Numerical simulations. From Sturm theorem we must find 0 < L < K to guarantee the existence of a conjugate time on [0, T]. Since along a singular trajectory ,y(.) in P we have z(t) - +oo when t -+ +oo, such a lowest bound does not exists in our case. We have to use numerical simulations to compute

conjugate points. If 3< Al they show the existence of conjugate point for singular arcs such that -Y(0) E {z < 0}, see Fig. 7.2.

Conclusion. Using the symmetries of the system we have computed the curvature of the system for 0 # 0 without using the normalization of Chap. 6. We showed the existence of conjugate points for some singular arcs. The arc which project on S is without conjugate point. Of course in our problem with a terminal manifold N of codimension one, the concept of conjugate point must be adapted into the concept of focal point, see [24] or [27] for such a discussion. In our case this concept will be not necessary because the

problem (P) is equivalent to the problem (P') for a planar system. Such problems are without conjugate or focal points.

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7 Chemical Batch Reactors

(af)r

p =u

A,

V

Fig. 7.2.

7.4 Time Minimal Synthesis for Planar Systems in the Neighborhood of a Terminal Manifold of Codimension One 7.4.1 Problem Statement Consider a system in R2 of the from

v(t) = X (v(t)) + u(t)Y(v(t)), J u(t) (< 1

(7.14)

where X and Y are smooth (C°° or C") vector fields. Let N be a smooth regular submanifold of R2 of codimension one. The set U of admissible con-

trols is the set of measurable functions with values in [-1, +1]. We study the following local problem: let vo E N, compute in a sufficiently small open neighborhood U of vo the closed loop optimal control (local synthesis) for the time minimal control problem with N as terminal manifold, the system being given by the restriction of (7.14) to U. This problem is well proved because Filippov theorem of Sect. 2.4 can be applied to ensure the existence of open loop optimal solutions. The aim of this section is to give the tools to analyze this problem for generic triplets (X, Y, N). To make this classification the following results of previous chapters are useful:

- Classification of extremals explained in Sect. 3.5; - Local bounds on the number of switchings for time optimal control problem with fixed extremities of Sect. 3.8. Our analysis is based on the local analysis of the solutions of the maximum principle using semi-normal forms.

7.4 Time Minimal Synthesis for Planar Systems

181

Definitions and notations. The system is denoted (X, Y) and v = (x, y) are the coordinates in R2. A coordinate system (U, v) is said adapted if the restriction of Y to U is gy . The point vo is identified to 0. The terminal manifold N is locally identified to the image of a neighborhood of 0 by an immersion f : R -+ R2 with f (0) = 0. The problem is said flat if the vector field Y is everywhere tangent to N (it is the case for batch reactors). A normal

n at N in v is denoted n(v). We lift N in the cotangent space using the transversality condition into Nl : { (v, p) E R2 x R2; v E N, (p, w) = 0 Vw E In this section an extremal (v, p, u) is supposed to be defined on (T, 01 with T < 0. It is called a BC-extremal if it satisfies the boundary conditions given by the transversality conditions: (v(0), p(0)) E N1. We denote by K the set of switching points of the BC-extremals and W the set of switching points of optimal trajectories. The splitting line L is the set of points where the optimal control is not unique, it will form the cut locus. Our program is to evaluate for the triplets (X, Y, f) the sets W and L for all the situations

of codimension 0 and 1. In particular we want to stratify W. Using the concepts of (103] we have two kinds of strata: a stratum is of first kind if the optimal trajectories are everywhere tangent and of second kind if they are everywhere transverse.

7.4.2 Assumption C1 We identify locally N to the segment x = 0, y El - e, cf. If we denote by C(v) the convex set {X (v)+uY(v); I u 1< 1}. The assumption Cl is the following:

we assume that X, Y are not collinear at 0 and that the convex set C(v) is contained in the half-space x > 0. Then the optimal synthesis is computed in the subspace x < 0. If n(v) denotes the normal oriented towards x > 0, then from the maximum principle for every BC-extremal (v, p, u) with v(0) near vo = 0, we can assume p(O) = n(v(0)).

7.4.3 The Generic Cl Case By assumption, the two vector fields X ± Y are not tangent to N at 0. Then with our convention we have (n(v), X (v)) > 0 for v near vo = 0, and p(0) = n(v(0)) for every BC-extremal. If we assume (vo, Y(vo)) non zero, the local optimal synthesis is deduced from the transversality condition: if (vo,Y(vo)) > 0 (reap. < 0) then the optimal control is u' = +1 (reap. -1) and maximizes the normal speed with respect to N.

7.4.4 The Generic C1 Flat Case In the flat case, Y is tangent everywhere to N and we have no information from the transversality condition. But differentiating the switch-

ing function #(t) = (p(t), Y(v(t))) and evaluating it at t = 0, one has

182

7 Chemical Batch Reactors

(0) = (p(0), [Y, X](vo)). Hence if (n(0), [X,Y](0)) 0 0 the arcs 'v+ (resp. 'y_) are BC-extremals if and only if (n(vo),1Y, X) (vo)) < 0 (resp. > 0) and the optimal synthesis is given by Fig. 7.3.

n

n

N

N <0

>0

Fig. 7.3.

7.4.5 The Case Cl of Codimension One We analyze now the situation where Y is tangent to N at 0 at a single point. We assume (n(0), [Y, X] (0)) # 0. Using assumption Cl, we have (n(0), X(0)) > 0. This situation is not encountered in the flat case, but the geometric discussion is relevant to analyze many situations.

Analysis. Our objective is to evaluate the switching loci K, W and the splitting line L. For this we make the following normalizations. Since X, Y

are transverse at 0 we can assume locally Y = g and that the trajectory corresponding to u - 0 and hitting the target at 0 is the curve t F.- (t, 0). Hence the system (7.14) can be written locally as +00

= 1 +

ati(x)y`

(7.15) +oo

y = u + E bi(x)y` Moreover if we change y into -y and u into -u if necessary, we can assume a = al (0) > 0 with a = (n(0), [Y, X] (0)) where n(0) = (1, 0) is the normal to N at 0. The terminal manifold is given locally as the image of s E- (c(s), s)

7.4 Time Minimal Synthesis for Planar Systems

183

with c(s) = ks2 + o(s2). Near 0, the normal n = (n1, n2) is n1 = 1, n2 = -c'(s) = -2ks + o(s). Hence for s small, if k < 0 we have n2 > 0 if s > 0 and n2 < 0 if s < 0 and conversely if k > 0. If k < 0, the maximum of H(v, n, u) = (n, X (v) + uY(v)) over I u I< 1 is reached for u = 1 ifs > 0 and u = -1 if s < 0 and conversely if k > 0. Hence we get the following

result: if k < 0 the BC-arcs y+, y_ can cut themselves, contrary to the case k > 0, see Fig. 7.4. Now using the results of Sect. 3.8, we have the

V

x

x

k
k>O

Fig. 7.4.

following. Near 0, every optimal solution for the time minimal problem with fixed extremities is of the form -1--y+. And clearly, each trajectory for the problem where N is the terminal manifold must be optimal for the problem with fixed extremities. The second step is to evaluate the switching set K for the BC-extremal -y-,y+. The adjoint system associated to (7.15) is +oo

Pi = -P1

+00 a',(x)y` - p2

i=1

b;(x)y`

i=1 (7.16)

+oo

P2 = -pi E iai(x)y`-1 - p2 i=1

+oo

E

ibi(x)yi-'

i=1

where p = (p1, p2) and a;, bi' are the derivatives of ai, b; with respect to x.

Integrating for t < 0 with v(0) E N and p(0) = n(v(0)) = (1, -2ks+o(s)) we obtain

184

7 Chemical Batch Reactors

pl (t) = 1 + o(1), p2(t) = -2ks - at + o(s, t).

are obtained by solving p2(t) = 0, t < 0. We get t = - 2k" + o(s). Moreover for k < 0, we must have s < 0 and for k > 0, s > 0. The switching points for BC-extremals -y+-y- are then (x(t), y(i)) _ The switching times

s(- 2-k, 1 + -k) + o(s). Hence we have proved the following result.

Lemma 29. Assume k # 0. Hence the switching points of BC-extremals y+y_ are located on a smooth curve K whose tangent space at 0 is the line R(- ak, I+ ak ).

0 and k Lemma 30. Assume k -S.. Then a BC-extremal -y+-y- is crossing K if k > 0 or - a < k < 0 and it is reflecting on K if k < - a Proof. At 0, the slope of the tangent to K is -1 - sk and the slope of 'y+ is +1. Hence if k > 0, the slope to K is less than -1 and if k < 0, -1 - -' > 1 if and only if - a < k. Hence the geometric situations are given in Fig. 7.5.

k>O

-a
k<-a4

Fig. 7.5.

Proposition 62. The optimal syntheses are given by Fig. 7.6 where the switching curve W and the splitting line L are smooth curves, the slope of their tangents being respectively -1 - ak and - 4°-Ej at 0.

Proof. In the first two cases, the situation is clear because in the domain x < 0 near 0, there exists only one BC-extremal -y+-y- to reach the target. Consider the case k < - S. In this case we prove that an optimal trajectory is .

without switching points. Indeed, assume that there exist optimal trajectories y+y_ accumulating near 0. Then we can construct an optimal trajectory of the form y+y_ y+'y_ where the arcs are not empty. This is absurd. Also we

7.4 Time Minimal Synthesis for Planar Systems

k>O

-a
185

k<-a 4

Fig. 7.6.

can use the result of [81 showing that an optimal trajectory in the plane cannot reflect himself on the switching locus. Let us define the splitting line L near 0 as follows:

L = {v = (x, y); v small, x < 0 s.t. both exp t(X ± Y)(v) intersect N}. Computing we get that L is a smooth curve whose tangent at 0 has a slope equal to - E[0,1[. Observe that if X, Y are C, L is a sub-analytic set. From our analysis we deduce that x = 1 + ay, y = u is the local model of the behaviors and the linear approximation of K and L are computed using the model.

7.4.6 Generic Hyperbolic Cases Assumptions. We assume that the system satisfies assumption C1 and moreover we suppose that Y is tangent to N at 0 and that both vector fields Y and [X,YJ are collinear at 0. It is a situation of codimension 2 for triplets (X, Y, N) but of codimension one in the flat case. Moreover, we assume that the set S : det(Y, [X, Y]) = 0 is a smooth one dimensional manifold near 0 and we suppose that ad2Y(X) is not collinear to Y at 0. Using the previous assumptions there exists an unique singular arc 'y (.) defined on [T, 01 with -t,(0) = 0 and the singular control is denoted by u(.). We shall assume that u(.) is admissible and not saturating, i.e. u(0) E] - 1,+1[. Moreover we analyze only the hyperbolic situation i.e. we assume (n(0),ad2Y(X)(0)) < 0. In particular using the analysis of Chap. 6, the reference trajectory y,(.) is time optimal for T small enough for the fixed extremities problem. Next we give a local model to study the problem.

186

7 Chemical Batch Reactors

Model. We use an adaptated coordinate system in which the singular arc y, is identified to t e-- (t, 0). Hence the system can be written th = 1 + a(x)y2 + ox(y2)

(7.17)

y = -ulv=o + yX2(v) + u

where a(0) = (n, ad2Y(X)(0)) < 0, n = (1, 0) being the normal to N at 0, and N being given as the image of the mapping s '-+ (ks2 + o(s2), s).

Let p = (pl, p2) be the adjoint vector with p(O) = n(v(0)) = (1, -2ks + o(s)). Computing with u = E, e = ±1 we get the evaluation: p2(t) = -2ks a(0) (e - u(0))t2 + klst + k2s2 + o((s,t)2), where k1, k2 are real coefficients which are unimportant for our discussion. Hence we deduce the following lemma.

Lemma 31. In the hyperbolic case, any BC-extremal which meets N at a point v # 0, v small has no switching point if k # 0. Now, let us evaluate the accessibility set along the reference singular trajectory.

Proposition 63. There exists a neighborhood U of 0 such that the reference singular trajectory ry, (0, T] --+ U is optimal for the time minimal problem on U, with fixed extremities conditions. Moreover, if U is sufficiently small, the :

accessibility set at time T is near y,(T) a closed convex set with nonempty interior whose boundary is a curve s d(s) with d(0) = y,(T), d'(0) E RY(y,(T)), C2 but not in general C3. Moreover in every adapted coordinates its curvature is zero. Proof. The first assertion is proved in Chap. 6. To compute the accessibility set, we use Chap. 3. Indeed, we have proved that if U is small enough each optimal trajectory starting from y,(0) is a singular arc y, followed by an arc y+ or -t-. Hence near y, (T) the boundary of the accessibility set in time T

and with initial point y,(0) is parametrized by s - d(s), where s > 0 and

d(s) = exps(X f Y)exp(T - s)X(y,(0)) where X = X + uY. Since y, (T) = exp TX (y, (0)) we get

d(s) = exps(X ± Y) exp -sX (y,(T )) and by using the Baker-Campbell-Hausdorff formula we have

d(s) = exp [s(±1 - u)Y + 2s2(X, X ± Y] + o(s2)J I (y, (T)). The curve d(s) can be evaluated using Chen formula

7.4 Time Minimal Synthesis for Planar Systems

187

k 8nZn exp sZ(v) =

n. n=o

(id) (v) + o(sk)

for s sufficiently small where Z is any vector field acting by Lie derivative on

the mappings and id is the identity. If Y = a we have Yn(id) = 0 if n > 1. Since along a singular trajectory Y and [X, Y) are collinear we get d(s) = -f. (T) + (s(±1 - u) + f (s))Y(ye(T)) + o(s2)

where f (s) = o(s). Using the parameter s' = (±1-u)(s+ f (s)), the boundary is given by d : s' +- (-s(T) + o(si2), s' + o(s12)). Hence it is C2, d'(0) is collinear to Y and d"(0) = 0. Higher-order expansions would tell us the nature of its singularity. For

instance if the system is given by x = 1 - y2, y = it and y8 (0) = 0, we get d(s) = (T - 3 , es) with e = f 1. Hence the boundary is the graph of

x=T-1.

Proposition 64. If k 0 0, the optimal syntheses are given by Fig. 7.7. Moreover in the flat case, the optimal synthesis is given as the case k > 0.

k>O

k
Fig. 7.7.

Proof. Let zo = (0, (1, 0)) E N1. Then according to the classification of Chap. 3, the point zo is an hyperbolic fold and every BC-extremals near 0 is of the form y f ys y f . The singular arc ys is time optimal for the fixed extremities problem. To decide if ys is optimal for the problem with N as

188

7 Chemical Batch Reactors

terminal condition we use the previous proposition. Let vl = (T, 0) with T < 0 small enough. Since the boundary of the accessibility set has zero curvature we have for k # 0, the two situations of Fig. 7.7. If k < 0, N meets the interior of the accessibility set at time T and y, is not optimal contrarily to the case k < 0. In the case k < 0, the BC-extremals cut themselves and there exists a splitting line L whose tangent at 0 has the slope -u(0). The analysis is similar in the flat case. Before to analyze the "tangent case" we establish the following result.

Lemma 32. Let y(.) a smooth solution defined on [0,T] and associated to a constant control uo, not contained in N and reaching N at vo. We note Z = X +uoY and A(5, P) = >k o bk adkP)(V0) - Z(vo), where 6 E R and P is any vector field. If y(.) is optimal, then for every b > 0 small enough and any P in the polysystem {X + uY; I u 1< 1} we must have (n,.1(b, P)) < 0 where n is the normal to N at vo outwardly oriented with respect to y(.) Proof. Our demonstration is based on the proof of the maximum principle. We construct along the reference trajectory -y(.) an approximation of the accessibility set. Since the terminal manifold is of codimension one, this approximation has not to be convex to decide about optimality. We proceed as follows. The arc y(.) associated to uo is such that -y(O) = v1, y(T) = vo and we have exp TZ(vi) = vo. Let b, e > 0 small and P be a vector field in { X + uY; I u I< 11. We fix b and we consider the curve

a(e) = exp5ZexprPexp(T- a - E)Z(vl). We have a(0) = vo and a(e) belongs to the accessibility set A(v1,T). Using the Baker-Campbell-Hausdorff formula we can write K ak

a(e) = exp (E(E k>O

kj

adkZ(P) + o(5K) - Z) + o(e)) (vo).

Hence a'(0) = Ek>0 4k ad1Z(P) - Z+o(SK). If (n, a(8, P)) > 0 the reference trajectory is not optimal because we can construct a trajectory reaching N in less time.

7.4.7 Generic Exceptional Case Assumptions. We analyze the case where the assumption Cl is not satisfied and one of the arcs y+ or y_ arriving at 0 (and denoted y+, y°) is tangent to N at 0. We may assume it is y°.. We make the following assumptions: the two vector fields Y and X - Y are not tangent to N and the contact of y+ with N is minimum.

7.4 Time Minimal Synthesis for Planar Systems

Model. We choose near 0 a coordinate system such that Y = the curve s i- (0, s). The system can be written

31-

189

and N is

x(t) = X1(x(t), y(t)) + u(t), y(t) = X2(x(t), y(t))

with X1(0) = -1,

-a 0 0 and X2(0) 36 0. Moreover we can assume

X2(0) > 0.

Proposition 65. Under the previous assumptions and normalizations, the optimal synthesis is represented on Fig. 7.8.

a>O

a
Proof. We consider first the case a > 0. The arc ry+ is a BC-extremal with -n = (1, 0) as adjoint variable at 0. We prove it is not optimal using Lemma 32. The normal to N at 0 oriented as in the lemma is n. We have Z = X + Y

ands we take P = X - Y. We get a(8, P) = -2Y(0) + o(1). Hence for 6 small, (n, -2Y(0) + o(1)) > 0. This proves the assertion. Moreover a simple computation shows the following. Assume we are at a distance a from N in

the domain x > 0. The time to reach the target N is of order f along ry+ and of order c along 'y (the contacts are different). In the domain x < 0, the optimal control is +1, the value function v F- T'(v) being not C°. When a < 0, the analysis is similar but the target N is not accessible from the points in the sector x < 0, above ry+.

7.4.8 Generic Flat Exceptional Case Assumptions and normalizations. The terminal manifold N is identified to the curve s I- (0, s) and Y is assumed everywhere tangent to N. We

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7 Chemical Batch Reactors

assume X tangent to N at 0. We suppose that Y, X ± Y are nonzero at 0 and that [X, Y] is not tangent to N at 0. Then locally the system can be written ax + by + o(x,y)

y=X2+u with b = (n, [X, Y](0)) 0 0. We may take the following normalizations: a =

O,b=land1+X2(0)>0. Proposition 66. Under the previous norm.alizations, the optimal synthesis is given by the Fig. 7.9.

X2(0)>1

X2(0)<1

Fig. 7.9.

Proof. According to the terminology of Chap. 3, the point 0 E N lifts into a normal switching point of NJ- contained in E : (p, Y(v)) = 0. Hence every BC-extremal near 0 is of the form y+y_ or -y--y+. Since Nl C E, the switching points are located on N. Hence each optimal trajectory is an arc y+ or -y-. The optimal syntheses follow easily. The difference between the two situations is the following. If X2(0) > 1 the target is not accessible from the domain located above y° in the domain x > 0. In the case X2(0) < 1 the origin is locally controllable.

7.5 Global Time Minimal Synthesis

191

7.5 Global Time Minimal Synthesis 7.5.1 Preliminaries Our objective is to reach the target N = {(z, v); z = d, d > 0 fixed} in minimum time. The system is t

dv

(t) = MO - Qv'(t)z(t) + v(t)z(t))'

(t) = h(v(t))u(t)

where u E [u_,u+], u_ < 0 < u+. We study the nontrivial case a > 1. Moreover we make the following assumption on the parameters set:

Al>p

,

<-h(

z 1)ad
where vl is defined by Pl = (vl, d) and Pl is the intersection of the singular set

S = {(z, v); z(a/iv°-1-1) = 1} with the target N. These assumptions allow to analyze the most complex synthesis and the other cases can be deduced easily from this analysis. The meaning of the assumptions are the following:

- the physical space contains BC-extremals where z = 0 (exceptional case); - the control u(.) corresponding to the singular arc is admissible on the target and is not saturating.

7.5.2 Singular Arc First in this problem it is interesting to observe that we have a straightforward interpretation of the singular control. Indeed if we assume that the control is the temperature i.e. v, the problem becomes a time optimal control problem

in R with fixed extremities: z(0) = zo, z(T) = d, where z = (v-/3v°z+vz) and v E [vn, vM] Hence the optimal control has to maximize z(.) over v E [Vm, vA1]. The system is nonlinear and the maximum is not in general reached

= 0 and we get z(a,8v°-1 - 1) = 1 i.e. (z,v) E S. If u = T is taken as the control, a singular arc belongs to S and the singular control is given by u(z, v) = - h v'°z . In order to be admissible, we must have u E [u_, u+]. We observe that u < 0 and " < 0. Moreover v decreases along the singular arc and u tends toward -oo when v -+ Al . By assumption Pl is the intersection point between S and N and at Pl the for v = v,,, or vM. The singular control is simply defined by

singular control satisfies u_ < u < u+. Hence there exists an unique P2 where the control u is saturating and equals to u_. We denote by y, the admissible singular are. All these results are resumed on Fig. 7.10.

192

7 Chemical Batch Reactors

Z

N Y

/S I

Al

(aW

V

Fig. 7.10.

7.5.3 Regular Arcs The behaviors of the arcs corresponding to u = u_ or u = u+ are the following. Since h(v) > 0 for 0 < v < A1, the sign of v is the sign of u_ or u+. Moreover z = 0 for z = - 1)-1 whose graph is denoted E. The singular points in the domain 0 < v < Al are located on v = 0. If (Qv°'-1

P = E fl {v1 = Al }, then P = (((3Ai -1 - 1)-1, A1) and P is located by assumption in the domain z > 0. We represent on Fig. 7.11 the phase portraits.

u=u+

Fig. 7.11.

7.5 Global Time Minimal Synthesis

193

7.5.4 Optimal Synthesis in the Neighborhood of N The analysis of Sect. 7.4 allows us to analyze the optimal synthesis in the neighborhood of N. It is represented on Fig. 7.12. We have two situations of codimension one:

- At P1, the hyperbolic singular arc meets the target and the problem is flat. The optimal synthesis is given by Proposition 63;

- Let P3 = E f P. Then in these points we are in the flat exceptional case and the optimal synthesis is deduced from Proposition 65.

Fig. 7.12.

Remark 20. In this problem z = d has to be considered as a varying parameter. Hence in our analysis we must also consider the situation where the singular control is saturating for z = d. This situation can be easily analyzed as in Sect. 7.4, see for instance [28]. In our analysis it will be a consequence of the global synthesis near P2, see also [98].

7.5.5 Switching Rules Switching function. One of the main problem to be solved in order to compute the global synthesis is to find global bounds on the number of switchings. In our case it is related to pseudo convexity properties of the switching functions. Let p = (PI, P2) be the adjoint vector for the system. We have 171 = plv(,3va-1 - 1)

02 =

pi(a/va-iz

- 1 - z) - p2h'(v)u

7 Chemical Batch Reactors

194

and the switching function P(t) = p2(t)h(v(t)) satisfies the following equation

fi(t) = plh(v)[z(af3va-1 - 1) - 1] fi(t) = p1h(v)[(3v°(a - 1) +ua(a - 1)pv°-2zh(v)] + uh`(v)-h(t). The set fi(t) = 0 plays an important role when the switching points are computed. If pl is nonzero, its projection on the state space is the curve S. The adjoint variable corresponding to a BC-extremal defined on [0, T] must satisfy p(T) E ]R(1, 0). Moreover from the analysis of Sect. 7.4, it can be oriented according to the principle p(T) = (1, 0). The switching rules are the consequence of the following lemmas. Lemma 33. If O(0) = P(T) = 0, then a smooth BC-extremal y(.) meets the set S in a time 0 < t < T. Proof. We have pi (T) = 1 and pl has a constant sign on [0, T], hence pi > 0. Since iP(0) = 45(T) = 0, using Rolle's theorem there exists t E]0, T[ such that z(a,l3va-1 -1) = 0. fi(t) = 0. Since pi is not vanishing on [0, T], we have at t: Hence y(t) E S.

Lemma 34. If (-y, p, u) is a BC-extremal with u = u+ defined on [0, T] and 4i(T) = 0 then P(0) is nonzero.

Proof. Let cp(t) = h a evaluated along the given extremal. We have cp = pi (a - 1)(I3v" + u+a13v°-2zh(v)). Hence signcp = sign pl > 0 and gyp(.) is a strictly increasing function. Since

(y, p, u+) is an extremal and O(T) = 0, we must have 4t(T) < 0. Hence V(0) < V(T) < 0. Let us assume 45(0) = 0, then again we must have 4t(t) > 0.

This contradicts (p(0) = h o < 0. Proposition 67. Every optimal trajectory is of the form y+y_y, where each arc of the sequence can be empty.

Proof. The proof follows from the two previous lemmas and from the following remark. An optimal arc of the form y,'y_ where -y, and y_ are not empty

is not possible because the subarc y_ must meet twice the set S, which is absurd.

7.5.6 Optimal Synthesis Using our analysis it is straightforward to construct the synthesis represented

on Fig. 7.13. The switching curve W is the union of y, with a curve WThe curve W_ is the set of switching points of the arcs y+y_. This curve is containing the point P2 as a consequence of the saturation of the singular control at P2.

7.6 State Constraints due to the Temperature

195

Fig. 7.13.

7.6 State Constraints due to the Temperature In the previous synthesis, we did not take into account constraints of the

temperature: 0 < Tm < T < TM which imply constraints on the state v = Al exp -b) of the form v E vM]. It can be done in general using the maximum principle for systems with state constraints, see (91). Here we present an analysis valid for planar systems and using the clock-form.

7.6.1 Preliminaries The system is written as

z(t) = f(z(t),v(t)), v(t) = h(v(t))u(t) with u(t) E [u_, u+], u_ < 0 < u+. We denote respectively Lm and LM the vertical lines v = vm and v = vM in the plane (v, z). The control which allows to stay in these lines is the control u =_ 0. We note E the set z = 0. Let 7b(.)

be a trajectory located in the boundary. It is oriented towards the target if z > 0 and with the inverse orientation if z < 0, see Fig. 7.14. A crucial step is to understand the time optimality status of a boundary trajectory -yb(.) (it is one of the main aspect of the maximum principle with state constraints). For this we can use the clock form w defined for a planar system (X, Y) by the relations: (w, X) = 1, (w, Y) = 0. Computing we get

f (zzv)

and dw = - fa(y}

v)

of dv n dz,

196

7 Chemical Batch Reactors

Fig. 7.14.

Fig. 7.15. the two form dw vanishes on the singular arc S and the sign of dw in the state domain is represented on Fig. 7.15. Using this clock form we can analyze the optimality status of a boundary trajectory for the fixed extremities problems

and state constraints and deduce the optimal synthesis with N as terminal manifold. We present the optimal synthesis when the constraints are given by Fig. 7.16, the others situations could be deduced from this analysis. We note Bl = (z, vM) the intersection of the line: v = vM with the switching

7.6 State Constraints due to the Temperature

197

V

Fig. 7.16.

curve W_ computed in Sect. 7.5. We design by B2 and B3 the respective intersections of S and E with the line v = vM. Lemma 35. The optimality of a boundary trajectory for the fixed extremities problem is represented on Fig. 7.17.

Z

N

P1

I-- non optimal

,,l S non optimal

B3

dw >O

non optimal

-- ---------dw
0

VM

Vm

Fig. 7.17.

B

2 optimal V

7 Chemical Batch Reactors

198

Proof. We show for instance that an arc yb defined on [0, tI] and contained III v = vm is not optimal. Let y2 be an other trajectory of the system, defined on [0, t2] and with same extremities (see Fig. 7.18). We have

J )_J6W=t2-t1 =InU--fbW= JDdw<0 where D is the domain limited by the closed curve 72 U -yb.

7.6.2 Optimal Synthesis Proposition 68. Under the previous assumptions on the parameter space, the optimal synthesis is given by Fig. 7.18.

w

Vm

L

M VM

V

Fig. 7.18.

Proof. We note respectively T+ and r- the arcs corresponding to u = u+ and u_ and passing through B1 = W n {v = vM } and L = 1'+ n {z = 0}. The optimal synthesis is similar to the problem without constraint excepted in the dashed domain below T+ where each optimal trajectory is of the form y+yb1'_

and where we must use an arc yb in the boundary. To prove this

result we use the notations of Fig. 7.18 and we observe that starting from the

point M on z = 0 an optimal trajectory must pass through B1. Indeed, the arc T+ = LBI is optimal for the problem without constraints and matches the constraints. Moreover using the clock form w we get that the time along the arc MKBI is smaller that the time along an are MNB1.

7.7 The Problem in Dimension 3

199

7.7 The Problem in Dimension 3 7.7.1 Preliminaries In this section we consider the time minimal problem for the system

:r = -vx vx-13v°y v = h(v)u

with u E [u_ , u+], u_ < 0 < u+ and the target N is now the set y = d and as previously we assume a > 1. Contrarily to the problem studied in the previous section this problem cannot be reduced to a planar problem and the

analysis is much more complicated. The complete solution which use numerical simulation is described in [27]. Here we shall indicate the main steps of the analysis.

First we must blow-up the system using the symmetries and use the following canonical coordinates

x=lnx, z= yX and the system is written

X= _V

z=v-f3v°z+vz i) = h(v)u

and the target N is z exp x = d. We denote by n the normal to N oriented towards the sector z > d. In the concentrations space n = (0, 1, 0) and in the canonical coordinates we have n = (d, exp x, 0). The main steps of the analysis are the following. First we stratify the target N by computing the optimal control feedback at the terminal points. Then we must compute the optimal synthesis near the terminal manifold. This results are the extension of the analysis of Sect. 7.4 for planar system. This extension is not straightforward and the results complex. It has justified several articles, see for instance [28] and [27]. Finally we generalize the results of Sect. 7.5 concerning a glgbal bounds on the number of switchings. These three steps allows to compute the optimal synthesis using numerical simulations, where the local analysis near N plays the role of the analysis of the singularities.

7.7.2 Stratification of N by the Optimal Feedback Synthesis We write the system as (X, Y) and we have two sets where the optimal synthesis is not straightforward.

200

7 Chemical Batch Reactors

Exceptional set. It is the set of points where X is tangent to N. It is given by the relation E : { (n, X) = 0} fl N. Computing in the canonical coordinates we get E : 1 - ,Ov'-1 z = 0, z exp x = d.

The optimal synthesis near E is complex and we can give a geometric explanation. Consider the system written in the space of concentrations. The terminal manifolds is y = d, where d < 1. If we consider the system we have th + y < 0. Hence if x(0) < d, the point (x(0), 0) cannot be sent to N. Hence the set of points which can be steered on the target has a boundary.

Singular set. The singular trajectories corresponding to BC-extremals are located on the set S : {(n, [X, Y]) = 0} n N. Computing in the canonical coordinates we get S : a)3v°-1 z - 1 = 0, z exp x = d.

If a > 1, the set S in the physical space is a simple smooth curve transverse to Y. The singular feedback is given by u = - ti v and its restriction has the following properties: it is strictly decreasing from 0- to -oo when v goes from 0+ to Al . There exists an unique point denoted P0 where the singular control is saturating: u(Po) = u_. Moreover we know from Sect. 7.3 that all the singular trajectories are hyperbolic. Hence using the terminology of Chap. 3 if P E S, then P is lifted into the point (n, P) of N' contained in E : (p, Y) = 0 and we have

a

- if u(P) E]u_, u+[, the point (n, P) is an hyperbolic switching point; - if u(P) V [u_, u+] the point (n, P) is parabolic. At Po, the singular control is saturating. We represent on Fig. 7.19 all these results, the target N being identified to the plane (v, z).

7.7.3 Orientation Principle Outside E, the adjoint vector at the terminal point can be taken as n, according to the maximum principle. At a point of E, the Hamiltonian H is zero and we need the following lemma to take p as n. Lemma 36. Let y+ or y_ be a BC-extremal defined on [0, T] and tangent to N at xo but not included in N. Let n be the normal to N oriented towards

the domain not containing y+ or -y-. Assume that (xo, n) is an ordinary ching point, i.e. (n, Y(xo)) = 0 and (n, [X, Y](xo)) # 0. Then if y+ or y_ is optimal, the adjoint vector at x0 is necessary n.

7.7 The Problem in Dimension 3

201

S

E:=O

hyperbolic

=0

q

0

V

parabolic Fig. 7.19.

Proof. We use lemma 32. Assume for instance that ry+ is a BC-extremal associated to -n at xO. Hence we must have (n, [X, Y](xo)) < 0. Applying the lemma with P = X - Y we get the necessary optimality condition (n, [X,Y](xo)) > 0. A contradiction.

7.7.4 Switching Rules We can generalize the results of Sect. 7.5 and we have the following.

Theorem 24. Assume a > 1. Then each optimal control has at most two switchings and each optimal trajectory is of the form ry+y_rya where each are of the sequence can be empty.

The proof follows from a sequence of lemmas. If p = (Pi, p2, p3) is the adjoint vector in the canonical coordinates we have the equations

pl =0 P2 = P2(3va - v)

03 = P1 +

p2(01/3v°-' x - 1 - z) - P3h'(v)u

Hence we get.

Lemma 37. The ajoint vector p has the following properties. The component p, is a first integral: pi (t) = Pl (0); p2(t) = (expf(/3v0 v)ds) p2(0) and either p2(t) __ 0 or its sign is constant. In particular along a singular arc we have p2(t) > 0 and p, (t) > 0.

-

202

7 Chemical Batch Reactors

Proof. The first assertion is a direct consequence of the equations. For the second part we use the previous orientation principle. Along an optimal arc we can choose at the terminal point p(t) = n = (d, exp x, 0).

Lemma 38. The switching function P(t) = (p, Y) defined on IT, 0], T < 0 along a BC-extremal satisfies the equations:

1. 4 = h(v)p3, 45(0) = 0 since the problem is flat. 2.

= h(v)[pi +p2(af3v°-1 z - 1 - z)], = h'fiu + hp2[(a - 1)/3v° + ca(a - 1)/3v°-2huz], a.e.

3. If tp = n we have cp = p2(a - 1)/3v°-2[v2 + uazh(v)], a.e.

Lemma 39. Let y, be a singular subarc of an optimal trajectory. Then y, is contained in the sector aj3v°-1 z -1- z < 0 and satisfies the condition z > 0.

Proof. By definition fi(t) = 0 along a singular arc and P1, P2 > 0. Hence z - 1 - z < 0 along the arc y,. Since a > 1, in the we must have physical space we have 0 > a,33v°- 1 z - I - z > ,3va-1 z - 1 - z and with a,3v°-1

z=v(1+z-/3v°-1z)wegetz>0. Lemma 40. Let y_ be an optimal arc defined on IT, 0]. Assume that 0 and z - 1 - z > 0. Hence. -y- must T are switching times and -y_ (0) E meet the locus: a0va-1z - 1 - z = 0. a/3v°-1

Proof. Since 0(0) = O(T) = 0 there exists t E IT, 01 such that fis(t) = 0. z - 1 - z < 0. By assumption Since p1 i p2 > 0, at time t we have af3v°-1

y_ (0) E a/3va-1 z - 1 - z > 0. The assertion follows. Lemma 41. Let 'y_ be an optimal arc defined on IT, 0]. Assume that 0, T are switching times and moreover i'(T) = 0. Then y_ must meet the saturation

setC: u=u_ defined byu_2V Proof. Since P(0) = O(T) = 0 there exists t E]T, 0[ such that rh(t) = 0. satisfies tp(t) = V(T) _ Moreover by assumption (T) = 0. Hence c' = 0 and there exists t' E]T, t[ such that V(t') = 0 i.e. p2(a - 1),Qv°-2(v2 + u_azh(v)) = 0. Since p2 0 0 we have v2 +u_azh(v) = 0, i.e. u = u_. Lemma 42. Let -Y+ be an optimal arc defined on IT, 0]. Assume that 0 is a switching time. Then T is not a switching time. Proof. See lemma 34.

7.7 The Problem in Dimension 3

203

Proof (Proof of the theorem 24). We can project a trajectory y(.) in the space (z, v) and the projection is denoted We denote by G' the set i = 0 i.e.

1 - $v"-lz + z = 0 and by C the projection of the saturating set u = u_ which is the graph z = - .u h v and is a strictly increasing function of v. First we consider the case AI < J3T where i > 0 in the physical space. To prove our result it is sufficient to prove that no subarc 'Y,-y+ where y y_ are non empty is optimal. If such an arc is optimal, using Lemma 41 the arc y_ must meet the saturation set C : u = u_. If it is the case the piece of y_ before the intersection is contained in the sector u < u_ and the connection with y, is in this sector. This is not possible because in this sector the singular control is not admissible. The case Al > ATh can be analyzed in a similar way.

7.7.5 Local Classification near the Target Hyperbolic case. Under generic assumptions the system can be written as th = 1 + a(x)z2 + 2b(x)yz + c(x)y2 + R1 y = d(x)y + e(0)z + R3 z = (u - fi(x)) + f(x)y + g(0)z + R3

where R1 (resp. R2, R3) are terms of order > 3 (resp. > 2) with respect to (y, z). In this semi-normal form Y is identified to , the set L where [X, Y]

is tangent to N to the axis Oy and the target N to the plane x = 0. The reference singular arc y, is identified to the curve t -4 (t, 0, 0). Moreover [X,Y]I,. = [X,Y](0) In the hyperbolic case we have a(0) < 0 and u(0) E] - 1,+1[. Using the previous normalization we can construct the optimal synthesis near an hyperbolic point and we get the following result.

Proposition 69. The BC-arcs y+ and y_ are not cutting themselves near 0. The optimal synthesis is C°-equivalent to the synthesis using the model x = 1 + a(0)z2, y = 0, i = (u - v.) where each plane y = constant is an invariant leaf and in each leaf the synthesis is represented on Fig. 7.20.

Parabolic case. At a parabolic point P the singular feedback control is such that u(P) V [-1,+1]. We can construct the optimal synthesis as previously.

There exists a CO invariant foliation and in each leaf y = constant, the synthesis is given by Fig. 7.21.

Saturated case. Near the point P° where 11(P°) = u_, the optimal synthesis is more complicated and is described in [28]. The geometric situation is the following. As in the previous situations, there exists a CO invariant foliation

204

7 Chemical Batch Reactors

Nn y=constant

Fig. 7.20.

N l1 y=constant

Fig. 7.21.

whose leaves are given by y = constant. The point Po represents a bifurcation between the hyperbolic and the parabolic situation and the synthesis is represented on Fig. 7.22. In the leaves associated to a saturated point or a parabolic point the synthesis are equivalent, every optimal trajectory is of the form -f+-y- and the switching points are located on W+. In the leaves associated to an hyperbolic point the situation is more complex. Every optimal trajectory has at most two switchings and is of the form 'y+ry_-y, where each arc of the sequence can be empty. The switching locus is the union of these curves denoted W+, W, and ry,. The set W+ is the switching locus of optimal policy of the form ry+'Y_ and W. is the set of first switching points of optimal trajectories of the form -y+-y--y,.

7.7 The Problem in Dimension 3

205

N A y=constant

;p

hyperbolic

saturated

parabolic

Fig. 7.22. Remark 21. In the theorem 24 we prove that every global optimal trajectory has at most two switchings and is of the form -y+-y--y.. In the previous analysis we show that near the saturating point Po we have such optimal trajectories.

Generic exceptional case. We can easily compute the optimal synthesis in a generic point of E where x is tangent to N. A semi-normal form is constructed as follows. The point is taken at 0, the target N is identified to the plane x = 0. Moreover we can assume that the locus of tangent points is a simple curve transverse to Y and it is identified to the axis Oy. We choose Y = YZ- and moreover we can assume [X,Y](0) = M-10. The system can be written

x=z+R b+o(1) c + u + 0(1)

where R is of order > 2, b and c are constants c + 1, c - 1 nonzero and we can assume c + 1 > 0. There exists a C°-invariant foliation identified to y = constant and the optimal synthesis in each leaf is as the planar case and is described on Fig. 7.23. In our problem there is a point of codimension 2 where the condition c # 1 is not satisfied. The analysis is complicated and is presented in [27]. At such a point the curve 7_ has a contact of order 2 with the target.

206

7 Chemical Batch Reactors

c<1

c>1

Fig. 7.23.

7.7.6 Focal Points To deal with the time minimal control problem where the terminal manifold N is not necessary a point we must generalize the concept of conjugate point along an hyperbolic singular arc into the concept of focal point. The extension is straightforward and is the following in our problem. Let 'y, defined on [T, 0] be an hyperbolic singular arc and assume that the singular control it satisfied the bounds u_ < u < u+ (i.e. is not saturating). Then the local synthesis described in proposition 69 can be extended in a tubular neighborhood of y, up to a point t1 f called the first focal point. It is defined as follows. Let W (O) be a tangent vector at y, (0) to the curve a where [X, Y] is tangent to the manifold N and let W (t) be a solution on [T, 01 of the variational equation (7.11) obtained by linearizing the singular flow x = S(x) along the singular trajectories. The first local point t1 f is defined as follows.

Definition 79. Let T < t1 f < 0 be the first time t < 0 such that det(W(t), Y(ys(t)), S(y9(t))) = 0. Then y,(t1f) is called the first focal point along y,.

In the C" case, W(t) as the Jacobi field V(t) associated to the concept of conjugate point is contained in Span{Y, [X, Y1}1,y,.

Its numerical computation is straightforward.

7.7 The Problem in Dimension 3

207

Notes and Sources All the results presented in this chapter are coming from a series of articles. In [24], the algorithm for computing the conjugate and focal points are given and the planar case is solved. The generic local classification near the terminal manifold is intrincated and is presented in two articles [28] and [75] which deals with the exceptional case. This case has a connection with subRiemannian-geometry which shall be explained in Chap. 9. The problem in dimension 3 for batch reactors, where the reactions are not necessary of firstorder is analyzed in full details in [27]. Also this article contains numerical simulations which solve the problem.

8 Generic Properties of Singular Trajectories

8.1 Introduction and Notations Let M be a o-compact COO manifold of dimension d > 3. Consider on M a single-input control system i(t) = Fo(x(t)) + u(t)F1(x(t)) where Fo, F1 are C°O-vector fields on M and the set of admissible controls U is the set of locally integrable mappings u : (0, Tu] - IR, Tu > 0. The aim of this chapter is to show that for an open dense subset of the set of pairs of vector fields endowed with the C°°-Whitney topology, all the singular trajectories are with minimal order and normal. It is based on the article [25]. Also we analyze in dimension 3 the singularities of the vector field whose solutions are the singular trajectories of minimal order. This study was initialized in (22] for quadratic systems connected to the attitude control problem of a rigid spacecraft. This analysis is related to two important control problems: first of all the feedback classification, see Chap. 4; secondly the existence of broken singular minimizers in optimal control e.g. in sub-Riemannian geometry.

Before to state our results we must introduce very precise notations and slighty extend the concept of singular trajectories to deal with the singularities.

We shall denote by M a c-compact C°° manifold of dimension d > 3. We shall use the following notations:

- TM: tangent space of M, TmM : tangent space at m E M. - T'M: cotangent space of M, T,, M: cotangent space at m E M. - The null section of T'M is denoted by 0 and (T*M)o = T'M\0, JPT'M: projectivized cotangent space, i.e. JPT'M = T*M\1R`, [z]: class of z in for any integer N > 1, JNTM: space of all N-jets of COO-vector fields: JIM : JNTM - M: canonical projection. - VF(M): vector space of all C°O-vector fields endowed with the Whitney topology.

- E xM F: fiber product of two fiber spaces (E, JIE, M) and (F, 17,F, M) on M. - H: given any COO-function defined on an open subset (1 C T' M, H will denote the Hamiltonian vector field defined by H on .R.

210

8 Generic Properties of Singular Trajectories

- (HI, H2 }: given any two C°°-functions on 17, {H1, H2 } will denote their Poisson bracket: { H1, H2 } = dH1(H2) -

- SpanA: if A is a subset of the vector space V, it is the vector subspace generated by A. To each couple (FO, F1) of C°°-vector fields on M, we associate the control system:

dt (t) = Fo(x(t)) + u(t)FI (x(t)),

u(t) E R.

(8.1)

The study of time minimal trajectories leads to the consideration of the following solutions of Pontryagin's maximum principle (z, u) : J -+ T'M x lR, J interval [TI, T2], T, < T2 and 1. z is an absolutely continuous curve and u(.) is locally Lebesgue integrable;

2. z(t) # 0 (0 = null section) for all t E J;

3. T, (t) =Ho (z(t)) + u(t) Hi (z(t)), for a.e. t E J where Hi : T`M R,i = 1, 2, Hi(z) = (Fi (17T. M (z)), z) (canonical Hamiltonian lift of Fi); vH1(z(t))}.

4. for a.e. t E J, Ho(z(t)) + u(t)H1(z(t)) =

We observe that 4 is equivalent to Ho(z(t)) = 0 for a.e. t E J, but since z(t) is C°, this is equivalent to 4'. Ho(z(t)) = 0, Vt E J. t

Definition 80. A curve (z, u)

:J

T*M x R satisfying the conditions

1.,2.,3.,4'. will be called a singular extremal and (IIT M (z), u) a singular trajectory. This definition is only a slight extension of the concept of singular trajectories in the preceedings chapters, valid because system (8.1) is affine. It will allow to include the singularities concerning singular trajectories. Next we give a general algorithm to compute the singular algorithm in

the single-input case.

8.2 Determination of the Singular Extremals with Minimal Order Let (z, u) : J -- T'M x R be a singular extremal. Using the chain rule and condition 4'. we get

. y (Ho(z(t))) = dH1(z(t))(Ho (z(t)) + u(t) Hi (z(t))) dt d

0

for almost every t E J. Since t i- z(t) is C°, this implies the relation 0 = {Hl, Ho}(z(t)). Using the chain rule again we get

8.4 Geometric Interpretation and the General Concept of Order

211

0 = dt {H1, Ho}(z(t)) = {{Hl, Ho}, Ho}(z(t)) + u(t){{H1, Ho}, H1 }(z(t))

for almost every t E J.

Definition 81. For any singular extremal (z, u) : J -. T*M x R, R(z, u) will denote the set {t E J; {IH1, Ho, }, Hl}(z(t)) 0}. A singular extremal (z, u) : J - T'M x R is called of minimal order if R(z, u) is dense in J. The following propositions are an immediate consequence of the above considerations.

Proposition 70. If (z, u) : J

T*M x IR is a singular extremal and 1Z(z, u)

is not empty, then 1. z is restricted to R(z, u) is C°°; z t for all t E R(x, u); 2. a(t) =Ho (z(t)) + (JH1,HoJ,H1)(z(t)) 3. u(t) = Ho,H1 ,Ho zztt for almost every t E R(x, u). JJff1,Ho),H,

Proposition 71. 1. Let (Fo, F1) E VF(M) x VF(M) be a pair such that the open subset 17 of all z E (T'M)o such that {{Ho,Hl},H1}(z) = 0 is not empty. If H: 17 -+ R is the vector field Ho + {Hj,Ho,HAH1, then any trajectory of H starting at t = 0 is a minimal order singular extremal

of (Fo,Fi) 2. There is an open dense subset of VF(M) x VF(M) such that for any couple (F0, F1) in that subset the set 17 is open dense in T* M.

8.3 Statement of the First Generic Property Theorem 25. There exists an open dense subset G of VF(M) x VF(M) such that for any couple (Fo, F1) E G the associated control system (8.1) has only minimal order singular extremals.

8.4 Geometric Interpretation and the General Concept of Order We shall give a geometric justification of the above theorem; moreover it will

lead to define the concept of order to compute in general the singular extremals in the single-input case. If (z, u) is not of minimal order then there exists a non empty open subinterval J' on which {{H1, Ho}, Hl}(z(t)) = 0.

Combining with the relation

8 Generic Properties of Singular Trajectories

212

{{Hl, Ho}, Ho}(z(t)) + u(t){{H1i Ho}, Hi}(z(t)) = 0 it shows that there exists a non empty open subinterval J" on which we have {{H1, Ho}, Hl }(z(t)) = {{Hi, Ho}, Ho}(z(t)) = 0.

Differentiating both relations with respect to t we get a.e. on J" {{{H1i Ho}, HI), Ho}(z(t)) + u(t){{{Hi, Ho}, H1}, Hl}(z(t)) = 0, {{{Hi, Ho), Ho}, Ho}(z(t)) + u(t){{{Hi, Ho}, Ho}, Hi}(z(t)) = 0. This is a compatibility relation which means geometrically the following: there exists a singular extremal arc contained in both surfaces: {{H1,Ho}, Ho} = 0 and {{H1iHo},H1} = 0.

An instant of reflexion shows that this property cannot be generic but it needs a complete proof.

Also the previous computation gives us the general stratification to computed the singular arcs at any order. Notation. If X, Y E VF(M), the Lie bracket is computed with the convention [X, YJ (m) = - ay(m)X(m) and the adjoint mapping is defined by adX(Y) = [X, Y]. For any multi-index a E {0,1 }", a = (a1, , a"), length of a: I a 1= n, we note I a 10= card{i; ai = 01, a I I= card{i; a; = I}. The Lie bracket denoted by Fe0 is defined inductively Fan]. Similarly the function HQ : T'M -+ R is defined by: F. = [F(01,...

inductively by H0 = {H(01,...,0 _,l, H,,,.). Observe that using the relation {Ho,Hi}(z) = ([Fo,F1](UT.M(z)),z) we have H. = Definition 82. The singulat arc (z, u) defined on the open non empty interval J is said of order q > 2 if 1. H0(z(t)) = 0 for all a = (1, i2, , ik) E {0,1}k+1, I a 1< q. 2. Then exists 3 = (1, ii, -, iq_1 i 1) E {0,1}9+1 such that

{t E J; Ha(z(t)) # 0) is dense in J. Algorithm This gives us the stratification to compute the singular arcs of any order

- Order 2 H1 = H1o = 0, H1o1

9& 0,

u = -H.

- Order 3: Hi = Hio = H1o1 = H1oo = 0, 1on or H1o01 # 0 and u is solution a.e. of both equations Hio1o + uH1o11 = 0 H1ooo + uH1oo1 = 0 etc...

8.5 Proof of Theorem 25

213

8.5 Proof of Theorem 25 To prove Theorem 25 we are going to define for each N sufficiently large,

a bad set B(N) in JNTM X JNTM having the following property: any couple (Fo,FI) E VF(M) x VF(M) such that (jNFo, js Fl) 0 B(N) for all x E M has only singular extremals of minimal order. Then we shall show using transversality theory that the set G of all couples (Fo, FI) E VF(M) x VF(M) such that (jz Fo, jz FI) 0 B(N) for all x E M is open dense in VF(M) x VF(M). To construct the stratification of the bad set we have to analyze two cases: the points x where FO, F1 are linearly dependent and the points x where FO, F1 are linealy independent.

Construction of the bad set. Definition 83. For N > 2d - 1, let Ba(N) be the subset of JNTM x M JNTM of all couples (jx Fo, j2 F1) such that

dim Span{adFo(FI)(x); 0 < i < 2d - 1} < d.

Definition 84. 1. For N > 1, BB (N) is the subset of JNTM x M JNTM of all couples (jx Fo, j. F1) such that dim Span {Fo(x), F1 (x), [Fo, Fi](x)} < 1.

2. For N > 2, let Bi'(N) be the subset of JNTM xM JNTM x R of all triples (jNF0, jx F1, a) such that a) Fi(x) 34 0;

b) Fo(x) = aFi(x),

c) dim Span{adGa(Fi)(x),0 < i < d- 1,[[FoF1],Fi](x)} < d where

Ga=Fo-aF1 S. Denote by Bj'(N)the canonical projection of BI"(N) onto JNTM xM JNTM. 4. B1(N) = Bi(N) U BI" (N).

Definition 85. Using the notations 8.4 define the following. (i) For any integers c >0, any

a

=1

a 0 101"-2) any integer N > n + c - 1, let

B(N, a, c, 0) (resp. B(N, a, c, 1)) be the subset of JNTM X M JNTM x m PT*M of all triples (jz Fo, jx F1i [z]) such that 1. Fo(x), F1 (x) are linearly independent;

2. HQ0(z) 0 0, Ho, (z) 0 0;

8 Generic Properties of Singular Trajectories

214

3. 9(ZQ)kHlo.,_1(z) = 0 (resp. 9(ZQ)kH101.,_2(z) = 0 for 0 < k< c, where Za is the vector field H01 Ho -H00 Hi on T'M, and 9 denotes the Lie derivative.

(ii) B(N, a, c, a) (a = 0, 1) will denote the canonical projection of B(N, a, c,

a) onto JNTM xM J^'TM. Definition 86. We set B(N) = Ba(N) u B1(N) u (uB(N, a, 2d, a));

3I
Lemma 43. Let (Fo, Fl) E VF(M)2 such that all x E M, (ji Fo, j 'FI) B1(N).

1. Let (z, u) : J -' T'M x R be a singular extremal such that for all t E J, Then x(t) is dim Span {Fo(Y(t)),Fi(x(t))} < 1 where x = constant. 2. Let x0 E M. If TT0M contains a singular extremal, then there exist a A E IR and a line l E Tx.M x R such that every singular extremal (z', u') J' T*.M x ]R is of the form z'(t) = exp(At)zo, z0 E I and u' is constant a.e. All these extremals are of minimal order.

Prof.

1. Call S the set of all t E J such that F1(Y(t)) = 0. S is closed and has empty interior: otherwise there exists an open non empty interval

Ji C J such that F1(Y(t)) = 0 for all t E Ji. Then -X(t) = Fo(x(t)) for all t E J1. This implies that [F1, Fo](T(t)) = 0 for all t E J1. Then dim Span {Fo(Y(t)), F1(Y(t)), [Fi, Fo] (Y(t))} < 1 fort E J1. This contradicts the assumption of Lemma 43. Since dim Span {Fo(Y(t)), F1(Y(t))} < 1 for all t E J, there exists an absolutely continuous function

a : J\S - JR such that Fo(Y(t)) = a(t)Fj(x(t)) for all t E J\S. This implies that for a.e. t E J\S, (a(t) + u(t))[Fo,F1](Y(t)) = a(t)Fi(T(t)) because x(t) = Fo(x(t)) + u(t)Fi(Y(t)) = (a(t) + u(t))Fi(Y(t)) for a.e. t E J\S. hence a(t) = a(t) + u(t) = 0 for a.e. t E J\S: since by the assumption of Lemma 43, dim Span {Fo(Y(t)), F1(x(t)), [Fo, F1](Y(t))} > 2 and since Fo(Y(t)) = a(t) F1 (Y(t)), F1 (2(t)) and [Fo, F1](T(t)) are linearly

independent for all t E J\S. Suppose that S 54 0. Then the open set J\S contains an interval ]a, p[=

{t; a < t < Q}, where either a E S or 3 E S. Assume that a E S (p E S is similar). Since a(t) = a(t) + u(t) = 0 for a.e. t E]a, 8[, a is constant on ]a,,3[ and -u(t) = -a for a.e. t E]a,1[. Hence x(t) = 0 for a.e. So Y(t) = xo for all t E]a, Q[. This leads to the contradiction t 0 = F1(x(t)) = limt.a,gEi«,13iFi(Y(t)) = F1(xo) 54 0. Hence S = 0 and Y(t) = xo for all t E J. This proves 1.

8.5 Proof of Theorem 25

215

2. Let ('z, u) : J -' TOM x R be a singular extremal such that z(t) E T0M for all t E J. The assumption of Lemma 43 implies that dim Span {Fo(xo), Fo(xo), [Fo, Fi](xo)} > 2.

If Fo(xo) = 0, then Fo(xo) # 0 but we have for a.e. t E J : 0 = _X(t) _ Fo(xo). We have a contradiction. Since 0 = i(t) = Fo(xo) + u(t)F1(xo) for a.e. t E J, there exists an a E IR such that Fo(xo) = aFi(xo) and

u(t) = -a for a.e. t E J.

Let C = Pb - aF1

:

G(xo) = 0 and set H(z) =

By definition z(t) =H (I(t)) for all t E J and Hi (z(t)) = 0. Hence 0 for all t E J adkH(Hi)(z(t)) = and all k E N. Since G(xo) = 0, Span{adcG(F1)(xo); k E N}= Span{adkG(Fi)(xo),0 < k < d - 1}. By assumption xo is a singular trajectory, hence this space is at least of codimension one. Since (j o Fo, j o Fl) V BI (N), it is exactly of codimension one, and moreover ([Fi, [Fo, F1]](xo), z(t)) 0 0. Therefore z(t) belongs to a line l E TT0M and (7(t),11(t)) is of minimal order. By definition z(t) is solution of a linear system and since G(xo) = 0, it is autonomous. Hence 2 is proved.

Proposition 72. Let a couple (Fo, F1) E VF(M) x VF(M) which satisfies the condition: there exists an integer N such that for all x E M (is Fo,j2 Fi) ¢ B(N). Then the control system associated to (Fo,FI) has only minimal order singular extremals.

TOM x ]R is a singular extremal not J of minimal order. This shows that there exists an open subinterval J0 of J,Jo not empty, such that {{Ho,H1},H1}(z(t)) = 0. Then the closed set has an empty inIt E Jo; dim{Fo(1(t)),Fi(Y(t))} < 1}, Y = terior: otherwise it would contain an open non empty interval Jot C Jo. Then Lemma 43 applies to the restriction (z', u') of ('z,-u) to Jol. But since Proof. Assume that (11,'11)

:

{I Ho, Hi }, H1 } (z') = 0 we get a contradiction. Replacing J by an open non empty subinterval we can assume that for alll t E J:

1. {{Ho, Hi}, Hi}(z(t)) = 0; 2. dim Span {Fo(r(t)), Fi (Y(t))} = 2.

Since If Ho,Hi},H1}(z) =0, then {{Ho, Hi }, Ho}(z) =

d

({Ho, Hi }(-z)) = 0.

We claim that there exists a multi-index a E {0,1}n, a = (a1, al = 1 such that

(i) 3
(iii) either HQo(z) 96 0 or HQ1(z) 34 0.

,13k),/31=1,1
,

an),

216

8 Generic Properties of Singular Trajectories

In fact were there no such a, then Ho(Y) = 0 for all /3 E {0,1}k, A = , /3k ), 1 <1 /3 1< 2d. In particular taking Q = 10k, 0 < k < 2d - 1, we get (adkFo(F1)(Y(t)),-z(t)) = 0, 0 < k < 2d-1, for all t E J. This shows that (,31 i

dim Span{ad'kFo(F1)('Y(t)),0 < k < 2d - 1} < d

for all t E J and contradicts the assumption that (jz Fo, j2 Fl) 0 B. (N) for all x E M. The assumption (iii) above can be replaced by the following: (iv) the set {t E J\HHo(z(t)) 0 0 and Ha1(z(t)) 0) is non empty. In fact, suppose we have either Ha1(z) = 0 or Hao(z) = 0. In the first case we get 0 = AHa(z) Hao(z). This would contradict (iii). In the second case Ha1(z) # 0 by (iii). On the non empty open set 0 = {t; t E J, Ha1(z) 0 0}, we have almost everywhere 0 = Tt(Ha(z)) = iHa1(z). Hence u = 0 a.e. on 0. Then z(t) =Ho (z(t)) on O. Since H, (T) = 0, we get 0= dt Hl(z(t)) = adJHo(H1)(z(t)) for all t E O. This implies that for any t E 0: (adkFo(F1)(T(t)),z(t))) = 0 for all k > 0. This contradicts the assumption that (jz Fo, j'FI) 0 B0(N) for all x E M. Finally replacing J by a subinterval we see that we can assume that:

1.3
Since Ha(z) = 0 we get for a.e. t E J : 0

dt(HH(z(t))) = Hao(z(t)) + u(t)Ha1(z(t))

So u(t) _ - ?(z(t)) for a.e. t E J. Since Hl(g) = 0, this shows that z is a trajectory of 1'{a where fa :.fl - R is the function Ho -X,, H1 and .fl = {z; Ha1(z) 01. Define now y to be lOn-1 if a # a = 10"-1. Since f -y 1= n, H.y(z) = 0 and

l0n-1 and 101"-2 if

k

0 = dtk (H.y(z)) = adkfa(H.,)(z)

for all k > 0.

It is easily seen that this is equivalent to (B(Za)k(Hy))(z) = 0 for all k > 0, and ?a and Za are collinear. This shows since adk9-la(Hy) = that for all t E J, (j=ctiFo, j(t)F1.[z(t)]) E B(N, a, c, a), a = 0 if a 76 and a = 1 if a = 10"-1, where x(t) = and [z(t)] denotes the class of z(t) in IPT*M. 101-1

Now we shall compute the codimension of the bad set. It can be done by using semi-algebraic sets, see [17].

8.5 Proof of Theorem 25

217

8.5.1 Partially Algebraic and Semi-Algebraic Fiber Bundles Definition 87. A VP bundle on M is a locally trivial fiber bundle on M

,W 1P(Wn), where V, WI, whose typival fiber is a product V x IP(W1) x , IP(Wn) the associated proare finite dimensional vector spaces, IP(W1), x jective space and whose structural group is Aut(V) x Aut(IP(WI)) x -

Aut(IP(Wn)) (Aut(V) = GL(V), Aut(IP(W;)) = GL(Wi)\1R').

Definition 88. A partially algebraic (resp. semi-algebraic) subbundle of a VP bundle on M is a locally trivial subbundle whose typical fiber A is an algebraic (resp. semi-algebraic) subset of the typical fiber V x 1P(W1) x . . . 1P(Wn )

of the VP bundle.

Lemma 44.

(i) JNTM xM JNTM, JNTM xM JNTM x R, JNTM XM JNTM xM 1PT*M are VP bundles on M whose typical fibers are respectively P(d, N)

x P(d, N) x, P(d, N) x P(d, N) x R, P(d, N) x P(d, N) x IP(Rd) where P(d, N) denotes the set o f all polynomial mappings P = (P', , Pd) Rd such that degPt < N for 1 < i < d. Rd (ii) Ba(N), Bi(N), b" (N), B(N, a, c, a) are partially algebraic (for the first two) and semi-algebraic (for the last two), subbundles of the VP bundles JNTM xM JNTM, JNTM XM JNTM, JNTM XM JNTM x R,

JNTM XM JNTM xM IPT'M. Their typical fibers .F.(N), .F,(N), f' (N), P(N, a, c, a) can be described as follows: X. (N) = {(Po, PI); dim Span {{adk Po (PI) (0), 0 < k < 2d -1} < d}; P, (N) = {(Po, Pi); dim Span {Po(0), P1(0), [Pi, Po](0)} < 1}.

XI" (N) is the set of all triples (Po, P1, a) E P(d, N)2 x R such that 1. P1(0) 0 0, Po(0) = aP1(0);

2. dim Span{adkRa(P1)(0),0 < k < d- 1} and [[Po,P1J,Pij(0)} < d where Ra = Po - aP1. For the definition of the last fiber we shall use the following notations:

for a E {0,1}n, a = (al,

, an) the function Ha : Rd x Rd* --I' R is defined inductively as follows: if i = 0 or 1, Hi (x, C) (Pi (x), t). If a = (al, , an), Ha = {H.Is...san-I, Han } where {, } denotes the

Poisson bracket.

For any integer c > 0, any a E {0,1 }n, n > 3, a = (al, , an), al = 1, N > n + c - 1, .F(N, a, c, a) is the set of all triples (Po, PI, [l;]) in P(d, N)2 x IP(Rd) such that (i) Po(0),P1(0) are linearly independent; ( i i ) Hao(0, e) 0 0, Hal (0, E) -A 0;

218

8 Generic Properties of Singular Trajectories

(iii) (O(Z.)k(H.y))(0,1;) = 0, 0 < k < c; where ry = 10s-1, a = 0 if a 0 10n-1 a n d

8.5.2 Coordinate Systems on P(d, N) First let us explain a few facts about coordinate systems on homogeneous polynomials. For m > 1, the space Pm(d) of all homogeneous polynomials of degree m in d variables can be identified with the space of m-multilinear symmetric mappings on Rd as follows. Let f E Pm(d), (1), , b(m) E Rd, define the total polarization of f as (P f) (ill i, . . ,(r,,)) = D{(,) . . DC(-) f where ed). Clearly f and Pf can be identified since DC f = , _ , xm ). Given a basis e1, f (x) = -1=, P f (xi, , ed of l[td we define a system

'

of coordinates {X,,; v E Im} as follows. The set Im is the set of sequences v= ik E [1,d] where (i1,.. , im) and (iolll) , i,lml are identified for any permutation a. Hence we can order with it < < im. Let I v 1= m denote the length of v and I v J;= the number of occurence of i. Define X as follows. For m = 0, set Im = {0} and define X (f) = f (0). If m > 1, (Pf)(e1...... eim) Now let the couple (A, B) E P(d, N). Let U be a neighborhood of (A, B)

in P(d, N) and let e : U - (Rd)d be a smooth mapping such that for any (Q, R) E U, e(Q, R) _ (el (Q, R)), , ed(Q, R)) form a basis of Rd. Then to e we can associate a coordinate system

as follows

X '(Q, R) = (PQL)(ei, (Q, R), ... ea.,, (Q, R)) (Q, R), ... , e;,.. (Q, R)) R) = where Q;n (resp. R;,i) is the ith component of the homogeneous part of degree m of Q (reap. R). This system of coordinate is a curvilinear system of course.

We set X _ (X,1

,

Xd) and Y _ (Y1,

Yi).

8.5.3 Evaluation of Codimension of the F(N) Each F(N) being semi-algebraic in their corresponding spaces, the concept of dimension is well defined. We shall estimate their codimensions.

Lemma 45. cod(Fa (N); P(d, N)2) = d + 1

cod(F1(N); P(d, N)2) = 2d - 2

8.5 Proof of Theorem 25

219

Proof.

.F. (N) _ .T'a(N) U.F (N) UY.'(N) .F.' (N) _ -PT(N) n {(P0, Pi)/Po(0) 0 0) .Pa (N) = FP.(N) n {(Po, P1 )/P1 (0)

0, Po(0) = 0}

.F."' (N) _ )1. (N) n {(Po, Pi)/Po(0) = P1(0) = 0).

It is easy to see that cod(.P4'(N); P(d, N)2) = 2d. To compute the codimensions of .Pa (N), .P.(N) let us introduce the following semi-algebraic sets

C C Rd x End(R'), C = J (v, A); v 0 0, dim { A"v/0 < n < d - 1) < d}

and D C (Rd)2d = {(v,.. . , v2d_ 1); vi E Rd}, dim D < d. Then clearly cod(C, Rd x End(lRd)) = 1 and it is well known that cod(D, (Rd)2d) = d + Rd x End(Rd) x Rd and p 1. Consider the mappings A : P(d, N)2 P(d, N)2 -+ (Rd)2d defined as follows MPo, PI) = (PI (0), Poi, Po(0))

where P01 E End(Rd) is the linear part of Po at 0,

µ(Po,PI) =

(Pi(0),adPo(Pi)(0),...,ad2d-'Po(Pi(0))

Then.Fa (N) =,\-'(C x {0}) and.PQ(N) = µ-' (D). Since \ is a projection, it is a submersion and hence

cod(F. (N); P(d, N)2) = cod(C x {0}; Rd x End(Rd) x Rd) = d + 1.

We prove that v restricted to the open semi-algebraic subset fl of P(d,N)2, fl = {(Po, P1)\Po(0) 34 0} is a submersion. Since .F.(N) _ µ-' (D n fl) it follows that cod(.P.(N); P(d, N)2) = d + 1. To study µ, take a couple (Qo, Qi) E fl. There exist vectors e2,

, ed E

Rd such that (Qo(0) = e1, e2, , ed) is a basis of Rd. Then on a neighborhood V of (Qo, Q 1) contained in fl the mapping

e : V -- Rd x ... x Rd, e(Po, Pi) =

(ei(Po,P1),...,ed(Po,Pl))

e1(Po,Pi)=Po(0), ei(Po,P1)=ei, 2
ado(Pl)(0) kP = Y1k + Rk where Rk is a function of X', I v 1< k, Y,', I v 1<

8 Generic Properties of Singular Trajectories

220

k - 1. This shows immediately that v is a submersion at each point in V. As (Qo, Qi) is arbitrary in Q, vin is a submersion. The proof that cod(.P' (N); P(d, N)2) = 2d - 2 is very similar.

fi (N) _

(3) (N) u (4)

(N)

i31(N) = .P (N) n {(Po, Pi)/Po(0) = P1(0) = 0} yj(4)(N)

= yi\yi3)(N)

Clearly cod(F(3)(N); P(d, N)2) = 2d, Let Q0,1 be the open set of P(d, N)2

of all couples (Po,P1) such that Po(O) 0 0 or P1(0) # 0. The mapping v : nol - (Rd)3, v(Po, Pi) = (Po(0), Pi (0)), Poi (Pi (0)) - P11(Po(0))) is a submersion and F14i(N) = v-1(D3), where D3 = {(v0, vl, v2); vi E 1R d, i = 0, 1, 2, dim Span (vo, V1, V2) :5

Clearly cod(D3; (Rd)3) = 2d - 2. this gives the second result of Lemma 45.

Lemma 46. P(d, N)2 x R) = d + 2, cod(.) '(N, a, c, a); P(d, N)2 x IP(JRd)) = c + 1.

Proof. Let Z01 = {(Po,P1ia)/P1(0) 0 0,Po(0) = aP1(0)}. Define the mapping X : Z01 - Rd x End(Rd) x Rd as follows X(Po, Pi, a) = (Pi (0), Pol - aP11, [[Po, P1], Pl](0))

Clearly X is a submersion and .*,"(N) = X-1(C1) where C1 = {(v, A, w)/v, w E Rd, A E End(Rd), v # 0

dim Span{A"(v), 0 < n < d - 1, w} < d}. Then cod(C1

i

R' x End(Rd) x Rd) = 2.

So

cod(P('(N); Zo1) = 2 and

cod(Fl'(N); P(d, N)2 x R)

=

Zo1) + cod(Zo1; P(d, N)2 x R) = 2 + d.

Now we shall consider the case of P(N, a, c, 0). The case of P(N, a, c, 1) is similar. Let 1201 = {(Po, P1i )/Po(0), P1(0) are linearly independent, 34 0, (P«1(0),.) 34

E JP(Rd)} and where

8.5 Proof of Theorem 25

Pa = Let (:.Rol x Rd

221

Pa..J

Rc+1 be the mapping

((Po, Pi,()

),e(za)H,(B,(),...,e(Za)`H7(0,()),(9k 0,

where -y = 10"' 1. Then P(N, a, c, 0) = (-1(0). If we show that ( is a submersion it will follow that cod()(N, a, c, 0); P(d, N)2 X IP(Rd)) = c + 1. Using the rule O(Za(FG)) = F[Hal {G, Ho} - Hao{G, H1 }J +G[Ha1(F, Ho) - Hao{F, HI }J an easy induction shows that 8(Za)kH.y = HQ1 H10,.+k-1 +fla,k, where lla,k is

a polynomial in Ha, where either 16 I< n+k or 16 I= n+k but 8 0 Take a (PO, P,) E Floe. There exists e3, - - e' in Rd such that (Po(0), Pl (0), e3, , ed) is a basis of Rd. Then one can find a neighborhood V of (PO, Pl ) 10n+k-1.

such that for all (Po, P1) E V, the d vectors el (Po, PI) = PO (0), e2 (PO, P1

P1(0), e1(Po, PI) = e;, 3 < i < d form a basis of Rd. Let XL, Y be the coordinate system on V associated to the mapping e = (e1, , ed). Now [F1,Fo](0) = 9Fo(Fi)(0) - BF,(Fo)(0). Hence H1o(0,() = ((,Y1 X2). Therefore computing by induction we get H10#(0,() _ (p,Ylk+12k'

-

where n=l/ 1,k=I0I1, k'=I/3I2,k+k'= n, and R,1 is a polynomial in (, X;,, Y,,, I v I< n. Then the functions O(Za)k(H7)(0,() can be expressed as follows in these coordinates d

Y1..+k-,(' + Rk,

6(Za)kH.,(0, () = Nal (0, () 1=1

0 < k < c,

C = (SI,...,Cd)

and Rk is a polynomial in (, X;,, I v IS n + k - 1, Y,,' with I v IS n + k - 1, v I,< n + k - 1. Hence ( is a submersion.

Corollary 16. cod(.F','(N); P(d, N)2) > d + 1,

cod(F(N,a,c,a);P(d,N)2) > c+1 -d.

Lemma 47. cod(B(N);JNTM XM JNTM) > min(d + 1, 2d - 2, c + 1 - d) = min(d + 1,2d - 2) (since we have chosen c = 2d in the definition of B(N)).

222

8 Generic Properties of Singular Trajectories

8.5.4 End of the Proof of Theorem 25 For d > 3, cod(B(N); JNTM xM JNTM) > d+ 1. Hence B(N) is a partially semi-algebraic closed subbundle of the vector bundle JNTM x JNTM of codimension > d + 1. The theorem in 51 shows that the set of all (Fo, Fl) E N Fl) VF2(M) such that UN Fo, jy B(N) for all x E M is open dense. This ends the proof of Theorem 25.

8.6 Genericity of Codimension One Singularity Consider the control system i(t) = Fo(x(t) + u(t)FI(x(t)). Let (x,u) be a singular trajectory defined on [0, T] and associated to a control u(.) E L°° ([0, T] ). We note k(T) the codimension of the image of L°° ([0, TI) by the

Frechet derivative of the end-point mapping evaluated on u. A very important question to investigate the singularities of the end-point mapping is the

codimension of the singularities for generic pairs (FO, FI). We shall prove that generically we encounter only singularities of codimension one. More precisely we have the following.

Theorem 26. There exists an open dense subset G1 in G such that for any couple (Fo, Fl) in GI if z; : J -p (T* M)o, i = 1, 2 are two singular extremals then of system (8.1) associated to (Fo,FI) and if LIT. M(zl) = there exists A E R' such that z2 = az1. Remark 22. The existence of singularities of codimension greater than one means that the system on IPT`M = T*M\R*

z(t) =Ho (z(t)) + u(t) HI (z(t))

constrained to HI (z) = 0 and observed with the observation mapping

zE1PT`MF.-xEM is not observable.

8.7 Proof of Theorem 26 We proceed as in the proof of Theorem 25. First we define a bad set and secondly we compute its codimension.

8.7 Proof of Theorem 26

223

8.7.1 The "Bad" Set for Theorem 26 Let (Fo, F1) E VF(M)2. We shall use the following notations. Hi : T'M -+ R, i = 1,2 is the function Hi(z) = z). If a E {0,1}", H , : T`M -+ R is defined inductively by Ha = The set $loll is the

open subset of all z E T'M such that Holl(z) 54 0. Let Z be the field Ho +y" Hi on Doll Definition 89. (i) For any integer q > 0, any integer N > q + 2, let Br(N, q) be the subset of JNTM X M JNTM x M PT* M x M 1PT * M of all quadruples UP '& if F1, [z1), [z2]) such that: 1. [z1] 36 [z2],

2. HolI (zip 0 0, i = 1, 2,

3. 9(Z)1( )(zl) = 9(Z)k( )(z2), 0 <_ k < q, i = 1, 2, 4. Fo(x), F1 tx) are linearly independent. (ii) BC(N, q) will denote the canonical projection of B,(N, q) onto JNTM x m

JNTM.

Lemma 48 (Fundamental lemma). Let (Fo,FI) E VF(M)2 be a couple such that for any x E M, (j.Fo, j.F1) f B(N)UB,,(N, q). Then every singular extremal of (Fo, F1) is of minimal order and there does not exist any two singular extremals (Ti, ui) : J -+ T* M x R such that 17T M (71) = HT' M ('12) and [71] # [72].

Proof. The first part is just a restatement of Lemma 48. As for the second part let ('zi, u;) : J - T' M x R, i = 1, 2 be two singular extremals of (Fo, F1) such that and [z1] # [272].By the first statement both (zi,'ui), i = 1, 2 are of minimal order. The set {t; [zi(t)] # [z2(t)]} is open and non empty. Since the sets R(zi, iii) are both open and dense (see definition 81) there exists an open non empty subinterval J' of J such that:

1. [zi(t)] 0 [z2(t)] fro all t E J', 2. Hol1(zi(t)) 0, i = 1,2 for all t E Y. The closed subset

{t E J'; dim Span{Fa(i(t)), Fl(i(t))} < 1, T= has an empty interior: otherwise on an open non empty subinterval J" of J' we would have dim Span Fo(T(t)), F1(T(t))} < 1. Applying Lemma 43 to the restrictions of (Ti, ui), i = 1, 2 to J" we get that [zl (t)] = [z2(t)) for all t E J" this contradicts 1. above. Replacing J by an open subinterval we can assume that:

8 Generic Properties of Singular Trajectories

224

1. [71(t)] # [72(t)] for all t E J,

2. Hol1(7i(t))#0,i=1,2foralltEJ, 3. Fo(T(t)), Fi(T(t)) are linearly independent for all t E J.

It follows from Proposition 70 that ii = Z(ii) a.e., i = 1, 2, and ui(t) _ H

Pi (0) almost everywhere. Projecting on M we get

=(t) = Fo(T(t)) +Ui (t)Fj (T(t)) = Fo(r(t)) + 112(t)F1(Y(t))

for a.e. t E J. Since by 3. above F1(T(t)) # 0 for all t E J, ui(t) = u2(t) for a.e. t E J. This implies that Hloo Ho11

(zl)

_

H100

Holl (12 )

Differentiating this relation with respect to t we get that for all k E N (O(Z)k(H1oo)(I1) = (O(Z)k(H1oo)(92) Hol1 Holl

Hence for all t E J the quadruple (jN=) Fo,j(t) F1, [71(t)], [Z2 (t)]) belongs to Bi (N, q). A contradiction.

8.7.2 Evaluation of the Codimension of Be(N, q) It is clear that &(N, q) is a partially semi-algebraic subbundle of the VP bundle JNTM X M JNTM x M IPT' M x M IPT* M, N > q + 2. Its typical fiber q) in P(d, N)2 x P(Rd)2 is the set of all (Po, P1, (1) 16154 [6],

(ii) dim Span{Po(0), Pl (0)} = 2, (iii) Hol1(0, ti) # 0, i = 1, 2,

(iv) e(Z)k(H))(0,6) = e(Z)k(H))(M2), 0 < k < q where

Hi(x,t) = (Pi(x),t),

i = 1,2

Hioo(x,e) = ([[P1,Po],Po](x),e), Holi(x,.) _ ([[Po,Pi],P1](x),t), Z =Ho + WHff, Hi on the open subset .floe i in P(d, N)2 X ]Rd of all (Po, P1,

such that Doll(0, ) 0 0.

Lemma 49. (i) For every k _> 0, there exists a polynomial function tAk : P(d, N)2 x Rd R such that on 0011 k

Hioo))

0k(Po,P1 ,

(0, ) = Hol1(0, ) \e(Z) `Hol t / J

8.7 Proof of Theorem 26

225

(ii) #0 = H10o and #k, k > 1, has the form Wk = H101112 - (

H100

Hol l

)k+1 H101112 + y,k

where Ok is a polynomial in the Has such that I a J< k+3 or I a and I a lo> 1, 1 a Ii> 1. The proof is an easy induction on k

k+3

Corollary 17. ,'',(N, d) is the set of all quadruples (P0, P1, [s1], [C2]) such that

(1) [6] T [6], (ii) dim Span{Po(0), Pi (0)} = 2, (iii) Ho11(0, .i) 0 0, i = 1, 2, (iv) H011(0,52)#k(PO,PI,6) = Ho11(0,6)#k(Po,Pi,6), 0 < k < q.

in the open set 0 C P(d, N)2 X IP(Rd)2, 0 = Take any (PO, PI', [ iJ, (ii) dim Span{Po(0), P1(0)} = 2, (iii) {(Po, Pi, [6], [61); (i) [C1] 76 Ho11(0, i) # 0, i = 1, 2}. Complete Po(0), PI '(0) into a basis Po(0), Pl (0), e3, , e of Rd and define the mapping e : 0 -,

e(P,Q) =

(Rd)d,

(e1(P,Q),...,ed(P,Q)), e1(PQ)

= Po(0), e2(P,Q) = P1(0),

ei(P, Q) = e;, 3 < i < d. For a small neighborhood V of (PO, PI') in P(d, N)2, e1v is a basis valued and we can associate to e, a coordinate system X,, Y. We get for k > 1 d

ttYik+s+Rk {=l

where Rk is a polynomial in t, X', I v 1_< k + 2,

I v 1:5 k + 2 and v

1 k+2

Hence for k > 1, (iv) can be written d

d

Ho11(0,.2)

Y1ik+2 = Ho11(0, C1) F CiYik+9 + Rk i=1

where R' is not depending upon

i=1

Ylk+2.

Therefore we have

(HOl l (0, 2)tl - Hol l (0, tl )t2, Y1k+l + RA;' = 0

where Rk is not depending upon Ylk+9 For 1 < k < q, these q relations define on 0 n V a smooth submanifold of

codimension q, since 1 and t2 are not collinear. This shows that in 0 n V, q) is of codimension at least q. Since the V's for different choices of (Po, Pl) cover 0, , 'c(N, q) is at least of codimension q. Consequently its projection Fc(N, q) into P(d, N)2 is a semi-algebraic subset of codimension

>-q+2- 2d.

226

8 Generic Properties of Singular Trajectories

Lemma 50. (i) cod(2 (N, q); P(d, N)2) > q + 2 - 2d;

(ii) The set of all couples (Fo, Fl) E VF(M)2 such that (jNFo, jz Fl) B(N) U BC(N, 3d - 1) is open dense in VF(M)2. Lemma 49 and Lemma 50 prove Theorem 26.

8.8 Singularities of the Singular Flow of Minimal Order Consider the singular extremals of minimal order solutions of the equations ti

dz

(t) =Ho (z(t)) + u(z(t)) Hi (z(t)) dt

Hi(z) = Hio(z) = 0,

u(z) = Hoio(z)

Hioi (z)

and denote by S the singular set H1 = H10 = H101 = 0. In this section we shall briefly study the behaviors of the singular extremals of minimal order in the neighborhood of S. This analysis is motivated by two reasons. First of all, outside S the singular flow is smooth and numerous feedback

invariants are located near S. Secondly a broken singular extremal has in general to cross S, hence the analysis of the flow near S is connected to the important question about the existence of broken optimal singular extremals. We shall restrict our analysis to the case d = 3 and since our study is local we may consider a system in R3.

8.8.1 Preliminaries (see Sect. 4.1.3) We consider the system on 1R3

T (t) = Fo(v(t)) + u(t)Fi (v(t))

where v = (x, y, z). Let p = (px, py, pa) be the adjoint vector. A singular extremal z(t) = (v(t), p(t)) on [0, T] satisfies the following relations: (p, F, (v)) = (p, [F'o, Fi](v)) = (p, [[F1, Fo], Fo](v) + u[[Fi, Fo], Fi]](v)) = 0.

We introduce

- D1 = det(Fi, [Fi, Fo], [[Fl, FO], Fj])

- D2 = det(Fi,

[Fi, Fo], [[Fi, Fo], Fo])

8.8 Singularities of the Singular Flow of Minimal Order

227

and outside the singular set S : D1 = 0, a singular control is of minimal order and is given by the feedback 11(v) = - o, v, . The singular set S is feedback

invariant and splits into two sets C and S\C, where C is the set where F1 and [Fo, F1] are linearly dependent; it is clearly a feedback invariant. Outside the set S, the singular trajectories are solutions of

Z(v(t)), dt (t) =

Z(v) = Fo(v)

-

Di

(v) Fl (v)

The singularities of the flow near D1 = 0 can be analyzed as follows, [22]. We write the equation as

D1(v(t)) dt (t) = D1(v(t))Fo(v(t)) - D2(v(t))Fl (v(t))

and we make the following time reparametrization: ds = ALL.. Hence s t v(s) is solution of ds =

Z(v),

Z(v) = D1(v)Fo(v) - D2(v)F1(v)

where 2 is smooth. from [21] since (Fo, F1) ,-. Dl is feedback semi-covariant

then the map (Fo, F1) .- Z is a semi-covariant which encodes most of the feedback invariants. We introduce the time optimality to the problem by taking - D3 = det(Fl, [Fi, Fo}, Fo)

The set D3 = 0 is a feedback invariant and from Sect. 4.1.3, the exceptional trajectories H(v, p, u) = (p, Fo(v) + uF1(v)) = 0 are contained in R3\D1 =

0. They foliate the set D3 = 0 outside S. The hyperbolic trajectories are contained in D1 D3 > 0 and the elliptic trajectories in D1 D3 < 0. The singular control is solution of D2+uD1 = 0 and the sets D2 = D1 = 0 is a feedback invariant. It corresponds to the equilibrium points of the smooth vector field Z in D1 = 0.

8.8.2 Local Classification near C The set C is the set where F1 and [Fo, F1 [ are collinear. At a generic point vo of C the behavior of the solutions of the vector field Z = D1Fo - D2F1 is the following.

1. Through vo there passes a line of singularities which form the center manifold of vo.

2. The transverse behaviors is a node with two equal eigenvalues and the associated stable or unstable manifold has RF0 ® RF1 as tangent space.

228

8 Generic Properties of Singular T ajectories

We deduce the phase portrait of the vector field Z outside S by reversing the orientation of the trajectories of Z in D1 < 0. Moreover at a point of C, F1 and IF,, FoJ are collinear and the set of adjoint vectors in IPR3 such that (p, F1) = (p, [F1, Fo]) = 0 is two dimensional. An instant of refiexion shows that a point vo of C a singular trajectory cross smoothly the set C and with a bounded control, see Fig. 8.1.

Fig. 8.1.

8.8.3 Local Classification near S\C We shall describe the behaviors of the singular trajectories near a generic point vo E S\C, for the details see [901. The main properties are the following:

1. Through vo there pass a line L of singularities. 2. The divergence of Z at vo is zero.

We denote by (a, -a, 0) the spectrum of case. We have two cases:

JV_

(vo), where a

74

0 in the generic

Case (i). a E R and L is a smooth center manifold at vo. The transverse linear behavior corresponds to a saddle point.

Case (ii). a = ±ia E iR and the transverse linear behavior to L is a center. In both cases the singularity is not isolated and we are not in the hyperbolic situation and we cannot apply Hartmann-Grobman theorem. There exists singular trajectories crossing S with a bounded control is case (i) and they corresponds to the stable-unstable manifold of each singularity vo, see Fig. 8.2. The broken arc S1U1 is a singular arc contained in the hyperbolic

8.8 Singularities of the Singular Flow of Minimal Order

229

Fig. 8.2.

domain. It is a candidate in the hyperbolic domain. It is a candidate as a broken singular minimizer. Nevertheless it is proved in [90] the following. Lemma 51. Under generic assumption. The arc S1 U1 is not time minimal for the Ll-topology. Remark 23. To prove this assertion it is crucial that the set of controls is not uniformly bounded. For further discussion see [90J.

8.8.4 The Quadratic Case In the previous discussion F1 is taken transverse to the surface D1 = 0. In this section we shall consider the case where F0 = Q an homogeneous quadratic vector field and F1 = b is a constant vector field. An easy computation gives us that F1 is tangent to D1 = 0. This situation is important for the applications because it is the situation encountered in the controlled Euler equation. The singular trajectories are classified in [21] near the set D1 in the codimension one and two cases. The analysis goes as follows.

Notations. The map v - [Q, b](v) is linear and is noted v " Av. We denote by A the classical co-adjoint of A, i.e. if A-1 exists we have A = (det A)A-1. We set w = Ab, L1 = Rb, L2 = Rw. Generically w and b are linearly independent. In this case D1 = 0 is a plane generated by b and w. The restriction of D2 to D1 = 0 is a cubic homogeneous polynomial in two variables. The R-solutions of D1 = D2 = 0 are the line Rw and eventually to lines denoted L3 and L4. We shall describe the generic classification where we assume that the lines Li, i = 1, 2, 3, 4 are distinct. It is founded on the following remark: in the quadratic case the vector field Z = D1Q - D2b is

a cubic homogeneous vector field on R3. It can be studied by projecting on the sphere S2. More precisely we set r = HJvll, v = * and the equation i,(t) = Z(v(t)) takes the form

230

8 Generic Properties of Singular Trajectories

Wi(t) _ (v(t), Z(v(t)))r

d (t) = Z(v(t)) - (v(t), Z(v(t)))v(t) where a is given by r2dt = da. The second equation defines the projection of Z on the sphere.

A ray is a solution of v(t) = Z(v(t)) contained in a straight-line and the projected equation describes the transverse behaviors with respect to ray solutions (which project onto singular points on S2). The ray solutions in the set D1 = 0 are precisely the line Li, i = 1, 2, 3, 4 and the behaviors of the singular trajectories near D1 = 0 is described by the following proposition.

Proposition 73. Assume the line Li all distinct. The only points, where singular trajectories can cross the plane D1 = 0 at finite distances are on the lines L2, L3, L4 (if L3, L4 exists). This is done transversally with continuous controls and in the following manner.

1. Let v # 0 on the line L2. Then there exists near v an analytic manifold V (v) of dimension 2 transverse to L2 and invariant for the singular trajectories. If X (v) point towards Dl > 0 (resp. Dl < 0) each singular trajectories in V (v) n {D1 < 0} (resp. Dl > 0) cross the plane Dl = 0 at v, the singular controls and the adjoint vectors being all distinct.

2. Let v 0 0, on the lines L3, L4. Then there exists exactly one singular trajectory crossing Dl = 0 at v. 3. We have the following blowing-up phenomenon. If Q(b) points towards D1 > 0 (resp. Dl < 0) there exists a conic neigborhood U of Ll such that each singular trajectory with initial condition in U and Dl < 0 (resp. D1 > 0) hits in finite time and with an infinite control the plane Dl = 0 at infinite distance and on a line parallel to L1. We represents on Fig. 8.3 the corresponding singularities of the projection of Z on the sphere. The classification is given with all the details in [22]. It is more precise than the one in [90] where the phase portrait is not known. The situation of codimension one where the lines Li collide is described in [21].

Notes and Sources The first result of genericity in system theory is due to Lobry [81] which shows

that the ad-condition is generic. All the results of genericity concerning the single-input affine systems of this chapter are coming from [26]. The same results are conjectured to be true in the multi-inputs case but the proof is not straightforward. The classification of the singularities of the singular flow has been initialized in [22] and the global classification concerning the singular flow in Euler equations is given in [21]. It can be applied to the attitude control problem.

8.8 Singularities of the Singular Flow of Minimal Order

231

W

E

L1: node

L2 : node

L3L4 :saddle

Behaviors of the projected system near D TO

Fig. 8.3.

Exercises 8.1 Consider the multi-input control system on M: de (t) = Fo(x(t)) +

E`

1

u;(t)F; (x(t)) where m > 2. A singular extremal (z, u) defined on [0, T]

is said to satisfy Goh condition if for each F, G E Span{Fl, , Fm}, y H {HF, HG) (z) = 0 where Hp, He are the Hamiltonian lifts. Show that if -

m > 3 there exists an open dense set of (m + 1)-tuple {Fo, there exists no singular extremals satisfying Goh condition.

,

Fn} such that

8.2 Let el, e2, e3 be R3 canonical basis, Q = (Q1, Q2, Q3) be a quadratic vector field, where Q1 = alx2+a2y2+a3z2+aaxy+a5xz+asyz and Q2, Q3 are respectively defined by changing a2 into b; and c; and let b the constant vector field normalized to b = e3. Let v1 = b, v2 = [(Q, VI 1, VI 1, v3 = [[Q, v1], v2] and assume that the constant vector field VI, v2, v3 are linearly independent.

1. Show that there exists a linear change of coordinates and a feedback u = a(x) + /3v, a(x) quadratic, /3 nonzero constant such that (Q, b) can be taken generically with the following normalization

Q3=0,b=e3,a3=a5=b5=0,b3= ,b6=0,b4 = 1 2. Show that in the above representation we have

(i) D1 =y

8 Generic Properties of Singular Trajectories

232

(ii) D2 = -y3+ 2 -2b1xy2-b1xz2+a4x2z+(a4-3b2)y2z+2(al-1)xyz (iii) D3 = -b2y3 - blx2y - xy2 + alx2z + a2y2z + yz2 + a4xyz 3. Show that the behaviors near L1, L2, L3, L4 of the2projected system are given by (i) near L2 = IRe1: node with two eigenvalues equal to b1 (ii) near L3, L4, L3 = -a4 + f, L4 = -a4 6 = a4 + 2b1 > 0: saddle with two eigenvalues (A, -A) where A 1=1 2b1 - a4L L = L3 or L4. (iii) near L1 = lRb: node with two eigenvalues 1 and Z.

-

8.3 Let M be a a-compact C°°-manifold. Prove that there exists an open dense set of pairs (F1, F2) of COO-vector fields such that for all q E M we have:

Span{adkFiF2i k = 0,

,

+oo} = TqM

(8.2)

8.4 Let M be a a-compact C°° connected manifold of dimension 5 or 6.

F3

Consider the following control system on M: q(t) _ 1 u;(t)F1(q(t)) where F1, F2, F3 are smooth vector fields. Prove that there exists an open dense set of triples (F1, F2, F3) such that every two pairs of points can be joined by a piecewise singular and C°°-curve.

9 Singular Trajectories in Sub-Riemannian Geometry

9.1 Introduction The problem posed by the existence of singular trajectories in the Lagrange problem of the calculus of variations, which can be formulated as the optimal control problem T

min U() 10

subject to

d (t) = f (x(t), u(t)) and the end-points conditions: x(0) E MO, x(T) E M1, was known since a long time, see for instance the discussion in Bliss [18]. It has in some extend slopped the post-war research in this area. It was rediscovered by R. Montgomery in the context of sub-Riemannian geometry (SR-geometry) which has attracted very recently a lot of researchers.

The aim of this chapter is to explain precisely the role of singular trajectories in SR geometry. In our opinion it must be understood as an example of the more general time minimal control problem. In this example the

singular trajectories are exceptional and have the C°-rigidity property analyzed in Chap. 7, if we consider the SR-problem with rank 2-distribution. In SR-geometry the value function is called the SR distance. The key contribution of our work is to make a precise analysis concerning the behaviors

of extremals and optimal trajectories near the singular trajectories. A consequence will be a precise description of the level sets of the distance

function in the abnormal directions showing in particular that it is not sub-analytic. This phenomenon was already pointed out by Lojasiewicz and Sussmann in their article [82) but it covers a lot of different phenomena. Here we give a precise analysis in SR-geometry. Also we show the connection

with the computation of Poincare-Dulac mapping in the problem of limit cycles for planar differential equations. This will provide a connection with recent branches of real analytic geometry concerning in particular the exp-log category and pfaffian mappings.

234

9 Singular Trajectories in Sub-Riemannian Geometry

9.2 Generalities About SR-Geometry (In all this section we are working in the C'-category)

9.2.1 Definition A SR-manifold is defined as a n-dimensional manifold together with a distribution D of constant rank m < n and a Riemannian metric g on D. An q(t), 0 < t < T is an absolutely continuous curve such admissible curve t that q(t) E D(q(t))\{0} for almost every t. The length of q(.) is e(9) =

f

'

T

(9(t), 9(t)) dt

where (,) is the scalar product defined by g on D. The SR-distance between p, q E M denoted dSR(p, q) is the infimum of the lengths of the admissible curves joining p to q.

9.2.2 Optimal Control Theory Formulation The problem can be locally formulated as the following optimal control problem. Let qo E M and choose a coordinate system (U, q) centered at qo such that there exist m (smooth) vector fields {F1,

,

Fm} on U which form

an orthonormal basis of D. q(t) on U is solution of the control

Hence each admissible curve t system TY&

4(t) = Eui(t)Fi(q(t))

(9.1)

i=1

and the length of q(.) is given by fT

e(q) _

(n uq (t)) i dt.

(9.2)

io 1

The length of a curve is not depending on its parametrization. Hence every

admissible curve t '- q(t) with finite length can be reparametrized into a lipschitzian curve s - q(s) parametrized by arc-length: (4(s), 4(s)) = 1

a.e.

If an admissible curve q : [0, T] - U is parametrized by arc-length we have almost everywhere m

m

9(t) = Eui(t)Fi(q(t)), Eu?(t) = 1 i=1

i=1

9.2 Generalities About SR-Geometry

235

and

e(q)=

f

(u(t))dt = T i=1

and the length minimization problem is equivalent to a time minimization problem for the symmetric system (9.1) where the control domain is defined by: Ein 1 u? (t) = 1. This problem is not convex and it is worth to observe that the time optimal control problem with the constraint Ei=1 u? = 1 is equivalent to the time optimal control problem with the u? < 1. Indeed if q(.) is an admissible curve such that the constraint associated control satisfied E' 1 u, < 1 at a Lebesgue time then it can be reparametrized into a curve parametrized by arc-length and with shortest EL`_`

1

length. We can resume this into a proposition.

Proposition 74. Let (U, q) be a chart on which D is generated by an orthonormal basis {F1, , Fm} then the SR-problem (U, D, g) is equivalent to the time optimal control problem for the system m

ui(t)F2(q(t))

4(t) i=1

and the control domain Ei i 1 u? < 1. This result is useful to apply Filippov existence theorem. To save computations when computing optimal trajectories we use the following result whose proof is based on Cauchy-Schwartz inequality and Maupertuis principle.

Proposition 75. Assume that the admissible curve are defined on the same interval [0, T] (e.g. T = 1). Then the length minimization problem is equivalent to the energy minimization problem where the energy of a curve is defined T

by E(q) = f (4(t), 4(t)) dt.

9.2.3 Computations of the Extremals and Exponential mapping Consider the symmetric system on U C 1Rn

4(t) _

i.1

ui(t)Fi(q(t)) = F(q(t))u(t)

where rank{ F, (q), , Fn(q)} = Tn for every q E U. Let T > 0 be fixed and consider the energy minimization problem

Tm

u?(t )dt.

min E(q), E(q) = j i=1

236

9 Singular Trajectories in Sub-Riemannian Geometry

The Hamiltonian associated to the problem is m

H(q, p, u) _

ui (P, Fi(q)) - Po

u?

where po is a constant > 0 which can be normalized as po = 0 or po = 1 The equations of the maximum principle are the following:

4(t) =

OH OP

OH

(q(t),P(t), u(t)), P = - 8q (q(t),P(t), u(t)), (9.3)

8 (q(t),P(t),u(t)) = 0. We shall adopt the terminology used in SR-geometry.

Definition 90. A solution (q, p, u) of the maximum principle is an abnormal (resp. normal) extremal if po = 0 (resp. po = The projection of an extremal in the state space is called a geodesic. A (normal or abnormal) geodesic is called strict if it is the projection of an unique extremal (q, (p, po), u) where 2).

(P,Po) E 1Pn+1.

Abnormal extremals. They correspond to po = 0 and are in fact the singular extremals, solutions of the maximum principle with m

HO(q,P,u) _

ui(P,Fi(q)) =

Hl,o.

They satisfy the constraints (P(t), Fi(q(t))) = 0, i = 1, ... m.

In particular they are exceptional. The singular extremals of minimal order can be computed with the algorithm of Chap. 4. They are smooth curves and their projections are smooth geodesics.

Normal extremals. The normal extremals are the solutions of the maximum principle corresponding to po = z . They are solutions of the following equations

4(t) _

(q(t), p(t), u(t)),

j(t) _ -

a

4 (q(t), p(t), u(t))

(9.4)

8a 1 (q(t),P(t), u(t)) = 0

(9.5)

9.2 Generalities About SR-Geometry

where H j. (q, p, u) _ m 1 ui (p, F1(q)) - 2 m t

237

.

The previous constrained Hamiltonian equations can be easily solved. We

introduce Pi = (p, F; (q)), i = 1,

Solving the linear equation (9.5)

, m.

the control is given by

ui(q,p)=P,, i=1,.m and plugging fi = (fit,

,u,n) into Hi we define the Hamilton function m

Hn (q, p) = Hj (q, p, u.) = 2

P?

(9.6)

,.=1

and the normal extremals are solutions of the Hamiltonian differential equation H 8H (9.7) q(t) = pn(q(t),p(t)), p(t) = --.gn(q(t),p(t))

They are smooth curves. If we use the parametrization by arc-length: Em 1 u, = 1 then the curves are contained on the level set: H,, = 2

Poincard equations. To compute easily the normal extremals it is convenient to use the follwing coordinate system on T'U. On U we can complete , F,,} of TU. the m-vector fields F1, , Fm to form a smooth basis {F1, The SR-metric g on U can be extended into a Riemannian metric on U by tak,n ing the F= as orthonormal vector fields. We set Pi = (p, Fj (q)) for i = 1, and let P = ( P i ,--- , Pn). In the coordinates system on T*U : (q, P) (in gen-

eral not symplectic for the canonical structure) the normal extremals are solutions of the equations m

m

i=1

i=1

Ed]

{Pi, H2 } _ E{Pi, Pj}P?. i=1

We observe that { Pi, P, } = (p, [F, F.jI (q))

and since the Fj are a basis of TU we can write n

c (q)Fk(q)

[Fi, F,[(q) _ k=1

where the ck.7 are smooth functions.

They are several choices to complete the distribution Die = Span {F1,

,

F,n} and some are canonical. This will be discussed later.

238

9 Singular Trajectories in Sub-Riemannian Geometry

Exponential mapping. Consider arc-length parametrized curves. We fix qo E U and let q(., qo, po), p(., qo, po) be the solution of (9.7) starting at t = 0

from (qo, po). It is contained in the level set H = 2. The exponential mapping is the map (9.10)

expgo : (po, t) '-' q(t, qo, po).

Observe that its domain is the set C x R where C : F,i I P? = 1 with q = qo. The important remark is the following. In the Riemannian case m = n and C is a sphere but in the SR-case with m < n, C is a cylinder and in particular

is non compact, see Fig. 9.1. As in the Riemannian case the exponential

cylinder C

C=fin

m=n (Riemannian case)

m
Fig. 9.1.

mapping can be defined by taking t = 1 and relaxing the constraint H = .1

Conjugate and cut loci - Sphere and wave front. - A conjugate point along a normal extremal is defined as follows. Let (po,ti), ti > 0 be a point where the exponential mapping exp, is not an immersion. Then ti is called a conjugate time along the geodesic and the image is called a conjugate point. The conjugate locus C(qo) is the set of first conjugate points when we consider all the normal geodesics starting from qo. If (q(.), p(.)) is a normal or abnormal extremal with q(0) = qo, the point where q(.) ceases to be minimizing is called the cut point and the set of such points when we consider all the extremals with q(0) = qo will form the cut locus L(qo).

- The sub-Riemannian sphere with radius r > 0 is the set S(qo, r) of points which are at SR-distance r from qo. The wave front of length r is the set W(qo, r) of end-points of geodesics with length r starting from qo.

9.3 Research Program in SR-Geometry

239

9.3 Research Program in SR-Geometry 9.3.1 Classification Given a local SR-geometry (U, D, g) which can be represented as the optimal control problem m

q(t) _

ui(t)Fi(q(t)), min icl

u0 Jo

u?(t)dt

/

i=1

there exists a pseudo-group of transfomations called the gauge group G which is a sub-group of the feedback group defined by the following tranformations: (i) local diffeomorphisms p : q p--' Q of the state space U, preserving qo; (ii) feedback transformations u = (3(q)v preserving the metric g i.e., /3(q) E O(m, R) (orthogonal group). The classification of local SR-geometries is

- Find a complete set of invariants. - Compute normal forms. It is worth to observe that a preliminary step in this classification is the

classification of distributions. In particular the singular trajectories are gauge-invariants. Other invariants are found using the normal extremals.

9.3.2 Singularity Theory of the Exponential Mapping and of the Distance Function In the Riemannian case the local situation is rather simple: there exists a neighborhood V of qo on which each geodesic starting from qo is globally minimizing. In particular the sphere of small radius is smooth. In the SRgeometry with m < n the local situation is intrincated: - the cut locus C(qo) accumulates at qo; - the sphere of small radius have singularities. Hence the remaining of this chapter will be devoted to analyze this singularity problem. We shall restrict our study to the problems where U C ii and D is a rank 2 distribution. More precisely we shall consider in details two situations.

Contact situation: D is a contact distribution. It is the situation for a generic point qo E R3 where the system has no singular trajectory near qo (see Chap. 4). Martinet situation: D is a Martinet type distribution and through qo passes a singular trajectory.

240

9 Singular Trajectories in Sub-Riemannian Geometry

The main difference between both cases is the following. In the contact situation the local situation is intrincated but can be handle with the tools of standard singularity theory; the generic situation can be analyzed and the models of the singularities are polynomial mappings. In the Martinet case this is no longer true and we need much more transcendence because the singularities are not sub-analytic. This phenomenon is due to the existence of abnormal (or singular) extremals.

9.4 Privileged Coordinates and Graded Normal Forms Lemma 52 (Chow theorem). Let (M, D, G) be a SR-structure. Assume that M is connected and D satisfies the rank condition, that is DAL(q) is of dimension n for each q E M. Then for each qo, q1 there exists an admissible curve associated to a piecewise constant control and joining qo to q1. Corollary 18. If (M, D, g) is a SR-structure satisfying the previous assumption one can define a distance dSR on M by setting: dsR(go, q1) = min{e(q(.)) where q is an admissible curve joining qo to q1 }.

Proposition 76 (Filippov theorem). Let (M, D, g) be a SR-structure and assume that D satisfies the rank condition. Then sufficiently near points can be joined by a minimizing geodesic. Hence for r > 0 small enough, S(qo, r) C W (qo, r)

Proof. Locally the problem can be written as a time minimal control problem

for a symmetric system, and the control satisfies the convex constraint: z" 1 u? < 1. Hence condition (iii) of Theorem 10 is satisfied. Since the system satisfies the rank condition, it is controllable and condition (i) is satisfied. The uniform bound condition (ii): I q(t) 1< b, 0 < t < T is satisfied for T small enough and the results follows by a local application of Filipov theorem.

9.4.1 Regular and Singular Points Let (U, q) be a coordinate system centered at qo. Let D = Span{F1 i , Fm} and assume that D satisfies the rank condition on U. We defined reccursively:

D1 = D and for p > 2

D' = Span {D" ' + (D', DP-1] }. Hence DP is generated by Lie brackets of vector fields IF,, , Fn} with length < p. At q E U we have an increasing sequence of vector subspaces {0} = D°(q) C D'(q) C ... C D*(9)(q) = TqM where r(q) is the smallest integer such that Dr(4) (q) = TQM.

9.4 Privileged Coordinates and Graded Normal Forms

241

Definition 91. We say that qo is a regular point if the integer np(q) = dim DP(q) remains constant for q in some neighborhood of qo. Otherwise we say that qo is a singular point. Example 6.

- Contact case: D = ker a, a = ydx + dz; D is generated by F1 = 0

and F2 = a -yes We have [F1, F2] = e and all the Lie brackets of .

length > 3 are 0. Hence DAL is a nilpotent algebra and it is isomorphic to the Heisenberg Lie algebra. The point 0 is a regular point and we have

n1(q)=2,n2(q)=3forallgER3. - Martinet case: D = ker w, w = dz - 2dx; D is generated by Fl = and F2 = 3x + 2 e. The point 0 is a singular point and we have n'(0) _ n2(0) = 2, n3(0) = 3.

Remark 24. At a regular point the sequence np(qo) is strictly increasing.

9.4.2 Adapted and Privileged Coordinates Definition 92. Assume that D satisfies the rank condition at qo where qo is a fixed (regular or singular point). Consider a system of coordinates (U, q) centered at qo such that dql, , dqn form a basis of Tqo M adapted to the flag {0} = D°(qo) C ... C D''(9o)(go) = TgoM.

The weight wj of the coordinate q, is the integer such that dq3 vanishes on Dw-, -1(qo) and does not vanishes identically on Dw-, (go).

Definition 93. Let (U, q) be a chart centered at qo and (D, g) be the SRstructure represented locally by the orthonormal vector fields {F1 i , Fm}. If f is a germ, of smooth function at qo, the order µ(f) of f at qo is: (i) if f (qo) 0 0, p(f) = 0, t(0) = +00(ii) µ(f) = inf p such that there exists v1 i

...oLva(f)(go)00.

, vp E {F1,

,

F.} with L,,, o

The germ f is called privileged if µ(f) = min{p; dfp,, (DP(qo)) # 0}. A coordinate system (qj, , q,) : U IR is said to be a privileged if all the coordinates q; are privileged at qo. We have the following result (See [16]).

Proposition 77. There exists a privileged coordinates system q at every point qo of M. If w; is the order (=weight) of the coordinate qi we have the following estimate for the SR-distance: dSR(0,(gl,...,gn))'

1 glIW

+...+gnI

9 Singular Trajectories in Sub-Riemannian Geometry

242

9.4.3 Nilpotent Approximation Definition 94. Let (U, q) be a privileged coordinate system for the SR structure given locally by the m orthonormal vector fields {Fl, , Fm }. If wj is

the weight of qj, the weight of -4 is taken by convention as -wi. Every vector field F. can be expanded into a Taylor series using the previous gradation and we denote by Ft the homogeneous term with lowest order -1. The , polysystem {Fl, is called the principal part of the SR-structure. We have the following, see [16].

Proposition 78. The vector fields Pi, i = 1,

, m generate a nilpotent Lie algebra which satisfies the rank condition. This Lie algebra is independent of the privileged system.

9.4.4 Graded Approximation Definition 95. Let (M, D, g) be a SR-structure represented locally in an adapted coordinate system (U, q) by m-orthonormal vector fields IF,,

,

Fm}.

The graded approximation of order p > -1 is the polysystem {FP, . , F, } obtained by truncating the vector fields Fi at order p using the weight system defined by the adapted coordinate system.

We shall now discuss the contact and the Martinet case using graded normal forms.

9.5 The Contact Case of Order -1 or the Heisenberg-Brockett Example 9.5.1 The Contact Case in 1R3 We consider

9(t) = ui(t)Fi (q(t)) + u2(t)F2(q(t)) where

Fl=a +y8z, F2=a T -xa

(9.11)

.

If we set F3 = , we get [Fl, F2] = F3. Moreover all the Lie brackets of lengths > 3 are zero. Hence DAL is a nilpotent Lie algebra of dimension 3 and is isomorphic to the Heisenberg Lie algebra h3. We observe that (x, y, z) is a coordinate system adapted to DAL(0) and the weight of x or y is one and the weight of z is 2. The local SR-contact case in R3 is: (U, D, g) where U is an open set containing 0, q = (x, y, z) are the coordinates, D is a contact distribution and g is a smooth SR-metric on

9.5 The Contact Case of Order -1 or the Heisenberg-Brockett Example

243

D. The contact distribution can be written in a smooth coordinate system as D = ker a, where a is the one form

a = dz + (xdy - ydx) and the metric on ker a can be written as g = a(q)dx2 + 2b(q)dxdy + c(q)dy2 where a, b, c are smooth functions. By making a change of coordinates Q = cp(q), preserving D = ker a, with ep = (tp1) cp2i
the following normalization: a(O) = c(O) = 1,

b(O) = 0.

If g = dx2+dy2 it corresponds to the contact case of order -1 or Heisenberg SR-geometry and the associated SR-geometry (U, D, g) is the Heisenberg-

Brockett example. It corresponds to a left invariant SR-geometry.

9.5.2 Symmetry Group in the Heisenberg Case Let S be the transformation group defined by the following affine diffeomorphisms X 911 912 0 x (-11) a Y = 921 022 0 y + a 13 1 z Z y

where 0= (011012 1 EO(2),a,/3,yER. \ 021 022 J

Then S is a symmetry group for the SR-Heisenberg geometry. Proof. We check easily that the previous transformations preserve the distribution. They clearly preserve the metric g = dx2+dy2, which is the Euclidean metric in the plane (x, y).

Corollary 19. - For every a E R3 the sphere S(a, r) is isometric to the sphere S(0, r). - The sphere S(0, r) is a surface of revolution with respect to the axis Oz.

9.5.3 Heisenberg SR-Geometry and the Dido Problem We observe that the problem can be written

i(t) = U1 (t) y(t) = u2(t) 1(t) = i(t)y(t) - y(t)x(t), min

Hence we have:

J0

T (i2(t)

+ y2(t))5dt.

9 Singular Trajectories in Sub-Riemannian Geometry

244

1. The length of a curve t '- q(t) = (x(t), y(t), z(t)) is the length of its projection in the plane (x, y). 2. Consider the problem of joining qo = (xo, yo, zo) to qi = (xl, yl, z1). We can write rT zi - zo = f (T(t), x(t))dt

J0

where x = (x, y) subject to rT

,

X(t) = g(x(t), u(t)), min / L(x(t), u(t))dt. Hence it corresponds to an isoperimetric problem in the calculus of variations, see [13]. Moreover in our case it has a precise geometric interpretation. Indeed zi - zo

=1

T

(x(t)y(t) -(t)x(t))dt

and the integral is propostional to the area swept by the curve t -+ (x(t),y(t)) in the plane (x, y). Hence our problem is dual to the Dido problem: among the closed curves in the plane whose length is fixed find those where the area enclosed by the curve is maximal. The solutions of such problems are well known and are circles. 9.5.4 Geodesics

We can compute the geodesics. Since D is a contact distribution there exists no singular trajectory. The normal extremals are computed using the Poincare coordinates with p = (px, py, p,,): P1 = (p,F1) = Px + yPz,

A = (P, F2) = Py - XPz, P3 = (p, F3) = P.-

and we get the following lemma.

Lemma 53. In the coordinates (q, P) the normal extremals are solutions of the following equations

NO = P2(t) i(t) = Ply - P2x P3(t) = 0 Pi(t) = 2P3P2(t) P2(t) = -2P3P1(t)

s(t) = Pi(t)

Integration. By setting P3(t) = pz(t) = i we get the equation of the linear pendulum: P1+.12P1 = 0. The equations are integrable by quadratures using

trigonometric functions. The integration is straightforward if we observe that

z- 2 and we get the following parametrization.

-dt(x2+y2)=0

9.5 The Contact Case of Order -1 or the Heisenberg-Brockett Example

245

Proposition 79. Let qo = (x(0), y(0), z(0)) = 0. Then the geodesics are as follows:

1. \ = 0: x(t) = At cos cp, y(t) = At sin cp, z(t) = 0 and their projections in the plane (x, y) are straight lines.

0: x(t) =

2. A

z(t) = ct -

[sin(.1t + cp) - since], y(t) _ A AT sink

with A =

cp) - coscp],

Pl + P2 and cp is the angle of the

vector (z, -0-

9.5.5 Conjugate Points The computation of conjugate points using the previous parametrization is straightforward. We have the following: - The geodesics whose projections in the plane (x, y) are lines and are without conjugate points. - The geodesics whose projections are circles have two series of conjugate points and one is given by tc = 2a* , k = 1, 2, . - and the first conjugate time is tlc = T. The first conjugate point is at the first intersection of the geodesic with the axis Oz. This imply the following.

Proposition 80. The conjugate locus C(0) contains the axis Oz minus the origin.

A strict geodesic cannot be minimizing beyond its first conjugate point. Also an easy computation shows the following:

- the geodesics whose projections are lines are globally minimizing (this is obvious because they are global minimizers for the Euclidean metric on (x, y));

- for a geodesic whose projection is a circle, the cut point is also the first conjugate point. At such a point, due to the symmetry of revolution around the axis Oz, there exists a one parameter family of minimizing geodesics ending at the cut point axis Oz minus the origin. Hence we get the following.

Proposition 81. The cut locus L(0) is the conjugate locus C(0).

9.5.6 Sphere and Wave Front By homogeneity we can take the radius as r = 1. Using the symmetry group S it is sufficient to represent their intersection with the plane y = 0. Moreover the SR-Heisenberg structure is symmetric with respect to the transformation

S'

:

(x, y, z) c--, (-x, -y, -z). Hence we can assume z > 0. To compute

S(0,1) and W (O, 1) we use the parametrization of proposition 79. The sphere

is obtained for A # 0 by imposing t < tl,.

246

9 Singular Trajectories in Sub-Riemannian Geometry

- Sphere S(O, 1). Its intersection with the plane y = 0 is represented on Fig. 9.2 in the domain z > 0. It is smooth except at the intersection with the axis Oz where it admits an algebraic singularity.

- Wave front W(0,1). Its intersection with the plane y = 0 is represented on Fig. 9.3 in the domain z _> 0. The wave front has two types of analytic singularities which are the singularities corresponding to conjugate points.

They are denoted M N,. The two series of both sequences M, and Ni accumulate at 0. 0.4

0.31

Z a2

0.1j

-1

-0.6-0.6 04-0.2

0.2

0.6

0.4

0.8

x -01

Fig. 9.2.

9.5.7 Conclusion About the SR-Heisenberg Geometry This example exhibits one of the main difference between the Riemannian geometry and the SR-Riemannian geometry with rank D = m < dim M.

When A - oo, the conjugate points tend to 0 and the small SR-sphere has singularities. This is due to the noncompactness of the domain of the exponential mapping. Hence one of the main research program is to investigate those singularities. In the Heisenberg case the singularities are analytic for the sphere and for the wave front we have an accumulation of analytic singularities. Clearly we have the following.

9.6 The Generic Contact Case

247

0.41

-1

-0.8 -0.6 -0.4 -0.2

0.2

0 6

0.4

0.8

1

x

Fig. 9.3.

Proposition 82. The Heisenberg sphere S(0, r) is semi-analytic but the wave front is not sub-analytic.

9.6 The Generic Contact Case The Heisenberg case is not a model to study the generic contact situation. Indeed its corresponds to a left invariant SR-structure on the Heisenberg group. The structure admits a nondiscrete symmetry group and the cut locus is precisely the conjugate locus. This situation is similar to the Euclidean structure induced on the sphere S2. The investigation of the general case is the object of several recent papers [6, 51. They contain the description of the small SR-sphere at a generic contact point and also the much more complex classification of all the small SR-spheres for all the generic contact SR-geometries in R3. We shall present briefly the small sphere at a generic point. The analysis and computations are based on graded normal forms.

9.6.1 Normal Forms in the Contact Case To compute a normal form we can use two approaches. First given a contact SR-metric (U, D, g) then D can be identified to ker a, a = dz + (xdy - ydx). The stabilizer of D, that is the set of germs of diffeomorphisms co preserving 0 and the distribution, that is cp * a = ha where h is a germ of smooth

248

9 Singular Trajectories in Sub-Riemannian Geometry

nonvanishing functions has been computed in the litterature devoted to contact geometry, see e.g. [14, 93], and can be used to compute a normal form by normalizing the metric g restricting to diffeomorphisms in the stabilizer. This point of view will be used in the Martinet situation to compute isothermal normal forms where g is a sum of squares. Here we shall present the classical approach to compute formal normal forms in singularity theory, see Martinet [83]. Recall that the gauge group is defined by the following transformations: 1. X =
2. v = e(x)u where e E 0(2). The infinitesimal action associated to the second action is e(x) = exp(i2(x)) where .R(x) is an antisymmetric matrix. The first result is due to Brockett [32].

Lemma 54. There exists a local coordinate system (U, q) centered at 0 such that the SR-problem in the contact case takes the form

i=u1+Rl =u2+R2

=uly-u2x+R3 where R1, R2 have vanishing first partial derivatives at 0 and R3 has vanishing

first partial derivative with respect to z and second partial derivative with respect to x, y.

Proof. Computations (see [32]).

Corollary 20. Using the following gradation for the variables: the weight of x, y is one and the weight of z is two, graded forms of order -1 and 0 are:

- order-1:

F1=3+yj,F2=-xj;

- order 0: as before.

General algorithm. An algorithm is described in [39] to construct at any order a normal form taking into account the previous gradation. This normal form is unique if we use a proper subgroup of the gauge group. (We cannot

expect to get an unique normal form for the gauge group action because the existence of a nondiscrete Lie symmetric subgroup for the Heisenberg SR-geometry)

9.6.2 Generic Conjugate Locus Using a graded normal form of order 1, the generic conjugate locus has been computed in [3, 39] and all the asymptotics are given. It required a proper time rescaling. It is represented on Fig. 9.4. On Fig. 9.5 we draw its intersection with a plane z = constant together with the cut locus; such trace has 4 cusps.

9.7 The Martinet Case

249

Fig. 9.4.

cut locus

conjugate locus

trace with z=constante

Fig. 9.5.

9.7 The Martinet Case 9.7.1 Preliminaries In this section we consider the local SR-Martinet geometry where U is an open set in R3 containing 0, D is a Martinet type distribution which can be smoothly identified to ker w, w being the one form: dz - 2 dx and g is a SRmetric on D which can be locally represented as g = a(q)dx2 + 2b(q)dxdy + c(q)dy2 where a, b, c are germs of real analytic functions at 0. The associated control system is 9(t) = ul(t)Fi (q(t)) + u2(t)F2(q(t))

where

250

9 Singular Trajectories in Sub-Riemannian Geometry 2

F1=- +2a, F2=ay and for the scalar product associated to g we have

(Fi, Fi) = a (F1, F2) = b (F2, F2) = c and (q, q) = u 2 a + 2u1u2b + u2c. We have [Fi, F21 =

ya

,

[[F1, F21, F1]= 0, [[F1, F21, F21 =

a

a and we note: F3 = 8z Hence 0 is a singular point and q = (x, y, z) is an adapted coordinates system where the weight of x, y is one and the weight of z is 3.

The abnormal extremals are contained in the plane y = 0 called the Martinet plane where w is not a contact form. Moreover they are straight lines: z = zo and the abnormal extremal passing through 0 is the line z = 0.

9.7.2 Normal Form The contact geometry where D = ker a, a = xdy + dz is rather simple and well investigated. In particular it has a characteristic direction identified here and da is a symplectic form on the quotient R3\ a . A Martinet type to distribution is more complicated. In particular the only obvious invariants are the singular trajectories. To compute a normal form in SR-Martinet geometry we shall use a different algorithm than the one presented in the contact case. Indeed we proceed as follows:

- compute the stabilizer of D; - normalize the metric using the stabilizer.

This approach has the advantage of clarifying the Martinet geometry and can be used to analyze others geometric control problems e.g. the feedback classification problem for affine control systems: 4(t) = Fo(q(t))+D(q(t))u(t), where D is a Martinet type distribution, see [931. We denote by D the set of germs of smooth diffeomorphisms of R3 preserving 0. The set D; is the subgroup of V preserving the line distribution Rte. Observe that

First we need the following lemma.

Lemma 55. Let cp E D : (x, y, z) ,-- (X, Y, Z) such that (i) Xz(0) 96 0 (such cp is called regular)

9.7 The Martinet Case

251

(ii) dZ- 2'dX =dz - 2dx Then p E Di and is defined by the equations 3

Y2=y2ax, x=a+2ay, Z=z- lay where a is any germ of a smooth function at 0, a : (y, X) - IR such that a(O) = 0 and ax > 0. Proof. Since

dZ -

y2

2

dX = dz - 2 dx,

2

then by differentiating we observe that p applies y = 0 onto Y = 0 (indeed they are the representation of the Martinet plane). Moreover if we introduce

S=Z+

one has

2 dX + dz - xydy. 2

dS =

If X2(0) 36 0, we can choose y, z, X as coordinates at 0 E IR3 and S(y, z, X is solution of

Sx= Y22 Sy=xy, S:=1. ,

Hence S can be written

2

S=z+ 2a(y,X) where ax (0) > 0 and x, Y are defined by the relations

Y2 = y2ax, x = a + ay 2

and Z is given by

Z

S_ x22

Definition 96. Without loosing any generality, we can choose the branch Y = yo-x-. Hence the regular diffeomorphism cp preserving w = dz - 2 dx is uniquely defined by a and a is called the generating function of W. We denote by Si,., the set of such diffeomorphisms parametrized in the previous proposition.

The following lemma is straightforward.

Lemma 56. Let so : (x, y, z) H-+ (X, Y, Z) be an element of Di preserving V i.e. sp * w = h.w where h is nonzero at 0. Then h is a constant.

Hence we have a complete parametrizations of the regular diffeomorphisms stabilizing V in the isoperimetric case. It can be extended to the general case and the following lemma is proved in [4].

9 Singular Trajectories in Sub-R.iemannian Geometry

252

Lemma 57. Let ,p E D : (x, y, z) - (X, Y, Z) such that: (i) (y, X, Z) is a coordinate system at 0 E 1R3 (such

(ii) cp preserves V = ket w that is there exists h : 1R3 - R, h(O) 76 0 with

dZ - 2' dX = h.(dz - edx). Then there exists two germs at 0 of smooth functions a : Z I.-# RI or (y, X, Z) i--+ R with a(0) = 0, a(0) = 0, ax (0)as (0) < 0 and cp is given by the following relations

x=-(a+yay), y2=_

yea;

az+ zaz

3

z=a(Z)-y-4ay Remark 25.

- The sP E V preserving D = ker w are infinitesimaly parametrized in [93]. - The Martinet geometry is clarified by the previous computations and the mappings a, a are the equivalent of the generating function in symplectic geometry.

We can now compute a normal form using the previous computations. We present the isoperimetric case, the general case being similar. Proposition 83. Let g = a(x, y)dx2 + 2b(x, y)dxdy + c(x, y)dy2. If cp E S,,, :

(x, y, z) ' (X, Y, Z) with generating function or is transforming g into a sum of squares A(X, Y)dX2 + C(X, Y)dY2, then a is solution of a partial differential equation of the form 3ay + yay2 = F(y, a, ax, yay, y, axx, yaxy),

when F is smooth. Proof. Easy computations show that we have 3ay + yayz

Ci

2A

with A = ayaX X

B = 4byoXX - a(2ax + yaxy)2 C = 4cyoXX - 2b(2ax + yaxy)2

Algorithm to compute a normal form. 1. We solve the Cauchy problem:

3a, + uay2 = F(y, a, ax, yay, yaxx, yaxy)

a(X,y=0)=ao(X) in order to get an isoperimetric metric: A(X, Y)dX2 + C(X, Y)dY2. 2. We van choose ao(X) such that A1Y=o is one.

9.7 The Martinet Case

253

Solvability of the Cauchy problem. We observe that the PDE is singular (it is of the Briot-Bouquet type). Nevertheless - an unique formal solution a (X, y) = E >o ° nX y' can be easily computed by reccurence by simply differentiating the equation; - using the result of [48], if oo(X) is real analytic, the previous power series is converging at 0 and we get an analytic solution. Hence our algorithm allows to compute easily an analytic solution which can be approximated at any order using a simple and implementable algorithm.

The same approach can be used in the general case and we get the following form.

Theorem 27. Every local SR-Martinet geometry (U, D, g) can be written in a local coordinates system as D = ker w, w = dz - dx, g = adx2 + cdy2 (sum of squares) with a = 1 + yF(q), c = 1 +G(q) with 2 0. Moreover if g is isoperimetric, the image of g can be chosen isoperimetric.

9.7.3 Orthonormal Frame

If we set F1 = 9 + 2 , F2 =

then with g = a(q)dx2 + c(q)dy2 we get

(Fi, Fi) = a, (F2, F2) = c, (FI, F2) = 0. If we introduce 1

G1 = LF1, G2 =

1

7

F2

we get an orthonormal frame {G1, G2}. It can be completed by G3 = 9 to define locally a Riemannian structure whose restriction at D is g.

9.7.4 Graded Normal Form It is constructed using the frame {G1, G2) and the weight given by the adapted coordinate system: 1 for x, y and 3 for z. One gets:

- Normal form of order -1: g = dx2 + dye :

flat case

- Normal form of order 0: g = (1 + ay)2dx2 + (1 +'8X + 7y)2dy2,

o,, 8, ry

constants

N.B. It contains terms of order 1 but by convention we identify two normal forms of order p > 1 whose Taylor expansion coincide at order p.

254

9 Singular Trajectories in Sub-Riemannian Geometry

Remark 26.

- At order {GP,, GZ} truncation of {GI, G2} generates the same distribution D = ker w. - The normal form of order 0 is isoperimetric. - Another normal form using the full gauge group has been computed in 151 using a different algorithm. It is similar at order -1 (as expected) and at order 0 contains the same number of parameters.

- Any normal form can be used but we shall give later the adapted coordinate system to study the SR Martinet geometry and related to the

transcendance we need to parametrize the exponential mapping. 9.7.5 Geodesics

We restrict our computations to the isoperimetric case. The problem is written 4(t)/= ui (t)Fi (9(t)) + u2(t)F2(9(t)) min

T (a(q(t))u(t) + c(q(t))u2(t))dt

Jo

and the Hamiltonian associated to normal extremals is 2

Hi Solving

-H 'OH

(aui + cua).

u: (P, F: (q)) -

= 0 we get 2

ul = a (Px + Pz 2 ), u2 =PV

and plugging u into Hj we obtain

2)2 + Hn = (uia + u2c) = 1((Ps + aP; 2

2

p2V

c

and the normal extremals are solutions of

th=a(Px+Pz

z

C

2) s 2 (Px +P-K2

2c

=P

Py - 2c Pz = 0

2a

+ (PsiP:

2a

4)za

- (p. +p.a

)Ps U

)

9.7 The Martinet Case

255

Using the Poincare coordinates (q, P) with P = (P,, P2, P3) associated to the orthonormal frame Gl

_F,

f'

GZ

_F2

f'

8

G3

8z

and given by Pi = (p, Gi) we get

Z Pl - px + p: 2 , Va

P2 =

f Py

a

P3 = P.

Hence we have H,, = 1(P12 + P2) and the following equations.

Lemma 58. In the Poincare coordinates (q, P) the normal extremals are solution of

2k A

P2=- n

(Y P3

2

-PI+2c,P2) a

yP3-2 QPi+2 Pa

P3 = 0 The following properties are straightforward.

Lemma 59. If ax = cx = 0, then x is a cyclic coordinate and the equations whose solutions are the normal extremals can be integrated by quadratures.

Lemma 60. The abnormal geodesic a : t '-- (±t, 0, 0) is not strictly abnormal if and only if the restrictions of ay to the Martinet plane y = 0 is zero. In this case this geodesic is also projections of a normal extremal where the choice of pz is arbitrary. In particular it can be immersed in the flow with pz = 0. Moreover all the axis y = z = 0 minus 0 is formed with conjugate points.

The geodesics parametrized by arc-length are located on HH = 1(P, + P2) = 1 and moreover P3(t) = A = constant. We must distinguish between two cases.

9.7.6 Riemannian Metric on the Plane (a, y) In the isoperimetric case we can define on the plane (x, y) understood as the quotient space R3\ 38 a Riemannian structure denoted 9R and given by gR = a(x, y)dx2 + c(x, y)dy2. The following result is clear.

256

9 Singular Trajectories in Sub-Riemannian Geometry

Lemma 61. The geodesics of the Riemannian structure 9R are the projections on the plane (x, y) of the SR-geodesics corresponding to A = 0. We deduce the following.

Proposition 84. There exists an open set V containing 0 such that each SR-geodesic corresponding to A = 0 is a minimizing curve if its projection in the plane (x, y) is contained in V.

Proof. According to standard Riemannian theory there exists such a neighborhood V of 0 on which the geodesics of the Riemannian structure 9R are global minimizers. Hence they are minimizers for the SR-structure. Remark 27. In the strict case the previous assertion shows that such geodesics play no role in the singularities of the SR-sphere with small radius. Never-

theless the metric gR gives informations about the invariants. In particular an easy computation gives the following.

Lemma 62. In the graded form of order 0 the Gaussian curvature K of gR is zero for every y if a = 0 = 0.

To analyze the geodesics when A # 0 and in particular when A -4 oo we introduce a foliation (F) which is the basic object in SR-Martinet geometry.

9.7.7 Asymptotic Foliation Associated to the Normal Form at Order 0 The geodesics are parametrized by arc-length. If 0 kir, we introduce the cylindric coordinates: P1 = cos 0, P2 = sin 0, P3 = p; = A 54 0. The geodesics equations take the from

f

cos 0

y=

sin 0

z=

v

y2 cos 0

2\/a-

(9.12)

6=

-

(yA - 2af cos B + 2; sin B)

where a,c are given in the graded normal form by: a = (1 + ay)2, c (1 +,Qx + -yy)2 and the last equation is

B= -

1

I

(yA - a cos 0 + 6 sin 0).

The equation defines a one dimensional foliation (F) in the plane (y, 0). Indeed if we use the parametrization

dt d

=

d

dr

(9.13)

9.7 The Martinet Case

257

and denoting ' the derivative with respect to r, the equation (9.12) can be projected onto y' = sin 0(1 + ay) (9.14)

0' = -(yA - a cos 0 +'3 sin 0) and differentiating the second equation we get

0"+Asin0+a2sin 0cos0-a/3sin20+/3cos00' = 0.

(9.15)

The parameters a, /3 are fixed by the metric g but A E JR represents the height of the cylinder on which the exponential mapping is defined. If we set e is small for A large and using s = r a the previous equation s v71-A1 ,

can be written dO

+I

ff

sin 0 + F3 cos 0 do + F2a sin 0(a cos 0 - /3 sin 0) = 0.

(9.16)

Notation. We denote by (F) the one dimensional foliation defined by (9.14) on the cylinder: (exp(i0), 0) and (.FE) the asymptotic foliation defined by (9.16) where s is a small parameter. Geodesic equations (9.12) are used to construct the sphere and integrability of those equations will be reduced to integrability of (F). The asymptotic foliation (.Fe) will allow to compute the sphere in the neigborhood of extremities of abnormal geodesics.

9.7.8 Properties of the Asymptotic Foliations In our analysis, we can assume A > 0 because changing A into -A we exchange only the role of the singularities.

20 Flat case. In this case the foliation is defined by d77 + sin B = 0. It is a pendulum which admits a global analytic first integral. The equation has two equilibrium points which are: a center if 0 = 0' = 0 and a saddle if 0 = ir,0' = 0. On the cylinder all the trajectories are periodic except two

separatrices El, E2 which are saddle connections, see Fig. 9.6. We have two types of periodic trajectories: those which are homotopic to 0 and those which are not homotopic to 0. They admit different parametrization using elliptic functions on R.

9 Singular Trajectories in Sub-Riemannian Geometry

258

Case /3 54 0. In this case neglecting the terms of order e2 we get the equation d$2

+sin0+e(3cosO- = 0. ds

(9.17)

One of the main property of equation (9.17) is the following.

Lemma 63. If /3 # 0, the asymptotic foliation has no global C°-first integral.

Proof. For 0 # 0, the origin becomes a focus and such a singular point has no local CO first integral.

Case 3 = 0. In this case the asymptotic foliation is given by d29

ds2

+ sin B(1 + e 2 a 2 cos 0) = 0 .

(9 . 18)

The main property is the following.

Lemma 64. The equation (9.18) is integrable using elliptic integrals of the first kind. We shall prove this result in the next section.

9.7.9 Integrable Case of Order 0 and Elliptic Integrals

Parametrization and properties. If the metric is of the form g = a(y)dx2 + c(y)dy2, the geodesics equations are integrable by quadratures

9.7 The Martinet Case

259

and the exponential mapping can be computed. In particular it will allow to describe analytically the behaviors of the geodesics near the Martinet plane y = 0 containing the abnormal geodesics. We proceed as follows. Using H,, (P? + P2) = 2 with Pl (y) = I (pz + p= z ), P2(y) = CL where p. and pz are first integrals because the metric doesn't depend upon x and z. We get

f

(f4)2+(Ps+P:2

)2=1

(9.19)

Definition 97. The previous equation is called the characteristic equation. Using the time parametrization dij = it describes the evolution of a one dimensional mechanical system with position y and in a potential field given by V(y)

P12(y)_

Let e(.) _ (x(.), y(.), z(.)) be a normal geodesic defined on [0, T], starting from 0 and parametrized by arc-length. If y is not zero, we denote by 0 < tl < < tN < T the successive times defined by y(t1) = 0. We introduce o

f sign(y(0)) if y(0) # 0 sign(y(0)) if y(0) = 0

Under mild assumptions on V, the motion of y is periodic with period P and we set

y+ =

tmaxx

y(t), y- = 9e

y(t)

The geodesics can be easily computed if we parametrize by y and they are solutions of the following equations cPl dz

y2 VC-P1

dy f P2 dy

2 f P2

dx

(9.20)

with

dt - fZdy

(9.21)

P2

andP2(y)=a 1-P1(y)fortE[0,t1]. If y(T) = 0, T = tN we obtain the following formulas:

- N odd. fP1(y)dy VcP1(y) x(T)=2v./.dy+(N-1) a 1-P y) fl-P (y) J y+

v°

JO

z(T)=2

V_

1yoaf2PPy(y)dy+(N-1)JY+ 0

V-

(9.22)

2f1P1Pi)(y)dy

260

9 Singular Trajectories in Sub-Riemannian Geometry

- N even. 1+

x(T)=NJ

cP1(y)

1-Pl(y)dy

-

(9.23)

y+

z(T) = N

cy2Pi (y)

Jy- 2 f 1 - Pl (y)

dy

and the period is given by y+

P=2

c

1 - PP (y)

y_

(9.24)

dy.

Hence we get explicit formulas to evaluate the exponential mappings.

Characteristic equation in normal form and elliptic integrals. The behaviors of normal geodesics near the abnormal direction in SR Martinet geometry can be understood as a property of elliptic integrals. Such standard objects in mechanics appear in the problem as follows. Assume that the metric is given by g = (1 + ay)2dx2 + (1 + ryy)2dy2 where a, ry are real parameters. We have P, = cos 0, P2 = sin 9, px = cos 9(0) and P3 = ps = A. By symmetry

we can assume \ > 0 and the characteristic equation can be written 2

(v v' )2+(P:+2)=a with a = (1 + ay)2, c = (1 +'yy)2. Using the parametrization dT = 73-7 we get l2

(

L

\ dT /

F(y),

where F(y) = (1 +ay)2 - (Ps +pz 2 )2. The analysis of the motion is related to the roots of F(y) = 0. The function F is a quartic which can be factorized into F = F1F2 where

Fl =(1+ay)-(px+pz 2 ), F2=(1+ay)+(px+\Y!). We can write

F(y) = where

(2m2

- 2 (y - 0)2) (2m" - A2 (y + A )2) a2

2m"=1+pz-

a2 2a

9.7 The Martinet Case

261

and m2+m"=1,m2>0ifa0 0 and m2>0if a#0and 00nrr.If we set y

and we get

v y+ a

a

2m/'

2m

'1

2m

2mf

F(y) = 4m2(1 - 772)(m" + m2Tj2).

(9.25)

The roots of F on C are 17 = f 1 and r = f mW7 . Definition 98. The geodesics are said critical when m" = 0. (The quartic F admits double roots) Lemma 65. If a # 0 in the graded normal form of order 0 then there exists (in the integrable case) critical geodesics starting from 0.

Geometric interpretation. If a 54 0, the abnormal geodesic is strict. The previous lemma gives us a clear geometric interpretation of the role of a. If

a = 0, the situation is similar to the flat case; in this case there exists no critical geodesics starting from 0, but when px -- -1, m" - 0 and we have geodesics starting from 0 which are nearby the critical case. The role of a

is to push critical geodesics has admissible geodesics starting from 0.

Normal form. The characteristic equation can be normalized as follows. Indeed if a # 0, there exists two distinct reals v1, v2 such that the pencil F1 + vF2 is a perfect square: K, (y - p)2 and K2 (y - q)2. Using the homographic

transformation

u=(y-p)(y-q)-1

(9.26)

the characteristic equation takes the normal form dy

F(y)

-

(p - q)-'du

(9.27)

(A1u2 + Bl)(A2u2 + B2)

and the trajectory u(.) can be computed using an integral of the form

f J

du

(Alu2 + B1)(A2u2 + B2)

which is called an elliptic integral of the first kind, see [76]. If a = 0, the result is still true. Indeed in this case 77 = 1 and F(y) is normalized as before using a simple dilatation: q =

Hence we get a neat interpretation of our graded normal form of order 0.

262

9 Singular Trajectories in Sub-Riemannian Geometry

Proposition 85. The graded form of order 0 is a coordinate system in which the oscillation of a normal geodesic with respect to the Martinet plane can be parametrized in the integrable case using elliptic integral of the first kind.

We shall now parametrize the geodesics in the flat case using elliptic integrals of the first and second kind. The previous reduction will show that the integrable case of order 0 has the same transcendence.

9.7.10 The Martinet Flat Case If g = dx2 + dye the geodesic equations take the form

Pi(t) = yP2(t)P3 P2(t) = -yPi(t)P3 P3(t) = 0

x(t) = P1(t)

y(t) = NO z(t) = z Pl(t)

Notations. We compute the geodesics starting at t = 0 from 0. We set Pl (0) = sin cp, P2(0) = cos p (i.e. cp = 0(0) + and A = P3(t) = P3(0). Let S 2) be the group generated by the two diffeomorphisms S1 : (x, y, z) i - (x, -y, z)

and S2 : (x,y,z)'-' (-x,y,-z).

Symmetries. In the flat case the SR-geometry is left invariant by S. Indeed

it is the symmetry group of the geometry. Observe it is a discrete group and the situation is different from the Heisenberg case where the symmetry

group is not discrete and the sphere is a surface of revolution. The symmetry group is related to the geodesic equations as follows. If we change p into ir - cp, the solution (x, y, z, P1, P2, P3) is changed into (x, -y, z, P1, -P2, P3) and if we change p into -cp and A into -A it is changed into (-x) y, -z, -P1i P2, -P3). Hence when computing the geodesics we can assume

Assumption (H1). V El - 2, +2[, A >- 0.

Quasi homogeneity. In the flat case the geodesic equations are invariant for the following transformations

x=eX Pi=Q1

y=EY P2=Q2

z=63Z P3=

and the length is changed as follows L(X, Y, Z) = s-1L(x, y, z).

9.7 The Martinet Case

263

Parametrization. We have the following particular solutions. When cp = x(t) = t, y(t) = z(t) = 0 and it corresponds to the abnormal geodesic. If A = 0, x(t) = t sin cp, y(t) = t cos cp, z(t) = sin cp cos2 cp and their projections on the plane (x, y) are straight lines (they are the geodesics of the Riemannian

s

metric gR = dx2 + dy2). The others solutions are computed using elliptic

integrals of the first and second kind. The characteristic equation is equivalent to

y2=(1-P1)(1+P1)=(1-Px- )(1 +Px + ipx). We introduce 0 < k, k' < 1, k2 + k'2 = 1 defined by k2 = sine

" (4

-

1 - sin cp

cp

2) =

k'2 _

1 + Sin cp

2

2

where cp E] - 2 , + [. Hence y satisfies 2

y2 = (2k2 - 2 A)(2k'2 + 2 A). Assume A > 0 and set 71 = y2k , we get the equation 2

,

=(1-r12)(k2+k2ii2) which has to be integrated with the initial condition 71(0) = y(O) = 0 and using the initial branch f1(0) > 0 since y(0) = P2(0) > 0. The solution can be written using the Jacobi elliptic function cn whose inverse mapping is defined by

cn-1(x,k) _

d71

x'

(1 -,i2)(k'2 + k2ii2)

and we set

r1(t) = -cn(K(k) + tf, k)

Vr-

where the period is 4K, K being the complete integral of the first kind 11

K(k)=J

o

d'1

(1 - r12)(k'2 + k2712)

f

=J (1-k2sin9)-4dO. o

77 cos(K(k)+t ' , k) and represents a periodic motion whose period is 4K and with amplitude 2. We have two limit behaviors. When cp - 2 and K(k) --+ 2 and cp -' 7r+ - - , then k' - 0 and K(k) ti Ink, . Hence K(k) --+ +oo, when k -+ 1 Hence y(t)

and admits a logarithmic singularity. In the pendulum representation we have the system

264

9 Singular Trajectories in Sub-Riemannian Geometry

y=sing, B= -Ay. Since y(O) = 0, we have 9(0) = 0 and we have only the oscillating solutions of the pendulum. The two limit behaviors k -b 0 corresponds to the linearized pendulum and k - 1 corresponds to the oscillating solutions tending to the separatrix realizing the saddles connections. The others components can be computed similarly. The only additional transcendence we use is the Jacobi epsilon function, see [76] defined by rt.

E(t, k) =

J0

dn2u du =

rent [1-- k2u2

J0

1-u2

du.

Endly we get the following parametrization.

Proposition 86. Arc-length parametrized geodesics starting from 0 are given by

x(t) = -t +

3(E(u) - E(K))

y(t) = -- 2k Z(t)

=3

((2k2 - 1)(e(u) - E(K)) +

k'2tV,-\)

+2k2snu cnu dnu)

where u = K + t \/'A-, A > 0, lP E] - 2 , + 2 [, snu , cnu dnu and E(u) being the Jacobi elliptic functions, K and E(K) being the complete integrals of the first and second kind, or the particular solutions when A = 0 x(t) = tsin ip y(t) = t cos ca z(t) = 11 sinwcos2

s

where P E) - M' j [ and the curves deduced from the previous ones using the symmetries Sl2 : (x, y, z) '- (x, -y, z) and S2 : (x, y, z) "- (-x, y, -z).

Return mapping and intersection of S(0, r) and W (O, r) with the Martinet plane. Definition 99. Let e(., gyp, A) be a geodesic parametrized by arc-length. We the map which associates to e(., cp, A) its first (resp. denote by Rl (resp.

n-th) intersection with the Martinet plane. Its domain is the set cp # n!2 , A # 0. They are called respectively the first and nd return mapping. Image of R1. We may assume by symmetry W E) - 2 , + [. Let r > 0, using 2 the parametrization of Proposition 86 we represent the image of Rl for t < r on Fig. 9.7. The important property is the following: RI maps the curve r,

defined by r/ = 2K onto the curve cl and preserves the orientation of the curves. The interior of the shaded domain is sent onto the interior of the shaded image.

9.7 The Martinet Case

265

+1

-1

Image R. , t
Domain R, with t
Fig. 9.7.

Corollary 21. A geodesic whose projection on the plane (x, y) is not a straight-line is optimal up to its first intersection with the Martinet plane.

Proof. We observe that due to the symmetry S2 : (x, y, z) '- (x, -y, z) at each point P1 of cl there exists two distinct geodesics with same lengths ending at P1. From the maximum principle, every minimizer is smooth and clearly a geodesic cannot be minimizing at any point beyond P1. To prove that a geodesics is optimal up to its first intersection with y = 0, we observe

that every point P of y = 0 in the domain z > 0 is the image by R1 of an unique (gyp, cpp), A > 0, cp El -' ,- [ in the parameter space. Moreover from the existence theorem there exists a minimizing geodesic joining 0 to P. Hence the geodesic corresponding to (Ap, tpp) is minimizing at P and therefore

minimizing up to P. The result is proved. The following theorem is a direct consequence of the previous corollary.

Theorem 28. The geodesics whose projection in the plane (x, y) is a line are minimizers. A geodesic whose projection is not a line admits a cut point at time tp = corresponding to its first intersection with the Martinet plane. Hence the intersection of the conjugate locus and the cut locus is empty.

T

266

9 Singular Trajectories in Sub-Riemannian Geometry

9.8 Estimates of the Sphere and of the Wave Front near the Abnormal Direction in the Flat Case and exp -In Category We can give precise estimates of the distance from 0 ti a point near the abnormal directions. We prooced as follows.

9.8.1 Intersection of S(0, r) with the Cut Locus Let r > 0 and setting ti, = r =', which corresponds to the first intersection with the Martinet plane, in the parametrization of the geodesics one gets

x(k) = -r + 2 (E(3K) - E(K)) (9.28)

z(k) = 3ai ((2k2 - 1)(E(3K) - E(K)) + 2Kk'2) (9.29)

Using the relation E(3K) = 3E(K) and the notation E for E(K), we get that the intersection of the cut locus with the sphere S(0, r) is a curve denoted k c(k) contained in the Martinet y = 0 which admits the following parametric representation

x(k) _ -r + 2ry z(k) - s ((2k2 - 1)E + k'2K) where k E]0,1[ and the curve deduced from c using the symmetry (x, z) H

(-x, -z). In order to understand the properties of this curve, the following properties are fundamental. First, by definition

K=

ri

J

o

Vfd

(1 - rtz)(k'2 + k2g2)

and

E=

rx

J

=

(1 - k2sin2o)-4d9

Jo

fI

dn2udu=(1-k2sin20)1do

0

z[1+(2)z

K

z

k4+...J k2+(2a) z 2 E=2[1-(2) k2-3(24) k4+...1

9.8 Estimates of the Sphere and of the Wave Front

267

In particular, when k -* 0+, the cut locus in the domain z > 0 is of the graph of an analytic function which can be represented by 3

2r3(x - r) + o(x - r)

Z

Hence it is a semi-analytic object. When k -+ 1, the situation is quite different. This is due to the following fact. We can extend the mapping k H K(k) to an analytic function on C\[1, +oo[ which presents a logarithmic singularity when k 1. More precisely we have a representation with k' = 1 - k K(k') = ul (k')in

+ u2(k')

where u1i u2 are analytic function of k' which can be easily computed by reccurence k42

ul = 1 +

+

o(k'3),

k42 + o(k 3)

u2(k')

The function E has a similar property and can be decomposed into

E(k') = u3(k')ln

+ u4(k')

where u3, u4 are analytic at 0 and moreover u3(k') =

k4+ o(k 3), u4(k') = 1 -

k82

+ o(k '3).

Introducing

Xi=k',

X2 =

14 In k,

we have X1 = 4e-3k. Such a graph is not analytic but is pfafflan because solution of X2dX1 - X1dX2 = 0.

Hence near k' - 0, the cut locus is the image of a pfafflan set and in general we cannot expect such an object to be semi-analytic. Indeed precise computations give the following.

Theorem 29. When k' - 0, the graph of c is given by

r3 x+rl3 F( x+r) 6 \ 2r / + 2r / where F is a flat function of the form F(X) = -4r3X 3e- * + o(X3e- 1). In particular the sphere is not sub-analytic.

268

9 Singular Trajectories in Sub-Riemannian Geometry

This is not a new result in optimal control, see [82], but here we can be more

precise using the exp -In category in real analytic geometry [78]. Indeed to compute the graph we must eliminate the parameter k' occuring with logarithmic complexity in the parametric equation. Using an implict function theorem in this category we can prove the following theorem, see [4].

Theorem 30. The intersection of the sphere S(0, r), r > 0 with the Martinet plane y = 0 in the domain z > 0 is near X = 0 with X = 2r' a graph of the form

z=F(X,eX%1 where X > 0 and F is an analytic mapping from a neigborhood of OR2 onto

Wave front. We can make the same analysis and compute the wave front represented in the domain y = 0, Z > 0 on Fig. 9.8. It contains an infinite

Fig. 9.8. where the curve cn corresponds to the n-th intersection of the geodesics with y = 0 and the curve cl belongs to the sphere. In particular it has an infinite number of intersections with the axis Oz and number of curves cl, c2i

belongs to no reasonable category of analytic geometry. Nevertheless each curve has the same properties than the curve cl, that it is not suban-

alytic but belongs to the exp-ln category.

9.9 Conclusion Deduced from the Martinet SR-Flat Geometry Concerning the role of Abnormal Geodesics in SR-Geometry 9.9.1 Behaviors of the Normal Geodesics near the Abnormal Direction - Geodesic C°-Rigidity We represent on Fig. 9.9 the behavior of a geodesic e(.) = (x(.), y(.), z(.)) near the abnormal direction (i.e. p - -1). ). The coordinate y oscillates pe-

9.9 Conclusion Deduced from the Martinet SR-Flat Geometry

x

Y

t

269

z

X

t -> y(t)

t

t -> x(t)

t t -> z(t)

Fig. 9.9.

riodically with period 4K and for t ' - x(t), z(t) there exists a shift. The average behavior can be obtained by taking the respective lines joining 0 to (2K, x(2K)), (2K, z(2K)). This shift is a property of the Jacobi epsilon function E. In particular at the first intersection z(k) =

_± ((2k2 - 1)E + k '2 K) 3A 22

and when o -+ -L,

k

1 the shift is given by

lim z(k) =

k-.1

4 3A

The numbers is a basic invariant and explain the C°-rigidity of the abnormal extremal, discussed in Chap. 5. More precisely we have the following

geodesic rigidity property. Proposition 87. For all M > 0, there exists e(M) such that if e

:

t-

(x(t), y(t), z(t)), t E (0,T]. is a normal geodesic whose image is not contained in the axis 0s corresponding to the abnormal direction and with length less than M then we cannot have simultanously y(T) = z(T) = 0 if I y 1:5 e(M)-

9.9.2 Nonproperness of the First Return Mapping Rl and Geometric Consequence We recall on Fig. 9.10 the first return mapping for p E] - 2 , + [ and with fixed length. We observe that the mapping R1 is not proper near 2to the point (-r, 0) and this is due to the logarithm branch of K(k) when k -, 1. If we use the pendulum representation this nonproperness is due to the behaviors

of the trajectories tending towards the separatrix, see Fig. 9.11. The

270

9 Singular Trajectories in Sub-Riemannian Geometry

a

P

ai x

+r

Fig. 9.10.

Fig. 9.11.

Fig. 9.12.

9.10 Cut Locus in Martinet SR-Geometry

271

phenomenon is similar to the one encountered in the evaluation of PoincareDulac return mapping with respect to a section of a separatrix cycle, see Fig. 9.12. This shows the connection between the SR-sphere in the Martinet case

with the problem of computing the Poincare return mapping in the 16-th Hilbert problem concerning the number of limit cycles for planar differential

equations. All techniques developped to study this mapping can be used to evaluate the SR-distance.

9.10 Cut Locus in Martinet SR-Geometry An important question briefly discussed in this section is to understand the role of abnormal direction in the cut locus. We represent on Fig. 9.13 the cut locus in the Martinet flat case, in the northern hemisphere A > 0 of the sphere. The two points A and -A are the intersections of the abnormal direction with the sphere. The abnormal geodesic is not strict and both points A and -A are contained in the equator A = 0. It can be understood as a deformation of

Fig. 9.13.

the contact case represented on Fig. 9.14. Next we shall conjecture the cut locus in the Liu-Sussmann example (79], where the SR-metric is given by 2

D = ker w, w = (1 + ey)dz - 2 dx 2 (1

y)2 + dye

The model is not generic because the orthonormal frame generates a nilpotent algebra. Moreover the geodesisc equations are integrable. They are the following in cylindric coordinates 2

± = (1 + ey) cos 8, y =sin 8, z = 2 cos 8,

272

9 Singular Trajectories in Sub-Riemannian Geometry

conjugate point

Heisenberg cut locus

generic cut locus in the contact case

Fig. 9.14. 6

= -(Pxe + pzy)

where px and p,z are first integrals and pZ = A. The angle evolution is the pendulum 8 + A sin 8 = 0 and the constraint y = 0 defines the section 6 = -pXE.

If E # 0, the abnormal line is strict and intersects the northern hemisphere A > 0 at a single point A. If e = 0, we are in the flat case and the cut locus is represented on Fig. 9.14: each point of y = 0 minus ±A is a cut point endpoint of two disctinct geodesics corresponding to oscillating trajectories of the pendulum. If E 96 0 the cut locus splits into two distinct branches ramifying at A, a branch Lc corresponding to oscillating solutions of the pendulum and a branch LD corresponding to rotating solutions of the pendulum. The extremity of each branch not ending at A are conjugate points, see Fig. 9.15.

cut locus in 1S-example

Fig. 9.15.

9.10 Cut Locus in Martinet SR-Geometry

273

Notes and Sources A good general presentation of SR-geometry is provided by (69]. See also the articles of Bellaiche and Gromov in [16]. A detailled analysis of the contact case is provided by the series of articles [6, 5] from Agrachev-Chakir-GauthierKupka; see also the article from [9] concerning the generalization of the Dido problem. The description of the Martinet case is given in details in the two articles [4, 23] and see [29] for a generalization.

Exercises 9.1 In Heisenberg SR-geometry compute the models of the singularities M; and Ni of the wave front. 9.2 Consider a SR-geometry (U, D, g) in R3 where D = ker w with 3

W = we = dy - (xy + 3 + xz2 + mx3z2)dz or

w = wh = dy - (xy + x2z + mx3z2)dz

and g = a(q)dx2 + 2b(q)dxdz + c(q)dz2. Compute in both cases the graded approximation of order -1. Compute in this approximation the normal and abnormal extremals. 9.3 Consider the flat Engel SR-geometry in R4 with coordinates (x, y) z, u),

D=Span {F1,F2}, F1 = F+yj+'j3U, F2 =

ZOFj

andg=dx2+dy2.

1. Compute the abnormal geodesics. 2. Compute the normal geodesics equations using the Poincare coordinates Pi = (p, F1) where F3 = [Fl, F2] and F4 = [[Fl, F2], F2]. Prove that P4

and C = P1P4 + 2 are first integrals. 3. Integrate the normal geodesics flow using elliptic integrals.

9.4 Consider the local SR-geometry (U, D, g) where D is a rank two COOdistribution and g is a C°°-metric. We can identify the SR-geometry to the set of orthonormal pairs of vectors fields. Show that for an open dense set of pairs (Fo, F1) for the Whithney topology each abnormal or normal extremal is strict.

9.5 Consider a smooth SR-structure (M, D, g) where g denote the scalar product on D. Given a smooth vector field X, we denote by exp tX its corresponding flow. The vector field is called an infinitesimal symmetry of the distribution if (exp tX) *,d = A and of the SR-structure if (exp tX) * 4 = A and (exptX) *g = g. The Lie algebra of the distribution (resp. SR-structure) is denoted SymD (resp. Sym(D, g)).

274

9 Singular Trajectories in Sub-Riemannian Geometry

1. Prove the following:

a) X E SymD if and only if adX (a) C A. b) X E Sym(D,g) if and only if adX E so(D(q)) for all q E M where so(D(q)) is the set of antisymmetric matrices over the space vector D(q) with scalar product gq. 2. Consider the Heisenberg SR-geometry with D = Span{Fi, F2}, F1 & + x a: , and g is defined by taking Fl, F2 orthonormal. F2 = a) Show that

SymD = {X; X = Pox +Qay

+ Rz},

(9.30)

where P = -fq-xfz, Q = f, R = xff- f with f being an arbitrary function.

b) Prove that the symmetries of (D, g) form the four dimensional Diamond Lie algebra: Span{X°, X1, X2, X3} where: 49

X0

a

1

19

+X-

- y2) 8z

(X 2

Xl

9

(9

TX +yOz

X2= 9 X3 8z

3. Consider the flat Engel case with D = Span{F1i F2), F1

+X 28v = ay +X 8z F2 =

8

(9.31)

Compute SymD and prove that Sym(D, g) is isomorphic to the Engel Lie algebra: {Fl, F2}AL.

9.6 [Grusin example] Consider the singular Riemannian problem:

i(t) = u(t) y(t) = x(t)v(t) min

J

tl (u2(t) + v2(t)) i dt

Define the length of a curve y(.) by e(y) = fro (v2(t) + v2(t))dt and the distance between two points qo, ql as d(qo, ql) = infe(y) where y(.) is an admissible curve joining qo to ql.

9.10 Cut Locus in Martinet SR-Geometry

275

1. Prove that outside the line x = 0, the problem is Riemannian and is defined by the metric ds2 = dx2 + 2. Show the following estimates:

x

dy2.

2(IxI+Iyl )<_d((0,0),(x,y))

- 3(IxI+Iyl4).

3. Prove that the geodesics starting from (0, 0) are the lines y = 0 and the family of curves given by:

x(t) = P(O) sin(pyt) y(t) = px(0)(t 2py

-

sin(2pyt))

2py

with py # 0.

4. Represents the geodesics starting from (0, 0) and the points at distance r from the origin. Compare with the Heisenberg sphere. 9.7 Consider the SR-problem in R3: 2

9(t) _

ul (t)F (q(t)) + u2(t)F2(q(t)) min

t (ui(t) + us(t)) 1dt

Jto

where

1. Prove that the Lie algebra generated by F1 and F2 is isomorphic to the Heisenberg Lie algebra h3 and deduce that the SR-problem is isometric to Heisenberg SR-geometry. 2. Prove the following estimates for the SR-distance:

3(IxI+IyI+IzI1)sd(0,(x,y,z))

- 4(IxI+IyI+IzIi).

9.8 1. Consider the contact SR-geometry in 1R3. Prove that a graded normal form of order 1 is defined by the two orthonormal vector fields:

F 1 = a + 2(1 +Q)a-, F2 = 8- - 2(1 +Q) 4 x

(9.32)

where Q is a quadratic form, the weight of x, y is one and the weight of z is two.

2. To get a frame, complete F1, F2 by F3 =

ffz-

and introduce P =

(P,, P2i P3) with P, = (p, F1 (q)), i = 1, 2, 3 and q = (x, y, z). Write the equation for the normal extremals in the coordinates (q, P).

276

9 Singular Trajectories in Sub-Riemannian Geometry

3. Let the geodesics be parametrized by arc-length and introduce the cylindric coordinates: P1 = cos 9, P2 = sin 9, P3 = A. Show that the geodesic equations are given by: th(t) = P1(t)

X0 = P2(t) z(t) = P1(t)y(t)(1 + Q) - P2(t)x(t)(1 + Q) 2

6(t) = (1 + 2Q)a

where P3 = A is a constant. 4. Assume A > 0 and make the following reparametrization: ds = (1 + 2Q) A, show that 9(s) = s + Bo, 9o =constant.

5. Assume A large enough, e = z small parameter. Set x = EX, Y = EY, z = E2Z. Prove that:

X =Xo+E2Xl+0(62) Y = Yo + E2Y1 + o(E2)

Z = Zo + o(E)

where go = sin(s + Oo) Yo = cos(s + 90)

Z0 = sins + 9o)Yo(s) - cos(s + 9o)Yo(s) 2

X1 = sin(s + 9o)Q(Xo, Yo) X2 = cos(s + 90)Q(X0, YO)

and Q is the quadratic form defined by 1+2

Cy2+...,

= 1+Q = 1+Ax2+2Bxy+

9.9 Consider the flat Engel SR-geometry in R4 with coordinates q = (ql, q2, q3, q4) and defined by the two orthonormal vector fields: a Fl-8ql+92

2 q2 a

a

aq3+ 2aq4 F2 - a aq2

and let L 1i L2 be the two matrices

0000 0010

L1=

0001 0000

_

,

0100 0000

L2- 0000 0000

9.10 Cut Locus in Martinet SR-Geometry

277

1. Show that the flat Engel SR-geometry can be lifted to a right-invariant SR-geometry:

R(t) _ (uI (t)LI + u2(t)L2)R(t),

min Ito (ui(t) + u2(t)) where R belongs to the Engel group Ge represented by the nilpotent 1 q2 q3 q4

0 1 q, 2 0 0 1 ql

matrix

(0

0

1

2. Prove that the Martinet flat case is isometric to (Ge/H, dqi +dq2) where H is the following sub-group of Ge : {exp t[L1, L2); t E R}. 3. Compute the flat Engel SR distance from 0 to the line (ql, q2, *, q4) (resp. (qi, q2, q3, *)) and compare with the flat Martinet distance from 0 to (qi, q2, q4) (resp. with the Heisenberg distance from 0 to (qi, q2, q3)). 4. Deduce that the flat Engel SR-sphere is not sub-analytic. 9.10 Consider the Martinet SR-geometry of order 0 with orthonormal frame

1(0 + 2 ), TO where a = (1 + ay)2, c = (1 + /3z +7y)2, a,,O,-y are real parameters and a > 0. Let X = 2r' , Z = rs, r > 0. Prove that near (-r, 0, 0) the trace of the sphere S(0, r) with the Martinet plane y = 0 in the

domain z < 0, is a graph of the form: Z = - 2 X2 + o(X2). 9.11 Prove that in SR-geometry each CI-local minimizer is a global minimizer if its length is small enough. 9.12 Let U be a neighborhood of qo in R" and consider the SR-problem:

q(t) _

M

min

T

fo

,-1

u?(t))idt

with q(0) = qo. Prove that if m < n, then in every neighborhood of qo there exists a point q qo where the SR-distance function to qo is not continuously differentiable.

9.13 Let U be a neighborhood of qo in R" and consider the following SRproblem: m

Q(t) _

min

u+(t)Fa(q(t))

/ T( u; (t)) i dt

278

9 Singular Trajectories in Sub-Riemannian Geometry

with q(O) = qo. The set of controls is endowed with the L2[0, 11 topology and this induces the Hl -topology on the space of trajectories. Prove that for any

small enough r, the set of minimizers of prescribed length r is compact in the Hl -topology. 9.14 Let D = Span.{Fl, , F,,,) be a m-dimensional distribution of a manifold M and consider the system m 4(t)

ui(t)Fi(q(t)).

(9.33)

1. The distribution D is called fat at a point q E M if for any vector field X with X (q) E D, X (q) 0, then we have: [X, D] (q) = D(q) + Span{ [X, YJ(q); Y E D} = TqM.

(9.34)

Prove that if D is a fat distribution at qo, then there does not exist a nontrivial singular trajectory starting at qo.

2. The distribution is called medium fat if for every vector field X E D, X (q) 54 0 then

[X, D') (q) = D2(q) + Span{(X,YJ(q); Y E D2) = TqM.

(9.35)

Prove that if D is a medium fat distribution at qo then there doest not exist a nontrivial singular trajectory starting at qo and satisfying Goh conditions.

9.15 Consider a smooth control system of the form: 4(t) = f (q(t), u(t)) where q(t) E U neighborhood of 0, u(t) E JIB'" and f (0, 0) = 0.

1. Show that a necessary condition for the existence of a C'-feedback control u(.) such that 0 is asymptotically stable for 4(t) = f (q(t), u(q(t)) is that the mapping (q, u) -- f (q, u) is onto on open set containing 0. 2. prove that the following system in R3: ±(t) = u(t)

W) = v(t)

z(t) _ (u(t)y(t) - v(t)x(t)) cannot be made asymptotically stable at 0 using as smooth feedback.

9.16 Let0
cn"1(i , k) n

(1 - t2)(k'2 + k2t2)

(9.36)

9.10 Cut Locus in Martinet SR-Geometry

279

where f -1 is the inverse of the function f . Prove that 77(t) = cn(t, k) is solution of the differential equation: i2(t) = (1

- 772(t))(k 2 + k2n2(t))

(9.37)

and is a periodic function of period 4K(k) where 1

K(k)

-

I

k2g2).

2. Define:

sn-'(77, k) =

(9.38)

(1 - q2)(k'2 +

dt

J

n

V,r(

- t2)(1 - k2t2)

(9.39)

Prove that sn(t, k) is a periodic function with period 4K(k) solution of the following equation:

i2(t) = (1 - rl2(t))(1 - k2r12(t)).

(9.40)

3. Let 0
= fn

1

(1 - t)(t2 - b2)

(9.41)

Prove that do is a periodic function of period 2K(k) solution of: t 2(t) = (1 - 7I2(t))(rl2(t) -

b2).

(9.42)

4. Let

E(t) =

J0

and

t

dn2(v, k)dv

(9.43)

K

E (K) =

dn2(v, k)dv.

(9.44)

fo

Prove that

E(t + 2K) = E(t) + 2E(K)

(9.45)

9.17 1. Prove the following relation:

dK_ dk

1

k'2k2

(E

kI2K)

(E-K) dk = 2. Show that K is solution of the equation: k1

k(1 - k2) dk2 + (1 - 3k2) dk - kw = 0.

(9.46)

280

9 Singular Trajectories in Sub-Riemannian Geometry

3. Prove that E is solution of.

k(1-k2)dkz +(1-k2)k+kw=0. 4. Deduce the following expansions:

a) When k - 0:

K(k)=' 1+ 4k2 + o(k3)) E(k) = 2( 1 - 4k2 + o(k3)) b) When k' -' 0: E(k) = ul (k')lnk, + u2(k') K(k) = u3(k')1n 4 + u4(k') where '2

ui(k') = 2 + o(k 3) '2

+ 0(0)

u2(k') = 1 -

4

'2

u3(k') = 1 + 4 + o(k'3) '2

ul(k')

4 +

o(k'3)

(9.47)

10 Micro-Local Resolution of the Singularity near a Singular Trajectory - Lagrangian Manifolds and Symplectic Stratifications

10.1 Introduction The aim of this chapter is to stratify the singularity of the end-point mapping near a singular trajectory using micro-local analysis and Lagrangian manifolds [57],[86]. We analyze two situations: the generic affine case, which corresponds to a situation of codimension one and the generic SR-case which

is a situation of codimension two. This leads to a stratification of the solu-

tions of Hamilton-Jacobi-Bellman equation viewed in the cotangent bundle. The analysis is not standard, the main difficulty is the non existence

in general of a preparation theorem to solve the equation near a singular trajectory.

10.2 Lagrangian Manifolds We introduce briefly the concept of Lagrangian manifold which is crucial in our analysis.

Definition 100. Let (V, w) be a linear symplectic manifold. A subspace L of V is called isotropic if wl L = 0. An isotropic space of maximal dimension= dim V is called Lagrangian. Let (M, w) be a smooth symplectic manifold and

L C M be a smooth regular manifold. We say that L is isotropic if the restriction of w to TL is 0 and if dim L =

d`

2 M, L is called Lagrangian.

Example 7. Let (q, p) be Darboux coordinates on M identified to IIt2n and let S : q - S(q) be a smooth function. If L is the graph: (q, p = (q)) then L is Lagrangian. Wasq-

Proof. We have n

n

dpi A d% =

since

82S 9 Uq-i

_ _

aS A dq; = d(-) 0%

"

i,k=l 80,19%

82S q

82S

'

More generally, we have the following, see [86].

dqk n dq; = 0,

10 Resolution of the Singularity near a Singular Trajectory

282

Proposition 88. Let L be a Lagrangian n-dimensional manifold. Then there exists Darboux local coordinates (q, p) and a smooth function S(qj, p-), where , n} such that L , m}, 7 = {m+ 1, , n} is a partition o f { 1, I = { 1, is given by the equations pi

as = agj

_ap: 19S

q=

Proof (For details see [86]). Our analysis is local and we may assume that M R1n, w = dp A dq and we are in a neighboorhood of 0 E R2". At 0, the tangent space T0L is Lagrangian and we consider the standard projection 7r (q, p) -4 q. In the previous example the projection zr restricted to L is a :

submersion. But in general it is not the case and we proceed as follows. There exists at 0 linear coordinates (qj, p) where 1, 7 form a partition of {1, , n} such that the projection 7r : ToL -+ (ql, pT) is an isomorphism.

Then we must find a mapping S(qj, pT) such that L is given locally by ,ql=- that is

pj=

dS = pjdgj - gTdpT.

(10.1)

Consider the one form a = pjdgj - q dpT,

it can be written a = pdq - d(p7gT)

and since L is Lagrangian da = 0 on L. The existence of S follows from the Poincare lemma. Definition 101. The mapping S which represents locally L is called the generating mapping of L.

10.3 Application to Classical Calculus of Variations We consider the minimization problem min

fT

J0

L(q(t), q(t))dt,

q(t) E M

where T is fixed and the boundary conditions are q(0) = q0, q(T) = qj. We assume that the strong Legendre condition 02 L > 0 is satisfied. The extremals are solutions of the Hamiltonian equations: 4(t) =

(q(t),p(t)),

P(t)

OH

(q(t),p(t))

10.4 Singularity Theory of the Generating Function - Generating Family

283

where H(q, p) is the Hamiltonian defined on T* M. We denote by Wt = exp t H the flow of H, Lqo = TqM is the fiber above qo and Lt = cpt(Lgo). Let '(.) be a reference extremal 7r(y(0)) = qo, defined on [0, T]. We have the following.

Proposition 89. 1. At t = 0, Lqo = TQ0M is a Lagrangian linear manifold (the fiber). 2. For each t > 0, Lt = Wt(Lo) is Lagrangian.

3. Along '(.), the vector field H is transverse to Lt for t > 0. 4. The time tc is a conjugate time along y(.) if and only if the projection is singular.

Proof. It is a reformulation of the results of section 2.1 using Lagrangian formalism. Indeed the fiber Lqo is Lagrangian by definition and Lt is a Lagrangian manifold because it is the image of Lqo by a symplectomorphism Wt. By definition the conjugate times are defined as the singularities of the mapping: t'- 7r(exp t H (' (0)), hence we deduce 4. Remark 28. Let 0 < t < tic where tic is the first conjugate point along ry"(.) Then locally near ir('(t)) the Lagrangian manifolds Lt are parametrized by

a generating function t - S(t, q). This is equivalent to solve HamiltonJacobi-Bellman equation. Indeed we integrate the extremal trajectories starting from q(0) = qo to construct a mapping t i- S(t, p) and p is eliminated

using the implicit function theorem by solving at t fixed the equation: S(t, p)

=q

This gives us the generating mapping S(t, q).

Beyond the first conjugate point the Lagrangian manifold Lt is still well defined but we must use singularity theory to analyze the structure of (Lt, ir). This leads to the next section.

10.4 Singularity Theory of the Generating Function Generating Family Locally Lt can be represented by a generating mapping S. Itt was a recent research activity of singularity theory to make a classification of pairs (L, 7r). All the simple singularities are classified in [15], they correspond to singularity up to codimension k = 6. Here we give the classification up to codimension k < 3, see [15] for details.

Proposition 90. Up to codimension k < 3 the generating function S have the following normal forms:

284

10 Resolution of the Singularity near a Singular Trajectory

k > 1: A2 : S = pi (fold) k > 2: Moreover we have, A3 : S = fpi + g2pi (cusp) k > 3: Moreover we have (swallowtail)

A4 : S = pi + g2p3l + gspi

(2 cases)

D4 : S = pi f plp2 + gspi

Remark 29. The two first cases are equivalent to Withney classification for applications from the plane onto the plane. Definition 102. Let L be a Lagrangian manifold. A caustic is the projection on M of the singularities of (L, 7r). Another way to parametrize Lagrangian manifolds is the concept of generating family coming from optics. Definition 103. Let N, M be two smooth manifolds, u E N, q E M being coordinates. The family F(u, q) is called a generating family for the Lagrangian manifold L C T * M if

L={(p,q); 3us.t.

OF(u, q)

au

OF

=O,p= a-

}.

Construction of F. (see [15]) The Lagrangian manifold L is defined locally by its generating function S(qj, pT) by the equations: PI =

as

as

q-I

agj

OPT.

We set

F(u, q) = S(qj, u) + (qT, u)

where (,) is the scalar product and dim u = dim 7 = m (u = pT). We define a Lagrangian manifold V on the extended cotangent bundle T*(IRrn X R'1) by setting

OF y = T U-

OF

OS

au + '

p = aq .

By cutting by y = 0, one gets as 19U

since u = pT.

as OPT,

10.5 Application to Optimal Control Theory and to SR-Geometry

285

10.5 Application to Optimal Control Theory and to SR Geometry Consider the following control problem

rT min J L(q(t), u(t))dt, T fixed, q(O) = qo, q(T) = q1 0

subject to 4(t) = f (q(t), u(t)). The previous theory can be applied to analyze the structure of the optimal synthesis excepted in two situations: 1. When the transversality condition 3 of Proposition 89 is not satisfied; 2. When the reference trajectory is singular.

The case 1 is encountered in SR-geometry. Indeed, the length of a curve does not depend on the parametrization and the optimal control problem is parametric. This induces a symmetry which has to be taken into account when writing Hamilton-Jacobi-Bellman equation. This will be handled in the next section.

10.5.1 The SR-Normal Case We shall restrict our analysis to the SR-case

4(t) = ul(t)F1(q(t)) + u2(t)F2(q(t)), q(t) E M min e(q) = fT (U2 1(t) + ua(t)) #dt

where F1, F2 are two smooth linearly independent vector fields. In this problem the Lagrangian has the following property L(q, A4) = L(q, 4)

for ). > 0

and the cost is not depending upon the parametrization. The normal case fits in classical Lagrangian singularity if we choose a distinguished

parametrization. Lagrangian manifolds in the normal case. Let P, = (p, F; (q)) i = 1, 2, the Hamiltonian associated to normal extremals is given by Hn = (Pi +P2 ). 2 the folLet t'-. -f(t), t E [0, T] be an injective reference geodesic. We assume lowing:

286

10 Resolution of the Singularity near a Singular Trajectory

Hypothesis. We suppose that ry(.) is strict, that is there exists an unique liftingt [5''] of ry in the projective bundle P'(TM). We use the following notations:

- exp,ytol is the exponential mapping. It is defined by t 1--4 Ir(q(t)) where t i-+ q(t) is solution of Hn starting from y(O) at t = O.We assume that the normal extremals are parametrized by arc-length, that is H = 2 - Lt = expt Hn (Ty(O)M). - Qt: intrinsic second-order derivative evaluated along -Y and indexed by t E]O, T].

The following results are standard.

Proposition 91. 1. Lo = T(O)M is a Lagrangian linear manifold and for each t > 0, Lt is a Lagrangian manifold.

2. The time t > 0 is conjugate along y if and only if the projection it : Lt M is singular at ry- (t).

3. Assume that the geodesics are parametrized by arc-length t and let W = Uo
Proposition 92. 1. The time t > 0 is conjugate if and only if Qt has a zero eigenvalue. 2. The intrinsic second-order derivative is elliptic, that is its spectrum a(Qt) is discrete:

a(Qt)={a;

<...
and accumulates at +oo and Qt can be diagonalized in a smooth Hilbert basis.

3. The Morse-Maslov index of Qt, that is the number of strictly negative eigenvalues counted with their multiplicity, is equal to the number of 0eigenvalues counted with their multiplicity contained in ]0, t[.

Remark 30. - The caustic of Lt can be analyzed using Lagrangian singularities and Proposition 90. - We represent locally W by an Hamilton-Jacobi wave function defined as follows. We integrate the normal flow starting from y(0) and parametrized

by arc-length: Pl + PZ = 1. By setting P1(0) = cos 9, this gives us the family of geodesics:

E : p=

(0,Al,...,An_2,t) E S' x Rn-' -+ M

(10.2)

10.5 Application to Optimal Control Theory and to SR-Geometry

287

and p is eliminated by solving the equation E(p) = q near y(t) using the implicit function theorem. Beyong t1 the computations need the preparation theorem and Legendrian singularity theory, see [15].

Application. The generic conjugate locus in the contact case (see Sect. 9.6) is an astroid and its intersection with a plane z = e is the analytic set represented in figure 10.1. The regular points correspond to folds (A1) and

Fig. 10.1.

the singular points correspond to cusps (A2). Here the analysis is localized near the terminal point ql = y(t1c) of the reference trajectory. More complicated is the germ of the singularity near 0, because the curves formed by the cusps of the astroid are interacting. The analysis is done in [7], where the generating family is computed at 0. The result is the following.

Proposition 93. 1. The germ at the origin of the astroid is not stable. 2. There exists a stable singular S-dimensional manifold W in M x R whose projection on M is the astroid.

10.5.2 Jacobi Fields Consider a smooth Hamiltonian vector field H defining the extremal trajectories of a minimization problem

4(t) =

49P

(q(t),P(t)),

p(t)

OH

(q (t), p(t)).

19q

We denote by V the associated variational equation dq

where

= aq (y(t))dq

(q, p) and' is a reference extremal with qo = a(y(0)).

(10.3)

288

10 Resolution of the Singularity near a Singular Trajectory

Definition 104. The Hamiltonian linear equation (10.3) is called Jacobi equation. A Jacobi field J(t) = (oq(t), op(t)) is a nontrivial solution of (10.3). It is called vertical if bq(0) = 0.

Proposition 94. Let Lt = expt H (TQoM), then the space of vertical Jacobi fields is the tangent space of Lt for t > 0.

Lemma 66. Assume we are in the Cw-category. Let J(.) be a vertical Jacobi field and e -+ a(e) be a C"-curve such that: &(0) = J(0). Let Y be

a C"-vector field on T*M such that Y(' (0)) = a(0). Then t -- J(t) = dexpt Hy (&(0)) is analytic and is given for t small enough by BakerCampbell-Hausdorff formula:

J(t) =

Wiad"

H (Y)(7(t))

n>o

Definition 105. In the previous construction Y can be choosen Hamiltonian. Let (ei) be the canonical basis of Tqo M and Het = -qei. The Jacobi fields can be computed with the space:

8={{He{,{H,Her},

};

which is called Jacobi observation space.

The following lemma is crucial, see [181 for the proof or Proposition 54. Lemma 67. Let 0 < t < t1 where ti is the first conjugate point along then there exists 2n vector fields Ci = (5qi, bpi), G7i = (bqi, bpi) i = 1, n solution of Jacobi equation and satisfying the boundary conditions

5gi(0)=b9i(t)=fi,

j74 i

bgi(t) = bgi(0) = 0, where fi is the canonical basis of R".

j=i

10.5.3 Reduction Algorithm in the SR-Case Assume H = H" = 1(Pl + PZ ). Then due to the independence of the length with respect to the parametrization, one of the vertical Jacobi field is trivial. This is given by the following lemma.

Lemma 68. 1. The solutions of H" satisfy the relations q(t, qi , Apl) = q(At, qi, pi) p(t, qi, Api) = Ap(At, qi, pi ),

A E R\0.

2. Let a(e) = (qo, po + epo) and let Jl be the associated vertical Jacobi field then: 7r(Jl (t)) = ty(t).

10.6 The Singular Case

289

10.6 The Singular Case 10.6.1 A Geometric Remark It is geometrically relevant to compare the Heisenberg case and the Martinet flat case.

10.6.2 Heisenberg Case (see Sect. 9.5)

D = ker a, a = dz + xdy - ydx, g = dx2 + dy2. The line Oz is formed with points where the linearized control system is not controllable: they are trivial singular trajectories. Here the normal geodesics are starting prependicular to Oz, see figure 10.2. The shortest are staying as far as possible from the line Oz.

oz

z=0

Fig. 10.2.

10.6.3 Martinet Flat Case 2

D = ker w, w = dz - 2 dx, g = dx2 + dye. The geodesic equations are the following:

:E(t) = cos0(t), y(t) =sin0(t), z(t) = y 2t) cos8(t) and 9 is given by

0(t) + Asin0(t) = 0.

290

10 Resolution of the Singularity near a Singular Trajectory

0

0(0)--71

0

0(0)-0

Fig. 10.3.

Fig. 10.4.

The projections of the geodesics starting from 0 in the plane (x, y) are the inflexional elastica. We have different shape depending upon 0(0) which can be taken in ] - it, 0[, see Fig. 10.3 with their respective contribution

to S(0, r) n {y = 0} (Fig. 10.4). The case 0(0) , 0 is exogenous to the geometry and is related to nonstrictness of the flat case, but the case 0(0) N -7r is relevant: the geodesics which contribute to the sphere near the end-

point of the abnormal direction are starting tangentially to the abnormal direction but have loops, hence a Martinet sector of S(0, r) is given by Fig. 10.5. Therefore contrarily to the contact situation where the geodesics can

10.6 The Singular Case

291

Fig. 10.5.

be parametrized by z, it is not possible to parametrize by x because the loops. Hence consider the system 9(t) = ur(q(t))Fl(q(t)) + u2(q(t))F2(q(t))

where Fl, F2 are smooth and linearly independent. We shall analyze the time minimal control problem under the following constraints on the controls:

A : affine case: ul = 1,

SR: sub-Riemannian case: ui + u2 = 1. We take a reference one-to-one singular trajectory t '-. y(t), t E [0, T] identified tot i- (t, 0, , 0) and we can assume that u2 = 0 along y. We work in a tubular neighborhood of y(.). Our analysis is based on the results of Chap. 6. Recall the following.

Definition 106. For 0 < t < T let K(t) = Span{adkFiF2k E N) where ir is the standard projection. The space K(t) is the first order Pontryagin cone along y and since y(.) is singular K(t) is of codimension greater or equal to one.

Assumption (Hr) for 0 < t < T, K(t) is of codimension one and generated by { F 2 ,. .

.

,

ad"-1

Fi (F2)1, }.

Rorn Chap. 6, we have the following.

Lemma 69. 1. For 0 < t < T, K(t) is the image of the Frechet derivative of the endpoint mapping at fixed time t, evaluated along the reference trajectory.

10 Resolution of the Singularity near a Singular Trajectory

292

2. If ry- = (py, ry) is an Hamiltonian lift then the adjoint vector p.r(t) is unique

up to a scalar and is orthogonal to K(t) for 0 < t < T.

Definition 107. The initial condition zo = (q(0), py(0)) E T, (o)M (resp. the reference singular extremal j(.)) is called of order two if the following condition holds

0 d2 OHQ

0 at zo (resp. a long-y) au dt2 au #

where H. = P1 + uP2 that is: Assumption (H2) [P2, [P1, P2]] # 0.

Definition 108. The reference singular trajectory y(.) is called exceptional if P1 = 0. It is called hyperbolic (resp. elliptic) if F1 and [F2, [F2, F1]] have the same orientation (resp. opposite orientation) with respect to K. In the previous chapters we proved the following.

Lemma 70. 1. In the affine case, only the hyperbolic or exceptional trajectories are candidate to be minimizers. 2. In the SR-case, only the exceptional trajectories are candidate to be minimizers.

10.7 Resolution of the Singularity in the Hyperbolic Case The resolution is based on the results of Chap. 6 interpreted in the symplectic formalism.

10.7.1 Normal Form From Chap. 6, the system (F1, F2) can be locally written in the hyperbolic case as: a

n-1

- F1 = - + aQ1

a Qi+1

i=2

a9

a

n

+

aij(g1)Qigj

ij=2

+R a91

8 - F2 = oQn

where R can be neglected in our analysis. The singular trajectory is identify to ry

:

t -+ (t, 0

,

0). Then by setting q1 = t + t;1 we can approximate

(replacing q1 by t) the system by

10.7 Resolution of the Singularity in the Hyperbolic Case

293

n

1(t) _ E aij

(t).j (t),

(10.4)

i,j=2 (10.5) 42(t) = 6(t), ... , S.-1 = Sn, Sn(t) = u(t). Equation (10.5) represents the linearized system along the reference trajectory and K(t) is identified to S p a n { Wq_2 , . . . , 3-q. Moreover p.y can be set to (e, 0, , 0) where s = -1 = HaI., and ea,,,,, = (p-f, ad' F2F1), a.,n < 0 in the hyperbolic case.

10.7.2 Intrinsic Second-Order Derivative The intrinsic second-order derivative denoted E" for 0 < t < T is given by t

s

J

q(s)ds,

q =

ai3(s)Si(s)ej(s)

i,j=2

with the following boundary conditions obtained by restricting to the kernel of E': WO)=b(t)_..._W0)G(t)=0.

We write q, with y = 2 as: = q(y)

n

1: bij (s)y(') (s)y(j) (s) i,j=2

where the bij are symmetric.

According to Chap. 6, the intrinsic derivative can be represented as a differential operator introduced below.

Definition 109. Let 0 < t < T and denote by Di the Euler-Lagrange-Jacobi operator: n-2

D1(y) = 2 E(-1)ia yQ)(y) It is a differential operator of order 2(n - 2) of the form: can,ny2(n-2) +

... = 0,

hence it is nonsingular. We denote by Di its restriction to smooth curves which satisfy the boundary conditions: Y(0) _

= y(n-2)(0) = y(t)

y(n-2)(t) = 0.

It is a self-adjoint operator which represents the intrinsic second-order derivative.

Remark 31. It is a representation in Euler-Lagrange form but everything can be translated into Hamiltonian formalism as in the normal case.

294

10 Resolution of the Singularity near a Singular Trajectory

10.7.3 Goh Transformation The direction RF2 is cheap and it is identified in the normal form to 4,,(t) = u(t). To resolve the singularity we take as a new control and we work

on the so-called reduced space which is the quotient space: M/RF2. We denote by Dlt the restriction of Di to the set of smooth curves which satisfy the 2(n - 2) boundary conditions: Y(O)

= ... = y(n-3)(0) = y(t) = ... = y(n-3)(t) = 0. is relaxed) We have the following.

(The boundary condition on

Lemma 71. 1. In the reduced space M/RF2 where x,a(.) is the control, the operator Dit is a regular differential of order 2(n - 2) and hence is elliptic. 2. In the initial space, the intrinsic second order derivative has a continuous spectrum which can be approximated by Dirac-type control u = na on [a,a+ n], -na on [b,b+ n], see Fig. 10.6.

__..__F[

b

10.8 Lagrangian Manifolds in the Hyperbolic Case Lemma 72. Under the assumption (H1), (H2), the singular arcs are solutions of the following equations

9=

8Ha 8p

,

P=- 8Ha 8q

where Ha = Pl + uP2, u = - P3, P_'P'

,

P2={Pi,P2}=0

10.8 Lagrangian Manifolds in the Hyperbolic Case

295

Notations. Let 0 < t < T we introduce the following. We denote by y'(.) the projection of y(.) on the reduced space M' = M/RF2 and q' = f (q', the associated Hamilthe projection of the afIlne system, HQ = (p', f (q', tonian lift. The Hamiltonian Ha is regular along -y' that is satisfies the strong Legendre condition and the singular flow is obtained by solving ee = 0 and

defined an Hamiltonian vector field denoted H. . We note

- L= : the image by expt H. of the fiber T. (O)M. 1

- Lt' : the image by expt H. of the fiber T7, (o) M. 71

-It :theimage byexptH.ofT7(o)+RF2Mn(P2={P1,P2}=0). - £t (y(0)) and £ta(y'(0)) the respective projections of I, and Lt'a on M and M'= M/RF2. Proposition 95. Let t E]0, T] then: 1. LL (resp. Lta) is a Lagrangian manifold in T'M (resp. in T`M/RF2) and It is an isotropic submanifold in T'M. 2. The reference singular trajectory y(.) (resp. y'(.)) is transverse to £, RF2 (resp. £ta(y'(0))) To construct the optimal synthesis along y we proceed as follows, see Chap. 6.

Definition 110. A generalized trajectory is a finite concatenation of jumps in the cheap direction RF2 and arcs solutions of the system.

Definition 111. A time tc is said to be conjugate to 0 along the reference extremal y if there exists a non trivial solution J of Dlt'° = 0.

Proposition 96. Let 0 < t < ti, where t1 is the first conjugate time along y (for the operator Di). T7ten:

1. The reference trajectory is C°-time optimal in the class of generalized trajectories. 2. In a tubular neighborhood of y(.) the time optimal synthesis is the con-

catenation of a singular arc and a jump at the initial time and at the final time. It is the limit of the time optimal synthesis with the constraint IIu211 <_ M when M - +oo, see Fig. 10.7. S. In the reduced space Ut>0£ta(y'(0)) is a time optimal central field in a C'-neighborhood of y'(.). 4. Assume that t > tlc, then the trajectory y'(.) is no more CI -time optimal.

296

10 Resolution of the Singularity near a Singular Trajectory

singular

arc

M=+ oo

I U) < M<+ 00

Fig. 10.7. Remark 32.

1. The Hamiltonian H. is linear with respect to p and an extremal is represented as a curve on the projectivized cotangent bundle P(T*M). 2. The concept of conjugate point along the singular arc is equivalent to the concept of focal point for the following problem: mint for system (F1, F2) subject to boundary conditions: q(0) E RF2,

q(tr) E RF2.

10.9 Examples The hyperbolic case corresponds to a singularity of codimension one and the geometric setting is clear: we must replace M by M/RF2 and we need to add to the family C;, Vi of lemma 67 two jumps Si, 3;, one at the initial time and

one at the final time, to match the boundary conditions with respect to the variable q = exp tF2i, . It is a boundary layer phenomenon. Hyperbolic trajectories play no role in SR geometry but they have a direct consequence when we consider SR problem with drift: rT ¢(t) = F1(q(t)) + u2(t)F2(q(t)),

min

u2(t)dt

J0 where T is fixed. This is illustrated by the following.

Example 1 (Working example 1). We consider the problem t(t) = 1 - y2(t), 1/(t) = u(t), min

10

T u2(t)dt.

10.9 Examples

297

Without loosing any generality, we may assume T = 1. The line t ' -, (t, 0) is singular. We note A(0,1) the accessibility set from 0 at time 1. It can be computed near (1,0) end-point of the singular line. If we impose the constraints Jul < 1 it is represented on Fig. 10.8, together with the limit case u E R. If Jul < 1, its boundary is the graph: x = 1 - iyi which is C2 but not C3 at (1, 0) and whose tangent space is Rev (jump direction). If u E R, it is closed near (1,0) in the class of generalized controls. Consider now the

s

x

Fig. 10.8.

problem: mn

j

u2(t)dt. The admissible controls are L' and if we restrict

ourself to L°O, the normal optimal trajectories are solutions of the maximum principle:

x(t) = 1 - y2(t) J(t) = pu(t)

P1 (t) = 0 Pu(t) = 2y(t)px

where the control is u(t) = pu(t). The abnormal line is not strict and can be immersed in the above flow by setting: pu =- 0. If we set px = 2 the extremal trajectories can be projected onto the linearized pendulum:

X0 - Ay(t) = 0. When A > 0, we are near a saddle point and the solutions are hyperbolic

functions. When A < 0, we are near a center and the solutions are trigonometric functions. Integrating with y(0) = 0 we get for A < 0: y(t) = A sin(vt) and for A > 0, y(t) = Ash(f t). Fixed t = 1, the limit between the two cases is given by A = 0 and form the parabola x = 1 - s see Fig. 10.7 (i). The end-points of minimizing trajectories are represented on Fig. 10.9 (ii) at fixed cost r. The base point S = (1, 0) of the pencil is obtained at the limit by starting tangentially to the abnormal direction and

298

10 Resolution of the Singularity near a Singular Trajectory

with A -+ +oo. Hence we are loosing properness due to saddle pass in the pendulum equation.

S

(1)

Fig. 10.9.

Proposition 97. Let T = 1 and r > 0 small enough. Then the level sets r of the value function are homeomorphic to a pencil of circles where the base

point S = (1,0) is excluded. In particular they are not closed. The closure of a level set has two singularities located on the abnormal direction x: SI and S; SI is semi-analytic and S is not sub-analytic. The abnormal line is formed by cut points.

Definition 112. The singularity of the level sets in S, associated to a one saddle pass is the simplest non sub-analytic singularity due to the existence of abnormal minimizers. It will be denoted: NPHI. Example 2. We consider a system (FI, F2) in 1R3, q = (x, y, z). Along a singular extremal we have (p, F2 (q)) = (p,[Fj, F2] (q)) = 0

and

(p, [[F1, F2), F2)) + u(p, [[F', F2), F2J) = 0.

We note D = det(F2, [F1, F21, [[F1, F2], F2]), D' = det(F2, [F1, F2), [[F1, F2), F1))

and

10.9 Examples

299

D" = det(Fl, F2, [F2, F1]).

The singular trajectories satisfying (H1), (H2) are solutions of q(t) = F,(q(t))

where F,(q) = Fj(q) + u,F2(q), u, _ -D'. The exceptional trajectories are contained in D" = 0. The vector field F3 is the characteristic vector field defined by the one form

(p, F, (q)) = 1, (p, F2 (q)) _ (p, IF,, F21 (q)) = 0

in the domain S2 = flt3\{D" - 0}. Given any singular arc, the two vectors F2 and [F1, F2] are generating the first order Pontryagin cone in the domain where both vectors are independent.

The Jacobi operator Dit is of order 2 and the vertical Jacobi field is generated by a single vector denoted J(t). By definition, along t i-s y(t) it is the derivative of the curve ,Q(s) = exptF, o expeF2(y(0)) evaluated at 0.

It can be written /3(E) = exp tF, o exp eF2 o exp -tF,(y(t)) = exp tF, * eF2(y(t)) = C E.>o ri ad"F,(F2(y(t))) for t small, in the analytic case.

Since F, is singular, Span{ad"F,(F1); n > 0} along y is the first order Pontryagin cone and is in the domain .f2 the space Span{F2i [F1, F2)). We represent J(t) along y on Fig. 10.10. At t = 0, J(0) = F2(-y(0)) and the first

IF,,F2] (Y (t) )

J(t)

Oft)

Fig. 10.10. conjugate time t1 is the first t > 0 where J(t) becomes collinear to F2(y(t)).

300

10 Resolution of the Singularity near a Singular Trajectory

The hyperbolic trajectories in .fl form a one-dimensional foliation of the

domain DD" > 0. If we denote by iP the angle of J we get an unitary representation of the Lagrangian manifold. The Jacobi equation can be written in the normal form: u"+K(ry(t))u = 0, where K corresponds to the curvature of the space in the hyperbolic sector (see Sect. 6.3) associated to the time optimal control problem. This analysis is a first step in analysing the higher-order singularity. We must distinguish two situations: - The case of exceptional trajectories; it is the geometric situation where F1 is contained in the first-order Pontryagin cone and the reference trajectory is not transverse to the surfaces EL introduced in Sect. 10.8. This singularity is invariant with respect to the singular flow and will be analyzed in Sect. 10.10.

- The case of broken singular trajectories (see Sect. 8.1); we meet this situation when the Lie bracket [F2, [F1, F2]] cross the first-order Pontryagin cone. Then D(q) = 0 and hence in order to have L°°-singular controls we must have D'(q) = 0. In the generic case the situation is the following: the set D fl D' = 0 can be locally identified to a segment and the behaviors of the singular trajectories is represented on Fig. 10.11. The singular arcs

hyperbolic sector

DA D'=0 elliptic sector

Fig. 10.11.

meeting D fl D' = 0 are the stable and unstable of a reparametrization of F,. The geometric situation is the following. An hyperbolic arc H1 can pass through Dfl D' = 0 and become elliptic: El or can be reflected and remain in the hyperbolic sector: H2. In both cases, since in the domain 1? the Pontryagin cone is generated by F2 and [F1, F2] both arcs H1H2 and H1 E1 are extremal. The curvature K can be evaluated and has a pole.

Its evaluation requires a time renormalization.

10.9 Examples

301

The operator D' associated to the intrinsic second order becomes singular. We can make some remarks concerning its spectrum. In the elliptic sector, the are is time maximal, and its index is infinite: it is a consequence

of our spectral analysis. Indeed along Hl El the second order derivative

has a spectrum which accumulates both at +oo and -oo. For the arc Hl H2 the optimality analysis is more complicated, see Sect. 8.8: the arc Hl H2 although it is contained in the hyperbolic sector is not time minimizing. The spectral analysis comes from singular operators of the form: C(t)U(k) (t) + = 0 where e(0) = 0. It is related to the following important phenomenon concerning the Jacobi

field associated to the problem. the Lagrangian manifold splits when passing the saddle and we get sectors, see Fig. 10.12. In particular there is

IF;,F21

J(t

\ /

J(t,)

1

J(t2)

F

J'(t2) Fig. 10.12.

a shift in the phase. Here the computation is simple but it is important to make a general theory and a general research program is the following. Problem. Make a general theory for the Jacobi fields when passing the corners. The invariants of this theory are: 1. The separatrix lines (C°° or 0'). 2. The speed along the separatrices which allow to evaluate the limit of the

curvature along each separatrix. This computation is related to 16-th

Hilbert problem. Example 3 - Open problems (Working example 2). In the hyperbolic case the model is given by

i(t) = 1 + a(t)z2(t) + 2b(t)y(t)z(t) + c(t)y2(t)

X0 = z(t) z(t) = u(t)

10 Resolution of the Singularity near a Singular Trajectory

302

where a(t) < 0. In order to make a Morse theory concerning the second variation we must enlarge the set of controls and add Dirac type controls to jump

in the direction RF2. Hence we get necessary and sufficient conditions in

this framework and the associated Lagrangian manifolds are smooth. An open question is to analyze what happens when we consider uniformly bounded controls. The problem is here to work in the L°°-topology. A simplified working example is to consider the model

-

±(t) = 1 - (z2(t) y2(t)),

y(t) = z(t),

z(t) = u(t).

Here the Jacobi equation is u" + u = 0 and conjugate points occurs at time t, = kv. The structure of extremals must be analyze for the problem: rT min

t

u2(t)dt, see (107).

0

10.10 Resolution of the Singularity in the Exceptional Case It is a situation of codimension two, met in the affine case or in SR-geometry. We need an additional assumption along the reference extremal:

Assumption (H3) We assume that F111 belongs to KI1 but not to Span{adkF1F2,.,;

10.10.1 Normal Form - Intrinsic Derivative We use the results of Chap. 6, in the exceptional case. The system (F1, F2) is written as n B n-2 - F1 = 8 Bqi +

i=1 4i+1 aq,

i,,=2 aii(g1)giqjq +R

- F2 = NO- I where an-1,n-1 > 0 on (0,T) and R can be neglected in our analysis (see Chap. 6).

In this representation the reference trajectory is identified to 'y

:

t --

(t, 0 , 0) and is associated to u2 = 0. We have two choices for the adjoint vector: p.y = (0, . - , 0, ±1). The first order Pontryagin cone along ry is KI.1 _ Span{

- , . . . , 8q

}and by definition F1 1, E K1,. The intrinsic second

order derivative is identified to wt n-1 JU

where £1(t) = 6 (t), -

E

i,,.)=2

, Sn_ 1(t)

= u2(t) and with zero boundary conditions.

10.10 Resolution of the Singularity in the Exceptional Case

303

10.10.2 The Operators The jump direction given by Goh transformation is here identified to Rte. with 41(t) We consider the quadratic form: q = En,,,=2 1;2(t),- -',4n-2(t) = n_1(t) written in a symmetric form n-1 q =

i,j=2

We denote by Di the associated Euler-Lagrange-Jacobi operator (see definition 109) and we denote by D1t the self-adjoint restriction to smooth curves which satisfies the boundary conditions:

Sl (0) _ ... = bn-2(0) = S1(t) _ ... = G-2(t) = 0. We observe that q is not depending upon 1;1 and hence we can introduce a differential operator D2 of order 2(n - 3) by setting

ed td

D'

dt

D2

dt

and we denote by Da the self-adjoint restriction of D2 to smooth curves which satisfy the boundary conditions:

Y(0) = ... = y(n-4)(o) = y(t) _ ... = y(n-4)(t) = 0. Both operators Al and DZ are regular. The concept of conjugate point in both affine and SR-cases are taken in the following sense, see Chap. 6.

Definition 113. A time tcc is said to be conjugate to 0 along the reference exceptional extremal y(.) if there exists a nontrivial solution J of D2 «J = 0.

10.10.3 Isotropic Manifolds in the Exceptional Case Lemma 73. Under the assumptions (H1), (H2), (H3) the exceptional extremals are solutions of the following constrained Hamiltonian equations:

q=

8Ha 8p ,

p=-

8Ha 8q

where H. = P1 + uP2, u = -

Pi =P2={P1,P2}=0

H,,Hz H2

10 Resolution of the Singularity near a Singular Trajectory

304

Notations. For t E]O, T] we set:

- 1a = exp t H. (7(0)) (here we take the restriction to P1 = P2 = [P1, P2] _ 0)

- Pt'E =7r(I='e(7(0))) We have the following lemma.

Lemma 74. For t E]O, T], I," is an isotropic manifold. The optimality status is the following.

Proposition 98. 1. In the affine case, the exceptional trajectories are C°-optimal up to the first conjugate time t1cc, in the class of generalized control. 2. In SR-geometry, there exists r > 0 such that an exceptional extremal of length < r is a global minimizer. Such an r is uniform for all exceptional trajectories contained in a tubular neignborhood of the reference trajectory. Remark 33.

1. Conjugate points along exceptional trajectories can occur only when

dimM>4.

2. The computation of conjugate points used in fact the second variation of

the time extended end-point mapping. 3. It is related to a concept of focal points where the intial and final manifolds are the two dimensional plane: D = Span{Fi, F2} and the constrained P1 = P2 = 0 are interpreted as transversality conditions with respect to D.

Example 4 (Working example 3). Let M = U neigborhood of 0 E R4 and let D = Span( F1, F2) be two dimensional distribution. We shall say that DE is of Engel-type if. - [DE, De] has dimension 3 on U, - [DE, [De1 De]] has dimension 4 on U. From [35] there exists local coordinates (w, x, y, z) such that the distribution DE is the span of the vectors fields G2

.

-+za +w -, z GI = y

a

.

Consider now the system (F1, F2) defined on a tubular neigborhood T of

'Y: t '- (t,0,0,0) by

10.10 Resolution of the Singularity in the Exceptional Case

F1 = (1 + q2)

a

a

2

305

a

2

(q3 - q2 ) aq4 agl + q3 a92 + F2

43

Then D = Span{Fi, F2) is locally isomorphic to De. We set qi (t) = t + fi(t), 4(t) = 92(t), 42 (t) = q3(t) and the operator Di is the Euler-Lagrange operator associated to e(t) = q2(t), 42(t) = q3(t) = u(t), min ZO T (u2(t) u(.)

- g2(t))dt.

H = p4g2 + p2u - 1(u2 - q2) and from

= 0 we get u = p2 and the Hamiltonian is

H=p{q2+p2u+ 2+ 2

22. 2

The Hamilton equations are

t(t)

= q2(t) 42(t) = q3(t) = p2(t)

pt(t) = 0 p2(t) = -pt - q2(t)

By differentiating we obtain that Dt is given by the operator

(4)(t) and D2 is computed using Di = as D2 We set q(t) = 4(t) and conjugate points are obtained by solving aL

tj(t) + ra(t) = 0.

They correspond to to = k7r and let i7(t) = c sin t be the solution which vanishes at tie = ir. Then fi(t) = -ecost + (0) and q2(t) = esint, q3(t) _ e cos t and u2(t) = 43(t) = -F sin t. Integrating we get ir

q4 (7r) =

r e2(cos2 t - sin2 t)dt. 0

Hence the corresponding value of the end-point mapping is

gl(7r)=7r-a q3 (r) _ -E

g2(ir)=0 q4 (7r) = 0

306

10 Resolution of the Singularity near a Singular 'fl ajectory

where ql(0) = 92(0) = q4(0) = 0, q3(0) = e and corresponds to u2(t) _ -e sin t. We have 43(t) = u2(t). Let 1 small enough and 1u21 (rl. Iff q3(0) = 0, we have q3 (t) of order rat and for t small, using 44(t) = q32 (t) - q22 (t) we have q4(t) is of order r12t3. Hence if tlt = E, q4(t) is of order £2t, where t is small. In particular one gets the following Lemma [35] (see also example 10.9).

Lemma 75. If t > ir, then the reference trajectory y : t .-+ (t, 0, 0, 0) is no more isolated in the C1-topology among the set of curves q(t) = D(q(t)).

Remark 34. If we consider system q(t) = ul(t)C1(q(t)) + u2(t)G2(q(t)) a simple computation proves that the singular trajectory -y : w = t, x = y = z is C1-rigid. But the normal form is only valid for local coordinates around 0 and hence it corresponds to a local Cl-rigidity result.

10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius - Lagrangian Stratification of the Martinet Sector 10.11.1 Preliminaries In order to construct the optimal synthesis along a reference trajectory we must use a symplectic stratification using both normal and abnormal extremals.

Abnormal extremals. They are solutions of the constrained Hamiltonian equations:

dgMHQ dp8Ha

8q dt 0p-' where H,, = u1P1 + u2P2i the singular control being defined by u1 {{P1, P2}, P1 } + u2{{ P1, P2}, P2} = 0

and they are contained in the set

P1=P2={P1,P2}=0. Let y(.) be a reference abnormal extremal. We identify the initial condition y(0) to 0. The abnormal trajectory is extended onto the abnormal line -y(t), t E [-T, +T]. We assume that assumptions (H1), (H2), (H3) are satisfied along -y and we denote by p.1(0) the associated adjoint vector taken in P(T'M). We can assume that -y(.) is parametrized by arc-length and we assume T small enough. Hence M can be identified to a small neigborhood U

10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius

307

of 0. Our analysis is micro-local and we choose a neighborhood V of p.,(0) in P'TOM such that all the abnormal extremals starting from 0 x V are satisfying (H1), (H2), (H3). According to Proposition 98, there exists r > 0 such

that each abnormal extremal starting from 0 x V are global minimizers. We denote by E, the sector of U covers by the projections of abnormal extremals with length < r and starting from 0 x V. This defines a mapping denoted Exp. The construction is represented on Fig. 10.13.

Exp Y

E

oXv Fig. 10.13.

Normal extremals. They are solutions of the Hamiltonian differential equations:

q(t) =

a -n(q(t),p(t)),

p(t) _

M-2 (q(t),p(t))

where H = 2(P, + P2) and we assume that they are parametrized by arc-

length. We consider the trajectories starting from qo = 0 and we denote by exp the associated exponential mapping.

10.11.2 The Smooth Abnormal Sector Lemma 76. Fort r small enough the set E,. is a smooth sector of the SR-ball homeomorphic to C U -C where C is a positive cone of dimension n - 3 if n > 4 and I if n = 3. Its trace on the sphere is formed by two smooth surfaces of dimension n - 4 if n > 4 and reduced to two points if n = 3.

10.11.3 The Smoothness of the Sphere in the Abnormal Direction Proposition 99. Let A be the end-point of the abnormal direction and let K(r) be the first order Pontryagin cone evaluated at A, r small enough. Then K(r) is of codimension one and y(.) is strict. Let a a(e) be a smooth curve on S(0, r), a(0) = A, e > 0; assume the following:

308

10 Resolution of the Singularity near a Singular Trajectory

1. a(e) C S(0, r)\E,. for E # 0. 2. a(E) fl L(0) = 0 where L is the cut locus.

Then the tangent space to the sphere evaluated at a(e) tends to K(r) when e -- 0. Proof (Sketch of the proof (for the details, see [107])). We reparametrized the geodesics on [0, 1] and to construct S(0, r) we restrict ourself to control u E L2([O,1]) such that IIuIIL3 = r.

let A(e) be a point of a(e), e # 0, by construction A(E) is the end-point of a normal geodesic qe : q,(1) = A(E). Let E = n, then we can associate to q,(.) denoted qn(.) an extremal normal control un(.) and an adjoint vector

We claim that Ilpn(1)II -' +oo when n - +oo. Indeed if not, taking a subsequence we can assume that pn(l) converges to p(l) as n - +oo. Let E be the end-point mapping. Then by construction we have the following equality on L2([0,1]): (10.6) pnE'(un) = un. Now the basic result that we need is the compacity of minimizing controls in L2([0,1]) (see [1]). Hence since A(e) tends to the end-point of the

abnormal geodesic associated to the abnormal control u.y(.) we can extract a subsequence such that un - u in L2([0,1)) as n +oo. Secondly the end-point mapping E is smooth if we endow the set of controls with the L2-norm. Then we get taking the limit in (10.6) the equality

p.E' = +u. Hence p is a normal adjoint vector associated to -y(.), which contradicts the fact that y(.) is strict. We end the proof as follows. We consider the end-point mapping k of the extended system 4(t) = ul (t)F1 (q(t)) + u2(t)F2(q(t)),

4°(t) = (ui(t) + uz(t))

and let A(E) be the end-points in the extended space. Then by definition the tangent space at A(E) of S(0,r) x r is Im E'(un) and gin(1) = (p,,(1), -2) is a vector normal to the previous space. Dividing by Ilpn (1) II we get that P. (1)

E'(un) _ +

I1pn(1)II

un (10.7)

IIpn(1)II

By the same reasoning as above we have p-(1) Ipn(1)II

--, p(1) IIp(1)II

as n - +oo

where p is the adjoint vector associated to the abnormal trajectory.

10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius

309

We have

p(1)J-Im E'(u) and since IIp,,(1) II - +oo as n - +oo we have from (10.7)

Im E'(u) In particular we have the following general result to be compared with the analysis of Chap. 9 of the Martinet case.

Corollary 22. Using the notations of the previous proposition, the curve a(e) is not the image by the exponential mapping of a compact subset of the cylinder: P12(0) + P22(0) = 1. In particular the exponential mapping is not proper in the abnormal direction.

10.11.4 The Martinet Sector In order to make the stratification of the Martinet sector, we need to use the pendulum representation of the geodesic equations. We shall restrict our analysis to the 3-dimensional case which is the Martinet case. We recall the following from Chap. 9.

Notations. We denote by q = (x, y, z) the coordinates in R3, D = ker w, w = dz - dx and the metric is taken in the normal form: g = a(q)dx2 + We shall restrict our c(q)dy2, a2= (1 + ay)2 + , c = (1 + 'OX + yy)2 + analysis to the case of order 0 but it can be generalized. The abnormal line

through 0isL:ti-+(±t,0,0)andletF1 = 1 completed by F3 =

to form on orthonormal frame with D = Span{F1i F2}.

(Pl + P2) and we parametrize by arc-length

Let Pi = (p, F2 (q)), H. = Hn _ -

a + 2j),F2= I a

2

12'

We introduce

P1 =cos9, P2=sin0, P3=A. The exponential mapping is defined on the cylinder (0, A) E S' x R and we can assume A > 0. The geodesic equations are reparametrized by

s = rOa- `fc- dtd _ drd where t is the arc-length. The angle 0 is solution of the pendulum equation oil

+sin 0+e0cos00'+e2asin0(acos0- ,3sin0)=0, e=

71.

Moreover by cutting by y = 0, this induces the following section S:

Of = e(acos0+Qsin 0). The pendulum equation defines a one dimensional foliation in the cylinder S1 x IR represented on Fig. 10.14 in the space (0, 0'). We recall the following.

310

10 Resolution of the Singularity near a Singular T ajectory

Proposition 100. 1. The abnormal line is strict if and only if a 34 0. 2. The pendulum equations are integrable (using Jacobi elliptic functions) if and only if /3 # 0.

+7t

+7L

00, E small

0=0

Fig. 10.14.

Definition 114. We call a Martinet sector C the intersection of the ball B(0, r) with any plane z 54 0 containing the abnormal line.

In the sequel we shall take the plane y = 0, this case being analyzed in full details in Chap. 9.

Proposition 101 (strict case: a# 0). 1. A Martinet sector is the image by the exponential mapping of a noncompact subset of the cylinder (A -' oo). 2. It is homeomorphic to a conic sector centered on the abnormal line L and in particular it is simply connected. 3. It is foliated by leaves Z1, c2 in the sphere S(0, e), e < r which glue along the abnormal line according to Fig. 10.15.

Proof. Construction of the Martinet sector. To understand the stratification 0, geodesics we use the pendulum representation. In the strict case or starting from 0 project onto (0, 0') onto: - oscillating trajectories,

- rotating trajectories,

10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius

311

C

L

Fig. 10.15.

Z

exp C31 C]

c2

Fig. 10.16. see Fig. 10.16 where S is the section induced by y = 0: 0' = e(a cos 0+,6 sin 0).

The curves c; ramifying at (-r, 0) end-point of the abnormal direction are constructed as follows.

- The curve c2 is the image of the curve C2 associated to small displacements of rotating trajectories localized near the saddle point. It corresponds to a singularity denoted NPH1 in Definition 112.

- The curves cl and c3 are associated to large displacements and we have to consider two curves respective image of C2 and C3 corresponding to oscillating and rotating trajectories. Among the two curves cl, c3 only one belongs to the sphere and is denoted cl (on Fig. 10.16 it is the curve c3). About contacts. From Chap. 9, the contact being the curves are computed

in the polynomic category. All curves ct are tangent to the abnormal direction. Here the Pontryagin cone K(r) at (-r, 0) along the abnormal line is the plane (x, y). The estimates are the following: Z = ., X = err, r small we have:

312

10 Resolution of the Singularity near a Singular Trajectory Cl, C3 : Z = (1 + 0(r))X 3 + o(X3)

C2 : Z = -r a2X2+O(X2)

Definition 115. A singularity of the sphere corresponding to two saddle passes (as in Cl and C3) will be denoted NPH2.

10.11.5 Symplectic Stratifications In order to describe the sector we need 3 strata: Stratum 1. It corresponds to the isotropic space 1, associated to abnormal extremals and is parametrized by

Sa=exptHo. (TTUn(Pi =P2={Pl,P2}=0)). Stratum 2. It corresponds to the Lagrangian manifold associated to normal extremals projecting onto rotating trajectories of the pendulum. It can be denoted

Sn = exp t Hn (T0'u n rotating). - Stratum 8. It corresponds to the Lagrangian manifold associated to normal extremals projecting onto oscillating trajectories. It can be denoted

Sn = exp t Hn (To U n oscillating).

The transition between rotating and oscillating trajectories corresponds to behaviors on the separatrices. Remark 85. In the previous stratification the Lagrangian spaces are not represented by the so-called generating family because the exponential mapping

is not sub-analytic and we do not have a preparation theorem to resolve the singularity in general.

10.11.6 Transcendence of the Sector A Martinet sector is a non sub-analytic sector and we need the hyperbolic functions in order to compute the SR-distance. In the integrable case of order 0, where f3 = 0 the computations used the exp-!n category which is an extension of the sub-analytic category. An important result is the existence

of a preparation theorem in this category, see [78] in the general case or [29] in the SR-case. The computations of the sphere and hence of the boundary of the sector is complicated because when cutting by t = r there is

a phenomenon of compensation which can be explained by the microlocal invariants of the sector.

10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius

313

Micro-local invariants of the Martinet sector. Consider the pendulum equation written as

0'=v V' = - (sin 0 + E/3 cos Ov + e2ce sin 0(a cos 0 - /3 sin 0))

where a is a small parameter. It has two singular points MI = (0, 0) and M2 = (0, 7r). The local analysis is a follows.

1. Near MI. The linearized system is a focus whose eigenvalues are

of =

-c o ± 2i 1 + e2(a2 - ,) 2

It is a perturbation of the linearized pendulum of the flat case 0" + 0 = 0. 2. Near M2. The linearized system is a saddle whose eigenvalues are

fi/3±2 V 1+E2(e -a2) of = 2

and the saddle is resonant i.e.

E Q if and only if /3 = 0, that is if the system is integrable. This spectrum can be written

nt=f1+2+o(1) and e = .

.

It is a perturbation of the flat case n = f1 which is reso-

nant. In order to compute the sector we use the whole spectrum band which is clearly stable by perturbation. But in order to compute the sphere we have to take into account an interaction between all the eigenvalues to compute an average and this is much more complex and unstable.

10.11.7 The Micro-Local Analysis of the Whole Martinet Sphere To conclude this section we discuss the description of the generic Martinet sphere. It is represented on Fig. 10.17, in the northern hemisphere, where the equator is formed by end-points of geodesics corresponding to A = 0. The cut locus is formed by two branches MCI, MC2 which ramify at M, end-point of the abnormal direction and CI, C2 are conjugate points. We observe in fact

three kinds of sectors: - Riemannian sector located near the equator A = 0, such a sector being represented near R. - Contact sectors located around C where C is a cut point.

314

10 Resolution of the Singularity near a Singular Trajectory

Fig. 10.17.

- A Martinet sector around M where M is the end-point of the abnormal line.

The micro-local invariants of the SR-ball are the following:

- Spectrum band corresponding to the focus M1. - Spectrum band corresponding to the saddle M2. - Invariants connected to the family of Riemannian structures a(q)dx2 + c(q)dy2 induced on the plane (x, y), where z is taken as a parameter.

Notes and Sources For the analysis of Lagrangian singularities, see Hormander [57] and Arnold's school [15]. The generalization to SR-geometry in the contact case is due to

[7]. For the concept of Jacobi fields in control theory see [26] and [2]. The stratification in the SR-case is based on computations from [23] and [29]. The smoothness of the sphere in the abnormal direction (Proposition 99) is a general property of the accessibility set along an abnormal direction, see [27] for examples and Trelat for a general statement [107].

Exercises 10.1 (Engel systems, see [34]) Let D = Span{Fl, F2) be a smooth two dimensional distribution defined on 0 E U C R4. Let us denote D2 = Span{D, [D, D]}, D3 = Span{D2, [D, D2]}.

10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius

315

We call D an Engel distribution if: dim D = 2, dim D2 = 3 and dimD4 = 4, for each q E U.

1. Prove that there exists local coordinates (w, x, y, z) such that D is the span of

X = ax

+z- +waz, W= y

5tv.

Such a coordinates system is called an Engel chart. Let

w, = dy - zdx, W2 = dz - wdx, prove that D is spanned by wi = w2 = 0.

2. Show that the singular trajectories are the leaves of S = . 3. (Symmetry group) a) Consider the two diffeomorphisms: V : (w, x, y, z) '-' (w, x + xo, y, z),

(w,x,y,z)'-'' (w+wo,x,y+yox+zox+ 2wox2,z+zo+wox). Show that for i = 1, 2:

cp*wi=wi, V) *wi=wi b) Let d be the dilation: (w, x, y, z) '-, (rw, x, ry, z). Show that d preserves D. 4. Let 7 = (t, x(t), y(t), z(t)) be a smooth curve defined on 10,1], tangent to D and such that x, y, z vanishes at the extremities.

a) Show that y' = zx', z' = tx'. b) Prove the existence of h, h(0) = h(1) = 0 such that x = h', z = th' - h. c) Show the following: it

y(t) = 2th 2 - hh' + 2 J h'2(s)ds 0

and h' = 0. d) Deduce that y = (t, 0, 0, 0). 5. Prove that for any p = (wo, xo, yo, zo), q = (wi, xo, yo, zo) the singular curve is up to reparainetrization the unique curve y(.) joining p to q, tangent to D such that y * dw > 0. 6. Let D be the distribution defined for x, y, z small by dy - zdx = cos 9dz - sin 8dx = 0.

a) Show that b is spanned by

316

10 Resolution of the Singularity near a Singular Trajectory

Xl

=cosOa +zcosOa +sin0az y

X2

= ae

b) Prove the following:

(i) D2 = ker (dy - zdx) (ii) D is an Engel system (iii) The singular trajectories are the leaves of j c) Let 7 : 0' - (0, x(9), y(0), z(0)) be a curve defined on [0, P] tangent to D and (x, y, z) vanishing at 0 and P. (i) Using 0 = cos 0dz - sin Odx show the existence of a smooth function h(0) defined on [0, PI which satisfies the equations:

zcos0-xsinO=h, zsinO+xcos0=-h'. with dy = zdx - xdz prove the relation y(O) = h(0)h'(0) +

fe

(ii) Assume that P < ir, prove that h = 0. (iii) Assume that P > 7r, construct a 1-parameter family of functions ha(0) such that the above formulae are valid. d) Deduce that the singular line y : 01-* (0, 0, 0, 0), 0 E [0, P] is C1-rigid if and only if P < Tr.

e) Compute along -y the differential operator DZ introduced in Sect. 10.10.2 and the corresponding conjugate points.

f) If T is a tubular neigborhood along y prove that DZ can be written as J" + K(y(t))J = 0 where -y(.) is a singular trajectory contained in T. The function K defined on T is called the curvature tensor (in the exceptional case).

10.2 (Goursat systems in dim 5) Consider the two-dimensional distribution D in 1R5 such that

dim D2 = 3, dim D3 = 4, dim D5 = 5. 1. Compute a local coordinates system. 2. Compute the singular trajectories.

10.3 (Cartan system in dim 5) Consider on 1R5 a two dimensional distribution and assume

dim D2 = 3, dim D3 = 5. 1. Compute a local coordinates system. 2. Compute the singular trajectories. 3. Compare Goursat and Cartan case.

10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius

317

10.4 Consider the problem:

x(t) = 1 - y2(t), y(t) = u(t), min J T u2(t)dt 0

where T is fixed to 1, x(0) = y(0) = 0. 1. Show that the normal extremals are solution of±(t) = 1 - y2(t) px(t) = 0 py(O) = 0

py(t) = 2pxy(t)

y(t) = py(t)

associated to the Hamiltonian Hn(4,p) = px(1 - y2) + Zp2y.

2. Set px(0) = 2 and integrate the previous equations. If A > 0, show that the solutions starting from x(0) = y(O) = 0 are given by

y(t) = Ash(v't)

pxt =

A

2

py(t) = Av5ch(V t) x(t) = t(1 + 22)

-

A2

sh(2v' t)

(10.8)

3. Let A > 0, y(0) = py(0) > 0. Show that the level set r1

Cr = min

u2(t)dt = r > 0

J0

is given by the relations 1

A

2r

(

- Lsn2f 2

y(l) = Ashv5

+1]

a

X(1) = 1 + 22 4. Deduce that near the end-point M = (1, 0) of the abnormal line the union of M with the level set C, is a graph of the form 4

X(1) = 1 - - + 3y2 exp( 4r) + o(y2 exp(

yr) ).

Prove that near M in the sector A > 0 the value function is given for

x ,-

1 by y4

y4

_y2 S(x,y)=41-x+1-xexp(1-x)+o(y

4 IexL) - x ).

318

10 Resolution of the Singularity near a Singular 'Ilajectory

5. Describe near M the stratification of the value function.

6. Compute the Jacobi equation and discuss the existence of conjugate point. 7. Represent the wave front and analyze its singularities.

10.5 Let L be a smooth Lagrangian such that L(x, >d, t) = \L(x, t, t), A E 1R\{0}.

1. Show that

Lz2.i = 0.

(i, L±) = L,

2. If p = e deduce the following:

H=(p,i)-L=0,

detLxx=0.

10.6 Consider the Riemannian problem in the plane

mint

4(t) = ul(t)FI(q(t)) +u2(t)F2(q(t)),

where q = (x, y), F1 =

7X

,

F2 =

TV-

T

I ui(t) + u2(t) I Idt

where q(0) E M: parabola s

(x(s),y(s))

1. Use the maximum principle to write the extremals of the problem. 2. Compute the caustic. 3. Represent the wave front. 4. Analyze the singularities of the caustic and the wave front. 5. Compute the optimal synthesis. 10.7 Consider the following optimal control problem: m ui(t)Fi(q(t))

4(t) = Fo(q(t)) + i=1

jT

min

m

> u?(t)dt i=1

where T is fixed and q(0) = qo.

1. Compute the intrinsic second order derivative of the end-point mapping of the cost extended system: 4(t) = Fo(q(t)) + E ui(t)Fi(q(t)) i=1

M

4°(t) _

u? (t) i=1

with (q(0), q°) = (qo, 0)

10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius

319

2. Consider the following optimal problem in 1R2:

i(t) = 1 - y2(t) y(t) = u(t) ri min J u2(t)dt 0

with x(0) = y(O) = 0. Compute the intrinsic derivative of the end-point mapping of the cost extended system. 10.8

1. Consider the Hamiltonian function:

H(q,p) = 1(q2 +P2),

(q,p) E R2.

(10.9)

Integrate the following equations:

4(t) = 8p (q(t),p(t)),

P(t)

aq

(q(t),p(t))

(10.10)

with q(0) = qo, p(0) = po.

2. Let Lo be the line p = po and Lt = expt H (Lo), L = Ut>oLt. Represents L in the time extended phase space (q, p, t). Show that L projects diffeomorphically onto the plane (q, t) if and only if t < 2 3. Prove that there exists a function S on the surface L which satisfy the condition

dS = pdq - Hdt.

(10.11)

4. Solve the Hamilton-Jacobi equation:

T + 1(q2+(aq)2' =0 with S(q, 0) = p(O)q.

(10.12)

11 Numerical Computations

11.1 Introduction In Chap. 9 and 10 we study the singularities of the exponential mapping near singular trajectories. You cannot get away from such analysis in order to determine for a given sub-Riemannian structure the analytic properties of the sphere, conjugate and cut loci. As these analytical computations are in general highly nontrivial problems, numerical computations provide a good alternative. In particular, as we have seen along this book the determination of conjugate points in optimal control theory is of crucial importance, but their formal computation is in general not possible. However, the algorithm to compute the conjugate points, and hence the conjugate locus, is simple and very easy to implement numerically. If in this chapter we restrict ourself to the sub-Riemannian case, the generalization to more general problem is straightforward. On the other side, if the definition of cut points is still simpler than the one for conjugate points, their computations is much harder even numerically and is not a topic of this book.

11.2 Numerical Algorithm In this section we describe algorithms and numerical methods used to compute the conjugate points and to obtain the sphere. The purpose of this chapter is not to develop new and sophisticated numerical methods but to use and adapt classical numerical methods to solve ordinary differential equations. In our opinion the results obtained with this simple approach are very satisfactory and the pictures significant. The advantage of the suggested method is that we can use existing and well tested numerical algorithms already implemented, and that the user does not have to make tedious computations but only need to provide the differential Hamiltonian equations governing the flow for the normal extremals as well as its linearization to compute the conjugate points. Let us start by reminding the algorithm to compute the conjugate points along normal extremals. Let U be an open subspace of R's and consider the following local sub-Riemannian problem on U:

322

11 Numerical Computations m

u:(t)F:(q(t))

4(t) _

min J(i(t) , 4(t))y dt

)

where the vector fields Fi are smooth and generate a distribution D of rank m, and (,).g describes the scalar product associated to the metric g. In the sequel we assume the vector fields Fi to be orthonormal for g. Hence we are minimizing:

sn JT(u(t))cit. 1: We have seen in section 9.2 that the normal

i=1

extremals are solutions of the following differential Hamiltonian equations:

q(t) =

P(t) _

n (q(t),P(t)),

-aqn (q(t),P(t))

(11.1)

where the Hamiltonian is given by Hn(q, p) = 1 Ei_1 Pit, Pi being the Poincare coordinates: Pi = (p, Fi(q))2. We assume the geodesics parametrized m

by arc-length, which is E P,2(t) = 1 for all t. Then we have Hn(q(t), p(t)) _

LetgoEUandr>0fixed and expgo:CxR -+U (Po, t) '-' q(t, qo, Po)

be the exponential mapping, the domain C being given by m

C={PoElRn; E(Po,F:(go))2=1}.

(11.2)

Because the possible existence of strictly abnormal geodesics, the wave front

W(qo, r) is the closure of the image of C by expQ0 evaluated at t = r. The sphere S(qo, r) centered at qo and of radius r defined as the set of points at a distance r from qo is a subset of the wave front. A point is said to be conjugate to qo along a normal extremal if it is the image of a point (po, t1) E C x IR such that the exponential mapping is not an immersion. The time ti is called a conjugate time. Numerical computation of conjugate points uses the variational equation corresponding to the normal Hamiltonian flow. Recall that the vertical (respectively horizontal) space are respectively subspace of T*U generated by the vectors of the form ayp- (respectively f3j ). If q(tl) is conjugate to qo = q(O) along a normal extremal, there exists a nontrivial Jacobi field J such that J(O) and J(ti) belongs to the vertical space (see Chap. 6). If z = (q, p) and bz = (Jq, bp), a Jacobi field is solution of the variational equation given by: bz(t) =

ae n(q(t),P(t))bz(t).

(11.3)

11.2 Numerical Algorithm

323

To find the conjugate points to qo, we proceed as follows. Let Ji(.) be the Jacobi fields defined by Ji(0) = ei, i = 1, , n, where the vectors e 1, - , en form a basis of the vertical space. Then, a time t,, is conjugate to 0 if there exists a = ( a1 ,--- , an) E Rn, a # 0 such that: - -

-

n

aiirh(Ji(tc)) = 0

(11.4)

i=1

where 7rh is the projection mapping on the horizontal space: 7rh(Ji(t)) _ bgn(t)). Numerically we apply the following procedure. (bg1(t), bQ2(t), Given an initial starting point qo and an initial value po, we integrate in parallel the flow for the normal extremal corresponding to these initial values

and the flow of the variational equation for the n initial values 6z(0) = ei. We then have n solutions of the variational equation: bzi(t) = (bqi (t), bpi (t)), linearly independent at the starting point qo. At a conjugate time, the projection of these n solutions on the state space are linearly dependent, see (11.4). Hence to find the conjugate points, we must check at each step of integration if the determinant vanishes:

bqi (t) ... bqi (t)

D=

(11.5)

bgn(t) ... bgn(t)

where bqi = (bqi , ... , 6q;).

The conjugate locus is the set of first conjugate points, then it can be computed using the previous method with po varying in C.

To integrate the flow corresponding to the normal extremals and the variational equation, any classical numerical method can be used. The pictures shown in this book have been computed using an explicit Runge-Kutta method of order 5(4), due to Dormand and Prince. Moreover, to increase the efficiency of the computations, the Runge-Kutta method is supplemented by

a dense-output. Indeed, the classical Runge-Kutta methods are not efficient when we need to approximate the solution in a dense set of points, for example to draw a graphic or to study the zeroes of a function involving the approximated solution (as it is the case to determine the conjugate points). By adding a dense output, we then obtain a numerical approximation with regularity properties (continuity, C1, we can impose some values of the function and its derivative, etc) depending on the choice of interpolation used for the dense output method. For our problem, the gain due to the dense-ouput is important as we need to find the zeroes of a determinant, see equation (11.5). Then, instead to refine the step of the Runge-Kutta method and start again the algorithm, we simply use a dichotomie to locate with a very good precision the zeroes. The dense-ouput used by the authors is of order 4 due to Shampine, it provides a globally C1 approximation of the solution. More details about this numerical method can be found in [53].

324

11 Numerical Computations

The wave front is computed using the same numerical method to evaluate

the flow exp Hn (qo) at t = r and by taking its adherence. The sphere S(qo, r) is the exterior envelop.

11.3 Applications In the next few sections, we present the results obtained when applying the algorithm and the numerical method described previously to some subRiemannian structures. We will start with the contact situation, pursue with the Martinet case and finally analyze the elliptic and hyperbolic cases.

11.3.1 Contact Case in Dimension 3 The contact case of order -1 and the generic one are discussed in respectively Sect. 9.5 and 9.6. The picture of the generic conjugate locus can be found in Sect. 9.6.2, Fig. 9.4.

11.3.2 The Martinet Flat Case It is the local sub-Riemannian structure (D, g) defined on a neighborhood U of 0 E R3, where D is the kernel of the Martinet one-form: w = dz - z dx, q = (x, y, z) are the coordinates of R3 and the metric is given by g = dx2+dy2. The Martinet flat case is analyzed in Sect. 9.7. If we take F1 = +I FX- , F2 = a as generators for the distribution D =ker w, then we have [F1, F2J = yWZ_ and the abnormal geodesics are contained in the analytic surface {y = 0} called the Martinet plane. In the sequel we assume the starting point of the geodesics qo to be the origin. The abnormal geodesics starting from 0 and parametrized by arc-length are the straight lines: x(t) = ±t, y =- 0, z = 0. The equations of the normal

geodesics are given in Proposition 86. In particular, as the equations are integrable in the flat situation, we have an analytic expression for the normal geodesics in terms of the elliptic functions. This means we can use softwares as Mathematica or Maple to draw the geodesics and even the sphere (as we know that along a normal geodesic, the cut point corresponds to its first intersection with the Martinet plane). However, two importants remarks have to be made.

First, if we do the computations using the analytical expressions with the elliptic functions it is very time consuming and there is no comparaison with the numerical computations. If succeded, the computations for the sphere would take several hours instead of few minutes. Moreover, the precision of the algorithm is such that we would not see a difference between the two methods of computation. The second remark concerns the conjugate points and the conjugate locus. Using the reduction procedure described in [41, we can compute a parametrization of the conjugate points as follows. If

11.3 Applications

325

,y(t) = (x(t), y(t), z(t)) is a normal geodesic parametrized by Proposition 86, a conjugate time along 'y(.) satisfies the following equation: c(t,'P,

A)

= 8J (t,,P, A)

! (t,
(11.6)

After tedious computations, it can be proved that: c(t, W, A) =

- A2dnv (cnv Gl (v) - snv dnv(E(v) - k'2v))

(11.7)

where v = tVX and Gi(v) = E2 (v) - 2k'2vE(v) + k'2v2. From this equation, the time variable t cannot be expressed as a function of the two variables cP and A. Hence we cannot deduce an analytic expression for the conjugate time. Numerical computations are very helpful because they allow to localize the conjugate points with a very good precision. In particular, from the numerical simulations, see Fig. 11.1, we observe that for each normal geodesic the quotient s where tic is the first conjugate point is approximately a constant equals to 0.97. The following result based on these simulations has 3K

.w

a

aN

."

Vlc

In

1.4

1.

In

Fig. 11.1. been proved in [4].

Theorem 31. Let y(.) be a normal geodesic starting at t = 0 from the origin and defined on [0, +oo[. If the projection of y(.) on the plane (x, y) is not a line, then there exists a first conjugate point along -y corresponding to a conjugate time ti satisfying the inequality: 2K < tl,,X < 3K. On Fig. 11.2, 11.3 and 11.4 we represent the projection of normal geodesics

on the (x, y) plane and we have stopped the numerical integration at the first conjugate point. The corresponding initial values for (gyp, A) are (0.3,50), (-0.4,35) and (-0.9,80). The respective values for the first conjugate time are

0.72, 1.02 and 0.53. On Fig. 11.5 and 11.6 we represent the conjugate locus

326

11 Numerical Computations curve a t->(x(t).y(t))

ylt)

x.0.

o.l `

T

x(t) .a

4.M

-1.n

Fig. 11.2.

curve

t->Ix(t).ylt))

a

y(t)

Fig. 11.3. curve

a

Fig. 11.4.

11.3 Applications

327

for values of Jai contains in [105, 4 * 106]. The shape of the conjugate locus i or if cp - -M. . Indeed, from Theorem 31 when cp -- 2 , is different if P

we have ti/ -

s2

and when w --4-!E we have t1/ - +oo. In fact the

astroid forming the conjugate locus around the origin is pinched along the Martinet plane and goes to infinity in the x-direction. This is not surprising as the same kind of phenomenon is true for the sphere. It is a consequence of the logarithmic singularity of the complete elliptic integral of the first kind, see Sect. 9.8.1. Hence, we can conjecture the non sub-analycity of the conjugate locus as well. The sphere centered at the origin and of radius

Fig. 11.5. r = 0.1 is represented on Fig. 11.7 as well as its projection on the Oxz plane on Fig. 11.8. The projection on the Oxy plane is a disk with radius r = 0.1 and is not represented here. We can notice that as for the conjugate locus, the sphere is pinched along the abnormal geodesic. In Chap. 9, we proved that in the Martinet flat case the exponential mapping is not proper in opposite

to the contact situation. In other words, the sphere is not obtained as the image of a compact subset of C x Jr} by expgo. This phenomenon appears very clearly while running the numerical computations.

328

11 Numerical Computations

Fig. 11.6.

Fig. 11.7.

11.4 The Tangential Case The tangential sub-Riemannian structure is as in the previous two examples a 2-rank distribution in R3. In the contact situation, the Lie brackets of length

< 2 generate the all tangent space everywhere. In the Martinet case, on the Martinet plane containing the abnormal geodesics, the tangent space is generated by the Lie brackets of length < 3. The tangential case corresponds to a distribution such that if Fl and F2 are two generators for the distribution,

11.4 The Tangential Case

329

Fig. 11.8. there exists q E det(Fl, F2, [F1, F2]) = 0 with det(Fl, F2, [[F1, F2], F;])(q) = 0 for i = 1, 2. The normal form is computed in [111], there is two cases:

1. The elliptic case: A = ker w, w = dy - (xy + 3 + xz2 + bx3z2)dz; 2. The hyperbolic case: A = ker w, w = dy - (xy + x2z + bx3z2)dz where b is a real constant. In these normal forms the point q is normalized to be the origin. In both cases they are abnormal geodesics, but as we will see on the pictures they do not play the same role for the regularity of the sphere.

11.4.1 The Elliptic Case Let us take F1 =

0 TX

3 ,

F2 = a + (xy + 3 + xz2 + bx3z2)a TZ y

as generators for the distribution. The abnormal geodesics are contained in the surface: {q E 1R3; [F1, F2] (q) = 0} given by -(y+x2+z2+3bx2z2)7 = 0. To determine the controls corresponding to the abnormal geodesics, we have to compute the Lie brackets of length 3. We have: ([F1, F2J, F1J -

[[Fl, F2], F2] = (-2z +

(2x + 6bxz2) ay

233 - 6bx2z + 2bx3z2)

ay.

It follows that, via a choice of parametrization, the abnormal controls are given by: ul = -(-2z + 233 - 6bx2z + 2bx3z2) and u2 = -(2x + 6bxz2). In the plane (x, z), the origin is for the abnormal flow a weak focus, see [111].

330

11 Numerical Computations

Let us consider the flat situation: g = dx2 + dz2. The Hamiltonian system corresponding to the flow of the normal extremals is then given by: 3

H,, = psul + ((xy +

+ xz2 + bx3z2)pv + ps)u2 - 2 (ul + u2)

and the constraint tu- = 0 is equivalent to 3

U1 = px, u2 = (xy + 3 +xz2 + bz3z2)pv + p=

In Poincare's coordinates: Pi = (p, G; (q)) for i = 1, 2 and P3 = py, we have H,, = (P1 + P2) and the normal extremals are solutions of. z

x=P1 x3

+ xz2 + bx3z2)P2

(xy + 3

i = P2 P1 = -(y+x2+z2+3bx2z2)P2P3 P2 = (y + x2 + z2 + 3bx2z2)P1P3 P3 = {P3, P1 }P1 + {P3i P2}P2 = -xP2P3

where P1 = ps and P2 = (zy + g +xz2 + bx3z2)py + p, The set C is a cylinder described by C = {P(0) E R3; p12(0) + p2(0) = 1, P3(0) E 1R}.

Notice that if P3(0) = 0, then P3 =- 0 and the projection of the corresponding normal geodesic on the plane (x, z) is a straight line. Indeed in the

case P3 - 0, we have P1 and P2 constant and as the starting point is the origin, hence x(t) = ±P1t, z(t) = ±P2t. This curve is minimizing for the flat metric. Then the sphere S(0, r) contains a curve corresponding to the endpoints of geodesics such that their normal lift on the cotangent space satisfy P3 - 0 and whose projection on the (x, z)-plane is a circle of radius r. The order at the origin of the coordinates (x, y, z) is given by: (x, z) of order 1 and y of order 4. On Fig. 11.9,11.10, we have represented, for b = 0, the sphere S(0, r) of small radius centered at the origin. From the numerical computations we can deduce that the sphere is obtained by considering only a compact subset of the cylinder C. Then, the sphere is subanalytic. This results is a consequence of the general one proved in [1J. Indeed, in the elliptic case the minimizers passing through the origin do not contain any piece of abnormal geodesics.

11.4.2 The Hyperbolic Case In this situation, the generators of the distribution can be taken as: F1 = a

,

F2 = 8 + (xy + x2z + bx3z2)8 y

.

(11.8)

11.4 The Tangential Case

Fig. 11.9.

Fig. 11.10.

331

332

11 Numerical Computations

Then, the singular surface containing the abnormal geodesics is given by y + 2xz + 3bx2z2 = 0. Computing the Lie brackets of length 3, it is easy to verify

that the abnormal controls are given by: ul = -(-2x+x2z(1-6b)+2bx3z2) and u2 = -(2z + 6bxz2). It follows that we have two abnormal geodesics crossing the origin: the x-axis and the z-axis. In the plane (x, z), the origin is a saddle. As before, we study the flat case: g = dx2 + dz2. The normal extremals are solutions of:

x=P1 (xy +x2z + bx3z2)P2

z = P2 P1 = -(y + 2xz + 3bx2z2)P2P3 P2 = (y + 2xz + 3bx2z2)P1 P3

P3 = -xP2P3

where P1 = p2 and P2 = (xy + x2z + bx3z2)pv + pz. As previously for the elliptic situation, the normal geodesics corresponding to P3 =- 0 are straight lines in the plane (x, z) and then are minimizing for the flat metric. Hence they belong to the flat hyperbolic sphere. Notice that the abnormal geodesics are not strictly abnormal (they are also projections of the normal extremal flow corresponding to P1 = ±1, P2 = 0 and P3 arbitrary for the Ox axis and P1 = 0, P2 = ±1, P3 arbitrary for the z-axis) but for our metric they are globally minimizing. From the numerical simulations, it appears that the exponential mapping restricted to minimizers is not proper. Indeed, during the computations we encountered the same kind of problems as for the computation of the flat Martinet sphere (non properness of the exponential map and logarithmic singularities). See Fig. 11.11,11.12 where we have represented the sphere of small radius centered at the origin.

Notes and Sources The reader interested by running his own simulations is strongly invited to visit http://www.math.hawaii.edu/-mchyba which provides programs, simulations of spheres and conjugate loci for several sub-Riemannian structures and is as well a good source of papers. The book [53] is an excellent reference for the reader looking for more details about the Runge-Kutta-methods and

the dense-output. In [7, 5, 9] we can find representations of conjugate loci in contact sub-Riemannian-geometry in dimension 3 using numerical simulations based on formal computations.

11.4 The Tangential Case

Fig. 11.11.

Fig. 11.12.

333

12 Conclusion and Perspectives

Our aim is to end this volume by indicating some open problems in the analysis of singular trajectories in optimal control and pointing con-

nections with others domains of systems and singularity analysis.

12.1 The Category of the Distance Function in SR-Geometry In SR-Martinet geometry the exponential mapping is not proper near a strict abnormal direction and the distance function is not subanalytic. An impor-

tant question is to find in which category is the sphere and the distance function in SR-geometry. This problem can be generalized to any optimal control problem and leads to the question of determining the

category of the value function near a singular minimizer.

12.2 SR-Distances and Control of the Oscillations of Solutions of Ordinary Differential Equations In SR-geometry with a 2-dimensional distribution, abnormal curves which are not depending on the metric are in general small length minimizers and tools

have been developped to analyze and estimate the distance function in the abnormal direction. Despite this singularity, SR geometry has two important properties:

Property 1 The value function is a distance and the SR-balls have a size which can be evaluated using their nilpotent approximations.

Property 2 The set of minimizing controls is compact for the L2-topology and more precisely a norm has been introduced by [10] to control the oscillations of minimizers based on their nilpotent approximations. On the other side many efforts has been devoted to control the oscilla-

tions of solutions of ordinary differential equations, in particular for the following problems:

336

12 Conclusion and Perspectives

- Gradient problem. Let X = gradV be an analytic gradient vector fields

in the plane (or in R') with X(O) = 0 and let t '- 7(t) be a trajectory tending to 0 when t - +oo. Analyze the behavior of the tangent vector 'y(t) when t -- +oo.

- Estimate the number of limit cycles for planar differential equation and 16-th Hilbert problem. Let X be a planar vector field in the plane. A limit cycle is an isolated fixed point of the Poincare mapping. The 16-th problem is the following: let X be a polynomial vector field

in the plane with degree k. Find an uniform bound N(k) on the number of limit cycles. Both problems are related to the transport of a section a which can be taken as a curve of a fixed vector field Y identified to e , by the flow of X. The transport can be evaluate using the Backer-Campbe Hausdorff formula: ri

(exp tX exp sY) (x) = ( exp 3(E - ad"X (Y))) (y), y = (exp tX) (x) n>O

n!

(12.1)

and can be lifted in the cotangent bundle using the Hamiltonian vector fields

with Hamiltonians: Hx = (p, X) and Hy = (p, Y). If y(.) is a trajectory of X tending when t - +oo to a singular point, there is a singularity which has to be analyzed. Let us examine in more details the problem of limit cycles. We take a polycycle P with vertices S1, , Sk corresponding to singular points of X represented on Fig. 12.1 and the problem is to evaluate the Poincard return mapping R along the polycycle. In our control theorical approach, to evaluate

Fig. 12.1.

R we consider the transport of the section a by the flow of X. Let -Y(.) be a

trajectory of X and let tl be the first return time. When ry(.) tends to the separatrix cycle P then tl -. +oo and we must analyze the singularity of tl r-- (exp ti X)(ar). For the system 4(t) = X(q(t)) + u(t)Y(q(t)), the singular trajectories are located at the points where X, Y are collinear and in particular at the vertices of P where X vanishes. The singularity encountered

12.2 SR-Distances and Control of Oscillations

337

when ry - P is similar to the singularity of the exponential mapping restricted to minimizers in singular Riemannian geometry, in the abnormal directions. They fit in the same geometric framework. Now, in the 16th Hilbert problem, we can consider the family of singular Riemannian problems whose orthonormal vector fields are (X, Y) and Y being fixed. They can be lifted on the sphere S2 using Poincare cornpactification and in particular we can cover S2 by SR balls of fixed radius. In SR-geometry, we know that the oscillations of minimizers can be controlled.

We conjecture a relation between such oscillations and the number of limit cycles (and here a link between N(k) and the number of SR-balls we need to cover S2).

Computing. To compute the Poincar6 return mapping we can use the tools introduced in SR geometry and in particular Lagrangian manifolds: the section a is lifted into Lo defined by (p, Y) = 0 and we transport Lo by the flow of Hx with Hx = (p, X). When visiting a singular point of X we observe two phenomena:

- A splitting of the Lagrangian manifold illustrated on Fig. 12.2, (i). - A rotation of the Lagrangian planes (Jacobi fields) represented on Fig. 12.2, (ii).

a (ii)

(+)

Fig. 12.2.

The problem is to relate the asymptotic expansion of the Poincar6 return mapping with the previous geometric properties. In any cases, besides the possible connection between the number of limit cycles and the oscillations

of minimizers, it is clear that the same tools and the same research program are needed to analyze all the problems: find the category of the value function or the Poincar6 return mapping and make the contact analysis that is compute asymptotic expansions. It is an extension to

nonproper situations of classical Lagrangian and Legendrian singularity theory.

338

12 Conclusion and Perspectives

12.3 Optimal Control Problems with Bounded State Variables Consider to simplify the time minimal control problem for a system on lR' of the form: 4(t) = X(q(t)) + u(t)Y(q(t)), I u(t) 1< 1 where X, Y are smooth vector fields. The extremals are solutions of 4(t) = eH (q(t), p(t), u(t)), p(t) _ - eH (q(t), p(t), u(t)) where H(q, p, u) = (p, X (q) + uY(q)) and the extremal control minimizes the Hamiltonian H. They split into: regular extremal where

u = -sign(p, Y) and singular extremal where u is chosen to satisfy the constraint: (p, Y(q)) = 0. To construct the optimal solution, we must:

- determine if the singular arcs are optimal; - classify the behaviors of regular extremals near the switching surface E (p, Y(q)) = 0, that is to classify the solutions of the pair of Hamiltonian vector fields defined by H+ = (X (q) + Y(q)) and H_ = (p, X (q) - Y(q)), near E.

Consider now the time minimal control problem, for the same system but with state constraints of the form c(q) < 0 where c : IIt" -+ IIt is a smooth function. Such problems appear in many applications and their analysis is a research area widely open. For such problems there exists a maximum principle that we are stating now.

12.3.1 Maximum Principle with State Constraints A boundary arc t F--+ ryb(t), t E [0, T) is an arc contained in the boundary: c(q) = 0. The boundary control can be generically computed by differentiating the constraint mapping t H c(q(t)) up to the first occurence of u: c(q)

... = c(-

)(q) = 0

c(m)(q) = a(q) + ubb(q) = 0

and m is called the generic order of the constraint. We make the assumption that b(yb(t)) 34 0 holds along the boundary arc and that the boundary control is admissible and not saturating for t E)0,T[:[ a(-yb(t)) 1< b(-yb(t)). The maximum principle is the following, see [841. Define the Hamiltonian by:

H(q, p, u, rl) = (p, X (q) + uY(q)) + ic(q),

(12.2)

where q is a Lagrange multiplier associated to the constraint. The necessary optimality conditions are: 1. There exists a scalar function ra(t) > 0 such that:

4(t) = X(q(t)) + u(t)Y(q(t)),

P(t) = -p(t) ((q(t)) (q(t)) + u(t)

q (q(t))) -,agq(t)

12.4 Singular Trajectories and Regularity of Stabilizing Feedback

339

2. The function ra(t) is zero in the interior of the domain c(q) < 0 and is continuous in the interior of the boundary arc. 3. There exists a jump condition at a contact point or a junction point with the boundary: ac

p(tI+) = P(ti -) - ui-,

vl > 0

(12.3)

4. The optimal control minimizes the Hamiltonian: u(p, Y(q) = 1m in (p, Y(q))v

(12.4)

and moreover (p, X (q) + uY(q)) < 0.

Now since I u6 1< 1 in the interior of the boundary are, we deduce that along a boundary arc: (p, Y(q)) = 0. Hence as observed by Maurer [841, there

exists an analogy between singular arcs and boundary arcs and in particular we must analyze the following problems:

- Determine when a boundary arc is optimal; - Analyze the behaviors of regular extremals near the switching surface. The research program is similar to the one associated to singular trajectories. It is related to the classification of triples (X, Y, c) and everything can

be expressed in terms of iterated Lie brackets of X, Y acting on c by Lie derivative. The main difference with the singular case is that the adjoint vector can suffer discontinuities when reaching the boundary. Also note that the maximum principle with state constraints allows to analyze optimal control problems where the system has discontinuities.

12.4 Singular Trajectories and Regularity of Stabilizing Feedback Consider a smooth control system in R" of the form: 4(t) = f (q(t), u(t)), u(t) E lR"', with f (0, 0) = 0. The stabilization problem is to find a smooth u(q) such that 0 is asymptotically stable for the differenfeedback q tial equation: 4(t) = f (q(t), u(q(t)). On the other side we can consider the family of optimal control problems where we minimize a cost of the form: fo f°(q(t),u(t))dt, T being fixed or not, or T = +oo, and fo is any smooth function. Singular trajectories of the system are not removable and their existence has consequences on the regularity of the value function and its level sets. The singularities of the value function can be analyzed by computing the conjugate and the cut locus of the origin. To be more precise consider the Heisenberg SR-geometry where the system is given by:

340

12 Conclusion and Perspectives

±(t) = u(t) y(t) = v(t)

i(t) = 2 (x(t)v(t) - y(t)u(t) and the cost is fo (u2(t) + v2(t)) idt. The singular trajectories are the points of the axis Oz and the conjugate points and the cut points are the line Oz. If we take a generic perturbation, the conjugate locus is an astroid centered

along z and on the sphere the extremities of the cut locus are conjugate points corresponding to cusps of the conjugate locus. They correspond to stable singularities. On the other side, such a system cannot be stabilized

using a smooth feedback. An interesting question is to analyze the link between the conjugate and cut loci, both connected to singular trajectories, and the non existence of smooth stabilizing feedback. A conjecture in dimension 3 is the following: the cusp singularity of the conjugate locus is responsible of the non existence of smooth stabilizing feedback.

12.5 Computations of Singular Trajectories Consider a system of the form: q(t) = X(q(t)) + Ei_1 u;(t)Y;(q(t)). The generic singular trajectories called of minimal order can be easily computed and if m = 1, we gave an algorithm to compute all the singular trajectories. If m > 1, the problem is open. Such a computation is important for instance to classify the distributions and also in many applications where we deal with nongeneric systems.

12.6 Computations of Conjugate Points The computations of conjugate points is a central problem in optimal control. It concerns both bang-bang extremals, singular extremals and in SRgeometry, normal geodesics. It allows to estimate the length of a minimizing curve and in SR-geometry, to classify the singularities of the exponential mapping. Efficient algorithms have to be found to compute such points, using both formal and numerical computations and based on Jacobi fields or the second order intrinsic derivative. Also the algorithms differ if we are in the bang-bang case, singular case or normal geodesics. An interesting question is to unify all the algorithms. To be more precise consider the time optimal control problem for a system of the form: q(t) = X(q(t)) + u(t)Y(q(t)), I u(t) 1< 1. Let -y(.) be a reference bang-bang extremal, here the concept of conjugate points is defined by taking variations of switchings along the reference extremal to cover a neighborhood of the end-point, see Fig. 12.3 and it is

12.6 Computations of Conjugate Points

341

not easy to implement. But we can regularize the system by adding a control v(.) and considering the system: 4(t) = X(q(t)) + u(t)Y(q(t)) + v(t)Z(q(t)), where Z is a vector field and u2 + s2v2 < 1. For such a system, the concept of conjugate point is standard: an extremal control is computed by minimizing the Hamiltonian over u2 + e2v2 < 1 and is given generically by a smooth function u(q,p) with values in u2 + e2v2 = 1. It defines an Hamiltonian vector field on the cotangent bundle and the computations of conjugate points is similar to the algorithm of the SR-case, based on the variational equation. Here we have replaced switchings by oscillations and both concepts have to

be related when a - 0.

Fig. 12.3.

Exercises 12.1 Consider the time optimal control for a planar system: 4(t) = X (q(t)) + u(t)Y(q(t)), u(t) 1< 1 with a state constraint of the form c(q) < 0. Let c(0) = 0 and assume the following: I

i) X (0), Y(0) and Y(0), [X, Y](0) are independent; ii) the constraint is of order 1 at 0: Lyc(O) 0; iii) the small time boundary arc ryb(.) through 0 is admissible and not saturating, i.e I ub(0) 1< 1 where ub() is the boundary control.

1. Compute the time optimal synthesis in a small neighbordhood of 0 (2 cases).

2. Prove that at a junction or contact point with the boundary, the adjoint vector is continuous.

12.2 Consider in It2 the optimal control problem: 4(t) = v(t)X(q(t)) + u(t)Y(q(t)), mind (u2(t)+v2(t))dt where Y = vector field: (Ml) X = (y+1)T'9X_

(M2) X = y

Ty-

and X is one of the following

342

12 Conclusion and Perspectives

(M3) X = (y2 + 1) 0 (M4) X = (y2 - 1Tx(Ni) X = (y2 + ax) I , a E ]R (N2) X = (y3 + xy + a(x)) , where a is a smooth function and a(O) # 0

Compute in each case the extremal trajectories equations and discuss the optimal problem with initial condition q(0) = 0.

12.3 Consider the time optimal control problem for a smooth system in R': q(t) = X(q(t)) + u1(t)Yi(q(t)) + u2(t)Y2(q(t)), with u; (t) + u2(t) < 1. Let 45 = (P1,'P2), fi; = (p,Y;(q)) be the switching mapping and E1 be the switching surface rh1 = IP2 = 0-

1. Let (q, p) E T* R"\El, prove that an extremal control corresponding to where the extremal passing through (p, q) is given by: (U1, u2) _ 401'0' 11411 II'PII =

---7

1+

'p2.

2. Let z(t) = (q(t),p(t)) be a smooth extremal. Compute i(z(t)) for i = 1, 2.

3. Assume [Y1, Y2] = 0. Prove that

1 = '2 = 0 is the surface E2

(p, [Y1, X1 (q)) = (p, [Y2, X] (q)) = 0. Let t '-4 z(t) be an extremal pass-

ing through a point (p, q) E E1\E2. Show that when crossing El, the extremal control is discontinuous and rotates of an angle 7r.

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Index

CO

- isolated, 156 - minimizer, 35, 36 - one side rigidity, 153 - optimal, 304

ad-formula, 90 adjoint

- equation, 12, 70 - system, 23, 42, 46, 183 - vector, 12, 42, 55, 147, 193, 200, 226,

- rigidity, 134, 156, 166, 169, 233, 268 - sufficient optimality conditions, 35 - time maximal, 143, 153 - time minimal, 143, 153

admissible - control, 1, 119, 209, 297 - curve, 234

- time optimal, 295

Agrachev, 113, 273

- topology, 35, 75, 143

arc-length, 234, 322

C'

Arnold, 314

- isolated, 168 - rigid, 132, 133, 156, 306 - topology, 27, 306

Arrhenius law, 172 asymptotic foliation, 256 attitude control, 209 augmented system, 37, 51

C°°-Whitney topology, 209 G-equivalent equations, 102 G'-classification, 104

L' - perturbation, 46 - topology, 229 L°°-topology, 302 16-th Hilbert problem, 271, 301 16-th Hilbert problem, 336 2-form, 66

- regular, 68 abnormal

- case, 43 - extremal, 236 - geodesic, 236, 268

absolutely continuous curve, 1, 234 accessibility set, 1, 12, 117, 119, 127, 145, 186

- at time T, 1, see ex.3.7, 119, 186 - small time, see ex.3.6 accessory problem, 33, 148, 161 activation energy, 172

292

bad set, 213, 213, 223 - codimension, 216 Baker-Campbell-Hausdorff formula, 118, 123, 127, 186, 288 Banach space, 10 bang-bang principle, 11, 24 batch reactor, 171 BC-extremal, 181, 184, 194 Bellaiche, 273 Bernouilli problem, see ex.2.7 Bliss, 55, 134, 233 blowing-up phenomenon, 230 Bolza, 55 boundary

- arc, 338 - layer phenomenon, 296 bounded measurable mapping, 1 Briot-Bouquet PDE, 253

Brockett, 24, 248 Brunovsky canonical form, 24, see ex.1.4, 146

350

Index

bundle

- cut locus, 248

- partially algebraic, 217 - semi-algebraic, 217, 224

- distribution, see ex.4.1

calculus of variations, 27, 82, 92, 94, 244, 282

Caratheodory, 55 Cartan formula, 108

- system, see ex.10.3 Cartan-Baker-Campbell-Hausdorff formula, 66 Cauchy problem, 84, 118, 253 Cauchy-Schwartz inequality, 235 caustic, 284, 286 center, 228, 297

Cesari, 24 chained system, 113

Chakir, 273 character, 102 characteristic - curve, 31, 68, 108 - equation, 110, 259, 260 - field, 108 - polynomial, 6 - vector field, 68, 299 chemical

- batch reactors, 171 - engineering, 171 kinetics, 172 reaction, 172 Chen formula, 186

Clarke, 55 clock-form, 85, see ex.3.1, 105, 195 closed loop, 18, 180 complete vector field, 118 conjugate - point, 33, 89, 107, 156, 160, 178, 321, 340

- point(first), 160, 283 - time, 149, 162, 238, 283, 286, 295, 303

- time(along an exceptional extremal), 155 - time(first), 145, 154, 178, 295, 299 constraint Hamiltonian system, 45 contact - case, 241, 324

- conjugate locus, 248, 287

- distribution stabilizer, 247 - normal form, 110, 247 - point, 110 - sector, 313

- stabilizer group, see ex.4.1 - sub-Riemannian geometry, 239, see ex.9.8 continuous spectrum, 294 control - constraint, 291

- jump, 143 control domain, 1, 52, 235 - compact-convex, 9 - convex hull, 9 - convex polyhedron, 14 control system, 119

- autonomous, 36 - nonautonomous, 38 - single-imput affine, 209 - single-input affine, 15, 76, 134, 143 controllable polysystem, 123 - weakly, 123 controllable system, 2, 3, 9, 119, 124

- at time T, 2, 119 - weakly, 119 controlled Euler equation, see ex.4.3, see ex5.5, 229 convexification, 75 coordinates system

- adapted, 181, 186, 242, 254 - privileged, 241 cost function, 37, 44, 52 cotangent - bundle, 117 - space, 209 covariant, 102, 107 critical - geodesic, 261 - point, 71 curvature, 107, see ex.5.22, 163, 169, 179, 186, 300 curvature tensor, see ex.10.1

cut

- locus, 89, 181 - point, 321 cyclic coordinate, 68, 255

Index

Darboux coordinates system, 66, 281 dense-output, 323 determinant, 323 Diamond Lie algebra, see ex.9.5 Dido problem, see ex.2.6, 243, 244, 273 distribution, 39, 97, 98, 119, 234, 273, 322

- classification, 108, 239 - equivalence, 98 - fat, see ex.9.14

- flat, 99 - involutive, 119 - medium fat, see ex.9.14 - rank condition, 121, 125, 240 - regular point, 241 - singular point, 241 - stabilizer, 103 drift, 98, 296 driftless system, 108 dynamic programming, 53

351

- case, 205

- set, 200 excess Weierstrass function, 36, see ex.2.8 exp-ln category, 233, 266, 268, 312 exponential mapping, 235, 238, 286, 322

- noncompactness, 246 exponential matrix, 2 extremal, 30, 35, 51, 70, 76, 131, 176 - abnormal, 250 - bang-bang, 76, 81, 340 - broken, 73 - classification, 85, 93, 171

- control, 12 - exceptional, 153 - field, 35, 164 - Fuller, 82 - normal, 286

- trajectory, 12 extremities conditions, 27

Ekeland, 65, 69, 93, 111, 113 elementary simplex cone, 48 elliptic - case, 105, 110 - function, 257 - integral, 258, 261, 263, 327

- point, 74 end-point mapping, 39, see ex.2.5, 97, 132, 148, 150, 153, 222, 281, 291, 304

energy, 235

- minimal problem, 235 Engel

- distribution, 304, see ex.10.1 - sub-Riemannian geometry, see ex.9.2, see ex.9.9

enlargement technique, 125, 132 equivalent

- polysystem, 125 - systems, 106 - vector fields, 103 Erdmann Weierstrass conditions, see ex.2.2

Euler-Lagrange - equation, 29, 161

- operator, 33, 149, 158, 305 Euler-Lagrange-Jacobi operator, 293, 303

exceptional

feedback - classification, 5, 97, 104, 108, 112, 160, 209, 250

control(singular), 162 - equivalent systems, 5, 16, 97, 104, 146

- group, 16, 98, 106, 239 - group(linear), 5 - invariant, 16, 88, 97, 105, 109, 160, 226

- linear, 5 - optimal(classification), 171 - semi-covariant, 227 time optimal, see ex.3.3 fiber

- bundle, 117 - product, 209 first integral, 68 first order Pontryagin cone, 48, 117, 127, 130, 144, 146, 299, 302 first order reaction, 175 fixed time problem, 38 flat function, 267 flat problem, 181, 187, 193

Fleming, 55 focal point, 179, 206, 296 - first, 206

352

Index

fold point, 80, 87, 108

Gulberg-Waage law, 172

- elliptic, 80, 81, 87, 108

- hyperbolic, 80, 81, 87, 108, 187 - parabolic, 80, 81, 87 Frechet derivative, 41, see ex.2.5, 132, 222, 291 Flrenet formula, see ex.5.22 frequency factor, 172 Fuller

- example, 82 - phenomenon, 65, 82, 83, 94 fundamental lemma, 223 fundamental matrix, 1 Gateaux derivative, 39

Gamkrelidze, 113 gas constant, 172 gauge group, 239, 248, 254 Gaussian curvature, 256

Gauthier, 273

Hormander, 314 Hamilton-Jacobi - equation, see ex.10.8 - theory, 53 - wave function, 286 Hamilton-Jacobi-Bellman - equation, 30, 54, 155, 281 - function, 30 Hamiltonian constraint equations, 162, 237, 303 constraint vector field, 101 critical, 112 equations, 30, 282, 322 function, 29, 70, 133, 236, 283, 322 lift, 68, 72, 210, 292 linearized system, 162 vector field, 67, 68, 209, 295 harmonic oscillator, see ex.1.5

generalized

Hausdorff

- control, 144, 304 - trajectory, 144, 164, 295 generating - family, 284 - function, 68, 251, 282

- metric, 9

generic classification, 73 geodesic - minimizing, 240

- optimal, 265 - strict, 236, 286

- strictly abnormal, 322 geometric optimal control, 65 Goh condition, 117, 131, 133, see ex.8.1 Goh transformation, 97, 98, 149, 161, 171, 175, 294, 303 Goursat

- form, see ex.4.7 - system, see ex.10.2 graded - approximation, 242 - normal form, 240 gradient, 31 - problem, 336

Gromov, 273 growth vector - big, see ex.4.7

- small, see ex.4.7 Grusin example, see ex.9.6

- topology, 11 heat exchanger, 174 heat transfert - endothermic, 174 - exothermic, 174 Heisenberg

- conjugate locus, 245 - conjugate point, 245 - cut locus, 245 - cut point, 245 - group, see ex.5.18 - Lie algebra, 242, see ex.9.7

- normal extremal, 244 - sub-Riemannian geometry, 243, see ex.9.1, see ex.9.5, 289 Hermes, 24, 94, 117, 127, 130, 134 higher order maximum principle, 117 higher order Pontryagin cone, 130 Hilbert - invariant integral theorem, 36 - invariant theory, 113 Hilbert-Cartan form, 35 Hirschorn, 134 holonomic mechanics, 27 homographic transformation, 261 homotopy method, 66 hyperbolic

Index

353

- case, 105, 110, 203

- transformation, 29

- point, 74

Legendre-Clebsch condition, 117, 129, 131, 133, 145 Legendrien singularity theory, 287 Leibnitz identity, 67 length, 234 - minimization problem, 235

infinitesimal symmetry, 273 inflexional elastics, 290 initial condition of order two, 292 integral manifold, 120 interior product, 67 intrinsic second variation, 133 intrinsic second-order derivative, 32, 286, 293, 302 invariant, 4, 5, 66, 97, 102, 239, 301 - micro-local, 312

- set, 107 isoperimetric

- metric, 252 - problem, 244, 250

Lie

- algebra, 100, 119, 119, 242 - bracket, 67, 77, 117, see ex.5.4, 212 - derivative, 16, 67, 117, 187

- group, 125, see ex. of chapter 5 limit cycles, 336 linear system, 1

- autonomous, 1, 3 - autonomous single-input, 4 - nonautonomous, 7

Jacobi

linearized system, 40 linearly equivalent systems, 3 Liouville form, 66

- curve, 33

Liu, 271

isotropic space, 65, 281, 295

- elliptic function, 263 - epsilon function, 264, 269

- equation, 33, 288 - field, 178, 206, 288, 322 - field(vertical)vertical, 288 - identity, 67, 118

Liu-Sussmann example, 164, 271

Lobry, 230 local group G(D), 119 local semi-group S(D), 119 logarithmic singularity, 263, 267, 327

Lojasiewicz, 233

- method, 69

LQ-problem, 38, 44, see ex.2.4, 82

- observation space, 288 - operator, 299 Jakubczyk, 111, 113 jet of vector field, 209 jump direction, 143 Jurdjevic, 125, 134

- cheap, 82, 145

Klok, 94 Krener, 117, 134 Kupka, 65, 94, 125, 134, 169, 273

Lagrange problem, 233 Lagrangian space, 65, 281, 294 Lasalle, 24, 94 Lee, 24, 45, 55 left invariant - control system, 165 - vector fields, see ex. of chapter 5 Legendre

- condition, 32 - strong condition, 33, 282, 295

Luenberger observer, see ex.1.3 Maple, 324

Markus, 24, 45, 55 Martinet - abnormal extremal, 306 - abnormal geodesic, 255, 261, 324 - case, 241 - distribution, 249 - foliation, 256 - Hamiltonian, 254 - invariant, 250, 269 - minimizer, 307 - normal extremal, 254, 307 - normal form, 110, 252 - one-form, 324 - plane, 250, 264, 268, 324

- point, 110 - sector, 290, 310, 314 - singular point, 250

354

Index

- sub-Riemannian geometry, 239, 249, 252, see ex.9.10, 289

- method, 321

Martinet, 113, 248

observable system, see ex.1.2 observation mapping, 222 one parameter group, 68, 84, 118 open loop, 18 open problems, 335 optimal

Martinet flat case, 257, 262, 324 - conjugate locus, 265 - cut locus, 265, 271 - sphere, 266, 268 - wave front, 268 - characteristic equation, 263 - conjugate point, 325

- cut point, 265 - minimizer, 265 mass conservation law, 173 Mathematica, 324 Maupertuis principle, 235 maximum principle, 55, 70, 84, 117, 171, 176, 236, 265, 297

- for linear time optimal problems, 12 Mayer field, 35, 36 mechanical system, 259 minimizer, 29, 292, 304

- abnormal, 298 - broken singular, 209, 229 - singular, 335 molar - concentration, 172

- control, 37, 51, 70, 191

- control problem, 175, 285 - control(existence), 52 - cost, 158

- problem, 321, 335 - synthesis, 83, 85, 88, 111, 171, 184, 189, 190, 198, 203, 285, 295

- trajectory, 51, 70, 85, 131, 194, 297 optimality - high order conditions, 117, 127 - second order conditions, 31 orbit, 119, 119 order of a function, 241 ordinary differential equations, 321 orientation principle, 200 oscillating trajectory, 310

- extent, 172

parabolic - case, 203

mole, 172

- point, 74

Montgomery, 169 Mormul, 113 Morse

- index, 133 - theory, 33 Morse-Maslov index, 286

Needle type approximation, 46 nilpotent approximation, 242 nonholonomic mechanics, 27 nonintegrable constraint, 27 nonsingular linear operator, 293 normal - case, 43

- extremal, 236 - geodesic, 236

normal form, 66, 94, 239, 283, 292 - graded, 242, 261 - isothermal, 248 numerical - computations, 321, 325

pendulum, 257, 263, 269, 272, 297, 309

- linear, 244 perturbation - additivity property, 47 pfaffian

graph, 267 - mapping, 233 - set, 267 Poincare

- coordinates, 255 - equations, 237 Poincare lemma, 67, 282 Poincare-Dulac return mapping, 233, 271

Poisson bracket, 67, 84, 210, 217 Poisson stable, 126

- point, 124 - vector field, 124 polysystem, 119 - normalizer, 126

Pontryagin, 55

Index

Pontryagin maximum principle, 45, 127, 210

potential field, 259 preparation theorem, 312 principal part of the SR-structure, 242 privileged coordinates, 240 projected system, 177 pseudo-Hamiltonian, 41

quadratic control system, 137 quasi homogeneity, 262

rational polynomial, 128

- positive, 128 ray, 230 reactant, 172

reaction temperature, 171 reduced - problem, 175

- space,294 - system, 161

reduction procedure, 324 regular

- control, 41 - extremal, 78 - system, 98 - time, 46 - trajectory, 41 relaxed problem, 75

Respondek, 113 return mapping, 264, 284, 269 - nonproperness, 269 Riccati equation, 34, see ex 2.4 Riemannian - metric, 234, 322 - sector, 313 right invariant - control system, 138 - vector field, 125 - vector fields, see ex. of chapter 5 rigid spacecraft, see ex.5.9, see ex.5.14,

209

rigidity, 132, 156

Rishel, 55 rotating trajectory, 310

Runge-Kutta method, 323 saddle point, 228, 297

Sarychev, 169

355

saturated - case, 203

- set, 125 saturating control, 191, 193, 200 saturation set, 202

Schattler, 93, 171 second kind, 181 semi-algebraic

- conditions, 84 - set, 216 semi-analytic, 247, 267, 298 semi-covariant, 102, 107 semi-direct product, 141 semi-invariant, 102 semi-normal form, 145, 147, 163, 180, 205

separatrix, 257, 264, 269, 301 - cycle, 271 singular - control, 41, 84, 100, 106, 130, 227 - set, 200, 227 - trajectory, 39, 41, see ex.3.4, 106, 143, 210, 335, 340

- trajectory(classification), 97 singular extremal, 76, 99, 105, 210, 222 - broken, 226, 300

- elliptic, 107, 145, 158, 227, 292 - exceptional, 107, 109, 133, 145, 158,

227, 233, 236, 292, 300 - higher order, 102 hyperbolic, 107, 145, 158, 177, 185, 227, 292 - minimal order, 77, 101, 109, 144, 176, 211, 223, 226, 236 order, 212 time maximizing, 105 time minimizing, 105 singularity - codimension, 71, 222

- resonant, 111 singularity theory, 27, 69, 93, 240, 248, 283

small time optimality, 107, 108, 117 species, 172 spectrum of a matrix, 6 speed vector, 72 splitting line, 181 stability, 18

356

Index

stabilizable

- form, 102 - group, 66 - manifold, 66, 281

- linear system, 6 stabilization problem, 339 stable

- map, 65

- matrix, 6

- space, 65

state constraint, 52, 195, 338 - generic order, 338 stoichiometric coefficients, 172 stratification, 71, 181, 212, 281, 310 stratum is of first kind, 181

- structure, 66

strong CO

- optimality, 143, 165 - topology, 143, 153 sub-analytic, 233, 240, 247, 267, 298, 312, 327

- set, 84, 185 sub-Riemannian - conjugate locus, 238, 323 - conjugate point, 238, 322 - cut locus, 238

- cut point, 238 - distance, 233, 234, 240, 271, see

ex.9.12, 335

- geometry, 168, 209, 233, 285, 321 - geometry(left invariant), 243 - normal case, 285

- sphere, 238, 322 - wave front, 238, 324 Sussmann, 55, 65, 89, 93, 113, 171, 233, 271

switching, 19 - function, 79, 85, 89, 181, 193, 202

- local bound, 94 - locus, 182, 204 - normal point, 86 set, 79 - surface, 91 - time, 79, 90, 184, 202 switching point, 19, 91, 184, 194 - hyperbolic, 200

- normal, 79, 190 - parabolic, 200 - regular, 74 Sylvester criterion, see ex.2.9 symbols, 112 symmetric polysystem, 123 symplectic - diffeomorphism, 88

symplectic stratification, 306 symplectomorphism, 102 tangent space, 209 tangential - elliptic abnormal geodesic, 329 - elliptic normal extremal, 330 - elliptic singular surface, 329 - elliptic sphere, 330 - hyperbolic abnormal geodesic, 332 - hyperbolic singular surface, 332 - hyperbolic sphere, 332 - normal form, 329 - sub-Riemannian geometry, 328 target, 13, 37, 52, 193, 199 terminal manifold, 175, 182 Theorems: - Banach-Alaoglu-Bourbaki, 10 - Brouwer fixed point, 48 - Caratheodory, 37 - Cayley-Hamilton, 2, 5 - Chow, 122, 240 - existence, 24 - Filippov, 240 - Filippov existence, 52, 84, 180, 235 - Filippov selection, 13 - Frobenius, 120

- Hahn-Banach, 49 Hartmann-Grobman, 228 - Hilbert-Schmidt, 149 - Krein-Milman, 11 - Liapounov, 11 - maximum principle, 51 - Nagano-Sussmann, 121 - open mapping, 41 - Poincare, 124 - preparation, 287 - Rolle, 194 - Sturm, 179 - Weierstrass preparation, 111 time minimal

-

problem, 76, 94, 132, 233, 235, 291, 338

Index

- synthesis, 18, 21, 160, 164 - trajectory, 210 time optimality, 117 - necessary conditions, 143 torsion, see ex.5.22 total polarization, 218

357

vector fields

- orthonormal, 322 vertical space, 322 volume form, see ex.2.5, 124 VP bundle, 217, 224

Trelat, 314 trajectory, 1 transversality conditions, 51, see ex 2.1, 181, 285, 304

transversality theory, 213 value function, 30, 158, 189, 335 variation, 28, 91 - second order, 146 variational equation, 47, 90, 163, 178, 287, 322

wave front, 238 weak maximum principle, 39, 42, 132 weak'-topology, 10, 11 Weierstrass equation, see ex.2.8 weight of a coordinate, 241 Whitney topology, 101, 104

Wonham, 24 Zhitomirskii, 113

La Societe de Mothematiques Appliquees et Industrielles (SMAI(, fondee en 1983, s'est fixe comme objectif la promotion des mathematiques appliquees. Dons cet esprit, la SMAI a tree une collection d'ouvrages Mothemotiques & Applications, dont voici un nouveau numero. Le but de cette collection est d'editer des textes de niveau 3eme cycle universitaire ou de derniere annee d'ecole d'ingenieurs. Les lecteurs concernes sont donc des etudiants, mais egalement des chercheurs et ingenieurs qui veulent s'initier aux methodes et aux resultats des mathematiques appliquees. Certains ouvrages ouront oinsi une vocation purement pedogogique alors que d'autres pourront constituer des textes de reference. Lo principale source des monuscrits reside dons les tres nombreux tours qui sons enseignes en France, compte tenu de la variete des Dipl6mes d'Etudes Approfondies (D.E.A.), des Dipl&mes d'Etudes Superieures Specialisees (D.E.S.S.) ou des options de mathematiques appliquees dons les ecoles d'ingenieurs. Mois ce n'est pas l'unique source: certains textes pourront avoir une autre origine.

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Bernard Bonnard - Monique Chyba Singular Trajectories and their Role in Control Theory The role of singular trajectories in control theory is analysed in this volume that contains about 60 exercises and problems A section is devoted to the applications of singular trajectories to the optimisotion of botch reactors. The theoretical part based on the Martinet case concerns the singularity analysis of singular trajectories in sub-Riemannian geometry. An algorithm is given to evaluate conjugate points and a final chapter discusses open problems. The volume will interest mathematicians and engineers.

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9°783540°008385 ISBN 3-540-00838-1

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