Contemporary Trends in Nonlinear Geometric Control Theory and Its Applications

dt

= -P • (AxQp - (lkAo) + - T r (M[fifc, AJ) In particular,

23

dP — • A0 = P • nkA0 at

= -ilkP

-

• A0 ,

and therefore, —- = -flkP. Equivalently, —- = [fifc,P]. at at It then follows that Ti(M[Slk,Ai]) = Ti([M,Qk]Ai) from which we get that

To obtain the equation for M(t) we need to recognize that Aa • b can be written as - -Trace ((a A b)A) for any vectors a and b, were a A b denotes the antisymmetric matrix defined by (a A b)x = (a • x)b — (b • x)a. Therefore,

-^Tr ( ^ f V ) =-\Tr((P*flp)A1) + lTt{[M,nl]A1) from which it follows that ^

= [nt,M] + P A O p ,

a n d ^ = [tt fc ,P]

(3.5)

are the Hamiltonian equations on the semi-direct product. Let us now look at the conservation laws that are due to the left-invariant symmetries. As already noted in 5 , the solutions of equations (3.1) evolve on co-adjoint orbits of G i.e., E(t) = Ad*(t)(^(0)). Therefore, any right invariant vector field on G is a symmetry for the Hamiltonian H. Indeed, let X(g) = Ag be any right-invariant field. Then its Hamiltonian Hx is given by Hx(t,g) = tig^Ag) = A d ; _ , ( 0 ( 4 ) , and hence Hx{l(t),g(t)) = Ad^Ad^O))^) = £(0)(A). Consequently, Hx is an integral of motion for H. For semi-simple Lie groups, the condition that l(g~1Ag) is constant for any A in g is expressed equivalently as g^Litfg-1

(t) = constant

(3.6)

for any integral curve of (3.1). In particular, (3.6) implies that the spectrum of L(t) is constant along the solutions.

24

The spectral invariants of L(t) are also known as Casimir functions on g. Evidently, they are conserved along the flow of any left-invariant Hamiltonian H. It is easy to see that Trace Lk, k > 1 is a Casimir function on any semisimple Lie group, because ^-Trace(L fc ) = k (Lk-\ at \

^ \ = k{Lk-\[dH,L}) at J

= k(dH, [Lk~\L\)

=0.

Of course, the trace of Lk is related to the coefficients in the characteristic polynomial of L(t). For semi-direct products the adjoint orbits are not in the same correspondence with co-adjoint orbits and hence condition (3.6) takes a slightly different form. We recall equation (3.5)

£ = m..*]+»Afl,. § together with — = gdH = g($lp + $lk), where 1 0

9=

[xRl-

Hence, dR dx ^ - 3 - = Rilk and — = Rilp dt dt It follows that

^-R{i)P(t) = -RtlkP + RQkP = 0 dt

and jiRMR-1

+xARP)=

R(nkM

- Af fifc)iT1 + R([ilk, M] + P A(lp)

+ Rtip A RP = 0 . Thus, R{t)P(t) -

constant, and R{t)M{t)R~l

(t) + x(t) A RP = constant

(3.7)

25

are the conservation laws on the semi-direct product due to left-invariance. Let us now exploit the conserved quantities. 3.3

The Conservation Laws

As already remarked the quantities Tr(L*) are conserved along the trajectories of a left-invariant Hamiltonian system on a semi-simple Lie algebra g. Remarkably, some of these conserved quantities also persist on semi-direct products as we shall presently see. Let us begin with non-Euclidean cases. The extremal curves L(t) are written as sums M + P with pn\

/ 0 -pj epi

P(t) =

0 \£Pn

and M(t) an n x n antisymmetric matrix. It will be convenient to write

£(*) = with P = (pi,...

,pn)T-

o -PT eP

M

Then,

||L|| 2 = (L,L) = - - T r L 2 = e | | P | | 2 + (M,M)

.

Let h = e||£|| 2 = ||P|| 2 + e(M,M). It follows from (3.7) that ||P(t)|| 2 is constant along the extremal curves on the semi-direct products. Hence, we may simply let e = 0 in the above expression to get the appropriate integral of motion on the semi-direct product. Let us now calculate the trace of L4. Then, 2 _ f-e\\P\\2 eMP

M

2

-PTM ~eP®P

with a b denoting the matrix (a ® b)x = (b • x)a. It follows that

26

TrL 4 = 2||P|| 4 + TraceM 4 + 4£||MP|| 2 But TraceM 4 = 2(M,M) 2 , and I\ = ||P|| 4 + <M, M>2 + 2e(M, M)\\P\\2 Hence £

2

I

r2 +, -I T TV i-K/t » M I I D I I 2 + , 11^6112 -/f r Lr 44 \) = -{M,M)\\Pf \\MP\\

It follows that I3 = -(M,M)\\P\\2

+ \\MP\\2

is an integral of motion independent of e. It is now an easy matter to verify that I3 is an integral of motion on the semi-direct product as well. Let us single out two cases that will be needed later on. In the three dimensional case, 0 - M 3 M2 M = I M3 0 -Mi 0 k - M 2 Mi 2

2

2

fpi V2 \P3J

2

2

and \\MP\\ = \\M x P\\ = ||M|| ||P|| - (M • P) . Therefore, h = - ( M i p i + M2P2 + M 3 p 3 ) 2 that is Mipi + M2P2 + M3P3 is also an integral of motion. For the problems of Euler-Grimths and Dubins-Delauney (

0

-Mx

Mn-i\

Mi

M =

0 \M„_i

as we shall see presently. Then,

/

27

/ Mx \ (M,M)

2

= ||M|| , where M = \M„-i/

and ||MP|| 2 = ||M|| 2 p 2 + ( £

Pi+1Mi)

. Hence,

I3 = -||M|| 2 ||F|| 2 + ||M||2p2 + ( $ > + i M i J . With these integrals of motion at our disposal we are now ready for the solutions of our problems. 3.4

Solutions for the problems of Griffiths and Dubins

Let us recall the decomposition of son(ffi) as the direct sum j + p with j isomorphic to so n _i(lR) and p the vector space in which the controls take their values. We shall write M — M\ + M2 for matrices M in so n (E) with Mi € p and M 2 € j . It is an immediate consequence of the Maximum Principle that

H = ±(M1,M1)+Pl is the appropriate Hamiltonian for the problem of minimizing

\f (U(t),U(t))dt over the trajectories of (2.3), and that extremal controls u = (m,..., are given by u = Mi. Hence ft* = Mi, f2p = E\ with / 0 -e 0 • • • 0 \ 1 0 0 Ei =

For the semi-simple case the extremal equations (3.2) become

w n -i)

28

^

= \MUM) + [EUP],

and

^

= [MUP] + [EUM] .

Since [j,p] C p, and [p,p] C j , it follows that

^ = t*.n ^ - » . f = w . i + (*.*]•

M

Therefore, M2(t) = constant. Because of the transversality conditions relative to manifolds Si and 52, the extremal curves l(t) annihilate the tangent vectors of Si and 52 at t = 0 and t = T. Since the tangent spaces of Si and 52 translated from the left to the group identity coincide with the subalgebra j , it follows that M2 = 0. Equations (3.8), written more explicitly, are then given by the following expressions: dMi -jj-=-pi+i(t), dp ~dt dp

t=

l,...,n-l

n-1 1=1

= -Mi(pi

-e),

i=

2,...,n

The reader may verify that the preceding equations are also valid in the Euclidean case. We shall now obtain the remarkable differential equation of Griffiths for the geodesic curvature K,(t). Recall that

J{t)=u\

+ --- + ul_i = Mt + -.- + Ml_i = \\M(t)\\2

Hence, ,

-JTJ

2/c k = 2M • —

71— 1

= - 2 ^2 MiPi+i . i=i

Therefore,

29 (n-\

«2K2= (^Mift+ij

=J3-||M|| 2 P l + ||P||2||M||2

2 = h K^H-\^\{h-e\\Mf)\\M\\

I3-K2{H-±KA

+{I2-EK2)K\

or, equivalently, , 2 _= - .- lKK 44 ++ ((i if f_-££))KK22 ++ (( // 2 -_i if72 2 K ) )++ 4- f 2

(3.10)

This expression coincides with the one obtained by Griffiths in case H = 0 and e = 0, ( 3 , p. 153). The time interval [0,T] is variable in Griffiths' formulation of the problem which explains H = 0 in this paper. We will now show the second remarkable discovery of Griffiths that K2(t)r(t) = constant, where r(t) denotes the torsion of the extremal curve in question. In the process we will also show that each extremal curve is confined to a three dimensional submanifold of M, a fact that was also noticed by Griffiths. To show these facts it will be necessary to recall the Serret-Frenet frames in three-spaces, consisting of the tangent, normal and binormal vectors denoted by t(i),n(t) and b(t) respectively. Their differential equations are the following

*; ^

=

K(i)n(t)

'

= -K(t)t(t)+r(t)h(t),

(3.11)

under the assumption that —— is linearly dependent on t , n, and b . at Since t(t) = x(t) = vi{t), the first leg of the Serret-Frenet frame and of the J-frame coincide. Since the deformations of the J-frame are given by

30

-j±

M

= - ^

i (*)«i+i (*) »

- ^ = Mt-! (t)vi (t)

and

t = 2,..., n

(3.12)

we have,

«(«>*(*) = £ ' = ^ = - ' £ ^ 1 . 8=1

and so +i

i=l

i=l

It then follows from equations (3.9) and (3.12) that dn

"

fi

iVi

« +K-^- = J2p i=2

/n-l

\

M v

=

- \'52 i) i \j=l

n

J2piVi - K 2 * W • i=2

/

Hence, -Dn

1^

-

k_

i=2

It follows from (3.11) that r b = —-—(- nt, which together with the equadt tion above implies that

rfi=^g(ft + ^ - i ) i ; « i=2

Then, 1 n r 2 = ^ ^ ( p ^ + ziM^i) 2 i=2 1 ™ »=2

31

Recall that

KK

T2K4

= — ^ Pi-^i-i, and therefore i=2

= £2 p2n2 — k2K2. i=2

But then

h = - | | M | | 2 | | P | | 2 + \\M\\Vi + (

^PiM^

M=l n

E K pi + K K lZ l ,

- 2i i2

,

i=2

therefore T^V = - J 3 Hence, /t 2 (£)r(t) = constant as claimed. I t rremains to show that —— is in the linear span of the Serret-Frenet triad It dt t, n, b . 2 ..Dri ,-.. „ . D n Since K2rb = n2 ——h K3t it suffices to show that ——— is in the linear dt at* span of the Serret-Frenet triad. To see this consider

kn + K-— = 2_^PiVi — K t dt i=2

Then, B

f

v^ .

j=2

i=2

Dvi

i=2

»=2

= (p\ — e)K,(t)n(t) — kt . 2 VZ

Dn

.

„

Therefore, is in the linear span of t, n and b . Having reduced the solutions to three dimensional problems, the remaining equations can be integrated by quadrature in terms of the geodesic curvature K,(t).

32

We now turn attention to the problem of Dubins. The passage to the extremal curves is straightforward thanks to the Maximum Principle. Nevertheless, there are some minor conceptual differences due to the constraints. The case n = 2 is slightly different because the boundary of u\ < 1 consists of isolated points ± 1 . In all other cases, the boundary is a smooth manifold Sn. Hence, the passage from the planar case to higher dimensional is more varied than for the problem of Griffiths. According to the Maximum Principle the optimal trajectories g(t) generated by the controls U(t) are the projections of integral curves of time varying Hamiltonians H\ of the form ra-l

Hx = -X + Pi +

Y^uiMi i=l

subject to the maximality condition that n—1

-\+Pl{t)

+ ^2ui{t)Mi{t)< i=\

n—1

-\+Pl{t)

+ Y,ViMiit)

>

i=l

for almost all t and for any vector v with v\-\ h v„-i < 1. The constant A is either equal to zero in the abnormal case, or equal to 1 in the regular case. In addition, H = 0 almost everywhere along the extremal curves. All of these conditions are dictated by the Maximum Principle. For time-optimal problems the only difference between the abnormal and regular extremal curves is that the Hamiltonians differ by a constant (i.e., they correspond to different energy levels). All points at which M(t) / 0 on an open interval (to,h) the maximality condition determines a single Hamiltonian H = p\ + \\M\\ with U(t) = —x——. For abnormal extremals H = 0, while for regular extremals

l|M(t)ll

H = \. For n = 2, M(t) is one dimensional, hence, M(t) is either positive or negative, and U(t) = sgnM(t). So on intervals {to,h) on which M(t) ^ 0, the extremal curves are circles (since the curvature is ±1). These circles concatenate at points t at which the curve M(t) crosses the switching surface M = 0 transversally. As is well known, the extremal curves for which M(t) = 0 on an interval project onto geodesies, which essentially explains the first part of the discovery of Dubins that optimal paths in E2 must be concatenations of circles of radius 1 and straight line segments.

33

The second, and perhaps more difficult part of Dubins' discovery, is that optimal arcs cannot contain more than three switchings. We shall not go into these details further. The solutions of Dubins' problem on 2-dimensional surfaces are well understood, and documented in full detail see 12 , 10 , and 15 . Instead, we will proceed to dimensions bigger than 2. Let us first assume that M[t) ^ 0 on an interval (to,h). We shall follow the notations of 13 and use (j,\(t) to denote the quantity ||M(£)|| with A denoting the dependence on the multiplier A that differentiates regular curves from abnormal curves. Otherwise we will follow the notations used in the Euler-Griffiths problem. The equations (3.3) and (3.5) are applicable. For e ^ 0 the extremal dM dP equations are — - = [n*,M] + [Q,P,P], and — = [fifc,-P] + [fi P) M] where ftk and Clp denote the appropriate projections of dH corresponding to H = pi + ||Mi ||. Therefore, fifc

= 177T7, = — M i ' a n d ttP = Ei . ||Mi|| MA As in the problem of Euler-Griffiths the projection of M on so„_i(R) is zero, and hence M = Mi. Therefore, ^

at

= [EltP] ,

and

^ = —[MUP] at /j,\

+ [E^M,]

.

The preceding equations can be written more explicitly in terms of the coordinates M i , . . . , M„_i of M as follows: dM; =-Pi+i dt

a}

P+ dt

l,...,n-l,

ra-1

dJ d i 1

i=

= ~Y,M^

and

1=1

** fP1 = - M i \fix

e

i - i i=

therefore -l

A«AAA =

-"Y^Pi+iMi. i=l

1 l,...,n-l

34

Now we can proceed much like in the problem of Euler-Griffiths: Since H^n

+ px,

J 2 = ||P|| 2 +

e/4

we have (MAAA)2

i 2 | | | | JI> ||22 , II JCT | | 22„ 2 = h - II^IIIIMiH + UMill ^

= h-A(H-&?

+ (I2-evl)til ^A 2

Therefore, (i\ = -(l + e)iJ,l + 2Hfix + I2-H2

+^

(3.13)

^A

where H = 0 for A = 0 and H = 1 for A = 1. Recall that equation (3.13) is derived under the assumption that M{t) ^ 0 on an interval. This assumption implies that the curvature of the associated extremal curve is identically equal to 1 in this interval. Hence, (3.13) contains the solutions for the problem of Delauney.This equation agrees with that of 13 , but differs from the one obtained by Griffiths on 3 p. 156 for the problem of Delauney. (Griffiths' variable A seems to be the same as the variable pi\ in this paper). We shall now show that the torsion r(t) of each extremal curve satisfies the relation Tfi\(t) — constant. Again we draw from the calculations obtained in the Euler-Grifnths case. It follows that the frame deformations in this problem differ from those in the problem of Euler-Griffiths by a factor fi\ i.e., Dvi —j— = at

1 ^ > MiVi fi\ *—'

and

Dvi 1 —j— = — M t v i , at n\

.

i=

2,...,n.

Since the curvature is equal to 1 in the interval (to, £i), it follows that the normal vector n(t) is given by n(t) =

^ MM, or H\ti(t) = - J2 MM. MA i=2

j=2

After the calculations analogous to the previous problem, with n replaced by fJ-\, we get that

(MAT)2

= YlPivl «=2

hence, (fj,\(t)) r(t) = constant.

~ PAAA = ~Is '

35

We shall omit the calculations which show that the derivative —r- of the dt binormal vector is in the linear span of the triad t, n, b , since they are the same as before. Consequently, we have recovered the result of D. Mittenhuber that each extremal curve is confined to a three dimensional submanifold of the ambient manifolds n . Of course, geodesic curves must also be the solutions to the problem of Dubins, hence there are solutions for which ||M(£)|| = 0. The algebraic surface S given by Mi = M 2 = • • • = M„_i = 0 is called the switching surface for this problem. We shall not go into any detailed analysis of the type of crossings of S, except to quote the following theorem 13 : A curve of minimal length for the problem of Dubins is necessarily one of the following: a.

A smooth arc with constant curvature equal to 1.

b.

A geodesic arc.

c.

A continuously differentiable curve that is a concatenation of geodesic arcs and arcs with curvature equal to 1.

It is still an open (but likely difficult) problem to determine the maximal number of switchings for an optimal curve. As already noticed in 3 any variational problem that involves functionals which depend on the first curvature only exhibits striking similarities to the preceding problems, although this observation has not been elaborated in full details yet. We now shift attention to the elastic problems. 3.5

The Elastic Problem and the Mechanical Tops

Recall that the elastic problem is defined through a positive definite quadratic form ( , ) on so„(E), and the parameters czi,... , a n . The Hamiltonian equations for this problem are most naturally expressed with the aid of the trace form (A,B)

= — -TrAB.

To begin with, the trace form is positive definite

on so n (M), and therefore (U, U) = (U, JU) for some positive definite (relative to the trace form) linear mapping J : so„(R) —> son(R). Each of the Lie algebra g in this paper is of the form g = 6 © p with p an n-dimensional vector space. We shall adhere with the notations established earlier and write

36

( 0 -pi epi

- p2

:

pn\

= P +M

M

\epn

/

for an arbitrary element L in Q with e ^ 0. In the Euclidean case /0

0--0\

Pi

L =

:

M

\pn

J

With these notations at our disposal, the Hamiltonian associated with an elastic problem is given by n

1 1

H=-(M,J- M) The parameters ai,...,an

+ Y,aiPi-

can be identified with a matrix A in Q given by ( 0 -eai

A=

ean\ 0

\an in which case (A, P) = ^ atPi, therefore the preceding Hamiltonian H can be also written as

H=-(M,J-1M)

+ (A,P) .

We remind the reader that this Hamiltonian is obtained through the maximally condition of the Maximum Principle. The maximum of -±(U,JU)

+ (A,P) + {U,M)

relative to the variables U occurs at U = J

M, and is equal to

^(M,J-1M) Then dH = A + J_1M. cases are ^

+ (A,P).

The extremal equations in the non-Euclidean

= [J-*M, M] + [A,P],

*£ = [J-'M, P] + [A, M)

whereas in the Euclidean case we have ^ - = [J-1M,M] +

PAA,

dp = dt

(-J~1M)P

(3.14)

These equations bear striking resemblance to the equations of motion of a rigid body under the influence of the gravitational force. To explain these somewhat mysterious connections we shall restrict our considerations to the Euclidean case and begin with the simplest case, the planar pendulum. Assume that a bob of mass m is attached to the pendulum of length I which swings under the gravitational force F = mge\. We have oriented the Euclidean plane by the right hand orthonormal frame ei, e-i that is centered at the fulcrum of the pendulum. We shall denote by (ai(£), 02(f)) the orthogonal frame attached to the pendulum and so adapted that the position q(t) of the free end is in the direction of Si(t) i.e., that q(t) = lai(t). See figure 1. Let R(t) denote the isometry defined by R{t)ei = Si(t),i = 1,2. Then each movement of the pendulum traces a curve R(t) in 502(M) given by q(t) = lai(t) = lR(t)ei. The kinetic energy T associated with each movement q(t) is given by dq(t) dt

- 2 ^

dR. ei dt

Since R(t) is a curve in S 0 2 ( dR(t) _

dt

R(t,

( 0

~H[T){w(t)

-w(t)

0

38

Figure 1. The pendulum

for some function w(t). The kinetic energy expressed in terms of w(t) is given by

T=]-ml2w2{t). The potential energy V in raising the pendulum from its equilibrium position q(0) = i€\ to an arbitrary position q(t) is given by

= mgl(l - (ei,.R(*)ei)) , where ( , ) denotes the Euclidean metric on E 2 . Therefore, the Lagrangian L — T — V is given by L(w,R) = -ml2w2

+ (ei,R(t)ei)

— mgl,

which we interpret as a function on the tangent bundle of S02(M) realized as the product so 2 (K) x S02(IR). According to the Principle of Least Action the movements of the pendulum minimize the integral / Jo

L{w(T),R(T))dr

39 over all paths *R(t) dt

( 0 = R{t) -^>\w(t)

"»(*) 0

subject to the appropriate boundary conditions. The Maximum Principle immediately yields the Hamiltonian H on T*S0 2 (E) of the form = ^pM2

H(M,R)

- mglieuRei)

+ mgl

where M denotes the Hamiltonian lift of the left-invariant vector field

i.e., M(p) = pA\, where A\ = I

1. In addition, the Maximum Prin-

ciple provides a relation between the angular velocity w(t) and the angular momentum M with w(t) =

—-rM(p(t)). ml2. Then, the integral curves of H satisfy

M(t) ml2

™'2

,

ft(A)=mgl(e1,RAe1)

(3.15)

)

for any A in 502(E), as can be verified through the formulas established in 5 . (Note that the above Hamiltonian is neither left nor right invariant.) To recognize these equations in their familiar form parameterize R(t) by an angle 6(t) defined by

n{>

/cos0(i)-sin0(i)\ ~ \sm6{t) cos6(t) J

Then, M

d_R-RH\

°

dt- \MifiL ml2

(t)\ -^nP\

r c ducc, tto redUCeS 0

n

J

*° -

M{t)

dt " mP

40

and

dt

= -77(^1) = m0/(ei,.R(i),4iei) = dt

-mglsinO

Hence, d20 _ 1 d M _ _mgl . dt2 ~ ml2 dt _ ~ r f S i n

'

or g

+

f s i „ » = 0.

(3.16)

While normally investigations of the planar pendulum end with equation (3.16), equation (3.15), which is hardly every noticed in the literature on mechanics, shows connections with the semi-direct product as follows. Let h{t) = mglR~1(t)ei, and let hi(t), h,2(t) denote its components. Then,

dh dt

= - " » < U ~ " o ( t ) ) *-'<<>«• =

and —— = (h(t),A\ei) dt Hamiltonian

H(huh2,M)

= h^U) constitute the equations of the left-invariant

= 7TV 2mt 2

M 2

+ ei> = T ^ 7J2 M 2 2m/

+ /l

i

on the semi-direct product G = E2 x S02(M). Of course, the mathematical pendulum is confined to the momentum level h\ + h\ = (mgl)2. So, the movements of the pendulum are subordinated to the problem of the elasticae. The same applies to the spherical pendulum but with minor modifications. The orthonormal frames are taken in the Euclidean space E 3 , and the moving frame a\(t), a2 (t), S3 (t) is adapted to the pendulum so that the position of the free end q(t) is given by q(t) = lai(t). Then each path of the pendulum traces a path R(t) on SOa(]R) defined by di(t) — R(t)ei i = 1,2,3. As usual, each path R(t) generates angular velocities w\(t), ui2(t), W3(t) denned by

41

0 dR(t) = R(t) | w3{t) dt ,-w2{t)

-w3(t) 0 (t) Wl

w2{t) -wi(t) 0

Since the pendulum does not rotate about its own axis we will impose the constraint that wi(t) = 0. That is, we will trace the movements of the pendulum by J-framed curves on S03(K). Then, the kinetic energy T is given by —W3 Wi

0

= ±-ml2 R(t) 2

T=-m

w3 -w2

0 0

0 0

ei

= -ml2{wl+wl) , while the potential energy V retains the same form as in the planar case and is given by V = mgl{l — (ei,i?ei)). The Principle of Least Action for

V)dt Jo

yields the Hamiltonian

H =

2ml2

{Ml + Ml) - mglieuRex)

+ mgl,

where M2(p) = p{A2),M3(p) = p(A3) for all p in so^R) with Ai,A2,A3 denoting the standard basis in so 3 (E). As a consequence of the Maximum Principle, the extremal angular velocities w2(t) and w3(t) are given by the usual relations w2(t) = ——M2(p(t))

and w3(t) =

TTvl

In a manner analogous to the one used in extremal equations dR = RQ{t) , ~dl with

——M3(p(t)). Tflil

and

5

(pp. 376-378) we get the

dM = M xQ, + P{t) x ei dt

42

/

o

M3(t) ml2 , M2(t) ml2 0 -M3 M(t) = ( Af3 0 , - M 2 Mi

n(t) =

M3(t) ml2

(3.17)

0 0

M2 -Mi 0

M(t) = ( M i , M 2 , M 3 ) T and P(f) = m ^ P " After identifying vectors A = (ai,a2,a3)T

(

M2(t)\ ml2

1

^. with antisymmetric matrices

0 -a3 a3 a3 0 — ai —a,2 ai 0

we can rewrite the extremal equations in the Lie bracket form: dR

„.,„,,

dM

= [Sl(t),M(t)]

+ [A1,P(t)]

.

The extension to the semi-direct product is straightforward: Each movement q(t) of the pendulum defines a curve g(t) in the group of 1 0 motions where g(t) = Since P(t) = mglR ei, lq(t) R(t) dP = P(t) x Cl(t) , dt Together with the equations ^^[n,M}

or

dP dt

= im,p(t)}-

+ [A1,p]

these equations coincide with equations (3.14). The equations of motion for arbitrary rigid body in E 3 are done in 5 , and for an n-dimensional rigid body in 14 . For convenience to the reader we shall quickly outline the procedure. Let e i , . . . , e„ denote an absolute frame, and let ai (£),..., an(t) denote a frame attached to the body with both frames centered at a fixed point O of the body. We may orient the absolute frame so that the gravitational force is of the form F = Ce\ where C is a gravitational constant.

43

Denote by R{t) the isometry that relates the moving frame to the absolute frame. If q = (qi,... ,qn)T denotes the coordinate vector of a point OP in the absolute space, and if Q = (Qi,... ,Qn)T denotes the coordinate vector of the same point relative to the moving frame then q = RQ . For each movement of the body a point q(t) on the body undergoes velocity —- = -r-Q. The kinetic energy of each such point is given by at at dq ~dt

1 2m

2

=

1

^m\\u(t)R(t)Q\\2

where u(t) denotes the antisymmetric matrix defined by dR(t) —^- =

,.. . . u(t)R(t).

The total kinetic energy is the aggregate of the contributions due to point masses and can be expressed as an n-dimensional integral in terms of the mass density p. Then, \\u(t)R(t)Q\\2p(Q)dQ

T=\f

L

* ./Body 1

Let U{t) = R~ (t)u(t)R(t).

\\R-1u(t)R(t)Q\\2p(Q)dQ

= \f ./Body

Then each motion of the body corresponds

to a curve R{t) on SOn(R) where — = u(t)R(t) = R(t)U(t). dt kinetic energy is given by

\ [ z

.

The associated

\\U(t)Qfp(Q)dQ

./Body ody

We now express kinetic energy in terms of the quadratic form {UuU2) = \f 1

(U1Q,U2Q)P(Q)dQ, ./Body

where (UiQ,U2Q) denotes the Euclidean inner product. The form ( , ) is positive definite on the space of n x n antisymmetric matrices, and T(U) = (U,U). Thus, the geometry of each rigid body defines its own left-invariant metric on SO„(E). To find the expression for the potential energy it is convenient to introduce the center of mass Q0 defined by

44

<9o /

p(Q)dQ = J

/Body •/Body

QP(Q)dQ.

./Body -/Body

Then the potential energy is V is given by

V = -C J J

/ e i • ™QSp(Q)dQdT = -Cm(euR{t)Q0) ,

where m stands of the total mass of the body. To get the equations of motion we shall assume that the Principle of Least Action is valid. That is, we shall assume that the motions of the body minimize

J (T-V)dr

=J

^(U(T),U(r))-Cm(e1,R(T)Q0)dT

over all curves R(t) in SO n (E) that are solutions of ^ ^ = R(t)U(t). at As in the case of elastic problems, the quadratic form ( , ) induced by kinetic energy can be expressed in terms of the trace form on so n (R), in which case, the Hamiltonian corresponding to the total energy of the body is given by H=^-(M, H

J~XM) + Cm{R-leuQ0)

.

To make the connection with equations (3.14) rename the variables P(t) = m C R - 1 e i , Qo = a, and identify

x(t) = / R{T)etdT Jo and R(t) with the matrix

»<*> = (*(t) R°(t)) Then,

45

/oo-

• o\

1 = 9(t) 0 il(t) dt

V

dM dt

[n(t),M(t)]

+ [A,P(t)},

7

and dp

= m),P(t)}

where ( P(t)

/o o

0 0 Pi(t) 0

0

\pn(*)0

•••0/

and

A =

0

ai 0 Van°

•••

°/

The preceding developments reveal a remarkable and unexpected phenomenon: The movements of an n— dimensional top always form an invariant subsystem of an elastic problem on a Euclidean space E™ For the top the extension to the semi-direct product is redundant. Indeed, once the angular momentum matrix M(t) is determined, the angular velocity U(t) is given by U(t) — J~lM(t), and hence the rotation matrix R(t), as the solution of the matrix differential equation —— = R(t)U(t),

is completely

determined. The translational momentum variable P(t), given by P(t) = CmR~1(t)ei, corresponds to the body coordinates of the fixed vector Cme\ and is determined by the rotation R(t). The same is not true for the elastic problems, even though R(t) and P(t) are intertwined by the conservation law R(t)P(t) — constant. Hence, the top is an invariant subsystem of the elastic problem denned by R(t)P(t) = Cme\. The central line of the top x(t) defined by x(t) = x0 + / R(s)eids, Jo which is never mentioned in the literature on mechanics, carries the geometric information which links the top to the geometric problems considered in this paper. To be more precise about this point we shall now focus on three dimensional tops, called heavy tops.

46

3.6

The top of Lagrange and the problem of Euler- Griffiths

For three dimensional problems the deformations of frames are described by K = SOs(]R), and the elastic quadratic form is a quadratic form on the Lie algebra k = 503(E). We shall continue with our conventions that matrix

(

0 - a 3 a2 \ a3 0 - a i -a2 ai 0 J

is identified with a vector A = (aia2,a,3)T. Then, [A, B] = B x A where x denotes the cross product in E 3 (Remember that [A,B] — BA — AB). Furthermore, (A,B) = A-B with A • B denoting the Euclidean metric in E 3 . We shall use Aj, A2, A3 to denote the eigenvalues of the form ( , ) relative to the trace form, called the principal moments of inertia in the literature on mechanics. Let Ai, A2, A3 denote an orthonormal basis in 503 (K) relative to the trace form of eigenvectors of J. We have that (A, A) = X1al + X2al + \za\ ,

for any matrix

A — o,\A\ + a2A2 + a3A^ .

Furthermore, we may assume that Ai, A2, A3 are positively oriented in the sense that i i x A2 = A3 ,

A3xAi=A2,

A2xA3=Ai,

or equivalently, we may assume that [AUA2] = -A3

,

[A1,A3] = A2,

and

[A2,A3] = -A1

.

For elastic problems on three dimensional space forms the Lie algebra of G is six dimensional, so we need to extend this basis in 503(E) to all of g. The most natural way to extend this basis is to define

Bl

/ 0 -an -U 0 ) '

(0 -eAf\ (0 S '{A2 0 ) ' * -U

B2

-SAT\

0 ) •

Then Ai, A2, A3, Bx, B2, B3 is an orthonormal basis on g relative to the trace form. The commuting relations for the preceding basis are given by table 1, as can be easily verified

47

Table 1. A2 ~A3

A! At A2 A3 Bx B2 B3

0

A3 A2 -Ax

0

^3

-A2

0

Bx

0

B3 -B2

0

B3 B2 -Bx

B2 -B3

0

0

A! -B3

B2

0

B1 -eA3

0

-Bi

eA3

0

eA2 -eAi

Bx

0

-EA2

eAi

0

B3 -B2

3

3

As before, we shall write L = M+P with M = £ MiAt and P = X] Pi#»1=1

8=1

In these coordinates the Hamiltonian H = — (M, JM) + (A, P) is equal to

2 \ Ai

A2

+ aipi + a2p2 + a3p3

A3

where a\, a2, a3 denote the coordinates of A with relative to Bi, B2, B3. Mi Mn M3 Then dH = -r^-Ax + -r~A2 +-r1 + axBx + a2B2 + a3B2 and the HamilAi A2 A3 tonian equations (3.3) and (3.5) expressed in terms of the vector products on R3 are as follows: dM

A

A

1

where M = (MxM2,M3)T,P

dP

A

A

, ,~

Is

r (3.18) — = P xtt + e(M x A) dt 1 1 1 = (Px,p2,p3)T,(l = (—M 1 ; — M 2 , — M 3 ) T

—— = M xQ + Px A , at

Ax

A2

A3

and A = {ax,a2,a3)T Equations (3.18) coincide with the fundamental equations in 5 except for the minor changes in notation. It is known, although not yet published, that two eigenvalues among Ai, A2, A3 must be equal to each other for (3.18) to be completely integrable. In the case that two eigenvalues are equal to each other, there are essentially only two cases that are completely integrable: the case of Lagrange and the case of Kowalewski 6 . The case of Kowalewski, considerably more challenging to understand and explain, will be treated in another paper. Here we focus on the case of Lagrange and relate its solutions to the solutions of Euler-Grifnths problem obtained earlier.

48

For Lagrange's elastic problem we assume that A2 = A3 and that a 2 = a3 = 0. Then it follows immediately from equations (3.18) that M\(t) is constant. Since a\ + a\ + a3 = 1 and a2 = a3 = 0, it follows that ai = ± 1 . We shall take a\ = 1 (The other choice ai = — 1 is isomorphic to this one). The control system that gives rise to this Hamiltonian is of the form: / 0 -e 0 0 \ 1 0 -u3 u2 0 «3 0 —«i

dg

di=9

\0 -u2

ui

0 J

dx If x(t) denotes the central line then x(t) = g(t)ei and — = <7(£)e2. Moreover, dt Ddx , , D_dx_ 2 -r—r- = uzge-z—U2gei, and therefore — (wj+wl). Thus, the geodesic at at ~dt~dl curvature n(t) associated with the central line x(t) satisfies K2 = u\ +u\, and hence

-J

{u,u)dt= ±J {K2+u\)dt.

Since the extremal controls are given by ui(t) = T-Mi(i) , Ai

u2{t) = -M2(t) A

,

u 3 (t) =

A

TM3(t),

with A denoting the common value A2 = A3, it follows that Ui(t) = constant and that M2 + M2 = \K2 . In particular when A = 1, then the geodesic curvature of the elastic curve x(t) is equal to y/M% + M | . Hence the Hamiltonian H corresponding to Lagrange case can be rewritten as

H

=r2

+

i-1Mi+pi

1 or in the reduced form as HQ = -K2 +pi, since M\ = constant. The equation

US) 2 - ( M * - 5^)" -.(. + »?))-(«.- (H- i«)«.)2 obtained on p. 460 in 5 can be easily translated into the notations in this paper. The constant of motion M\ = Hihi + H2h,2 + H3h3 is related to the

49

integral of motion I3 in this paper by Mi = ^fh. The other integral of motion \\h\\2 +e\\H\\2 is the same as J2 in this paper. Moreover, v2 = K2. Hence, the above equation becomes

K2k2 = K2 (l2 - (H0 - \K2)2 It was also shown in is given by 2 K

4

- S(K2 + M2)) - (-y/h-

(Ho -

^

2

)^i)

that the associated torsion r(t) of the central line

(t)r(t)

= Hi (H0 + i « 2 ) - Mi ,

which rephrased in the notations of this paper becomes

K2T

= Mi

Thus K2T = constant whenever Mi(t) = 0. Then K2K2 = K2 (h -{H-

\K2)2

- eA

+ h •

Remarkably, the preceding differential equation for K coincides with the equations (3.10) for the problem of Euler-GrifBths. We may summarize some of our findings in the following way: The elastic curves for the Euler-Grimths problem for n > 2 coincide with the elastic curves for the Lagrange's elastic problem whenever the latter are subjected to the same boundary conditions (Mi = 0). We may also rephrase the preceding observation by saying that the top of Lagrange generates a central line which is an elastic curve. More specifically we may relate the elastic curves of Euler-Grifnths to the movements of the spherical pendulum via the equation (3.18). In this context, the position q(t) of the pendulum dx and the corresponding elastic curve x(t) are related by — = q(t). Thus, the pendulum traces the tangent line of the elastic curve. This observation fits nicely with 4 which relates the planar elasticae to the movements of the planar pendulum. The analogy with the top further illuminates the significance of elastic curves. As we have just seen the elastic problem naturally extends to all space forms, while the top does not (at least not in some convincing way).

50

As a result, the elastic problems reorient our understanding of completely integrable Hamiltonian systems associated with the tops and suggest natural extensions to arbitrary Lie groups. Among many compelling and intriguing questions that can be asked about elastic curves, we shall single out the ones pertaining to closed elastic curves. Langer and Singer in 7 showed that all possible toral knots occur as the solutions of Euler-Grifnths problem for the Euclidean case. In view of the fact that the Euler-Grifnths problem leads to a completely integrable Hamiltonian system, the conclusions of Langer and Singer are not surprising. However, could the result of Langer and Singer be made more transparent through suitable action-angle arguments? More generally, does the same conclusion hold for the elastic curves on non-Euclidean spaces. Finally, passing to the case of Lagrange, we may ask for the types of knotted curves which occur as the solutions of the elastic problem for which K2T is not necessarily constant. References 1. C. Caratheodory, Calculus of Variations and Partial Differential Equations of the First Order, (Reprinted 1982, Chelsea Publishing Co., New York, Berlin, Teubner (1935)). 2. L.E. Dubins, On curves of minimal length with a constraint on average curvature and with prescribed initial positions and tangents, Amer. J. Math, 79, 497-516 (1957). 3. P. Griffiths, Exterior Differential Systems and the Calculus of Variations, (Birkhauser, Boston, 1983). 4. V. Jurdjevic, Non-Euclidean Elastica, Amer. J. Math., 117, 93-125 (1995). 5. V. Jurdjevic, Geometric Control Theory, (Advanced Studies in Mathematics, 52, Cambridge Univ. Press 1997). 6. V. Jurdjevic, Hamiltonian Systems on Lie Groups: Kowalewski type, Annals of Math,150, 1-40 (1999). 7. J. Langer and D. Singer, Knotted elastic curves in R3, J. London Math. Soc, 2, No. 30, 512-534 (1984). 8. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, (4th Edition, Dover, New York 1927). 9. D. Mittenhuber, Dubins' Problem in hyperbolic space, in Geometric Control and Non-holonomic Mechanics, Canadian Math. Soc. Conference Proceedings, Eds. V. Jurdjevic and R. Sharpe, 25, 101-115 (1998). 10. D. Mittenhuber, Dubins' Problem in hyperbolic plane using the open disk

51

11. 12. 13.

14. 15.

model in Geometric Control and Non-holonomic Mechanics, Canadian Math. Soc. Conference Proceedings, Eds. V. Jurdjevic and R. Sharpe, 25, 115-153 (1998). D. Mittenhuber, Dubins' Problem is intrinsically three-dimensional, ESAIM: Controle, Optimisation et Calcul des Variations, 3,1-22 (1998). F. Monroy-Perez, Non-Euclidean Dubins problem, A Control Theoretical Approach, (Ph.D. Thesis, University of Toronto, 1995). F. Monroy-Perez, Three dimensional non-Euclidean Dubins' Problem in Geometric Control and Non-holonomic Mechanics, Canadian Math. Soc. Conference Proceedings, Eds. V. Jurdjevic and R. Sharpe, 25, 153-183 (1998). T. Ratiu, The Motion of the Free n-Dimensional Rigid Body, Indiana Univ. Math. J., 29, No. 4, 602-629 (1980). H. Sussmann, Shortest 3-Dimensional Paths with a Prescribed Curvature Bound, Proceedings of the IEE Conference on Decision and Control, 33063312 (1995).

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53

CONTROLLABILITY OF LIE SYSTEMS J. D. LAWSON Deparment of Mathematics of Louisiana State University E-mail: [email protected] D. MITTENHUBER FB Mathematik, AG 5, TV Darmstadt, Darmstadt Germany E-mail: [email protected] The purpose of this paper is to present conditions that allow one to determine the controllability of a Lie system on a connected Lie group G by pushing the controllability question forward to smaller dimensional coset spaces. Since noncontrollable systems may become controllable on smaller dimensional coset spaces, the problem considered is the determination of closed subgroups H such that a system is controllable on G/H if and only if it is controllable on G.

1

Introduction

In recent years there has been an increasing interest in geometric control problems that can be formulated on Lie groups with control set a subset of either the left or the right invariant vector fields. For example, V. Jurdjevic has successfully treated various aspects of certain classical problems in mechanics by interpreting them in the context of control problems on Lie groups (see, for example, Chapter 6 of 5 ) . The advantages of interpreting problems in this framework include having available both the machinery of geometric control theory and the rich arsenal of highly developed theoretical tools from the theory of Lie groups and Lie algebras. The primary analysis in such problems generally involves application of the Maximum Principle, but one typically needs to check that the system is controllable at the initial stage of the investigation. The primary aim of this paper is to present some tools that are helpful for the determination of controllability for systems on Lie groups. The main technique involves pushing the controllability question forward to a potentially easier one on a smaller dimensional coset space. In this context one seeks to be able to recognize those subgroups H of a Lie group G with the property that a system is controllable on G/H if and only if it is controllable on G. For the analysis of such problems we find the consideration of subsemigroups of Lie groups with non-empty interior to be a useful tool. In the first sections we introduce and develop the basic concepts with which we work. The main focus of this paper is the setting of solvable Lie groups and we consider those in some detail in the main sections of the pa-

54

per. We close with brief remarks concerning the work of L. San Martin and collaborators in the semisimple setting. 2

Control systems on Lie groups

Let G denote a Lie group with Lie algebra g (one may think matrix Lie groups, the primary example, although the results of this section are valid for more general Lie groups). We henceforth identify g with the set of right invariant vector fields on G. For X 6 g and g € G, let X{g) € TgG denote the value of the vector field X in the tangent space TgG at g. Let M be a smooth (= C°°) manifold and suppose that $ : G x M -»• M is a smooth action of the Lie group G on M. We typically denote $(g,x) by gx or g.x. Let V°°{M) denote the Lie algebra of smooth vector fields on M. For x S M, the smooth mapping <J>X : G -> M given by §x{g) = g.x has derivative at e, c ^ : TeG -> TXM; alternately d$x(v), v G TeG, is given by » 4 G is the exponential mapping and X(e) = v. The mappings d$x give rise to a Lie algebra homomorphism d$ : g —> V°°(M) given by <M>(X)(x) = d$ x (X(e)) (note that the appropropriate match-up to obtain a Lie algebra homomorphism is right invariant vector fields with left actions). We denote the vector field d$(X) by X. Reference 21 provides a detailed treatment of transformation Lie groups. Let SI be a subset of 0. We consider the Basic Group Control System (BGCS) on G given by the differential equation g(t) = U(t) (g(t)),

(BGCS)

where U(-) is called a steering or control function and belongs to the class U(ft) of measurable functions from R+ = [0,co) into SI which are locally bounded, that is, bounded on every finite subinterval. (A measurable set in g is one whose image is measurable in the euclidean space which is vector space isomorphic to g; equivalently the set is measurable with respect to Haar measure on (g, +).) In the case of a matrix group G with matrix Lie algebra g, the Basic Group Control System becomes the control matrix differential equation g(t) = U(t) • g(t). A solution of (BGCS), called a trajectory, is an absolutely continuous function x(-) from E + into G such that the equation (BGCS) holds a.e. Control systems such as the one just described are called right invariant, since right translates of solutions of (BGCS) are again solutions. Lemma 2.1. Let U : R + —> ft C g be locally bounded and measurable. Then there exists a unique solution x(-) on G of the initial value problem x(t) = U(t)(x(t)),

x(0)=g.

55

This solution is globally defined in the sense that x(t) is defined for all t > 0. Proof. By passing to a chart, one obtains a system of ordinary differential equations satisfying the Caratheodory conditions; hence there exist unique local solutions. The homogeneity of G is used for extending local solutions to global solutions. See Lemma 2.1 of 6 for details. • If U is a locally bounded measurable steering function, we denote the solution x(-) of (BGCS) which satisfies x(0) = g by n(g, -,U), i.e., 7r(g,0, U) — g and 7r(<7,£, U) = x(t) for all t > 0. If h = ir(g,t,U) for some t > 0, we say that U steers g to h in t units of time. If there exists U £ U(Q) such that h = Tr(g,t,U), then we say that h is attainable from g at time t for the system SI. The set of all such elements attainable from g at time t is denoted A(g,t,Q). We also employ the notation

A(g,
(J

A(g,t,0),

0
A( fl ,fi)=

(J

A(g,t,n).

0
The set A(g,Cl) is called the attainable set from g. We use the subscript "pc" to denote the appropriately modified versions of the preceding notions for the case that only piecewise constant controls into ft are used. For example, the set of elements attainable from g at time t using only piecewise constant controls into fi is denoted Apc (g,t,il). Note that the equality A(g, Q) = A(e, Cl)g and analogs for other attainable sets follow from the right invariance of the control system. Thus attainable sets at the identity determine the attainable sets at other points, and we can restrict our attention primarily to the former. The following theorem is a summary of results surveyed in n . Theorem 2.2. Let 0 ^ tl C g. (i) A p c ( e , n ) C A(e,H) are both subsemigroups of (exph^), the subgroup generated by exp(hfj), where h n is the subalgebra of g generated by fL (ii) The subsemigroup Apc(e, Q.) is equal to the subsemigroup generated by exp(fi). (hi) The semigroups Apc(e,£l)

and A(e, Q) of G have the same closure in G.

(iv) The semigroups Apc(e,il) and A(e, J7) have non-empty interior if and only if Si generates (as a subalgebra) g. In this case the interior of each semigroup is dense in that semigroup.

56

(v) IfS = A(e,fl)~,

then

L(5) := {X eg: exp(tX) £ S for all

t>0},

called the tangent wedge of S, is a closed convex cone, and Jj(S) fl — L(5) is a subalgebra of g. Furthermore, h(S) is the largest subset of g having attainable set contained in A(e,fl)~. That fl generates g is the "full-rank" condition in the Lie group context. This motivates our focus on semigroups with non-empty interior. It is convenient to also denote the semigroup A(e, fl) by S(fl) and Apc(e, fl) by Spc(fl). Note that since the attainable or reachable set for g 6 G is S(fl)g, it follows that fl is controllable on G if and only if S(fl) = G. Since a dense subsemigroup with non-empty interior of a connected group G must be G (see 7 ) , we have that if 5(f)) is dense, then Spc(fl) = S(fl) = G, and the system is controllable and piecewise-constant controllable. These control theoretic notions on G have analogues for a smooth Gtransformation group $ : G x M -t M. For fl C g, we consider the basic manifold control system (BMCS) on M given by the control differential equation x(t) = U(t)

(x(t)),

where £/(•) : K+ -> fl is locally bounded and U(t) = d$ (U(t)). Proposition 2.3. The solution to the (BMCS) ±(t) = U(t) {x(t)),

x(0) = x

on M is given by x(t) = g{t).x, where g(-) is the solution of (BGCS) with steering function U(-) and initial condition e. The basic control differential equation on M has a global solution for any initial value and A(x, fl) = S(fl).x for all x € M. Proof. The first assertion follows from x(t) = d^xg(t)

= d$x (U(t)(g(t))) =

= d* f l ( 0 . I (l/(t)e) = U(t)(g(t).x)

d$x°dpg{t)(U(t)e) = U (x(t)),

where pg (h) = hg is right translation in G. That global solutions exist now follows from the corresponding assertion for the (BGCS). Also the last assertion follows readily from the first. •

57

3

Groups irrelevant for transitivity

Prom the preceding section we have the following general principle for controllability of invariant systems on Lie groups: A subset il of the Lie algebra g of a connected Lie group G determines a controllable system on G if and only if the attainable set at the identity e is all of G, i.e., if and only if S(fi) = A(e, CI) = G. It can be a quite difficult problem to determine whether or not this is the case. Sometimes the recently developed Lie theory of semigroups (as presented in 1 or 2 , for example) can be helpful in this regard. In this paper we consider the strategy of reducing the problem to a lower dimensional, hopefully simpler, problem. Homogeneous spaces are often good candidates for such reduction. Suppose that a topological group G acts continuously on a space M. The space M is a homogeneous space if the action is transitive (G.x = M for all x £ M) and if further the mapping g i-> gx : G —> M is an open mapping for each x £ M. Let S be a subsemigroup with non-empty interior (a general setting that includes the case of the A(e, 0) for fi generating). Let M be a homogeneous space for an action of G. We say that the semigroup S acts transitively on M if Sm = M for all m £ M. We would like to know conditions on G, M, and perhaps S that ensure that S = G if 5 acts transitively on M. We recall a result of San Martin and Tonelli 25 . Proposition 3.1. Let G be a topological group acting on the homogeneous space M. Let S C G be a subsemigroup with nonempty interior. For p £ M, let Gp denote its stabilizer subgroup. If S acts transitively on M, then Gp D int(S) ^ 0 for all p £ M. Conversely if M is connected and Gp D int(S) jt 0 for all p 6 M, then S acts transitively on M. Proof. Let p 6 M, let s0 € int(5), and let q = sQp. If S is transitive on M, there exists t e S such that tq = p. Then ts0 £ S (int(S)) C int(S') (since S (int(S)) is an open subset of S), and (ts0)p = p, i.e., tsQ 6 int(S) f~l Gp. Conversely suppose that p 6 (intS)p for all p e M. Let T := int(S). Then T and T _ 1 are open subsemigroups of G, p € Tp for all p e M, and hence p £ T~lp for all p £ T. Let x £ M, let U = Tx and let V = U{ T _ 1 2/ : V & U}. Then U and V are open since M i s a homogeneous space, and U U V = M since y £ T~ly. If U D V ^ 0, then there exists sx = t~xy for some s,t £ T and y £ U. Thus y = tsx £ Sx = U, a contradiction. Hence U f~l V = 0. We conclude fromn connectivity that V = 0, i.e., that Tx = M. Thus S acts transitively. • Corollary 3.2. Let G be a connected group acting on the homogeneous space

58

M and let S C G be a subsemigroup with nonempty interior. If S acts transitively on M and if Gp is compact for some p € M, then S = G. Proof. By transitivity of the action M = Gp is connected. By the previous proposition there exists s e Gp n int(5) ^ 0. Pick U = U'1 open containing the identity e of G such that sll C int(S). Since Gp is compact, we can find n such that sn € U~\ hence e G snU = sn-1sll C 5 (int(S)) C int(S). Since G is connected and int(5) is an open subsemigroup containing e, it follows that int(S) is all of G, and hence S = G. • For the following definition, we have in mind closed subgroups H and the corresponding homogeneous spaces G/H, but the definition applies to arbitrary subgroups H and the corresponding set-theoretic action of G on the cosets G/H. Definition 3.3. A subgroup H < G is irrelevant for transitivity (IFT) if for every subsemigroup S C G with int(S) ^ 0 the conclusion: S transitive on G/H (S.x = G/H for all x £ G/H) = • S = G holds true. If a group H is not (IFT), we say that it is (NIFT). Corollary 3.4. Let G be a connected Lie group and let H be a compact subgroup. Then H is irrelevant for transitivity. Proof. Suppose that 5 is a subsemigroup with non-empty interior. If 5 acts transitively on G/H, then by Corollary 3.2, S = G, since the isotropy group at e = eH = H is H. D Let G be a connected Lie group and let H be a closed subgroup. Then the homogeneous space M := G/H admits a (unique) smooth structure such that the action $ : G x M -> M is smooth. Let U be a generating subset of g, and let S(il) be the semigroup of elements reachable from the identity. If H is irrelevant for transitivity, then (from the previous section) the control system (BGCS) determined by Q, on G is controllable <3> S(fi) = G <3> S(fl).x = M for each x 6 M -O the basic control system (BMCS) on M determined by fl is controllable. Thus we have the following Proposition 3.5. Let H be a closed subgroup of a connected Lie group G that is irrelevant for transitivity. If Q be a generating subset of g, then the basic control system (BGCS) on G is controllable if and only if the basic control system (BMCS) on M = G/H is controllable. The preceding proposition provides good motivation for identifying (IFT) subgroups, since they allow the study of controllability questions on the lower dimensional manifold G/H.

59 The following criterion turns out to be useful so we state it for future reference: Proposition 3.6. Let H C G be a subgroup and S C G a subsemigroup. Then S is transitive on G/H

-v=>

SH = HS = G,

and

H is (IFT) -£=> for every proper S with nonvoid interior SH n HS ^ G. Conversely, H is (NIFT) <£=> there exists a proper S C G with 'mt(S) ^ 0 such that SH = HS = G. Proof. One sees directly that if S is transitive on G/H, then SH = G. If sgH = H, then g £ gH = s _ 1 ff C S^H, so S^H = G. Taking inverses of both sides gives HS = G. Conversely suppose that HS = SH = G. Then for cosets g\H and <72#, pick S\,s? 6 S and hi,hi € H such that g^1 = /i2s2 and gi = sihi. One computes directly that (siS2)(
Exploiting compactness and irrelevancy

Having equivalent statements on different levels can be exploited in two ways: if one wants to prove that a system on a big group G is controllable, one might hope that it is easier to show this for the induced system on the homogeneous space M = G/H. Conversely, noncontrollability "downstairs" follows from noncontrollability "upstairs." Having the possibility to switch back and forth has the simple advantage that it is not always clear in advance which framework is more suitable for the problem under consideration. In particular results that were obtained on one level become suddenly available on the other level, too. As an example highlighting these advantages we reconsider the proof given in 15 of non-controllability of Dubins' problem in hyperbolic space HP when the curvature bound does not allow overcompensation for the intrinsic (constant, negative) curvature of the underlying space. Dubins' problem can be stated as follows: we consider an n-dimensional manifold M of constant curvature e 6 {-1,0,1}. So for e = 0 or 1 we have euclidean space E" or § n whereas e = — 1 leads to hyperbolic space HP. Our task now is the following: Given initial and terminal points XQ,XI € M as well as initial and terminal tangent vectors vt £ TXiM, i = 1,2, find a length-minimizing C 1 curve 7: [0,L] -> M matching these boundary conditions (or Dubins' data) such that 7 has curvature (almost everywhere) bounded by a fixed constant KQ > 0, see FIGURE 1.

60

Y

<

M

XQ Figure 1. Dubins' problem

There are many nice interpretations for this problem: one may think of a car driving in the plane at constant speed that cannot make arbitrarily sharp turns; laying railroad tracks or a plumber laying pipes are other possibilities. The spherical problem one can interpret as a path planning problem for planes or ships. This variational problem can be formulated as an optimal control problem on a homogeneous space of a Lie group, the motion group G™ of M™. For the classical geometries we have: Rn S S E ( n ) / S O ( n ) ,

S" = SO(n + l ) / S O ( n ) ,

and

H" = S O o ( l , n ) / S O ( n ) .

The associated control system on G™ comes from the Serret-Frenet-equations for moving frames. In matrix notation one obtains the system: ' 0 1 0 -e 0 uT 0 -u X

ue

||U||2
Xeso(n-l).

(4.1)

So g g G? C GL(n + 1,R), and M£" = G?/K with K s SO(n). In the euclidean as well as the spherical case one immediately obtains that admissible paths for Dubins' problem always exist because the reachable semigroup has interior points and neither SE(n) nor SO(n) has proper subsemigroups with nonempty interior. This changes in hyperbolic space. Monroy observed that Dubins' problem in the hyperbolic plane is not controllable if the curvature bound «o is too small, namely if KQ < 1. This means that there are Dubins' data xo, £1, "o, «i for which there does not even exist an admissible arc. His argument however, construction of a bounded region B: H 2 —> R invariant under the flow of the control system, did not generalize immediately to higher dimensions. Nevertheless, we have the T h e o r e m 4 . 1 . Dubins' problem in HP is non-controllable iff KO < 1. Geometrically this is a nice theorem because it says that we must be able to overcompensate the curvature of the underlying manifold (constant —1). The proof

61 in 15 exploits irrelevancy and results from Lie semigroup theory. Lawson's investigations of the Mobius semigroups—certain geometrically interesting subsemigroups of 0 ( 1 , n)—and his results on their tangent wedges just happened to provide the answer modulo irrelevancy: Proposition 4.2. Let G = 0 ( l , n ) , fl = L(G) = s o ( l , n ) , and consider the wedge W = t) + C Q & with the cone C = i I r 0 v1 I | r 6 R, v 6 R n _ 1 , r >

\\v\\ 1

and the subalgebra

«6R"_I, X e s o ( n - l ) > = so(l,n-l). Then S{W) is a Mobius semigroup. In particular, S{W) C G is proper with nonempty interior. The important information, of course, is that S(WO ^ G because it immediately implies that S(C) / G. Since if «o < 1, C contains the admissible controls for the system (4.1) on the connected group SOo(l,n), the system is non-controllable. And since the compact group SO(n) C 0 ( l , n ) is irrelevant for control (Corollary 3.4), we immediately obtain the 'if'-part of Theorem 4.1. The 'only-if'-part also follows from a "compactness" argument, but a much simpler one. For v £ R n _ 1 consider Yv Then Y„3 = (1 - |M| 2 )y„. So if ||v|| > 1, then exp(RY„) is a circle group, hence exp(K + V„) = exp(Ry„). Since {{{Yv : \\v\\ = «o})) = 0, this immediately implies controllability of the system (4.1).

5

Irrelevant groups and algebras

We extend the notion of irrelevant for transitivity to an analogous control-theoretic notion, abbreviated (IFC) and also extend the notions to Lie subalgebras. Definition 5.1. A subgroup H of G is irrelevant for controllabilty (IFC) if for all flCg, 5(fi) acts transitively on G/H implies S(Q) = G. A subalgebra h < fl is (IFT) if (exp h) C G is (IFT). Similarly, f) < fl is (IFC) if (exp t)) C G is (IFC). If a group or subalgebra is not (IFT), we say it is (NIFT). L e m m a 5.2. (i) A closed subgroup H of a Lie group G is irrelevant for controllability if and only if for every SJ C g the following statements are equivalent: (a) The system (BMCS) p{t) = U(t) (p(t)), p(t) e M = G/H, U(t) G Q is controllable on M.

62 (ii) The system (BGCS) g(t) = U(t) (g(t)), g(t) 6 G, U(t) € Q is controllable on G. (b) A subgroup H is (IFC) if and only if for all fi C g, S(£l)H = HS(il) S(Q) = G.

= G =>

Proof. We know from Proposition 2.3 that the attainable set at x € M = G/H is 5(fi).x. Thus the system (BMCS) is controllable if and only if S(Q).x = M for all x 6 M and the system (BGCS) is controllable if and only if S(Q) = G. Part (a) is now immediate. The second part follows from Proposition 3.6. • We observe that (IFC) and (IFT) are related, but distinct. One difference is that for (IFT) we only look at semigroups S with int(S) ^ 0. This means in the case of systems Q C g that we only look at systems satisfying the local accesssibility property: int(S.p) ^ 0 for all p e M. For (IFC), on the other hand, we look at arbitrary n C g, so we do not assume in advance that the reachable sets have interior points, i.e., local accessibility. We note that local accessibility of the system induced by Q C g on G/H, H closed, is actually equivalent to ((H)) + t) = g, since the latter is the "full-rank" condition in this context. We will see an example that is (IFT) but not (IFC) in a later section. On the other hand for (IFT) we consider arbitrary semigroups with interior, not just those that arise as the accessibility set from the identity for il C g. Although we do not know whether (IFC) implies (IFT) in general for closed subgroups H, this is the case for a large class of Lie groups. Proposition 5.3. Let G be a connected finite-dimensional real Lie group such that G / R a d G is compact, where R a d G denotes the radical ofG. Then a subgroup resp. subalgebra is (IFT) if it is (IFC). Proof. Let S be a subsemigroup with non-empty interior. By standard maximality arguments (see 7 ) S is contained in a maximal subsemigroup T ^ G, which again must have interior, and which must also be closed. Furthermore, T is generated by exp(L(T)), where L(T) is a closed half-space in g, namely the tangent wedge of T (see Theorem 12.2 and the first part of the proof of Theorem 12.5 in 7 ) . Thus T = 5(fi) for Q := L(T) (see sections 4 and 6 of " ) . Now suppose that the subgroup H is (IFC) and suppose that there existed S with non-empty interior transitive on the set of cosets G/H, but S ^ G. Then the maximal semigroup T containing S is also transitive on G/H (since S C. T). But then by (IFC), T = S(fl) = G, a contradition. Since the proposition holds for all subgroups, it also holds for subalgebras. • Note that this proposition applies to solvable and nilpotent Lie groups since G = Rad(G) in this case. We consider now in more detail the (IFC) property. L e m m a 5.4. Let G be a connected Lie group, let Q C g, and suppose that the (BGCS) determined by il is not controllable. Then Q. is contained in a maximal

63 non-controllable system A and either A is a proper subalgebra of g or A is a closed Lie wedge (i.e., a closed convex cone that is the tangent cone of a closed semigroup) that generates g and hence 5(A) has interior. Proof. Order all subsets of g by inclusion and consider a maximal chain ( = totally ordered set) C of subsets each of which is non-controllable (such maximal chains exist by the Hausdorff Maximality Principle, a variant of Zorn's Lemma). Then their union either generates g or it doesn't. If it does not, then it generates a proper subalgebra o. Then the orbits for the system A = o are the cosets of the analytic group (exp a), and hence the system is not controllable. By maximality of the chain, o belongs to the chain and is its largest element. Again by maximality of the chain, any larger subset generates all of G. Suppose that A, the union of the chain, generates g. Then since g if finite dimensional, there are finitely many elements of A that generate g. Since A is the union of the chain, there must be some element Q in the chain that contains these finitely many elements, and hence fi generates g. Thus S(Q) has interior; let V ^ 0 be an open set contained in 5(fi). Suppose that T is another member of the chain such that T contains fi. Then S(T) misses V-1, since otherwise e € v~1V C 5 ( r ) for w _ 1 £ V-1 (~l S ( r ) . Since G is connected, the open set v_1V generates (as a semigroup) all of G, but this contradicts the fact that 5 ( r ) is proper. One verifies directly that

5Pc(A) = (J{s p c (n): n is in the maximal chain C}, and thus 5 pc (A) also misses V~l. Hence its closure also misses V-1, and hence 5(A) misses V-1 (Theorem 2.2). Thus we conclude that 5(A) is proper, and hence A in not controllable. Again it follows from the maximality of the chain C that A is in the chain and is a maximal non-controllable subset of g. Let 5 be the semigroup 5 ( A ) - , the closure of 5(A). Then it is standard from the Lie theory of semigroups C, 2 ) that the tangent wedge of 5 , L(5) = {X 6 g : exp(tX) e 5 for all t > 0} is a closed convex cone, and that it is the largest subset Q of g such that S(il) C 5 (see Theorem 2.2). It follows that A C L(5), and hence A = L(5) by maximality. D T h e o r e m 5.5. Let G be a connected Lie group with Lie algebra g. Then g contains a largest (lFC)-ideal, denoted IFC(g), i.e., IFC(g) is (IFC), and every other ideal i that is (IFC) is contained in IFC(g). Proof. i,From the preceding lemma we know that maximal non-controllable subsets of g exist, and that each such is either a subalgebra or a tangent wedge L(5) of a closed semigroup 5. Since X —> —X is an anti-isomorphism on g corresponding to inversion on G, it follows that — L(5) is also a maximal non-controllable subset. Thus in the intersection of all maximal non-controllable sets we can replace L(5) and — L(5) in the intersection by the single set L(5) l~l — L(5), a subalgebra (Theorem 2.2). We conclude that the intersection of all maximal non-controllable subsets of g is a subalgebra q, the largest subspace contained in all of the maximal non-controllable

64 sets. Any automorphism Ad(g), g e G, of g must carry a maximal non-controllable subset to a maximal non-controllable subset. Hence their intersection q is Ad(G)invariant; it follows that q is an ideal in g. Suppose that 5(fi)Q = QS(Q) = G for ft C g, where Q = (expq). We want to show that 5(ft) = G and apply Lemma 5.2(b) to show that q is (IFC). Suppose that 5(ft) ^ G; then ft is not controllable and hence by Lemma 5.4 is contained in a maximal non-controllable subset A. But by definition q C A, and thus G = 5(ft)Q n QS(Q) C 5(A)Q n Q5(A) C S(A) # G, a contradition. Thus q is (IFC). Suppose that i is an ideal that is not contained in q. Then there must be some maximal non-controllable subset A of g such that t is not contained in A. Since i is an ideal, the group I := (exp i) is normal, and hence 5 ( A ) / = 7 5 ( A ) / is a subsemgroup. It clearly contains exp(i)Uexp(A) and hence the semigroup 5 p c (iUA). By the maximality of A, the system i U A is controllable. It follows from part (i) of Theorem 2.2 that iUA generates g, and hence that 5 pc (iUA) is a dense subsemigroup of 5(i U A) = G with non-empty interior, and hence all of G. But again by Theorem 2.2, 5 p c (t U A) is the semigroup generated by exp(i U A), and hence is contained in 5(A)/. It follows that G = 5 ( A ) / = / 5 ( A ) , but 5(A) # G. Thus the ideal i is not (IFC). Thus the ideal q satisfies all the assertions of the theorem. • The following proposition if frequently useful for reductions in working with the property (IFC). Proposition 5.6. Let G be a connected Lie group such that the subgroup K := {exp IFC) (g) is closed (this will automatically be the case if G is simply connected). Then the group H = G/K has a trivial (lFC)-ideal. Proof. Note that K is a normal subgroup since IFC(g) is an ideal. Define a : G —> H by a(g) = gK. PicK F # 0 in h = g/IFC(g), and pick X € g such that da(X) = Y. Since the kernel of da is IFC(g), we have X is not in IFC(g). Thus there exists a maximal non-controllable set A in g missing X. Then IFC(g) C A, so if C 5(A). Since 5(A) is proper in G and contains K, it follows that its image is proper in H. Since a(expA) = exp(da(A)), we conclude that a (5(A)) = 5 ( d a ( A ) ) . Hence T := da(A) is not controllable. Since A contains IFC(g) and X 0 A, it follows that Y £ T. Furthermore, F is maximal non-controllable, since any set that properly contains T will have inverse properly containing A, hence be controllable on G and hence its image, the original set, will be controllable on H. Thus Y & T implies Y £ IFC(h). It follows that IFC(h) is trivial. • 6

Irrelevant groups and algebras: t h e solvable case

For solvable Lie algebras we are in the fortunate situation of having a classification of the maximal subsemigroups with non-empty interior, which makes possible a detailed treatment of irrelevant subgroups. Given a Lie algebra g the natural setting for treating controllability and transitivity questions is to work in the corresponding simply connected group G because

65 every other (connected) Lie group G with Lie algebra g is a quotient of the simply connected group: General A s s u m p t i o n . Throughout this section we assume that G is a simply connected solvable group. In the sequel we will discuss the notions (IFT) and (IFC) separately. We will see that every solvable Lie algebra g contains two special ideals, a maximal (IFT)ideal as well as a maximal (IFC)-ideal (already introduced). The first one is also maximal (IFT) as a subalgebra. Reduction modulo these ideals leads to Lie algebras having a very simple structure. On this level one can find easily the answers to many interesting questions. For example, it is easy to figure out the minimum number of elements necessary to generate g as a Lie algebra. 6.1

Discussion

of

(IFT)

For solvable groups virtually all questions we consider can be answered satisfactorily because we have a complete classification of their maximal subsemigroups with interior points 7 . Definition 6.1. Let g be a Lie algebra. A subalgebra f) of codimension 1 is called a hyperplane subalgebra. A halfspace bounded by such a hyperplane subalgebra is called a halfspace semialgebra. T h e o r e m 6.2. Let a be a solvable Lie algebra and G an associated connected group. (i) If S C G is a maximal subsemigroup with int(S) ^ 0, then L(5) = {X £ g | exp(R + X) C S } is a halfspace semialgebra and S is the semigroup generated by exp(L(5)). Every semigroup with non-empty interior is contained in some such maximal semigroup. (ii) Conversely, if G is simply connected and f)+ C g is a halfspace semialgebra, then S(f) + ) = (expl) + ) C G is a maximal subsemigroup. Thus in the simply connected case, the maximal subsemigroups of G with nonempty interior are in one-to-one correspondence with the halfspace semialgebras in g. Hyperplane subalgebras are well-understood and fully classified, cf. 3 . Let us write 7i(g) for the set of all hyperplane subalgebras of g. Then the so-called Aradical

A(g) = f)^(fl) plays a crucial role. In fact A(g) is an ideal, and due to Theorem 6.2 we have: Corollary 6.3. The ideal A(g) is irrelevant for transitivity (IFT). Proof. Indeed, let A(G) C G denote the corresponding connected normal subgroup. If S C G is a proper subsemigroup with nonempty interior, then S is contained in a maximal subsemigroup S(f) + )- As A(g) C f)+ we obtain S A(G) C S(f) + ) / GThus S is not transitive on G/A(G). D As we already mentioned, A(g) need not be (IFC), in general.

66 Example 6.4. The smallest dimensional example for which A(g) is not (IFC) is the algebra of euclidean motions of the plane: g

= sp&n{A,X,Y},

[A,X] = Y,

[A,Y] =

-X,

and all other brackets vanishing. Here A(g) = g' = span{X, Y}. If VI — RA, then the induced system on G/A(G) ~ R is controllable but the system g = ug, u G RA, is not controllable. Since A(g) is an ideal, g/A(g) is a Lie algebra (in 3 such Lie algebras are called A-reduced), and the homogeneous space G/A(G) is actually a Lie group. Our next goal is to show that there are no (IFT)-subalgebras beyond A(g). This will be carried out in several steps. The basic idea is to be found in the proof of the lemma below. L e m m a 6.5. Ift) £ g' then H := (expf)) is (NIFT). Thus ift) is (IFT), then t) C g'. Proof. Pick X G I) \ g'. We want to find a semigroup S ^= G such that int(S) ^ 0,

and

5exp(RX) = exp(RX)S = G

in order to prove that RX and hence H is (NIFT) (see Proposition 3.6). So let us pick a hyperplane t) D g' such that f) + RX = g and consider the halfspace semialgebra t)+ = t) +R+X, the semigroup S(f) + ) = (exp(fj + )\, and the subgroup H = (expljY Then fj is an ideal because of g' C fj, so H is a normal subgroup. Hence e x p ( R X ) ^ = .ffexp(lIL\r) = G, whence exp(RX) S(fj) = S(I))exp(RX) = G. So exp(RX) is (NIFT) which implies that H is (NIFT). • It is crucial in the previous proof that t) is an ideal. It is not difficult to find subgroups Hi, H2 of a group G such that f)i + ()2 = g but H\Hi ^ G. E x a m p l e 6.6. Let L be the two-dimensional non-abelian Lie group:

L=

{(oi)|a>0'

beR

\-

We may identify L with the upper halfplane: L = {(b, a) | a > 0} C R2. This allows to describe all interesting objects like one-parameter groups, cosets, etc. nicely in purely geometric terms. So we don't need do write down lengthy computations which are elementary but not very elucidating. The group identity is e = (0,1), the commutator group L' is the horizontal line R x {1} whereas an arbitrary one-parameter group is just an ordinary line—better said its part in the upper halfplane—through e. For g G L and a one-dimensional subgroup H the left coset gH is simply the line through g parallel to H. If H ^ L', then the line representing H intersects the horizontal axis at some point (v,0). The right coset Hg is nothing but the line through g and (v,Q), cf. the FIGURE 2. If we take two subgroups Hi ^ H' ^ H2 then H1H2 ^ L will hold, a fact one can best "see" in the picture FIGURE 3.

67

G

y/gH \ / /

/

/ X X

VV

^

Pi

Figure 2. The right coset Hg

Figure 3. HiH2 # L

Eventually we will use an argument of the above type, but it requires some refinement. L e m m a 8.7. Let_M = G/H, w: G -* M the canonical map, g 6 G, p = ir(g) € M and X € g. Let X denote the vector field induced by X on M. Then X(p) = 0 iff X 6 Ad( fl )h. Proof. Let po = w(e) denote the base point in M. Then its stabilizer algebra is gj 0 = h. For p = n(g) = g.po the stabilizer group is gp = Ad(g)QP0 = Ad(g)h. As X(p) = 0 iff X 6 flP, our claim follows. D

68 Proposition 6.8. Assume that g = hi + 1)2 with subalgebras fh>f)2 such that Ad(G)f)2 n hi = {0}. Then G = HXH2 with Hi = (exp) h<, i = 1,2. Proof. Let M = G/H2.

Forp € M let D{p) - | x ( p ) \ X € I n } . Then D(p) = TPM

for all p € M follows from the previous lemma because the map X i-» X ( p ) : f)i —> T P M has kernel C Ad(G)h 2 D hi = 0 and dim hi = dimT P M. Thus Hi.p is a neighborhood of p for every p 6 M. As M is connected, M = H\.n(e) follows, i.e., HiH2 = G. • Proposition 6.9. Let X € g \ A(g). Then there exists a hyperplane subalgebra t) such that Ad(G)X n f) = 0. So the subalgebra EX is (NIFT). Proof. First we pass to the quotient g/A(g), so we assume w.l.o.g. that Q is Areduced. This implies, cf. [KHH], that g is a so-called basic metabehan Lie algebra which means it has the following structure:

0=oe®8\

(t)

where o is a vector space, A is a set of nonzero linear forms from o —> R, the a A 's are vector spaces, too, and [A, Xx] = \(A) Xx

for A e 0, Xx 6 £|A.

All other brackets vanish: [flA,flM] = 0 for all A, fi. Moreover g' = © A e A f l A . Our goal now is to show that for every nonzero X in a basic metabehan $j we find a hyperplane subalgebra t) such that f) + MX — g and Ad(G)X n f) = 0. But this is a relatively easy task. For X £ g' this has already been proved in Lemma 6.5. For X 6 g' \ {0} we decompose X as X = ^2X X\ and pick a /i such that XM ^ 0. Next we fix a hyperplane V^ C gM with X^ 0 VM and let h= o + ^ g

A

+

V

ThenAd(G)Xnf) = 0 because of Ad(G)X C (0, oo)X„+g' and RX M nV M = {0}.

•

Thus we can conclude our discussion of (IFT)-subalgebras with the following theorem, which typically allows us to reduce transitivity questions on G to G/A(G). T h e o r e m 6.10. If a subalgebra t) C g is (IFT), then t) C A(g). In particular A(g) is maximal (IFT) not only among ideals but also among subalgebras. One can give an alternate more direct proof based on results of 7 . Suppose that f)i is (IFT), but is not a subset of A(g). Then there exists a subalgebra f) of codimension 1 such that hi \ f) / 0. Pick X such that X 6 f)i \ f). Note that g = h + RX. By Lemma 6.5 we must have that l)i C g', and hence X 6 g'. Set h + = f) + RX; by Theorem 6.2 S(l} + ) = (exp l) + ) is a maximal semigroup of G.

69 By results of Section 11 of 7 , there exists a surjective homomorphism with connected kernel (f> : G —f R or ^ : G -> L, the group of Example 6.6 such that 0(S(h+)) = R+ or, respectively, L+ := {(b,a) € L : b > 0}. The first case is impossible, since (d0) _1 (O) is an ideal of a containing the derived algebra, hence X, and containing I) since exp(h) must map to a subgroup of R + , hence be 0. Thus $j, and hence G, maps to 0, contradicting surjectivity. Thus there exists : G —> L such that ( s ( h + ) ) = L+. The subgroup H = (exph) must map into L\ := {(b, a) £ L : b = 0}, the largest subgroup of L+ and exp(X) must map to L' := exp(£') = {(6, a) £ L : a = 1}, the image of the derived algebra (! of the Lie algebra I of L. Suppose that the image of H is trivial. Then the image of f) is trivial, the image of g under d(j> is contained in the image of RX, which is most one-dimensional, and hence the image of G must be one-dimensional, contradicting surjectivity. In a similar way one argues that (expMX) is not the identity. Thus (f>(exp MX) — L' and <j) o exp = exp^ od<j> restricted to MX must be injective, since e x p t is and d(\> restricted to UX must be. Finally note the the kernel K of is a closed connected subgroup, hence a Lie subgroup with some Lie algebra t. If there existed Y € t \ h, then (expRY) = e implies d<j>{Y) = 0, and since R F + fj = fl, it follows that d(o) = d0(h) is one-dimensional, again leading to a contradiction of surjectivity. Thus EC ^ and hence K = (expt) C (exp h) = H. One verifies directly that L = L\L' = L'L\ and that the factorization is unique in each case. Let g 6 G. Then (exp tX)<j)(h) 6 L'L\ for some t 6 R, h £ H. Thus g = (exptX)hk for some k € K C H; thus g £ exp(RX)H. Suppose that g = exp(tiX)hi = exp(t2X)h,2. Then <j>(g) = {exptiX)4>(h{) = (expt2X)4>(h,2)By unique factorization in L, exp{d(p(tiX))

= 4>(exptiX) = 0(expt 2 X) = exp (d(j>(t2X)).

and by injectivity of expod<^> on MX, we conclude t\ = ti- Then by cancellation h\ =b,2, so the representation is unique. The same argument works for .H'exp(RX'). Thus by Proposition 3.6 f) is not (IFT). A reason for presenting the alternate proof is that this second proof gives us the following new characterization of maximal semigroups. T h e o r e m 6.11. Let G be a simply connected solvable Lie group and let S := S(h + ) = ( e x p h + ) be a maximal semigroup of G, where t)+ is a half-space semialgebra, and f) is a codimension one subalgebra which is not an ideal. Then there exists X £ (h + \ h) f~l g' and for any such X, S = H(expRX) = (expMX)H with unique factorization in each case, where H := (exp h) We remark that the proof remains valid in the more general connected case. 6.2

Considering

(IFC)

The notion of irrelevance for controllability (IFC) is the natural one to look at if one's starting point is control systems on manifolds rather than the semigroups of smooth invertible maps. So we start with the vector fields generating flow maps

70 rather than the flow maps themselves. Actually this was our original motivation to introduce the irrelevancy notion, which proved more broadly powerful. For detailed proofs we refer to 17 . As a consequence of Theorem 5.5 and Proposition 5.6, controllability questions on g can be reduced to a problem on the factor algebra JJ/ IFC(g). Of course whether this information is very useful or not depends on the following two aspects: (D) Determination: How hard is the problem of determining IFC(g)? (S) Simplification: How complicated is the structure of the quotient g/IFC(g)? Is it much simpler that that of a general solvable g? For nilpotent g we obtain the best possible answers one can hope for: T h e o r e m 6.12. If g is nilpotent, then IFC(g) = A(g) = g'. A subalgebra f) C g is QFC) iff t) C g,'. Proof. An (IFT)-subalgebra f) is contained in g' by Lemma 6.5, and thus A(g) C g' by Corollary 6.3. By Proposition 5.3 we conclude that every (IFC)-subalgebra is contained in g', in particular IFC(g) C g'. The converse containments in the first statement are proved in 17 , and the converse implication in the second statement then follows from Theorem 5.5. • For g solvable, not nilpotent, the answer to (D) is, roughly speaking, that it is as hard as finding the Jordan normal form of a linear map. And concerning (S) one finds that g/ IFC(g) is only slightly more complicated than the basic-metabelian algebras mentioned in the previous section. The structure of these algebras (in terms of brackets) is not very complicated and actually easily understood, the major task one has to perform is introducing a well-suited notation. One might be disappointed by our anwswer to (D), in particular after one finds out that it is also necessary to pick a "generic" element in g, so this is not a straightforward procedure for finding IFC(g). But actually finding IFC(g) itself is not necessarily the task that must be solved. There are smaller ideals which are easily determined and which already provide most of the possible reduction: Proposition 6.13. If Q is solvable, then D2Q = [g',g'] is an (IFC)-ideal. See 17 for a proof. The factor algebra g = Q/D2Q is metabelian, i.e., its bracket algebra [g, g] is abelian. For practical purposes Proposition 5.6 can be quite useful. It allows one to proceed inductively: having found some (IFC)-ideal, pass to the quotient and look for (IFC)-ideals there. Finally there is a nice characterization of (IFC)-ideals in algebraic terms. Definition 6.14. We call a subalgebra f) C g an (NCA)-subalgebTa if it has the following property: If a C Q is a subalgebra and a + I) = g, then a = g. The acronym (NCA) stands for no complementary algebra: f) being an (NCA)subalgebra means there is no proper subalgebra o with 0 + f) = g. Of course it makes sense to speak of (NCA)-ideals, too. Proposition 6.15. An ideal i o/g is (IFC) if and only if it is an (NCA)-ideal.

71 We cannot expect that a sum of (IFC)-algebras fji,h2 is (IFC). Nevertheless, we obtain an analogue of the previous proposition. Its proof requires a few extra arguments, though. Proposition 6.16. If the subalgebra I J C J generates a closed subgroup (exp f)) and is (NCA), then it is (IFC). Every (NCA)-algebra t) is contained in A(g). Proof. Suppose that t) is (NCA). Then h C A(fl) must hold because if I) \ A(g) ^ 0 we can actually find a hyperplane subalgebra f) such that i) + t) = 0, cf. Lemma 6.5. We claim that f) is (IFC), so we consider M = G/H and $ C g such that the system on M given by {x | x e $} is controllable. We must show that S ( $ ) = G holds true. As usual 7r: G —> M denotes the canonical map, pa = 7r(e) the base point, and pr,, = d7r(e): a —> fl/f) is the projection with kernel I). Now let a = (($)) C 0, the subalgebra generated by $ . Then controllability on M implies TP0M = < X(po) | X 6 o !• which is equivalent to pr1)(o) = 0, hence a+ f) = 0. As f) is (NCA), o = £| follows, so int(5(*)) # 0. Since f) C A(g), H C A(G). Thus 5 ( $ ) is transitive on G/A(G). As A(G) is (IFT) and int(S($)) # 0, we deduce S ( $ ) = G, finishing the proof. • Now we ask if the reverse implication holds true, so is every (IFC)-algebra (NCA)? Example 6.6 and the discussion leading to Proposition 6.8 show that this is a nontrivial problem. Take an t) which is not (NCA). Take a proper subalgebra o 7^ g such that a + t) = Q. Then it is clear that the system induced by $ = a on M = G/H will be transitive at least on a neighborhood of the base point po = 7r(e) of M, but we cannot expect in general that the semigroup S = A = (exp o) is transitive on M, for that is equivalent to G = HA, and the latter we know for sure only if Ad(G)fj n o = {0} is trivial. Example 6.17. Again we consider the algebra of motions of the euclidean plane 0 = s p a n { A , X , Y } with brackets: [A,X] = Y, [A,Y] = -X. Which subalgebras f) of fl are not (NCA)? Let us look at dim h: 3-dim: Of course 0 is not (NCA), but it isn't (IFC) either. 2-dim: There's only one hyperplane subalgebra: 0'. Of course 0' is not (NCA), but it is not (IFC) either. 1-dim: The line RX is not (NCA) iff X £ g' because 0' is the only possible choice we have for a complementary subalgebra. As Ad(G)0 ; = 0', RX n Ad(G)$' = {0}, so G = exp(RX)G' holds true, i.e., RX is not (IFC). So in this case -i(NCA) implies -i(IFC). In particular we observe that, for example, the subalgebra f) = RX is (NCA), so it is (IFC), too. But we also observe that f) g IFC(0) = {0}. So the inclusion f) C A(o) if t) is (IFC) cannot be improved any further. An (IFC)-algebra need not be contained in the ideal IFC(0). We conjecture that-.(NCA) =$• -.(IFC), resp., (IFC) =$> (NCA) so we state this as an open

72 Problem 6.18. Prove or disprove: an (IFC)-algebra is also (NCA). Now we return to our discussions of the benefits of passing to the quotient g/IFC(g). 6.3

The controllability

normal

form

Reduction modulo A(g) led us to A-reduced algebras which actually turned out to be the ones we called basic-metabelian. The algebra g/IFC(g) also deserves a special name, we call it the controllability normal form (CNF)o/g. If IFC(g) = {0} we say that g is in (CNF). Although the structure (bracket-wise) of such a g in (CNF) is not really complicated, having a reasonable terminology is indispensable for analyzing it. Roughly speaking, such a g is made out of two components, a real vector space V and a set (actually a vector subspace) o C g£(V) of linear maps V -*V such that (1) every A G a is diagonalizable over C, and (2) any two maps A,A'(Ea

commute: A A' = A' A.

These conditions are tantamount to saying that a is an abelian subalgebra of g£(V), and V may be decomposed into common eigenspaces of A, A'. Formalizing this statement it turns out that one can decompose g in the following way:

fl = h e 0 g A © ® g u , AeA

(6.1)

wen

which has to be interpreted as follows: • h is an abelian subalgebra of g corresponding to o plus the center 3(g). • The sets A, Q stand for the so-called roots signifying the common eigenvalues and g , g" are common eigenspaces also called root-spaces. • A is the set of real roots; each A G A is a nonzero linear form A: f) —> R, and [H,XX] = \{H)XX,

holds true for all H G h, Xx G g A .

• Q is the set of complex roots—actually only one half of it since complex eigenvalues always appear in conjugate pairs—so an w G fi is an R-linear map from h -> C such that w(h) % R. The root space g" is isomorphic (as a vector space) to some Cr considered as a real vector space, and for H € h, Xu G gw

[H,Xu]=u(H)Xu. • Each root space gA, g" is an abelian ideal and [g a ,g^] = 0 for all a,/3 G A u Q . At this point one clearly sees the difference between the (CNF) g/IFC(g) and the basic metabelian g/A(g): in the latter case there are no complex roots (eigenvalues) .

73 Proposition 6.19. Let g be as in (CNF). Then

fl' = ©8 A ©©8 W > A€A

"* A(g) = © g " .

wgn

wen

A hyperplane subalgebra a in g has the form 0- = cii, + 8 ,

wi£/i a hyperplane Of, C h, or

a = f) © ^ f t g a © a,,

6.^

Generic controllability

un£/a a hyperplane oM C g^.

rank and type

Now we reap some of the benefits of reduction because we will be able to answer the following questions: • What is the minimum number of elements Xi,... ,Xr necessary to generate a given g as a Lie algebra, i.e., {{X\,... ,Xr)) = 8? • Given a Lie algebra g, some "drift vector fields" Ai,... ,Aj and some "input vector fields" Bi,... ,Bi, what are the minimal numbers (i,d) such that U = {Ai ,...,Ad, ±B%,..., ±Bi) is controllable? • Will controllability or noncontrollability of the system given by U = {Ai ,...,Ad, ± B i , . . . , ±Bi} C jj be stable under perturbations (to Aj, Bk € fl)? Is it a generic property? For the first two questions we observe that existence of such minimal numbers r, resp (i,d) is not really an issue, but it is not at all clear how to compute them. We call r the generic controllability rank of g, GCR(g), and (i,d) the generic controllability type of g, GCT(g). This choice of terminology has something to do with the last question because once certain conditions can be expressed in terms of some polynomials (say in Xi,..., Xr) not vanishing (like a rank condtion), they are either void or generic, so they hold either never or in almost every case. Looking at the (CNF) of a solvable g we can immediately read off its controllability rank and type. With the notation of (CNF) we let d\ = max < diniR g | A 6 A > ,

and

du = max {dime g" | u; € fi} ,

and obtain: Proposition 6.20. Let g be in (CNF).

Then

GCR(g) = max {dim t), 1 + dA, 1 + dn} , and GCT(g) = (i, d) with i + d = GCR(g), i = max {dim f), 1 + ^ A } ,

and

d = max{0,1 + dn — i) •

74 The proof requires two separate arguments. One must find a set of generators of the appropriate size, and one must show that one cannot generate g with less elements. In both cases the root decomposition as in (6.1) is instrumental. The details can be found in 1T. A particularly nice result is the next theorem which simplifies Sachkov's work on controllability of solvable Lie algebras, cf. 2 2 , also 2 3 . Definition 6.21. We call a Lie algebra g SID-controllable, if there exist A,B € g such that S(A,±B) = G, i.e., the single input plus drift system g = g(A + uB), u 6 R is controllable. It turns out that this is a generic property of the Lie algebra g rather than the particular pair (A,B): the set of such pairs is either empty or open and dense in g x g. T h e o r e m 6.22. For a solvable Lie algebra the following statements are equivalent: (i) g is SID-controllable. (ii) every factor algebra g/i (i C g an ideal) is SID-controllable. (hi) g/£>2g is SID-controllable (iv) GCT( 0 ) = (1,1), (v) g/IFC(g) is isomorphic to a semidirect product Ck x$j R with Q diag(a>i,...,uik) 6 g£{k, C) and w i , . . . ,Wfc € C \ R pairwise distinct.

=

If g is SID-controllable, then the foloowing statements hold true: (vi) $j has no ideals of codimension 2. (vii) Q' is a hyperplane and A(g) = $)'. Alternatively, in view of (iii), one can reduce the problem of classifying SIDcontrollable algebras to the metabelian case [fl',g'] = 0. This is a basic case in Sachkov's work which allows a very short classification where the conditions involved are similar to familiar criteria from linear control theory. Proposition 6.23. A metabelian g is SID-controllable iff it is isomorphic to a semidirect product M.k x ^ R where A € g£(k,M) has nonreal spectrum spec(A) C C\R and every eigenvalue has geometric multiplicity 1. The latter is equivalent to iank(A + XI) > k - 1 for all A 6 C. Comparison with the previous theorem reveals how far away from g/ IFC(g) the possibly bigger algebra Q/D2Q actually may be. The metabelian g/D 2 g has a root decomposition similar to the (CNF):

g/D2g = h © 0 g A e 0 g „ . A€A

u6d

The difference is that the root spaces Q\, gw need not be eigenspaces, they may be generalized eigenspaces; and the subalgebra t) is nilpotent but not necessarily abelian.

75

7

Irrelevant groups and algebras: the semisimple case

The highly developed structure theory for semisimple Lie groups and algebras allows one to derive quite powerful results concerning irrelevancy in the semisimple setting. This project has been carried out by L. San Martin and co-workers over a period of several years. We do not undertake to explain the details of these developments here, but only highlight a few principle results. Let G be a connected semisimple Lie group with finite center and let a denote its Lie algebra. Then the Lie group and the Lie algebra admit Iwasawa decompositions of the form G = KAN, g = t + a + n, where K = exp(t), A = exp(a), and N = exp(n), and where the product factorization of each element is unique and the Lie algebra decomposition is into direct summands. The Iwasawa decompositions arise by taking an Cartan decomposition t + p of 5, picking a maximal abelian subalgebra o of p, letting II be a set of positive roots for a, and setting n equal to the sum of the root spaces for the positive roots. Set M equal to the centralizer of A in K. Then P = MAN is a group, called a minimal parabolic group. The coset space G/P is called a maximal boundary of G. The first major result of San Martin appeared in 2 4 . We cite it in the language of this paper, not the language that he used. T h e o r e m 7 . 1 . Let G be a connected semisimple Lie group with finite center, and let P be a minimal parabolic subgroup. Then P is (IFT). Much stronger results are available if G is simple 25 . T h e o r e m 7.2. Let G be a simple Lie group and let H he a closed subgroup such that 0 < dim(H) < dim(G) and the coset space G/H is compact. Then H is (IFT).

References 1. J. Hilgert, K. H. Hofmann, and J. D. Lawson, Lie Groups, Convex Cones, and Semigroups, (Oxford Press, Oxford, 1989). 2. J. Hilgert and K.-H. Neeb, Basic Theory of Lie Semigroups and Applications, (Springer Lecture Notes in Mathematics 1552, Springer, 1993). 3. —, "Hyperplane subalgebras in real Lie algebras," Geom. Dedicata 36, 207-224 (1990). 4. K. H. Hofmann, J. D. Lawson, and E. B. Vinberg, Editors, Semigroups in Algebra, Geometry and Analysis, (de Gruyter, Berlin, 1995). 5. V. Jurdjevic, Geometric Control Theory, (Cambridge Press, Cambridge, 1997). 6. V. Jurdjevic and H. Sussmann, Control systems on Lie groups J. Diff. Eq. 12, 313-329 (1972). 7. J. D. Lawson, Maximal subsemigroups of Lie groups that are total, Proceedings of the Edinburgh Math. Soc. 87 , 479-501 (1987). 8. —, "Ordered Manifolds, Invariant Cone Fields, and Semigroups," Forum Math. 1 , 273-308 (1989). 9. —, "Semigroups of Ol'shanskii type," Semigroups in Algebra, Geometry and

76 Analysis, (de Gruyter, Berlin, 1994). 10. —, "Semigroups in Mobius and Lorentzian geometry," Geom. Dedicata 70, 139-180 (1998). 11. —, "Geometric control and Lie semigroup theory," In: Differential Geometry and Control, Proceedings of Symposia in Pure Mathematics, 64, Editors: G. Ferreyra, R. Gardner, H. Hermes, H. Sussmann, (Amer. Math. Soc, 207-221 1999). 12. C. Loewner, "On some transformation semigroups," J. Rat. Mech. and Anal. 5, 791-804 (1956). 13. —, "On some transformation semigroups invariant under Euclidean or nonEuclidean isometries," J. Math, and Mech. 8, 393-409(1959) . 14. —, "On semigroups in analysis and geometry," Bulletin A.M.S. 70, 1-15 (1964). 15. D. Mittenhuber, "Dubins' problem inhyperbolic space," In: Geometric Control and Non-holonomic Mechanics, CMS Conference Proceedings, Volume 25, Editors: V. Jurdjevic, R.W. Sharpe, (Can. Math. Soc, 101-114 1998). 16. —, "Controllability of solvable Lie algebras," J. Dyn. Contr. Systems 6, 453459 (2001). 17. —, "Controllability of systems on solvable Lie groups: the generic case," J. Dyn. Contr. Systems 7:1, 61-75 (2001) . 18. F. Monroy-Perez, "Non-euclidean Dubins' problem," J. Dyn. Contr. Systems 4:2, 249-272 (1998). 19. G. I. Ol'shanskii, "Convex cones in symmetric Lie algebras, Lie semigroups, and invariant causal structures on pseudo-Riemannian symmetric spaces," Sov. Math. Dokl. 26 , 97-101 (1982). 20. —/'Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series," Fund. Anal, and Appl. 15 , 275-285 (1982). 21. R. S. Palais, "A global formulation of the Lie theory of transformation groups," (Memoirs of the Amer. Math. Soc, Providence, 1957). 22. Y. Sachkov, "Controllability of right-invariant systems on solvable Lie groups," J. Dyn. Contr. Systems 3:4, 531-564 (1997). 23. —, "Survey on Controllability of Invariant Systems on Solvable Lie Groups," In: Differential Geometry and Control, Proceedings of Symposia in Pure Mathematics, Volume 64, Editors: G. Ferreyra, R. Gardner, H. Hermes, H. Sussmann, (Amer. Math. Soc, 297-317 1999). 24. L. San Martin, "Invariant control sets on flag manifolds," Math, of Control, Signals, and Systems 6 , 41-61 (1993). 25. L. San Martin and P. Tonelli, "Transitive actions of semigroups in semisimple Lie groups," Semigroup Forum 58, 142-151 (1999). 26. H. Sussmann, "The bang-bang problem for certain control systems in GL(ra, R), SIAM J. Control 10, 470-476 (1972).

77 CANONICAL CONTACT SYSTEMS FOR CURVES: A SURVEY

WITOLD RESPONDEK Laboratoire de mathematiques de VINSA de 16 131 Mont Saint Aignan, France E-mail: [email protected]

Rouen

WILLIAM PASILLAS-LEPINE Laboratoire des signaux et systemes, Supelec 91 129 Gif-sur-Yvette, France E-mail: [email protected] We start with a brief overview of results characterizing distributions (equivalently Pfaffian systems) that are locally equivalent to a canonical contact system. Then we concentrate on distributions equivalent to the canonical contact system on 7 n (IR,]R m ), that is the canonical contact system for curves. We give various necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact systems for curves. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistributions contained in the successive elements of the derived flag. We give a new characterization of distributions locally equivalent to the canonical contact systems for curves that uses only one involutive subdistribution in which all information about the geometry and about the regularity is encoded. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points, at which we propose a new normal form that generalizes the canonical contact system on Jn(R, R m ) in a way analogous to that how Kumpera-Ruiz normal form generalizes Goursat normal form, that is, the canonical contact system on Jn(R, M.). Finally, we briefly discuss how the geometry of distributions equivalent to the canonical contact systems for curves is reflected by their flatness and illustrate some of our results by a model of a simple nonholonomic control system.

1

Introduction

Consider Jn(Rk ,Rm), and denote by (qi,...

,qk,ui,...

the space of n-jets of smooth maps from Kfc into E m , ,um,Pi),

for 1 < i < m and for 1 < \a\ < n,

the canonical coordinates, also called natural coordinates, on this space (see e.g. 3 , 25 , and 3 2 ) , where qj, for 1 < j < k, represent independent variables and Ui, for 1 < i < m, represent dependent variables; the vector of non-negative integers a = (<TI, . . . ,o~k) is a multi-index such that \o~\ = o~\ + • • • + o~k
78

Denote p° — ut, for 1 < i < m. Any smooth map ip from Efc into Rm defines a submanifold in Jn(Rk ,Rm) by the relations Pi = - ^ — ( g i , - - - , % ) ,

for 1 < i < m and 0 < \a\ < n. This submanifold is called the n-graph of ip. It turns out that all n-graphs are integral submanifolds, of dimension k, of a distribution called the canonical contact system on Jn (Rk, Rm) or the Cartan distribution 32 on J n (M f c ,E m ). The Pfaffian system that anihilates this distribution, which is also called the canonical contact system (see e.g 3 and 2 5 ) , is given in the canonical coordinates of Jn(Rk, Rm) by ft

dp? - YjPi

.j.

'dqj = 0,

for 1 < i < m and for 0 < |cr| < n — 1,

where a + \j = {a\,... , Oj + 1 , . . . ,ak). The above description explains the importance of contact systems in geometric theory of (partial) differential equations and in differential geometry. In the former, a (partial) differential equation is interpreted as a submanifold in J"(K fc , IRm) and thus it is natural to study the geometry of pairs consisting of a contact system and a submanifold, see e.g. 3 , 19 , and 3 2 . In the latter, contact systems allow to describe, for instance, diffeomorphisms which preserve the n-graphs of appplications (for example, n-graphs of curves in the case k = 1), see e.g. 3 and 2 5 . A natural problem which arises is to characterize those distributions which are (locally) equivalent to a canonical contact system. This problem was posed by Pfaff 29 in 1814 and seems still to be open in its full generality although many important particular solutions have been obtained. In the case n = 1, with m = 1 and an arbitrary k the final solution has been obtained by Darboux 7 in his famous theorem generalizing earlier results of Pfaff 29 and Frobenius n . The case n = 2, m = 1 and A; = 1 was solved by Engel in 8 . The case n > 2, m = 1 and k = 1 was solved by E. von Weber 33 , Cartan 5 and Goursat 14 (at generic points) and by Libermann 18 , Kumpera and Ruiz 17 , and Murray 24 (at an arbitrary point). The case n = 1, with k and m arbitrary has been studied and solved by Bryant 2 (see also 3 ) . This paper is devoted to the problem of when a given distribution is locally equivalent to the canonical contact system in the case k = 1, n and m arbitrary, that is to the canonical contact system for curves. This problem has been studied by Gardner and Shadwick 12 , Murray 24 , and Tilbury and Sastry 31 (as a particular case of the the problem of equivalence to the

79

so-called extended Goursat normal form). Their solutions are based on a result of 12 that assures the equivalence provided that a certain differential form satisfies precise congruence relations. The problem of how to verify the existence of such a form had apparently remained open. This difficulty was solved by Aranda-Bricaire and P o m e t 1 , who proposed an algorithm which determines the existence of such a form. Their solution, although being elegant and checkable, uses the formalism of infinite dimensional manifolds and thus goes away from classical results characterizing contact systems. Recently, the authors 27 gave a geometric characterization of canonical contact systems in the case k = 1, and n and m arbitrary. That characterization of the canonical contact system on Jn(R,Rm), recalled as Theorem 2.1, turns out to be a natural combination of that given for J 1 (E f c ,R m ) by Bryant (see 2 and 3 ) and that given for J"(M,M) by Murray 2 4 . The aim of this paper is to analyze various possible characterizations of the canonical contact system on J"(M, K m ) . We start with stating, as Theorem 2.1, the result of 2 7 . That result involves two conditions: an essential geometric condition stating that each element 2)W of the derived flag (see the definition in the next section) contains an involutive subdistribution of corank one and the regularity condition T>^ = T>i (see the definition of the Lie flag in the next section). We will discuss other equivalent formulations of both conditions using, in particular, the notion of characteristic distributions (Theorem 5.4) and the language of Pfaffian systems (Theorem 3.7 and Corollary 5.5). All those descriptions lead to the following characterization, stated as 5.6, which is the main result of the paper. Main Theorem A rank m + 1 distribution V, with m > 2, on a manifold M of dimension (n + l)m + 1 is equivalent, in a small enough neighborhood of any point p in M, to the canonical contact system on J"(M, M.m) if and only if the following conditions hold. (i) £>(") = TM. (ii) £>(™-1) is of constant rank nm + 1 and contains an involutive subdistribution Cn-i that has constant corank one in (iii) 2>(°>(p) g £„_!(?). This result implies that all information containing both the geometry and the regularity of the problem is completely encoded in the involutive subdistribution £„_i of corank one in D(™-1). In particular, its existence and the regular intersection with 2?(°) (p) imply the existence of an involutive subdistribution of corank 1 in each X>W and, moreover, imply all regularity conditions X>W — T>i. It is important to observe that if conditions (i) and (ii) of the theorem are satisfied then the involutive subdistribution jCn-i is necessarily

80

unique (see Lemma 4.2), which allows to check easily condition (iii). In the case of Goursat structures, that is for m = 1, we give Theorem 5.7, an analogue of the above result, in which all information is, this time, encoded in the characteristic distribution Cn-\ of £>(" - 1 '. The paper is organized as follows. In Section 2 we define the canonical system for curves and we give Theorem 2.1, a first characterization of distributions that are locally equivalent to the canonical contact system for curves. In Section 3 we sketch briefly the history of results describing various canonical contact systems. In Section 4 we discuss the problem of whether a given distribution possesses a corank one involutive subdistribution and we recall Bryant's solution of that problem. Section 5 contains other characterizations of the canonical contact systems for curves. One of them, Theorem 5.4, is based on the notion of characteristic distributions while another, Theorem 5.6, uses only one involutive sudistribution in which all information about the geometry and about the regularity is encoded. The presented proof gives a method of constructing canonical coordinates for any distribution equivalent to the canonical contact system. Still in Section 5, we present an analogue of Theorem 5.6 for Goursat structures that uses, this time, only one characteristic distribution. Flatness of control systems equivalent to a canonical contact system for curves is discussed in Section 6. In particular, we show how the geometry of that class of systems helps to describe their flat outputs. We give a model of a mechanical nonholonomic system equivalent to a canonical contact system in Section 7. In Section 8 we introduce extended Kumpera-Ruiz normal forms and give a result, Theorem 8.1, stating that any contact system, regular or singular, is locally equivalent to an extended Kumpera-Ruiz normal form.

2

The Canonical Contact System for Curves

A rank k distribution V on a smooth manifold M is a map that assigns smoothly to each point pin M a, linear subspace V(p) C Tp M of dimension k. In other words, a rank A; distribution is a smooth rank k subbundle of the tangent bundle TM. Such a field of tangent fc-planes is spanned locally by k pointwise linearly independent smooth vector fields / i , . . . , fk on M, which will be denoted by X> = ( / i , . . . , / * ) . Two distributions V and V denned on two manifolds M and M, respectively, are equivalent if there exists a smooth diffeomorphism ip between M and M such that ()(p) = V(p), for each point p in M. The derived flag of a distribution V is the sequence of modules of vector

81

fields D<°) C 2? (1) C • • • denned inductively by D(0)=£>

P< i + 1 ) = X>(i) + [ D ( i ) , P w ] ,

and

for t > 0.

(2.1)

The Lie /Jag is the sequence of modules of vector fields Do C T>\ C • • • defined inductively by V0=V

and

P i + i =Vt + [T>o,T>i], for t > 0.

(2.2)

In general, the derived and Lie flags are different; though for any point p in the underlying manifold the inclusion T>i(p) C X>W(p) clearly holds, for i > 0. For a given distribution V, defined on a manifold M, we will say that a point p of M is a regular point of V if all the elements T>i of its Lie flag have constant rank in a small enough neighborhood of p. A distribution V is said to be completely nonholonomic if, for each point p i n M , there exists an integer N(p) such that VN^(p) = TPM. A distribution V is said to be involutive if its first derived system satisfies V^ = £>(°'. An alternative description of the above defined objects can also be given using the dual language of differential forms. A Pfaffian system 1 of rank s on a smooth manifold M is a map that assigns smoothly to each point p in M a linear subspace l(p) c T*M of dimension s. In other words, a Pfaffian system of rank s is a smooth subbundle of rank s of the cotangent bundle T*M. Such a field of cotangent s-planes is spanned locally by s pointwise linearly independent smooth differential 1-forms w i , . . . ,ws on M, which will be denoted by I = (UJI,... ,w s ). Sometimes we will denote the Pfaffian system I = ( w i , . . . ,CJ S ) by ui = 0 , . . . , ws = 0. In the case of a rank one Paffian system I = (w) we will speak also about a Pfaffian equation u = 0. Two Pfaffian systems I and 1 defined on two manifolds M and M, respectively, are equivalent if there exists a smooth diffeomorphism ip between M and M such that X(p) = (ip*l)(p), for each point pin M. For a Pfaffian system I , we can define its derived flag 1^ D 2' 1 ) D • • • by the relations I<°) = I and l( i + 1 > = {a e I ( i ) : da = 0 mod ! « } , for t > 0, provided that each element X^ of this sequence has constant rank. In this case, it is immediate to see that the derived flag of the distribution V = 1± coincides with the sequence of distributions that anihilate the elements of the derived flag of I , that is v(i)

=

(i«)-L ;

for i > o.

For a given Pfaffian system X, we will say that a point p of M is a regular point if p is a regular point for the distribution V =2±, that is if all elements T>i of the Lie flag are of constant rank in a small enough neighborhood of p.

82

Consider the space J n ( E , R m ) of jets of order n > 1 of functions from R into R m . This space is diffeomorphic to R ( " + 1 ) m + 1 . The canonical coordinates associated to R (denoted by x°) and to R m (denoted by x^,... , x ^ ) can be used to define the canonical coordinates on J n (R, R m ) , which will be denoted by 0 T° r° r1 r1 T" J-OJ-^-I)--- j i f n j J - j , . . . , x m , . . . , x j , . . . x

T"

,x

m

,

with obvious indentifications org = 9 and x° = p° = u*, for 1 < i < m, and xj = p3i, for 1 < i < m and 1 < j < n (see the beginning of the Introduction). Observe that any smooth map tp from R into R m defines a curve in J"(R, R m ) by the relations x{ = tpy\x°), for 0 < i < m and 0 < j < n, where ip^ denotes the j - t h derivative with respect to zfj °f the z'-th component of (p. This curve is called the n-graph of (p. It is clear that not all curves in J"(R, R m ) are n-graphs of maps. In order to distinguish the "good" curves from the "bad" ones, we should introduce a set of constraints on the velocities of curves in J n (R, R m ) . In other words, we should endow J"(R, R m ) with a nonholonomic structure. The canonical contact system on J n ( R , R m ) is the completely nonholonomic distribution spanned by the following family of vector fields: /

n—1 m

( d^f > • • • > 5 i ^ " » 9 4 + E T,xi

\

^7) •

(2.3)

In a dual way, we will also call the canonical contact system on J " ( R , R m ) the Pfaffian system (dx{ - x{+1dx%, 0 < j < n - 1, 1 < i < m) annihilating the above distribution. For the canonical contact system on J"(R, R m ) we will also use the name canonical contact system for curves. By definition, if a curve in J"(R, R m ) is the n-graph of some map then it is an integral curve of the canonical contact system. More precisely, a section a : R -> J n (R, R m ) is the n-graph of a curve ip : R -> RTO if and only if it is an integral curve of the canonical contact system on J"(R, R m ) (see e.g. 3 and 2 5 ) . The aim of our paper is to analyze various possible answers to the question "Which distributions are locally equivalent to the canonical contact system on J"(R, R m ) ? " . The following result, obtained by the authors 2 7 , will be the starting point of our study. Theorem 2.1 (regular contact systems, first version). A rank m + 1 distribution V on a manifold M of dimension (n + l)m + 1 is equivalent,

83

in a small enough neighborhood of any point p in M, to the canonical contact system on Jn(IR, E m ) if and only if the two following conditions hold, for 0^1' that has constant corank one in (ii) Each element T>i of the Lie flag has constant rank (i + \)m + 1. Remarks 1. This result yields a constructive test for the local equivalence to the canonical contact system for curves, provided that we know how to check whether or not a given distribution admits a corank one involutive subdistribution. We give in Section 4 a checkable necessary and sufficient condition for the existence of such a distribution. 2. Item (i) describes the essential geometric property of distributions equivalent to canonical contact systems for curves while the condition (ii) distiguishes regular points p at which X>'*'(p) = T^i(p) from singular points, where this last condition is violated. Different roles of (i) and (ii) justify Definition 2.2 below. 3. Theorem 2.1 was proved in 27 , as a special case of a more general result that gives a normal form for any system satisfying (i) (see Remark 2 above). In Section 5 we will present a direct proof of Theorem 5.6, which is another version, under the weakest possible assumptions, of Theorem 2.1. Definition 2.2 (contact systems). A rank m + 1 distribution T> on a manifold M of dimension (n + \)m + 1 is said to be a contact system if, for any 0 < i < n, each element T>^ of the derived flag, has constant rank (i + l)m + 1 and contains an involutive subdistribution Ci C X>W that has constant corank one in A contact system is said to be a regular contact system at p G M if it is, locally at p, equivalent to the canonical contact system on Jn(R, Rm). Otherwise it will be called a singular contact system at p e M. Theorem 2.1 implies that a contact system is a regular contact system at a point p if and only if the point p is regular. Other, than that of Remark 2 above, equivalent characterizations of regular points of contact systems will be given in Theorems 5.4 and 5.6 , and in Corollary 5.5. 3

History

In this section we give a brief histry of the following problem: Which Pfaffian systems are locally equivalent to the canonical contact system on the space Jn(Rk,Wn) of n-jets of maps from Rk to Mm ? This problem is a natural

84

generalization of the follwing classical problem stated in 1814 by Pfaff: Which Pfaffian systems are locally equivalent to the canonical contact system on the space J1 (Rk,Rm) of 1-jets of maps from Rk to R? Studies of that problem have mainly concentrated on the four following cases of equivalence to the canonical contact system on the space • J^Rfc.R): Pfaff (1814), Frobenius (1877), and Darboux (1882); • J"(M,M): Engel (1890), E. von Weber (1898), Cartan (1914), Goursat (1923), Giaro, Kumpera and Ruiz (1978), Libermann (1978), Murray (1994), Mormul (1998-2001), Pasillas-Lepine and Respondek (1998-2001), Montgomery and Zhitomirskii (1999), • J ^ I f ^ i r ) : Gardner (1972), Bryant (1979); • Jn(R,Rm): Garnder and Shadwick (1992), Murray (1994), Tilbury and Sastry (1995), Aranda-Bricaire and Pomet (1995), Pasillas-Lepine and Respondek (1999-2001). In the following subsections we will describe solutions in each of the four above cases. Throughout this section we will use the notation p° = Uj, for 1 < i < m (see Introduction). 3.1

Pfaff-Darboux normal form

In this subsection we recall the classical solution given by Frobenius and Darboux to the original Pfaff question: which Pfaffian systems are locally equivalent to the canonical contact system on the space J 1 (Rk, E) ? We will need the following notion introduced by Frobenius n and Darboux 7 Definition 3.1. The class, at a point p, of the Pfaffian equation u = 0 is the largest integer p such that

Using the notion of class, Darboux 7 gave the following celebrated answer to the problem. Theorem 3.2 (Darboux, 1882). A Pfaffian equation w = 0, defined on a manifold of dimension 2k + 1, is locally equivalent to the equation given by the following canonical contact form dp0 - pldqi

phdqk = 0

on J1(Rk, R), if and only if its class is constant and equals k. The above canonical contact form is called Pfaff-Darboux normal form.

85

3.2

Bryant normal form

In this subsection we will recall a solution to the generalization of Pfaff's problem to the space of 1-jets of arbitrary maps. The following question has been studied by Gardner since 1972, and by Bryant (who gave a solution in 1979): which Pfaffian systems are locally equivalent to the canonical contact system on the space Jl(Wk, W1) ? In order to give an answer we will need the following generalization of the notion of class of a Pfaffian equation. Definition 3.3. The Engel rank, at a point p, of a Pfaffian system I — [uii,... ,uis) is the largest integer p such that there exists a 1-form a inX for which {{da)p A w i A - - - A w s ) ( p ) / 0. Obviously, in the case of a single Pfaffian equation u = 0, the notion of Engel rank reduces to that of class. A characteristic vector field of a distribution P is a vector field / that belongs to V and satisfies [/, V] C T>. The characteristic distribution of V, which will be denoted by C, is the module spanned by all its characteristic vector fields. It follows directly from the Jacobi identity that the characteristic distribution is always involutive but, in general, it needs not be of constant rank. If for a constant rank Pfaffian system I , the characteristic distribution C of the distribution V = l1- is of constant rank, then its anihilator (C)1is a Pfaffian system, called the Cartan system of the Pfaffian system I and denoted by C(X). We refer the reader to 3 for a definition of the Cartan system given in the language of Pfaffian systems. The following result was proved by Bryant in his Ph.D. Thesis 2 (see also 3 ) . Theorem 3.4 (Bryant, 1979). Assume that m ^ 2. A Pfaffian system 2=

(oJi,...,ojm),

defined on a manifold of dimension m + k + mk and such that I^1' = 0, is locally equivalent to the canonical contact system p\dqk = 0

Pm ~ Pmdqi

Pkmdqk = 0

d

dp\ - p\dqi

on J1 (Rk, lRm), if and only if its Engel rank is constant and equals k, and the rank of its Cartan system is constant and equals m + k + mk. The above normal form of a Pfaffian system is called Bryant normal form. Reasons for which the case m = 2 is excluded will be explained in Section 4, see Lemma 4.2 and Corollary 4.3.

86

3.3

Goursat normal form

Starting with the work of Engel 8 and von Weber 33 , the following question has been intensively studied: which Pfaffian systems are locally equivalent to the canonical contact system on the space J"(R, R) ? The case n — 1 was solved by Pfaff in his pioneering paper 29 ; it is also the simplest case of Darboux theorem given above. The case n = 2 was solved by Engel 8 . In order to give a solution for an arbitrary n, recall, see Introduction, the following fundamental notion. The derived system I' of a Pfaffian system 1 = (UJI, ...,u!s) is the Pfaffian system generated by all 1-forms a in I such that da AOJI A • • • A ws = 0.

Starting from a Pfaffian system 1 and its (first) derived system I ' we define the flag of derived systems 1^ D 1^ D • • • D 1^ D • • • inductively by I<°) = J a n d I < i + 1 ) = ( ! « ) ' . The following characterization of Pfaffian systems, that are locally equivalent to the canonical contact system on the space J"(R, R), was found by von Weber and has been intensively studied by Cartan. Theorem 3.5 (van Weber, 1898 ; Cartan, 1914). A Pfaffian system X = (uii,...,LJn) of rank n, defined on a manifold of dimension n + 2, is is locally equivalent (on an open and dense subset) to the canonical contact system dp0 - pldq = 0 , • • • , dpn~l - pndq = 0 on Jn(M., E) if and only if the elements 1^ of the flag of derived systems are of constant rank n — i, for 0 < i < n. The canonical form present in the above theorem is called Goursat normal form, mainly because it was intensively used by Goursat in his book 14 . We will call a Goursat structure any distribution V (resp. any Pfaffian system I) on a manifold of dimension n + 2 such that the elements of the derived flag X>(") are of constant rank i + 2 (resp. the elements j M 0 f the flag of derived systems are of constant rank n — i, for 0 < i < n). Therefore the above theorem says that a Goursat structure is locally equivalent to a Goursat normal form on an open and dense subset of the underlying manifold. Notice, see Proposition 5.1, that a Goursat structure is just a contact system of rank 2 (a contact system of corank 2, if we use the language of Pfaffian systems). Theorem 3.5 is thus a special case of a more general Theorem 2.1 stating that a contact system is equivalent to the canonical contact system on an open and dense subset. Applying the latter theorem to the case of Goursat

87

structures, the open and dense subset of Theorem 3.5 can be described as points p at which X>W(p) = T>i(p) (as shown earlier by Murray 24 directly for Goursat structures). Other equivalent descriptions of the open and dense set of regular points of Goursat structures will follow from characterizations of regular points of contact systems given in Theorems 5.4 and 5.6 and in Corollary 5.5, stated in Section 5. Here we would like only to emphasize that in the cases n = 1 and n = 2 we do not have to restrict ourselves to an open and dense set. In fact, the conditions dim X^\q) — 0, in the case n = 1, and dim l{1)(q) = 1, dim l(2)(q) = 0, in the case n = 2, satisfied for any q in a neighborhood of a given point p, guarantee that the Pfaffian system I on a manifold, respectively of dimension 3 and 4, is locally equivalent around p to Goursat normal form. Giaro, Kumpera, and Ruiz 13 were the first ones who observed that starting from n = 3, singularities may appear for Goursat structures. Example 3.6 (Giaro et al, 1978). The following Pfaffian system of five variables dx2 — xzdxi = 0 dx$ — Xidxi = 0 dx\ — x^dxi = 0 is a Goursat structure, that is, it satisfies the conditions of the above theorem but is not locally equivalent to the canonical contact system on J 3 (E, E) if x5=0. It was this example with which an increasing interest in the problem of singularities of Goursat structures started (for more results see e.g. 17 , 6 , 22 , 23

28 \

21

!

3.4

>

I'

Canonical contact systems for curves

In this section we will discuss the following question: which Pfaffian systems are locally equivalent to the canonical contact system on the space J"(M, E m ) , that is, to the canonical contact system for curves? We will put the answer, given to that question as Theorem 2.1 in Section 2, in a form analogous to that of Theorems 3.4 and refthm-goursat. Theorem 3.7 (regular contact systems, second version). A Pfaffian system I of constant rank nm, where m ^ 2, defined on a manifold of dimension (n + \)m + 1, is locally equivalent (on an open and dense subset)

88

to the canonical contact system dp0! - p\dq = 0 , • • • , dp7?'1 - p^dq = 0 dp°m -pldq

= 0 , • • • , dp^-1 -p^dq

=0

on J"(E, R m ) if and only if each element j W of the flag of derived systems satisfies the following conditions, for 0 < i < n. (i) 1^ is of constant rank (n — i)m; (ii) The Engel rank ofl^ equals one; (iii) The rank of the Cartan system of I^1' is constant and equals (n — i + l)m + l. The above theorem is given, as Corollary 2.4, in 2 7 . Its proof follows directly from Theorem 2.1 and a characterization, via the notion of Engel rank, of the existence of an involutive subdistribution of corank one in a given distribution. Such a characterization was obtained by Bryant 2 and will be discussed in Section 4. Notice that the formulation of Theorem 3.7 can be considered as a natural combination of, on the one hand, Weber-Cartan's Thereom 3.5 and, on the other hand, of Bryant's Theorem 3.4. Indeed, like Thereom 3.5, it involves the derived flag and requires to exclude some singular points. On other hand, like Theorem 3.4, it expresses the basic geometry of the problem through the notion of Engel rank. As we said, we will discuss links between the geometry and Engel rank in Section 4. What concerns regular points, one characterization as points p such that V^ (p) = T>i (p) was already given in Theorem 2.1. We would like, however, to express that regularity condition in terms of the Pfaffian system I ; but is not clear how to represent the elements T>i of the derived flag in terms of X. For that reason, we will present in Section 5 other equivalent descriptions of regular points, in particular, in terms of Pfaffian systems. 4

Involutive s u b d i s t r i b u t i o n s of corank one

The aim of this Section is to provide checkable conditions to verify (i) of Theorem 2.1 and to justify the statement of Theorem 3.7. This will be done by answering the following question: "When does a given constant rank distribution V contain an involutive subdistribution C C T> that has constant corank one in T>T\ In fact, the answer to this question is an immediate consequence of a result contained in Bryant's Ph.D. thesis 2 . Links between Bryant's result and the characterization of the canonical contact system for

89 curves have also been observed by Aranda-Bricaire and Pomet 1. Recall that in Subsection 3.2 we defined the Engel rank of a Pfaffian system X, at a point p, as the largest integer p such that there exists a 1-form u in X for which we have (dw)p(p) ^ OmodX. The Engel rank of a constant rank distribution V will be, by definition, the Engel rank of its anihilator V-1. Obviously, the Engel rank p of a distribution equals zero at each point if and only if the distribution is involutive. We will now give an equivalent definition of the Engel rank, see 2 7 , in the language of vector fields, in the particular case when p — 1, which will be important in the paper. Let V be a distribution such that Z>(°) and T>^ have constant ranks d0 and di, respectively, and denote ro(p) — d\ (p) — do(p). Assume that d0 > 2 and r 0 > 1. Take a family of vector fields ( / l , - - • , / d 0 ) 5 l ) - - - >#r 0 )

such that 2>(°) = ( / ! , . . . , / d o ) and V& = ( A , . . . ,fdo,gu . . . ,gro). The structure functions ck- associated to those generators, for 1 < i < j < d0 and 1 < k < ro, are the smooth functions defined by the following relations: [/* . / > ] = £ 4 9k mod P<°>,

for 1 < i < j < d0.

It is important to point out that the structure functions are not invariantly related to the distribution V, since they depend on the choice of generators. Assume that the Engel rank p of D is constant and that ro > 1. It is easy to check that p = 1 if and only if either do = 2, or do = 3, or do > 4 and the structure functions satisfy the relations

4-c« - <44 + &U + %A - ,& + « , • = o-

(4-i)

for each sextuple (i,j,k,l,p,q) of integers such that l 1. Then the two following conditions are equivalent: (i) The characteristic distribution CofT> has constant rank CQ = do—ro — 1 and the Engel rank p of V is constant and equals 1; (ii) The distribution T> contains a subdistribution B CD that has constant corank one in T> and satisfies [B,B] C T>.

90

Observe that if the first condition is satisfied then we must necessarily have r 0 < do — 1. The following result is included in the proof of Bryant's 2 normal form Theorem (see also 3 ) . An alternative proof of its surprising item (iii) is given in Appendix A of 27 , where the role of the assumption ro > 3 is explained by relating it to the Jacobi identity. Lemma 4.2 (Bryant). LetV be a distribution such thatV^ andV^ have constant ranks do and d\, respectively. Assume that the distribution T> contains a subdistribution B C T> that has constant corank one in T> and satisfies

[B,B]cV. (i) If ro = 1 then the distribution T> contains an involutive subdistribution C C T> that has constant corank one in T>; (ii) Ifr0>2 then B is unique; (iii) If ro > 3 then B is involutive. Observe that, in the first item of the above Lemma, the involutive subdistribution C can be different from B, which is not necessarily involutive. The following result is a direct consequence of Bryant's work. In order to avoid the trivial case ro = 0, for which the existence of a corank one involutive subdistribution is obvious, we will assume that r 0 > 1. Corollary 4.3 (corank one involutive subdistributions). Let V be a distribution such that X>'0' and T>^ have constant ranks do and d\, respectively. Assume that ro > 1. Then, the distribution V contains an involutive subdistribution C C T> that has constant corank one in D if and only if the three following conditions hold: (i) The characteristic distribution C ofT> has constant rank Co — do—ro — 1; (ii) The Engel rank pofDis constant and equals 1; (iii) If ro = 2 then, additionally, the unique corank one subdistribution B C T> such that [B, B] C D must be involutive. We would like to to emphasize that the above conditions are easy to be checked, and, as a consequence, so are the conditions of Theorems 2.1 and 3.7. Indeed, for any distribution, or the corresponding Pfafnan system, we can compute the characteristic distribution C and check whether or not the Engel rank equals 1 using, respectively, the formula (4.2) and the condition (4.3) below. This gives the solution if ro ^ 2 (equivalently, if m / 2). If r 0 = 2 we have additionally to check the involutivness of the unique distribution B satisfying [B, B] C V, whose explicit construction is also given by (4.4) below. In the sequel, we will show, following Bryant 2 , how to verify whether or not the Engel rank equals 1, and how to construct explicitly — if it exists — the unique corank one subdistribution B C V satisfying [B, B] C V (which, in the case r 0 > 2, is the unique involutive subdistribution of corank one in V).

91 Consider a distribution T> of constant rank d0, defined on a manifold of dimension N. Let w i , . . . ,w So , where so = N — do, be differential 1-forms locally spanning VL, the annihilator of V, that is V± =

(ui,...,uao).

We will denote by I the Pffafian system generated by u\,..., For any form ui 6 V-1, we put W(w) = {/GZ> :

wSo.

/jdweP1}.

Clearly, the characteristic distribution C of V is given by C=f)W(oJi).

(4.2)

Now assume that V^ is of constant rank di > do, that is ro > 1, or, equivalently, that the first derived system 2^ is of constant rank smaller than soBy a direct calculation we can check (see e.g. 3 ) that the Engel rank of the distribution V, or of the corresponding Pfaffian system I, equals 1 at p if and only if (dui A db}j)(p) = 0

modi,

(4.3)

for any 1 < i < j < soNow let us choose a family of differential 1-forms LJi,...,u)ro,Wr0+i,...,uSo such that I = (V^)1= (OJX,. .. ,wSo) and i W = (D^)1= (uv 0 +i,... ,w So ). When condition (i) of Lemma 4.1 is satisfied, independently of the value of ro > 2, the unique distribution B satisfying [B, B] C V is given, as shown by Bryant 2 , by B=Y:\V(LJi).

(4.4)

i=l

In fact, Bryant has also proved that it is enough to take in the above sum only two terms corresponding to any 1 < i < j < ro. In order to verify, in the case ro = 2, the conditions of Corollary 4.3 we have additionally to check the involutivity of this explicitly calculable distribution. Notice that, if the distribution V satisfies Lemma 4.1 (i) and r 0 > 3, then the formula (4.4) gives the unique involutive subdistribution C of corank one in V.

92 5

5.1

Contact systems, characteristic distributions, and involutive subdistributions Contact systems and characteristic

distributions

Let V be a contact system, that is, according to Definition 2.2, each distribution Z?W contains a unique involutive subdistribution d of corank one. It turns out that those involutive subdistributions are unique and, moreover, can be expressed in terms of characteristic distributions. Indeed, we have the following result. Proposition 5.1. Let T> be a rank m + 1 contact system, defined on a manifold of dimension (n + l)m + 1. Each distribution P ^ ' — either for 0 < i < n — 2, if m = 1, or for 0 2 — contains a unique involutive subdistribution Ci C P ' 4 ' that has constant corank one in P « and that, for 0 < i < n — 2, coincides with the characteristic distribution Cj+i o/p( J + 1 >. The following picture summarizes the above proposition. £>(°) c P ^ C • • • C X>("-2) C Pf"- 1 ) C P ( n ) = TM U U U U Co C C\ C • • • C Cn-2 C Cn-\ n n ii C\ C C2 C • • • C C n - i

n £>(i)

n c

(5.1)

n

£>(2) C • • • C D^""1)

The inclusions between corresponding successive elements of the first and second row are those of the definition of contact systems. The inclusions between the elements of the third and fourth row express simply the fact that Ci, being a a characteristic distribution of V^\ is contained in 2?W. The equalities between the corresponding elements of the second and third row express a basic property of the sequence of involutive subdistributions Ci. In the case m = 1, that is for Goursat structures, Proposition 5.1 seems to be already known to Cartan 5 . For a proof we refer the reader to Montgomery and Zhitomirskii 21 (see also 4 , 17 , and 2 0 ) . Incidence relations of distributions V^'s and Cj's have been intensively used by the authors 28 and by Montgomery and Zhitomirskii 21 to study the geometry of Goursat structures at singular points. In the case m > 2, to the best of our knowledge, Proposition 5.1, has never been stated. Notice a substantial difference between the cases m = 1 and m > 2, which is the presence of the canonical involutive subdistribution Cn-i in the second row for m > 2. In the case of Goursat structures, that

93 is if m = 1, the distribution p ( " - 1 ) contains many different involutive sudistributions of corank one but non of them has any canonical properties and therefore in the case m = 1, the second row of (5.1) terminates with £„_2This basic difference between the cases m = 1 and m > 2 will be reflected in different statements of Theorems 5.6 and 5.7, that describe, respectively, regular contact systems for m > 2 in terms of £„_i and, for m = 1, in terms of C n _i = £„-2Definition 5.2. We will call the corank one involutive subdistribution d given in the previous result — either for 0 < i < n — 2, if m = 1, or for 0 2 — the canonical subdistribution o / p W . When proving Proposition 5.1 we will use the following result proved in 2 7 . Lemma 5.3 {Co C CI). Let T> be a distribution such that p ( ° ' , T>^\ and T>W have constant ranks do, di > do + 2, and c?2 > d\ + 2, respectively. Assume that each distribution p W , for i = 0 and 1, contains an involutive subdistribution Ci C P ^ that has constant corank one in p W . Then Co CCiProof: (of Proposition 5.1). In the case m = 1, that is for Goursat structures, the result was proven in 21 (see also 4 , 17 , and 2 0 ) . Consider the case m > 2. By Lemma 4.2, we conclude that all involutive distributions d, for 0 < i < n—1, are unique. In order to show that d = Cj+i, for 0 < i < n — 2, apply Lemma 5.3, with T>(°\ T>^1\ and V^2\ being replaced, respectively, by V^\ T>(i+1\ and X>('+2). We conclude that d C Ci+1. On the one hand, the rank of d equals m{i +1) while, on the other hand, the rank of Ci+i is also equal to m(i +1), because V^+1^ contains a subdistribution C,+i that satisfies [Ci+i,d+i] C X>('+1) (see Lemma 4.1). We thus have d = Ci+iD In the case of Goursat structures, that is m = 1, using the canonical subdistributions we can distinguish singular points from regular ones, see 21 and 2 8 . Below we show that the geometry of incidence of canonical subdistributions describes completely also regular contact systems in the general case m > 2. Indeed, we have the following result. Theorem 5.4 (regular contact systems, third version). A rank m + 1 distribution T>, with m > 2, on a manifold M of dimension (n + l)m + 1 is equivalent, in a small enough neighborhood of a point p in M, to the canonical contact system on J"(R, E m ) if and only if the following conditions hold. (i) Each element P ^ of the derived flag, for 0 < i < n, has constant rank (i + \)m + 1. (ii) Each element T>^> of the derived flag, for 0 < i < n — 2, contains the characteristic distribution C,+i of V^l+1' that has constant corank one in P « .

94 (iii) X>(™-1) contains an involutive subdistribution £n-i that has constant corank one in Z)( ra_1 ). (iv) For any 0 < i < n — 3, we have 1)W (p) is not contained in Ci+2 (p) and, moreover, X>(n_2)(p) is not contained in £„_i(p). Remarks 1. Let us look at items (i)-(iv) in the case of of Goursat structures, that is m = 1. Firstly, item (i) implies, according to Proposition 5.1, item (ii). Secondly, in the case m = 1, as shown in 21 and 2 8 , a Goursat structure (that is a distribution of rank two satisfying item (i)) is locally at p equivalent to Goursat normal form under the regularity conditions V^l\p) ^ d+2(p), for 0 < i < n — 3. Thirdly, being equivalent to Goursat normal form, it clearly satisfies (iii) because it admits infinitely many involutive subdistributions of corank one in Z>(™_1), most of which fulfilling the second part of item (iv). 2. Notice that any distribution T> satisfying (i), (ii), and (iii) is a contact system. Indeed, since the characteristic distributions C; are involutive, we put d = Ci+i and thus each distribution 2?W, for 0 < i < n — 2, contains an involutive distribution £j C V^ of corank one because of (i) and (ii). The one before the last distribution 2?(™_1) contains, by (iii), an involutive subdistribution £ n - i , which is unique and it is the canonical involutive subdistribution in P ^ - 1 ) . 3. As we argued in Remark 2 above, (i)-(iii) imply that T> is a contact system. Item (iv) is thus another way of expressing the regularity condition 2?W(p) = T>i(p). Contrary to this last condition, however, item (iv), as well as items (i)-(iii), can be immediately expressed in terms of Pfaffian systems, which we will state below as Corollary 5.5. 4. We will omit a proof of Theorem 5.4 because it follows from a more general result, Theorem 5.6, which will be stated in Subsection 5.2 and proven in Subsection 5.3. Corollary 5.5 (regular contact systems, fourth version). Let I be a Pfaffian system of rank nm, defined on a manifold M of dimension (n + l)m + 1. If m ^ 2, the Pfaffian system I is locally equivalent, at a given point p of M, to the canonical contact system on J n (M, E m ) if and only if (i) The rank of each derived system ZW is constant and equals (n — i)m, for 0 < i < n. (ii) The rank of each Cartan system C(I^) is constant and equals (n + 1 — i)m + 1, for 0 < i < n — 1. (iii) The Engel rank ofl^n~^ is constant and equals 1. (iv) For any 0 < i < n - 3, C(l^+2^){p) is not contained in l^{p) and, moreover, Jn-\{p) is not contained in T^n~2^[p), where Jn-\ is the unique integrable Pfaffian system such that Z(™_1) is of constant corank

95

one in Jn-\, and whose existence is given by (ii) and (iii). Notice two differences between Corollary 5.5 and Theorem 3.7. Firstly, as we have mentioned in Remark 3 just above, item (iv) of Corollary 5.5 gives a description of the open and dense set of regular points of Theorem 3.7. Secondly, when using Corollary 5.5 we have to calculate the Engel rank of l ' " - 1 ) only (equivalently, to verify the existence of an involutive subdistribution of corank one in the distribution X>(™_1) only). 5.2

Contact systems and involutive subdistribution

£n-i

The aim of this subsection is to show that all information about a distribution V, which is locally equivalent to the canonical contact system on J n (K, R m ) , that is, about a regular contact system, is encoded completely in the canonical involutive subdistribution £ n _ i . This information concerns, simultaneously, the existence of all canonical distributions d = C;+i, for 0 < i < n — 2, the regularity of the point p, as well as the desired ranks of all intermediate distributions 2?W. In fact, we have the following result. Theorem 5.6 (regular contact systems, fifth version). A distribution V of rank m + 1, with m >2, on a manifold M of dimension (n + l)m + 1 is equivalent, in a small enough neighborhood of a point p in M, to the canonical contact system on Jn(R, K m ) if and only if the following conditions hold. (i) £>(") = TM. (ii) XH" -1 ' is of constant rank nm + 1 and contains an involutive subdistribution £n-i that has constant corank one in (iii) T>(°)(p) is not contained in Cn-i{p). If the conditions (i)-(iii) are satisfied then, of course, the distribution £ „ _ i , whose existence is claimed by (ii), is unique and it is with the canonical subdistibution of p f " " 1 ) . It is interesting to see how much information is encoded just in the existence of the canonical involutive subdistribution Cn-i and its regular intersection with Z>(°). In this case, the distribution £n-i "knows" about the existence of all canonical distributions £, = Cj+i, as well as about the regularity of all intersections of Z>(*) with C i + 2 . Notice, however, that although Cn-\ "knows" how to distinguish regular points from singular ones, it does not "know" about the geometry of the incidence of various DW's and C,-'s at singular points corresponding to different extended Kumpera-Ruiz normal forms (see Section 8). Theorem 5.6 was stated for m > 2, thus excluding m = 1, for the obvious reason: in the case of Goursat structures, that is for m = 1, we do not have the

96 canonical involutive subdistribution £ „ - i . The second row of (5.1) terminates at £„_2 = Cn-\. It can be noticed, however, that a "non-canonical" version of Theorem 5.6 holds also for m = 1: a rank-two distribution is equivalent to a Goursat normal form if and only if there exists a distribution £„_i satisfying the conditions (i),(ii), and (iii) of Theorem 5.6. Necessity is seen immediately from Goursat normal form while sufficiency follows from the sufficiency part of the presented proof of Theorem 5.6, which is the same independently of the value of m. What is important to emphasize, however, is that in the case m = 1, the condition (iii) is, in general, non veryfiable. In fact, if ranks of D( n _ 1 ) and T>^ equal, respectively, n + 1 and n + 2 then X>("-1) contains an infinity of involutive subdistributions of corank one. In order to check condition (iii) we have to calculate (some of) them and it is not clear how to do it because the method presented in Section 4 and based on Engel rank does not apply. Moreover, even if we would be able to find one such an involutive subdistribution, it may happen that it does not satisfy (iii) while others do. We will show now that an analogue of Theorem 5.6 holds for Goursat structures, that is for m = 1, with Hn-\ replaced by Cn-\. In fact we have the following result. Theorem 5.7 (Goursat normal form). A distribution T> of constant rank two, on a manifold M of dimension n + 2, is equivalent, in a small enough neighborhood of a point p in M, to Goursat normal form, that is to the canonical contact system on J n (R, E), if and only if the following conditions hold. (i) £>(") = TM. (ii) J)(n~2> is of constant rank n; moreover, the characteristic distribution Cn-i of X>(ra_1) is contained in £>(" -2 ) and has constant corank one in T>(n~2K (iii) p(°)(p) is not contained in Cn-i(p). Notice that we assume neither that T>^, for 0 < i < n — 3, are of constant rank nor that X>(n_1) is of constant rank nor that the ranks of 2?W grow by one at any point: the characteristic distribution C n _i "knows" about all that. 5.3

Proof of Theorem 5.6

Proof: Necessity is obvious. We prove sufficiency by giving a sequence of local diffeomorphisms that bring the distribution V to the canonical contact system on J"(E, R m ) . For simplicity, we will use the notation

dx>'

K

dx{,'"'dxL

97

Since £ n - i is an involutive distribution of constant corank m + 1 in TM we can choose m + 1 smooth functions o, <j>i, . . . ,0 m such that (d(j)0,d(l)i,...,d(f>m) = (^n-i)- 1 in a neighborhood of p. Item (iii) implies that there exists a vector field g~o € T>(°> such that go(p) £ £n-i(p). Moreover, there exists a function fa such that LgQ0(p) ^ 0. Define 1 Introduce coordinates x% = fa, and x° = 0j, for 1 < i < m, and complete them to a coordinate system (x^,x°,... ,xn), where xj = (x[,... ,x3m), for 0 < j < n. Denote \ = Lgo4>°, for 1 < i < m. We have

r

- JL

( ~ - - dz1'"m'dx»)

J n 1 (

A-)

and d ''O

^

A l

d

,. d

d .

i=l

Since p(™ ^ is of constant rank and corank of £ n _ i in 2?(" ^ equals one, item (iii) implies that 2?( n_1 ) = £ „ _ i + (#0)- Hence Z>(") = pf"" 1 ) + [ZXn-^ZX"-!)] is given by X > ( n ) = £ „ _ i + [9d,£„_i] + (S0). This relation and item (i) imply that we can choose among - ^ j , for 1 < i < m, 1 < j < n, vector fields / i , . . . , fm, for such that P<"> = £ „ _ i + ([po, /i], 1 < » < m) + (go). In particular, this implies that the differentials dfa^,, d\, for 0 < j < 1 and 1 < i < m, are linearly indpendent at p. By relabelling x\, for 1 < i < m, 1 < J < n i if necassary, we can assume that /j = g|r, for 1 < J < m. Introduce new coordinates by replacing xj by fa\. In new coordinates, denoted still by x\, we have

98

and d

V^

i

d

.,

d

d ^

We claim that

Suppose that there exists a vector field / £ X>(n_2) such that / ^ J"„-2- Since go € D("-2> and P< n - 2 ) c X*^"1) = ( 5o ) + £ „ - i , we can suppose that

for suitable smooth functions /3j, 1 < i < m, where at least one function, say /3k, is not identically zero in a neighborhood of p. Since / € X>(n_2) and 2 but [/, 5o ] = E " i Agftr ffo G Z><°) C P ( " - ) , it follows that [f,g0] € V^V _1 mod£„_!, which clearly is not in X>(™ ) = £ n _ i + (g0) thus yielding a contradiction. Hence it follows that V(n~2) C ( § § * , . . . , ^ ) + (
A =4>\ iorO<j
=Lgoi~1,

— l and 1 < i < m. Suppose, moreover, that in those coordinates

as well as 2 ? ( " - * ) c ^ „ _ ? ) = ( ^ , . . . , ^ ) + («*,)

(5.3)

and ^-

+ 1 )

=(5o)

+

(^zr)mod(A,...,^_).

(5.4)

Notice that we do not assume the ranks of £>(""«) and of 1)("-
99

We will prove that equations (5.2), (5.3), and (5.4) hold with q being replaced by q + 1. To start with, observe that X>("-«+1) = Z>(n-«) + [£>("-»),£>("-«)] together with (5.3) imply that we can find m vector fields h, • • •, fm G X» (n_9) such that P<"-' + 1 > = &>) + ([/<,»>], l < i < m )

mod(^,...,^).

After relabelling xj, for q < j < n and 1 < i < m, if necessary, we can suppose, because of (5.3) that d d d t A< ^ /i = ^ ™ ° d ( ^ T I , . . . , — ) .

G T>(n~9^, the above expression implies that

Since fi,...,fm ^

^

M

+ ^ m o d ^ , . . . , ^ ) .

(5.5)

Moreover, it follows that the differentials d(j>Q, dcjyf, for 0 < j < q and 1 < i < m, are linearly indpendent at p. Introduce new coordinates by replacing x\ by

for 1 < i < m. In new coordinates, denoted still by x\, the distribution (— — ) v d a ; « ' ' " ' dxn' is preserved while the vector field g0 becomes

for some smooth functions 4>J+1, 1
<m.

T

, . . . , A ) + (»,).

(5.7)

Suppose that there exists a vector field / e X>("-«_1) such that / ^ ^ r ( n _,_ 1 ). Since y0 G £>("-(""«), we can suppose, because of (5.5), that ^

Q

Q

Q

100

for suitable smooth functions /?,, 1 < i < m, where at least one function, say Pk, is not identically zero in a neighborhood of p. Since / e D t " - ? - 1 ) and g0 6 E H i f t ^ P (o) c 2?(n-9-i), it follows that [f,go] E 2>("-»> but [/, 5o ] = m o d ( g f 7 , . . . , gf^-), which clearly is not in p("-») because of (5.5). This yields a contradiction thus proving that (5.7) holds. Now observe that the three just proved conditions (5.6), (5.7), and (5.5) correspond, respectively, to (5.2), (5.3), and (5.4), with q being replaced by q + 1. Hence by an induction argument we arrive after n steps at Vi0)

C

*» =

(

}

^

+

(5o)

'

where r±

w=

n—1

m

p.

+

+

r\

a4 gS^ 'S|

mod( )

^-

Since r>(0) is of constant rank m + 1 it follows that X>(°) = T0 = ( ^ ) + (g0), which ends the proof. • 5.4

Constructing canonical coordinates

The presented proof gives a method to calculate a diffeomprphism bringing a regular contact system V of rank m + 1 to the canonical contact system on Jn(Rk, Rm), starting from m + 1 independent functions (j>%, 0 ° , . . . , £, whose differentials annihilate £ n _ i , that is

(d0g,d0;,...,d^) = (£n_1)-L. Choose a vector field gQ G V^ such that go(p) $ Cn-\. Without loss of generality (see the beginning of the proof), we can suppose that Lgo(f>Q(p) / 0. Define x° = <$ and

J

= &

L

9Q^i

=

for 0 < j < n and 1 < i < m. It follows from the presented proof that in the coordinates (XQ,X°, ..., xn), the contact system T> takes the form

T>-(JL

—

A

rv

i+'i)

101

5.5

Proof of Theorem 5.7

Proof: Necessity is obvious. In order to prove sufficiency, choose a local coordinate system (xi,... ,xn+2) such that x(p) = 0 £ R™+2 and that

c

-(d

-?-)

ax\

oxn-i

We will identify the point p with 0 € R"+ 2 . Since Z>(°) (0) is not contained in C„_i(0), choose a vector field g0 6 V^ such that gQ(0) ^ C„_i(0). By multiplying go, if necessary, by a nonvanishing function, we can assume that 9o = 9o 5I7 + 9o+1 asf^T + s^f^ m o d c « - i • Because C„_i is of corank one in X>("-2) and g0(0) $ C n _i(0), it follows that D<"- 2 ) = C n _! + (50). Hence

*~" = <»•£•••• • 5 b k ' s r ' - - ^ s ^ » -

(5 8)

-

We claim that dim Z>(n_1)(g) = n + 1 for any q in some neighborhood of p = 0. In order to prove it, we will firstly show that dim V^n~1'(q) < n + 1 for any q in some neighborhood of p = 0. If not, there is a sequence of points qi, i = 1,2,... such that qi -> p and such that dim V^n~1\qi) = n + 2. Take a sequence of open neighorhoods Ui, such that qi € Ui and dim Vl^n~1\q) = n + 2 for any q £ \J Ui. Hence on any Ui, the characteristic distribution C„_i of X^"" 1 ) coincides with D*""1) = TM, which contradicts the fact that C„_i is of rank n — 1. Now w are going to prove that dim V^n~l\q) = n +1 in a neighborhod of p = 0. Suppose that dim V^n~1\p) = n. Denote Cj = g | - , for 1 < j < n — 1. There exist an integer 1 < k < n — 1 and a sequence of points qt, i = 1,2,... such that qi -» p and such that [ck,9o](qi) £ 1>(n~2\qi); otherwise, in view of (5.8), we would have, in a neighborhood of p, that £>(™_1) = £>("~2), which contradicts V^ = TM. Take a sequence of open neighborhoods Ui, such that qi 6 t/j and [ck,go](l) $ X^ n _ 2 '(g) for any q &\JUi. From the fact that dim V^n~1\q) < n + 1 it follows that there exist smooth functions a_,-, for 1 < j < n — 1, defined on |J Ui and such that [CJ > 9o] = aj [cfc, go] mod 2?("- 2) .

(5.9)

on (JUi. Now recall that the vector fields Cj, for 1 < j < n - 1, are characteristic vector fields of Z>("_1) and hence [a, [cj,go]] 6 Z>' n_1 ), for any 1 < i, j , < n — 1. Thus we have 2>(n) = 2?(»-D +

( [ 5 o , [c

^

1 < j < n - 1).

102

Notice that (5.9) implies that [30, [9,So]] = ctj[g0, [ck,go]]mod'D<-n~1^ on \jUi. Since dim V^n-^(p) = n, it follows that dim X>(") = TM. We thus have proved that £>("-2) c p f " - 1 ) c £>(n) = T M is a sequence of distributions of constant rank equal to, respectively, n, n + 1, and n + 2. It follows, see e.g. 3 4 , 20 , 2 1 , 28 , that we can choose local ccordinates (a;i,...,ar n + 2 ) such that £>("-2) = (g0) + C„_i, where (g0) = arrf-igf^ + x « 9 ^ 7 + 9 i f n modC„_i and ax\

oxn-i

We define an involutive distribution £„_i = (gf~, • • • , gf-) and the remaining part of the proof follows the same line as that of Theorem 5.6. • 6

Flatness of contact systems

In this Section we will discuss flatness of nonholonomic control systems, that are feedback equivalent to a canonical contact system for curves. Consider two control systems, with m + 1 controls, m

S:

x — 2_]fi(x)ui

= f(x)u

and

(6.1)

i=0 m

t : x = ^2fi(x)

Ui = f(i) u,

(6.2)

i=0

evolving respectively on two open sets X and X of RN, where u = (u0,...,umy, u - (u0,...,um), / = (fo, • • •, fmf, and / = (fo,...,fm). We say that the systems E and E are feedback equivalent if there exits a smooth diffeomorphism ip : X -> X and a feedback u = /3u, where the matrix P, whose entries /3j are smooth functions, is invertible at any x 6 X, such that

MfP) = f, where for any vector field g on X we denote by
103

Since the distribution V remains invariant under any invertible transformation u = (iu, all objects that we construct with the help of V can be considered as feedback invariant objects attached to E. The concept of flatness was introduced to control theory by Fliess, Levine, Martin, and Rouchon 9 , 10 in order to characterize control systems that are linearizable by endogenous feedback. This concept goes back to Hilbert 1 5 and Cartan 5 It is easy to observe that contact systems are flat. In this section we explain how the geometry of contact systems, described in previous Sections, provides additional information on various aspects of the flatness of contact systems. Consider a nonlinear control system E of the form (6.1). We say that E is x-flat at (x*,u*) E M.N x E m + 1 (see 10 , 16 , 30 ) if there exits a neighborhood V0 = X0 x Uo of (x*,u*) and smooth functions (pt on X0, for 0 < i < m, called x-flat outputs, having the following property: there exits an integer k and smooth functions •ji, 1 < i < N, and Si, 0 < i < m, such that we have Xi = 7. (,¥>, •••,V (fe) ), Ui = Si(tp,tp,...,ip^),

l
where

• • •, 2; in the case m = 1, canonical contact systems for curves are Goursat structures and their flatness is well understood, see e.g. . Assume that the system E is feedback equivalent to a canonical contact system for curves. Then there exists an invertible feedback u = f3u such that the unique involutive subdistribution Co of corank 1 contained in I?' 0 ) = (/o, • • •, fm) is spanned by the m last columns of the matrix f(3. Any matrix (3 defining a feedback having this property will be denoted by /?,„„. By Ufnv{x) we denote the codimension one subspace of Rm+1 such that f(x)u € Co{x) for any u € Ufnv(x). It is clear that a control u belongs to Ufnv(x) if and only if the zeroth-component (uP~^v(x))° of the feedback modified control u = uf3^v(x) is identically zero. Put u°nv(x) = (u[3^v(x))°. Although f3inv is not unique, the expression for u°inv is uniquely defined up to multiplication by a nonvanishing function and its interpretation is clear: if we apply to the system E any control u such that u°inv = 0, then the system is governed by lRm -valued contol and all its trajectories remain tangent to the involutive sudistribution £0. It is u®nv which is crucial in dynamic linearization of E as shows the following result given in 26 Proposition 6.1 (flatness). Consider the control system E given by (6.1) and assume that it is feedback equivalent, in a neighborhood of x* € X, to the

104

canonical contact system on J"(K,M m ). Then we have. (i) £ is x-flat at any (x*,u*) such that (u*)°nv(x*) ^ 0. (ii) The dynamic precompensator zx

= z2 ;

Zn

=

(6.3) V0

linked with the system via the dynamic feedback law « — Hmv I

vfh

J J

where vm = ( « i , . . . ,vmY and 7 is a suitably chosen nonvanishing function, renders the overall system, controlled by {vo,vi,...,vm)t, static feedback linearizable at any point (x*,z*) such that z\ ^ 0. (iii) As x-flat outputs we can choose any m + 1 smooth functions (pa, ipi,..., (°) can be explicitely calculated (see 2 and 2 7 ) . Therefore we can calculate, up to multiplication by a nonvanishing function of the state, the single control which needs to be preintegrated: it is the control u°nv, which is the zeroth-component of the new control u = /3~^vu. Secondly, it is also the involutive distribution £ 0 that determines the controls at which the system ceases to be flat. Thirdly, the canonical subdistribution £ n - i , that is the last involutive distributions £ „ _ i in the sequence £o C • • • C Cn-\ determines flat outputs. We would like to stress that functions whose differentials aninihilate £ n _ i are not the only flat outputs of a system equivalent to the canonical contact system. They have, however, a certain minimality property. Indeed, when determining — with their help — the state and the control, we need to take the minimal possible number of derivatives. Equivalently, among all choices of (m + 1) smooth functions which, treated as pseudo-outputs, define a dynamically input-ouput linearizable system without zero-dynamics, a linearizing precompensator of the minimal dimension n is needed in the case of pseudooutputs whose differentials anihilate Cn-\-

105

7

A n Example

In this Section we will consider a simple example given by a kinematical model of a car for which both the front and rear wheels are steerable, see 31 . We denote by (xo,yo) (resp. by (xi,j/i)) the coordinates of the midpoint between the two front (resp. rear) wheels and by a0 (resp. by c*i) the angle of the front (resp. rear) axle with respect to the horizontal. Finally, by tp we denote the angle of the bar connecting the two axles with respect to the horizontal. The two nonholonomic constraints reflecting the fact that the front and rear wheels roll without sleeping are given, respectively, by fi0 = 0 and Qi = 0 where differential 1-forms ilo = sin aodxo — cos aodyo J7i = smaidxi — cosaidyo. The holonomic constraints %i —

XQ

+ c o s ij}

2/i = 2/o +sinV> define a map tp from (S 1 ) 3 x E 2 , equipped with the coordinates (tp,a0,ai,xo,yo) into (5 1 ) 3 x R 4 , equipped with the coordinates (ip,ao,a{^xo,,yo,xi,yi). The pull-backs wo =
= 7r/2 + kn, k 6 Z} C X

and

S2 = {ct\ — ao = kit, k € Z} C X. The differential 1-forms UJQ and uj\ are linearly independent at p 6 X if and only if p ^ S\ n S2- Therefore outside Si fl S2, the nonholonomic constraints define a control system of the form e = /o(0«o + / i ( 0 « i + / 2 ( 0 « 2 , where f = (^,oio,oii,xo,yo) are coordinates on X = (S 1 ) 3 x R 2 , and the distribution V = ( / 0 , / i , / 2 ) , where / 0 = -£-, h = 3^7, and / 2 = cos(ai - ip) ( c o s ( a 0 ) ^ + s i n ( a 0 ) ^ J + sin(ai - a 0 ) ^ .

106 It is interesting to notice that at points of S\ D S2, integral curves of the distribution V = (wo,Wi)x do not represent all trajectories of the system subject to the nonholonomic constraints (uo,uii). Indeed, through any point p £ Si f)S2 there pass trajectories of the system whose velocities do not belong to V(p). We thus consider the model of a car outside Si fl S2. Clearly, the distribution V contains the involutive subdistribution *

I da0 ' Sen / "

of corank one in V. Since n = 1, the distribution C = (~^-, ~^-) is the canonical involutive subdistribution £ n _ i . By a direct calculation we check that V^(p) is of rank five if and only if p £ Si U 53, where S 3 = {a0 - ip = TT/2 + fcTr, k € Z} C X. Therefore at any configuration point outside Si n S3, the model is feedback equivalent to the canonical contact system for curves on J X (E, R 2 ). It is interesting to observe that at points of S2 that are outside Si U S3, the system, although not being equivalent to a cononical contact system, is flat. It can, moreover, be transformed via a local feedback to the following normal form ±1 = x 5 u 0 ±2 =

X1U1

X3 =

U0

X4 =

Ui

x5 = u2, which generates a nilpotent Lie algebra. As flat outputs we can choose X2, X3, and Xi. Notice that the form of the set of singular controls at which the system ceases to be flat changes: it is formed by the union of the planes u0 = 0 and «i = 0 in E 3 . Finally, we would like to mention the following difference between kinematical model of a car towing trailers and that of a multisteerable car. The former defines a Goursat structure (that is a contact system of rank two) at any point of its congfiguration space and, conversly, any Goursat structure can be locally equivalent to that kinematical model at a well choosen point of its configuration space (see 28 and 2 1 ) . This correspondence is no longer valid for multisteerable cars. They define a contact system of rank m +1 > 3 at almost all configurations but, in general, they fail to do so at some configurations (like at p € Si U S3 for the considered example).

107

8

Singular points and extended Kumpera-Ruiz normal forms

The aim of this Section is to study the class of distributions that satisfy condition (i) of Theorem 2.1 but fail to satisfy condition (ii) of that theorem. The fist condition describes the geometry of the canonical contact system while the second condition characterizes regular points. Recall that a distribution that satisfies the former but fails to satisfy the latter at a point p is called a singular contact system at p (see Definition 2.2). In this Section we will recall a result of 27 stating that any such distribution can be brought to a normal form for which we propose the name extended Kumpera-Ruiz normal form. Those forms generalize the canonical contact system on J 1 (R, Rm) in a way analogous to that how Kumpera-Ruiz normal forms (see e.g. 17 , 2 1 , 2 3 , and 28 ) generalize the canonical system on J1(M,1R), which is also called Goursat normal form. Consider the family of vector fields K1 = (K\, ... , K^, KJ) that span the canonical contact system on JX(IR, M.m), where K1

i

- -AK1 - -Aaxp • • • ' "-m — aT£

—

and the family of vector fields AC2 = (K 2 , . . . , K^, KQ) that span the canonical contact system on J 2 (M, K m ) , where 2

-

K

a -A K-2 — H£[' • • • > ^m — o ^

"'l

—

K2

_ ji

a

,

, 2

a_ ,

a

I

,

,

d_ ,

I

a

Loosely speaking, we can write 2 9 d *2 K *"1 — 5 x f ' • • • ' ^m — a ^ T K0 = XxKt + |- XmKm + KQ.

In order to make this precise we will adopt the following natural notation. Consider an arbitrary vector field / given on Jn~1(R, E m ) by n —1 m

/ = E E / / ( ^ - 1 ) ^ + /o°(^-1)4> j=0i=l

u

i

where J " " 1 denotes the coordinates XQ,X°,... , x°m, x\,... , xlm,... , a;" - 1 ,... n_1 m , x ^ of J ( E , E ) . We can lift the vector field / to a vector field on Jn(R, Rm), which we also denote by / , by taking n—1 m f=

v w ^ - ^ j=0i=l

+^ n - ^ a

+Q

.

a +-.. + 0 - 4 - .

108

That is, we lift / by translating it along the directions -^,...

, -J^r-

Notation(lifts of vector fields) . From now on, in any expression of the form n% = S f c o a « ( x ) K r _ 1 ' t h e vector fields K ^ - 1 , . . . , / C ^ - 1 should be considered as their above defined lifts. Let K n _ 1 = ( K ? - 1 , . . . , K ^ - 1 , K ™ _ 1 ) denote a family of vector fields defined on Jn~1(M.,Mm). A regular prolongation, with a parameter c", of K n _ 1 , denoted by Kn = Rc„ ( K " _ 1 ) , is a family of vector fields nn = « , . . . , K £ , K£) defined on J " ( E , E m ) by „ra _

d

n

_

d

(8.1) where cn = ( c " , . . . , cJJ,) is a vector of m real constants. A singular prolongation, with a parameter c™, of /c n _ 1 , denoted by K™ = S C n (/e n - 1 ), is a family of vector fields Kn = (K?, . . . , KJJ,, K#) defined on J n (R, E™) by a

„n _

„n

«0" = (*? + W *

_

+ • • • + « - l + C_ 1 )«»-_ 1 1 + K""1 + XnmK%-\

n

^ " ^

where c = ( c " , . . . , c|J l _ 1 ,0) is a vector of m real constants, the last one being zero. A family of vector fields K" on J " ( E , R m ) , for n > 1, will be called an extended Kumpera-Ruiz normal form if K" = CT„ O • • • o ^ ( K 1 ) , where for each 2 < i < n the map Oi equals either Rct or Sci, for some vector parameters c1. In other words, a Kumpera-Ruiz normal form is a family of vector fields obtained by successive prolongations from the family of vector fields that spans the canonical contact system on J 1 (M, R m ) . The above defined prolongations and prolongations-based definition of extended Kumpera-Ruiz normal forms generalize for contact systems analogous operations introduced by the authors 28 for Goursat structures. Let x : M -> R(»+i)™+i =* J " ( R , R m ) be a local coordinate system on a manifold M, in a neighborhood of a given point pin M. We will say that an extended Kumpera-Ruiz normal form on J"(R, Rm), defined in x-coordinates, is centered at p if we have x(p) = 0. For example, on J 2 (R, E 2 ), we have the two following extended Kumpera-Ruiz normal forms -r2 d.

( JL-

-2,

\d^'

8if ) Xl 8^

I r2 +

a^

9

.

I -r1

9

.. I T 1

-+• X2 \X1 gjs

9

.. I

-TX2Q^

d "\ -+- Q^)

J ,

109

defined by R^^K1) and 5(0,o)(«*), respectively. These two normal forms are obviously centered at zero. Recall, see Definition 2.2 that we define a contact system as a distribution V such that each element 2?W of its derived flag has constant rank (i + l)m + l and contains an involutive subdistribution A C V^ that has constant corank one in PW, for 0 < i < n, that is as a distribution satisfyng the condition (i) of Theorem 2.1. Moreover, we say that a contact system is regular at p if it satisfies at p the regularity condition (ii) of that theorem and is singular at p if it fails to satify that condition. The following theorem from 27 asserts that extended Kumpera-Ruiz normal forms serve as local normal forms for all, regular or singular, contact systems for curves. Theorem 8.1 (extended Kumpera-Ruiz normal forms). A contact system T> of rankm+1 on a manifold M of dimension (n + l)m + 1 is equivalent, in a small enough neighborhood of any point p in M, to a distribution spanned by an extended Kumpera-Ruiz normal form, centered at p and defined on a suitably chosen neighborhood of zero. Acknowledgments This survey is based on lectures delivered by the authors at the Conference on Geometric Control Theory and Applications, organized by Alfonso Anzaido Meneses, Bernard Bonnard, Jean-Paul Gauthier, and Felipe Monroy Perez, at Universidad Autonoma Metropolitana (Unidad Azcapotzalco), Mexico City. The authors are grateful to the local organizers, in particular to Alfonso Anzaido Meneses and Felipe Monroy Perez, for having organized the conference and for their hospitality. Moreover, they are grateful to Jean-Baptiste Pomet for discussions on his results 1. References 1. E.Aranda-Bricaire and J.-B. Pomet, Some explicit conditions for a control system to be feedback equivalent to extended Goursat normal form. In Proceedings of the IF AC Nonlinear Control Systems Design Symposium, (Tahoe City California, 1995). 2. R.Bryant, Some aspects of the local and global theory of Pfaffian systems. Ph.D. thesis (University of North Carolina, Chapel Hill, 1979). 3. R.Bryant, S-S. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths, Exterior Differential Systems. Mathematical Sciences Research Institute Publications. (Springer-Verlag, New York, 1991).

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4. M.Canadas-Pinedo and C.Ruiz, Pfaffian systems with derived length one. The class of flag systems. Transactions of the American Mathematical Society, 353, 1755-1766, (2001). 5. E.Cartan. Sur I'equivalence absolue de certains systemes d'equations differentielles et sur certaines families de courbes. Bulletin de la Societe Mathematique de France, 42, 12-48. (Euvres completes, Part. II, Vol. 2, (Gauthiers-Villars, Paris, 1914). 6. M.Cheaito and P.Mormul, Rank-2 distributions satisfying the Goursat condition: All their local models in dimension 7 and 8. ESAIM Control, Optimisation, and Calculus of Variations, 4, 137-158 (1999). 7. G.Darboux, Sur le probleme de Pfaff. Bulletin des Sciences mathematiques, 2(6), 14-36, 49-68 (1882). 8. F.Engel, Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen. Berichte Verhandlungen der Koniglich Sachsischen Gesellschaft der Wissenschaften Mathematisch-Physikalische Klasse, Leipzig, 41,42, 157-176; 192-207 (1889), (1890). 9. M.Fliess, J.Levine, P.Martin, and P. Rouchon, Sur les systemes non lineaires differentiellement plats. Comptes Rendus de I'Academie des Sciences, 315, 619-624 (1992). 10. M.Fliess, J.Levine, P.Martin, and P.Rouchon, Flatness and defect of nonlinear systems: Introductory theory and examples. International Journal of Control, 6 1 , 6, 1327-1361 (1995). 11. G.Frobenius, Uber das Pfaff'sche problem. Journal fiir die reine und angewandte Mathematik, 82, 230-315 (1877). 12. R.Gardner and W.Shadwick, The GS algorithm for exact linearization to Brunovsky normal form. IEEE Transactions on Automatic Control, 37, 2, 224-230 (1992). 13. A.Giaro, A.Kumpera, and C.Ruiz, Sur la lecture correcte d'un resultat d'Elie Cartan. Comptes Rendus de I'Academie des Sciences de Paris, 287, 241-244 (1978). 14. E.Goursat, Lecons sur le probleme de Pfaff, ( Hermann, Paris, 1923). 15. D. Hilbert, Uber den Begriff der Klasse von Differentialgleichungen. Mathematische Annalen, 73, 95-108 (1912). 16. B. Jakubczyk, Invariants of dynamic feedback and free systems. In Proceedings of the European Control Conference, Groningen, (1993). 17. A.Kumpera and C.Ruiz, Sur I'equivalence locale des systemes de Pfaff en drapeau. In F. Gherardelli, editor, Monge-Ampere equations and related topics, (Instituto Nazionale di Alta Matematica Francesco Severi, Rome, 201-247, 1982). 18. P.Libermann, Sur le probleme d'equivalence des systemes de Pfaff non

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completement integrables. Publications Paris VII, 3, 73-110 (1977). 19. V. Lychagin, Local classification of non-linear first order partial differential equations. Russian Mathematical Surveys, 30, 1, 105-175 (1975). 20. P.Martin and P.Rouchon, Feedback linearization and driftless systems. Mathematics of Control, Signals, and Systems, 7, 235-254 (1994). 21. R.Montgomery and M.Zhitomirski Geometric approach to Goursat flags. (Preprint, University of California Santa Cruz 1999). 22. P.Mormul, Rank-2 distributions satisfying the Goursat condition: All their local models in dimension 9. (Preprint, Institute of Mathematics Polish Academy of Sciences, 1997). 23. P.Mormul, Goursat flags: classification of codimension-one singularities. (Preprint, Warsaw University, 1999). 24. R.Murray, Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Mathematics of Control, Signals, and Systems, 7, 58-75 (1994). 25. P.Olver, Equivalence, Invariants, and Symmetry. (Cambridge University Press, 1995). 26. W.Pasillas-Lepine and W.Respondek, On the geometry of control systems equivalent to canonical contact systems: regular points,singular points, and flatness, in Proceedings of the 39th IEEE Conference on Decision and Control, 5151-5156 (Sydney, Australia, 2000). 27. W.Pasillas-Lepine and W.Respondek, Contact systems and corank one involutive subdistributions. To appear in Acta Applicand&Mathematicas (2001). 28. W.Pasillas-Lepine and W.Respondek, On the geometry of Goursat structures. ESAIM Control, Optimisation, and Calculus of Variations, 6, 119-181 (2001). 29. J-F. Pfaff, Methodus generalis, aequationes differentiarum partialum, nee non aequationes differentiales vulgares, utrasque primi ordinis, inter quotcunque variabiles, completi integrandi. Abhandlungen der KoniglichPreufiischen Akademie der Wissenschaften zu Berlin, Mathematische Klasse, 76-136 (1814-1815). 30. J.-B. Pomet, A differential geometric setting for dynamic equivalence and dynamic linearization. In B. Jakubczyk, W. Respondek, and T. Rzezuchowski, editors, Geometry in Nonlinear Control and Differential Inclusions, 32, 319-339. (Banach Center Publications, Warszawa, 1995). 31. D.Tilbury and S. Sastry, The multi-steering n-trailer system: A case study of Goursat normal forms and prolongations. International Journal of Robust and Nonlinear Control, 5, 4, 343-364 (1995). 32. A.Vinogradov, I.Krasil'shchik, and V.Lychagin, Geometry of Jet Spaces

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and Nonlinear Partial Differential Equations. (Gordon and Breach, New York, 1986). 33. E. von Weber, Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen. Berichte Verhandlungen der Koniglich Sachsischen Gesellschaft der Wissenschaften Mathematisch-Physikalische Klasse, Leipzig, 50, 207-229 (1898). 34. M.Zhitomirski, Normal forms of germs of distributions with a fixed segment of growth vector. Leningrad Mathematics Journal, (English translation), (2), 1043-1065 (1991).

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THE BRACHISTOCHRONE PROBLEM A N D MODERN CONTROL THEORY HECTOR J. SUSSMANN Department of Mathematics Rutgers University Hill center, Busch Campus Piscataway, NJ 08854, USA E-mail: [email protected]. edu URL: http://www.math.rutgers.edu/~ sussmann JAN C. WILLEMS Department of Mathematics University of Groningen P.O. Box 800, 9700 AV Groningen, The Netherlands URL: http://wvjw.math.rug.nI/~ willems Dedicated to Velimir Jurdjevic on his 60"1 birthday. 1

Introduction

The purpose of this paper is to show that modern control theory, both in the form of the "classical" ideas developed in the 1950s and 1960s, and in that of later, more recent methods such as the "nonsmooth," "very nonsmooth" and "differential-geometric" approaches, provides the best and mathematically most natural setting to do justice to Johann Bernoulli's famous 1696 "brachistochrone problem." (For the classical theory, especially the smooth version of the Pontryagin Maximum Principle, see, e.g., Pontryagin et al. 31 , Lee and Markus 2 6 , Berkovitz 3 ; for the nonsmooth approach, and the version of the Maximum Principle for locally Lipschitz vector fields, cf., e.g., Clarke 12 '', Clarke et al. 14 ; for very nonsmooth versions of the Maximum Principle, see Sussmann 35-36,39,4o,4i,42,43,44. for t n e differential-geometric approach, cf., e.g., Jurdjevic 24 , Isidori 19 , Nijmeijer and van der Schaft 3 0 , Jakubczyk and Respondek 2 0 , Sussmann 37 - 40 .) We will make our case in favor of modern control theory in two main ways. First, we will look at four approaches to the brachistochrone problem, presenting them in chronological order, and comparing them. Second, we will look at several "variations on the theme of the brachistochrone," that is, at several problems closely related to the one of Johann Bernoulli. The first of our two lines of inquiry will be pursued in sections 3, 4, 5, and 6, devoted, respectively, to

114

1. Johann Bernoulli's own solution based on an analogy with geometrical optics, 2. the solution based on the classical calculus of variations, 3. the optimal control method, and, finally, 4. the differential-geometric approach. We will show that each of the three transitions from one of the methods in the above list to the next one leads to real progress for the brachistochrone problem, by making it possible to derive stronger conclusions about it than those that were obtainable by means of previously existing techniques. Specifically, we will point out that l.a the differential equation derived by Johann Bernoulli has spurious solutions (i.e., solutions other than the cycloids), as was first noticed by Taylor in 1715, but l.b the application of the Euler-Lagrange condition of the calculus of variations eliminates these solutions and leaves only the cycloids, and that 2.a with the calculus of variations approach, the existence of solutions is a delicate problem, due to the non-coercivity of the Lagrangian, but 2.b the optimal control control method renders this question trivial, reducing the proof of existence to a straightforward application of the Ascoli-Arzela theorem, and, moreover, 3.a the usual calculus of variations formulations require that one postulate that the optimal curve is the graph of a function x >-> y(x), but 3.b with the optimal control control method this postulate becomes a provable conclusion.

115

These observations show that the first two of the three transitions brought improvements to the analysis of Johann Bernoulli's problem. But the core of our argument is 6, where we look at the third transition and the advent of differential-geometric control theory. We will argue that, on the one hand 4.a previous methods require that we assume extra knowledge of physics (in the form of the law of conservation of energy) in order even to be able to translate Johann's problem into mathematics, and 5.a with those methods an anomalous situation appears to arise when one looks for a state space formulation of the problem and asks for its "Hamilton principal function" V (that is, the value function regarded as a function of both the initial and terminal states), since V ought to be a function of an even number v of variables, but for the brachistochrone problem the best choice of v seems to be five; but on the other hand 4.b in the differential-geometric framework one does not need to bring in conservation of energy in order to formulate the problem; instead, one writes the equations as a control system in ffi4, and then computes the accessibility distribution, which turns out to be three-dimensional at each point; this shows that a nontrivial conserved quantity exists locally, and direct computation shows that it exists globally and is equal to the energy, and 5.b the anomaly described in 5.a disappears, and the principal function V turns out to be defined on E 4 x K4 (although V(q, q') < +00 if and only if q and q' belong to the same leaf of the foliation of R4 by the level submanifolds of the energy function); the method provides a full explanation for the apparent anomaly, by showing that the full optimal synthesis requires impulse controls that lead to instantaneous jumps. The discussion in 6 will also show that the differential-geometric approach brings into the problem interesting connections with other mathematical ideas. To begin with, the analysis leads naturally to a minimum time optimal control problem with a four-dimensional state space and an unbounded scalar control entering linearly into the dynamics. Moreover, this problem is quite degenerate, because its accessibility distribution happens to be three-dimensional, as

116

pointed out above. Since the problem is affine in the control but not linear, complete controllability on each leaf is in principle problematic. The accessibility Lie algebra turns out to be the "diamond Lie algebra," a well known four-dimensional solvable Lie algebra that occurs, for example, in the study of the quantized harmonic oscillator. To establish controllability on each leaf one needs additional arguments, and it turns out that the issue can be settled by computing the Jurdjevic-Kupka Lie saturate (cf. Bonnard et al. 8 ' 9 , Jurdjevic-Kupka 21>22; Jurdjevic 24 ) of the family of system vector fields. The minimum time problem then makes sense in principle for every pair of points belonging to the same leaf. But, since the control is unbounded, it is not at all obvious that an optimal control joining A to B exists for every pair (^4, B) of points belonging to a given leaf L. An application of the Maximum Principle yields a vector field Z tangent to the leaves such that all optimal arcs are integral curves of Z. Since, in general, two arbitrarily chosen points A, B of a leaf L do not lie on the same integral curve of Z, it follows that for general pairs (A, B) € L x L an optimal control joining A to B does not exist. On the other hand, when a control system is completely controllable and a suitable properness condition is satisfied a , then most mathematicians would agree that optimal controls ought to exist, and if they do not then this can only happen because we have not made a good choice of our space U of admissible controls, so U has to be enlarged by adding suitably defined "generalized controls." (For a trivial example, consider the problem of moving from A to B in minimum time with the dynamics x = u, u arbitrary, so in particular no bound is prescribed for the velocity u. Clearly, the "minimum time," is 0, and to attain it one needs to allow instantaneous jumps, which means that the class of ordinary controls has to be enlarged to allow for Delta functions. We will show in 6 that something like this happens for the brachistochrone.) In our case, the true complete optimal synthesis of the problem turns out to require impulse controls, a fact known to be common in problems with "cheap" unbounded controls, and studied in other (linear quadratic) contexts by Jurdjevic and Kupka, cf. Jurdjevic-Kupka 23 , Kupka 2 5 . Our second group of arguments in favor of modern control is presented in 7, and consists of "five modern variations on the theme of the brachistochrone." Following a suggestion made by M. Hestenes in 1956, we "speculate on what type of brachistochrone problem the Bernoulli brothers might have formulated had they lived in modern times" (Hestenes 18 , p. 64), and we "See Definition 6.3 for a precise statement of the properness condition, and Proposition 6.4 for the proof that the condition is satisfied for the brachistochrone problem.

117

extend Hestenes' idea by speculating not only on "what type of . . . problem the . . . brothers might have formulated" but also on what type of answer they might have liked for such problems. We deal with five questions. The first one is Hestenes' own "variation of the brachistochrone problem," which turns out to be a problem solvable by optimal control but not by classical calculus of variations techniques. The second "variation" is the "reflected brachistochrone," a very natural extension of Johann's problem, in which the state space is the whole plane rather than a half plane. This problem is no longer solvable by means of traditional smooth or nonsmooth optimal control methods—because it is a minimum time optimal control problem with a non-Lipschitzian right-hand side—but turns out to lend itself to the "very nonsmooth" approach, which yields a solution with a simple geometric interpretation that, in our view, the Bernoulli brothers would have liked. The third "variation" is the brachistochrone problem with friction, which corresponds, mathematically, to "unfolding" the original Johann Bernoulli question by embedding it in a one-parameter family of problems, the parameter being, of course, the friction coefficient p. These problems turns out to be nondegenerate, in the sense that, for example, their accessibility distribution has rank four at each point. Consequently, their time-optimal controls behave qualitatively like the time-optimal controls for generic control-affine systems in R4 with scalar unbounded controls. In particular, through every point there pass a one-parameter family of optimal curves, since the optimal control is given in feedback form as a function of the state and of an additional scalar parameter that stays constant along each trajectory. The analysis clarifies the way in which the original problem of Johann Bernoulli is degenerate: the accessibility Lie algebra is eight-dimensional for p ^ 0 but four-dimensional for p = 0; the accessibility distribution has rank four everywhere if p ^ 0 but has rank three everywhere if p = 0; the optimal "feedback" control is a function of the state and an extra parameter, but this parameter only occurs in a term that is multiplied by p, so that when p = 0 the optimal control is a function of the state only. The fourth "variation" deals with the derivation of Snell's law of refraction from general necessary conditions for an optimum. Since Snell's law plays a crucial role in Johann's solution of the brachistochrone problem, we speculate that a natural question for him to have asked "if he had lived in modern times" is whether the general theories about minimizing curves that were created in the three centuries after Johann's work actually cover the Snell law case. Our answer is negative if only the classical calculus of variations and pre-1990 smooth and nonsmooth optimal control are considered, but becomes positive

118

if we add to the list our recent very nonsmooth methods To conclude the paper, we take the liberty of including a fifth "variation" that we find amusing but does not really have much to do with control theory. In this "variation"—which is a digression on a topic of great importance and related to geometry, to the brachistochrone, and to Johann Bernoulli—we "speculate" that Johann Bernoulli would have found it very interesting to know b that with his work on the brachistochrone he had been extremely close to solving one of the most important open problems in Geometry, namely, that of the independence of Euclid's fifth postulate from the other ones. (The problem had intrigued mathematicians since ancient times—cf., e.g., Bonola 10 —and was very much in the forefront of research in geometry in Johann's times, as shown by the publication of Saccheri's book 32 in 1733. A definitive solution was found by Beltrami in 1868, cf. Beltrami 2 and Stillwell 33 .) Remark 1.1. An earlier, related version of the argument presented here appeared in 3 8 . That paper also dealt with the evolution from the brachistochrone to the calculus of variations and optimal control, but focused on a different set of issues, especially on the search for the correct Hamiltonian formulation of the necessary condition for an optimum and how opportunities were missed before the "control Hamiltonian" was finally discovered in the 1950s. Q 2

Johann Bernoulli and the brachistochrone problem

In the June 1696 issue of the journal Acta Eruditorum, Johann Bernoulli—a Swiss-born professor of mathematics at the University of Groningen, in the Netherlands0—challenged mathematicians to solve the "problema novum" of finding the curve that—following Leibniz' suggestion—he named the "brachistochrone," from the Greek /3paxicroq: shortest, and xP^vcx;: time d . The journal's May 1697 issue contains articles on this problem by six of the most renowned mathematicians of the time: Johann Bernoulli, Johann's elder brother Jakob, Leibniz, Newton, the marquis de l'Hopital, and Tschirnhaus. Leibniz called Johann Bernoulli's question a "splendid" problem. The fact that he and five other thinkers of such unparalleled distinction chose to * Although he probably would not have thought of asking the question. c T h e Bernoulli family was originally from Antwerp in Flanders, where they lived until 1583. Like many other Flemish Protestants, they fled Flanders to escape religious oppression by the Spanish rulers. They spent some time in Frankfurt, and finally settled in Basel, Switzerland, early in the seventeenth century. d So the brachistochrone is the solution of a minimum time problem, i.e., of a genuine optimal control problem in the modern sense of the term!

119 become actively engaged in the search for its solution shows that at the time of its formulation the problem was perceived by the leaders of the mathematical community as a major challenge, lying at the very frontiers of their research. It would not have been easy at that point to predict the future impact on science of Johann Bernoulli's challenge. Yet, the six famous authors of the 1697 papers must have had their own reasons to sense that an exceptionally promising line of inquiry was being opened, and the developments of the 300 years that followed have lent ample support to their intuition. Today, it is clear that the events of 1696-97 represented a critical turning point in the history of mathematics, and it is widely agreed that with them an important new field, later to be called the "calculus of variations," was born. R e m a r k 2 . 1 . The words "calculus of variations" were first used by Euler in a 1760 paper, cf. 1 5 . Euler had received, in 1755, a letter from a 19-year-old mathematician from Turin, called Ludovico de la Grange Tournier, where a new method was proposed for the study of variational problems, by using what we would now call "variations" (cf. Goldstine ie, p. 110.). This was quite different from the approach Euler had been using until then, and presented in his 1744 book entitled Methodus inveniendi maximi minimeve proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti, that is "A method for discovering curved lines having a maximum or minimum property or the solution of the isoperimetric problem taken in its widest sense." Euler's earlier method was based on a time-discretization followed by a passage to the limit?, as the size h of the time intervals goes to 0. Euler was so enthusiastic about the new idea of the unknown youngster from Turin that he dropped his own method, adopted that of Lagrange instead, and renamed the subject the calculus of variations. In the summary to his first paper using variations, Euler says "Even though the author of this [Euler] had meditated a long time and had revealed to friends his desire yet the glory of first discovery was reserved to the very penetrating geometer of Turin LA GRANGE, who having used analysis alone, has clearly attained the same solution which the author had deduced by geometrical considerations." £> T h e newborn subject was destined t o become a major area of m a t h e matics. Moreover, its implications would soon extend t o physics, whose very foundations would b e shown in t h e 18th century—by Euler, Maupertuis, Lagrange, and others—to depend on principles formulated in t h e language of the calculus of variations. So it is not surprising t h a t t h e story of t h e brachistochrone is told in e

Euler himself talked about "letting h be infinitesimal."

120

many books, and that the importance of this curve is stressed in most histories of mathematics. The brachistochrone has received honors not commonly bestowed on mathematical curves. For example, it is depicted in a stainedglass window of the Academy building (the main venue of the university) in Groningen. Moreover, the brachistochrone is probably the only mathematical curve that has been deemed worthy of a monument in its honor. In 1994, on the occasion of the 375th anniversary of its foundation, the University of Groningen decided to honor in various ways the most famous former members of its faculty. In the case of Johann Bernoulli, who had been a professor from 1695 to 1705, a decision was made to honor him by erecting a monument to the brachistochrone, one of the most significant discoveries—and one of which he was particularly proud—that he made during his Groningen period. The unveiling of the brachistochrone monument, located in the Zernike complex of the University, took place in 1996, coinciding with the 300th anniversary of the publication of Johann's challenge in Acta Eruditorum. 2.1

Johann Bernoulli's challenge

The June 1696 challenge took the form of an "Invitation to all Mathematicians to solve a new problem": If in a vertical plane two points A and B are given, then it is required to specify the orbit AMB of the movable point M, along which it, starting from A, and under the influence of its own weight, arrives at B in the shortest possible time. So that those who are keen of such matters wiJJ be tempted to solve this problem, is it good to know that it is not, as it may seem, purely speculative and without practical use. Rather it even appears, and this may be hard to believe, that it is very useful also for other branches of science than mechanics. In order to avoid a hasty conclusion, it should be remarked that the straight line is certainly the line of shortest distance between A and B, but it is not the one which is traveled in the shortest time. However, the curve AMB—which I shall disclose if by the end of this year nobody else has found it—is very well known among geometers. Later, Johann Bernoulli changed his original plan to disclose the solution by the end of 1696 and, following a suggestion by Leibniz, extended the deadline until Easter 1697.

121

Responses came from the best mathematical minds of the time. In addition to Johann's own solution, there was one by Leibniz, who had communicated it to Johann in a letter dated June 16, 1696; another one by Jakob; one by Tschirnhaus; one by l'Hopital and, finally, one by Newton. Newton's solution was presented to the Royal Society on February 24, 1697 and published, anonymously and without proof, in the Philosophical Transactions. (The identity of the anonymous author was clear to Johann Bernoulli, since he thought that you can know ex ungue leonem, i.e., that you can know "the lion from its claws.") The May 1697 issue of Acta Eruditorum contains articles with the solutions by Johann 5 and Jakob 6 , as well as contributions by Tschirnhaus and l'Hopital, a short note by Leibniz, remarking that he would not reproduce his own solution, since it was similar to that of the Bernoullis, and Newton's anonymous paper, reprinted from the Philosophical Transactions. 2.2

The cycloid's double role as the brachistochrone and the tautochrone

As is apparent from Johann Bernoulli's announcement, he was under the impression that the problem was new. However, Leibniz knew better: Galileo in his book on the Two New Sciences in 1638 formulated the brachistochrone problem and even suggested that the solution was a circle. (Galileo had showed—correctly—that an arc of a circle always did better than a straight line—except, of course, when B lies on the same vertical line as A—but he then wrongly concluded from this that the circle is optimal.) Johann Bernoulli considered the fact that Galileo had been wrong on two counts, in thinking that the catenary was a parabola, and that the brachistochrone was a circle, as definitive evidence of the superiority of differential calculus (or the Nova Methodus, as he called this new set of techniques, following Leibniz). He was thrilled by his discovery that the brachistochrone was a cycloid. This curve had been introduced by Galileo who gave it its name: related to the circle. Huygens had discovered a remarkable fact about the cycloid: it is the only curve with the property that, when a body oscillates by falling under its own weight while being guided by this curve, then the period of the oscillations is independent of the initial point where the body is releasedJ Therefore Huygens called this curve, the cycloid, the tautochrone (from TOLVTCX;: equal, and xpovcx;: time). Johann Bernoulli was amazed and somewhat puzzled, it seems, by the fact that the same curve had these two remarkable properties related to the 'Contrary to what Galileo thought, the circle has this property only approximately: the period of oscillation of a pendulum is a function of its amplitude.

122

time traveled on it by a body falling under its own weight: Before I end I must voice once more the admiration that I feel for the unexpected identity of Huygens' tautochrone and my brachistochrone. I consider it especially remarkable that that this coincidence can take place only under the hypothesis of Galilei [HJS
2.3

Why was the brachistochrone so important in 1696-97?

Perhaps the crucial fact about Johann Bernoulli's question is that he is telling us very clearly that the solution is a curve that everybody knows, and yet there appears to be no way to guess which curve it is by means of some intuitive geometric argument. This situation should be compared with that arising in another famous and much older question about curve minimization, the isoperimetric problem. In this problem, we are asked to find the simple closed rectifiable curve that encloses the largest possible area among all simple closed rectifiable curves of a given length, and the answer also turns out to be a curve that is certainly "very well known among geometers," namely, a circle. But in this case it is fairly easy to guess directly that the answer is a circle, even though it is a delicate matter to give a rigorous proof. For example, it suffices to observe that the question is rotationally symmetric, so the answer ought to be a rotationally symmetric curve, and the only such curve is the circle. Although this argument does not amount to a completely rigorous proof9, it is undoubtedly rather convincing, and makes it quite obvious that the solution must be a circle. Therefore, if Johann Bernoulli had challenged "the best mathematical minds" of his time to solve the isoperimetric problem, it would not have made s

T h e argument assumes the existence of a solution, as well as its uniqueness up to translations. Indeed, it might conceivably happen that the solution is not unique, and in that case the only conclusion one can draw from rotational symmetry is that every rotation will map a solution 5 to a solution S', which may be different from S.

123

sense for him to say, as he did in his brachistochrone challenge, that "the curve—which I shall disclose if by the end of this year nobody else has found it—is very well known among geometers," since everybo,y would have guessed immediately that the mysterious curve was the circle. In the case of the brachistochrone, there is no way to guess h. The problem was posed and solved at a time when the Calculus was being invented, and cannot be solved without using Calculus. So it provides an impressive illustration of the power of the Calculus to solve problems that cannot be solved in any other way, and to discover truths about the world that can be stated and understood in "well known" terms ', but can only be arrived at by using this new mathematical tool. 3

The standard formulation and Johann Bernoulli's solution

We now discuss the "standard formulation" of the brachistochrone problem in modern mathematical language, as it is usually found in modern books, and Johann Bernoulli's ingenious solution based on an analogy with geometrical optics. In our view, the standard formulation is questionable. We will explain our objections in 6 below, where we will make a case for a different approach, based on differential-geometric control. But first it is necessary to present the classical point of view in detail. The key fact is that the "movable point M" is supposed to move "under the influence of its own weight," which presumably means "under the influence of its own weight and nothing else other than whatever force is needed to keep M on the given curve." This means, in particular, that energy is conserved. 3.1

Formulation of the brachistochrone problem

We choose x and y axes in the plane with the y axis pointing downwards, use (a, a) and (b, j3) to denote, respectively the coordinates of the endpoints A fe Even now, 300 years later, we are not aware of any simple geometric argument that might lead to the cycloid without using Calculus. ' T h i s is an important point. Compare the brachistochrone, for example, with Newton's 1685 work on the shape of a body with minimal drag. In this case, the solution also requires the Calculus, but once we have solved the problem all we find is a curve defined by some strange formula. This curve was not "well known to geometers," and would not have had any particular significance to mathematicians, who would only retain from going through Newton's solution the fact that the solution is "some curve." In the case of the brachistochrone, one does not need the Calculus to explain what the solution is, since cycloids were already well understood curves, but one does need it to discover that the solution is a cycloid.

124

and B of our curve, and fix a number B e l , thought of as the total energy of the particle. The condition characterizing the feasible paths [0,T] B t H- ^(t) = (*(*)»!/(*)) e K2

(3.3.1)

will then be—using m, g to denote, respectively, the particle's mass and the gravitational acceleration—that the sum of the kinetic energy ^{xty)2 +y(t)2) and the potential energy —gmy(t) is equal to E. We can certainly choose units so that m = 1. The condition then becomes l

-(i(t)2

+ y{t)2)-gy{t)

= E.

So we formally define an {A, B, E)-feasible trajectory (or (A, B, E)-feasible path) to be a map (3.3.1), defined on some interval [0, T], such that (i) t(0) = A, £(T) = B, (ii) £ is absolutely continuous, (iii) \{x{t)2 + y(t)2) = E + gy{t) for almost all t G [0,T]. Notice that Condition (iii) implies in particular that the derivatives x and y are bounded, so a feasible path is in fact Lipschitz continuous. Condition (i) states that the path £ must start at A and end at B. The number E + ga is the initial kinetic energy of the body. (Often, only the case E + ga = 0 is considered, corresponding to the body being dropped at time 0 with zero initial speed.) Condition (ii) is of course a modern technicality that would have made no sense to Johann Bernoulli, who was far from imagining that there could exist curves that cannot be recovered by integrating their derivative. Condition (iii) reflects conservation of energy. (The law that the kinetic energy of a body which has fallen from a height h is increased by an amount proportional to h was due to Galileo, and was well known in Bernoulli's time.) An (A, B, .E)-feasible path £ : [0, T] -> E 2 is said to be optimal if there exists no (A, B, £)-feasible path f: [0, f ] -> E 2 for which f < T. An (A, B, E)brachistochrone is either (a) an optimal (A, B, E)-feasib\e path, or (b) the geometric locus in ffi2 of such a path, that is, a subset B of E 2 such that B = {(x,y) G E 2 : there exists t G [0,T*], such that (x,y) = £*(*)} , for some optimal (A, B, .E)-ieasible path £* : [0, T*] i-> E 2 . A brachistochrone is an object which is an (A, i?,.E)-brachistochrone for some A,B,E.

125

Obviously, a feasible path" (3.3.1) must be entirely contained in the closed half-plane H

+(v) = {(x,y) : y > -n] ,

where

n= —,

(3.3.2)

since, for y(t) to be < — n, the kinetic energy would have to be negative, which is of course impossible. It is clear that the solution cannot always be a straight line, as Johann Bernoulli rightly points out in his "Invitation." For example, consider the extreme case when E = 0 and a = ct = (3 = Qj£b. (That is, when the endpoints of the curve are at the same height but do not coincide, and the initial kinetic energy vanishes.) We will show that in this situation there is a feasible path that goes from A to B in finite time. Since the straight-line segment S from A to B is horizontal, (iii) implies that the speed of motion along S vanishes, and then it will follow that S cannot be an optimal path, because the motion along S takes infinite time. To construct a feasible path from A to B we follow a circle C with center on the x axis. The circle can be parametrized by x, taking values in the interval [0, b]. The speed v of motion along C will be proportional to ^/y, and hence bounded away from 0 as long as x is bounded away from 0 and b\. Near A, we can also use y as a parameter, and then y \, 0 as x ], 0, and v behaves like -^ and also like y/y, so ^ ~ -j=, showing that t stays finite as x and y approach 0, since the function y H-> -4= is integrable near y = 0. A similar argument shows that t stays finite near B. So C joins A to B in finite time. 0 3.2

Johann Bernoulli's

solution

It turns out that the brachistochrone is a cycloid. When E = 0, it is the curve described by a point P in a circle that rolls without slipping on on the x axis, in such a way that P passes through A and then through B, without hitting the x axis in between. (It is easy to see that this defines the cycloid uniquely.) Johann Bernoulli's ingenious derivation of the brachistochrone has been the subject of numerous accounts, but since this event plays a crucial role in our own tale, we will briefly tell the story again. Johann Bernoulli based his derivation on Fermat's minimum time principle. If we imagine for a moment that instead of dealing with the motion of a moving body we are dealing with a light ray, Condition (iii) above gives us a °FVom now on, as long as A, B and E are fixed, we just talk about "feasible paths" and "brachistochrones," without explicitly specifying A, B and E.

126

formula for the "speed of light" c(x,y) as a function of the position (x,y): c(xiy)

= ^/2E + 2gy.

We now change coordinates so that E = 0. (This amounts to moving the x axis upward to a height rj.) Then r) is now equal to 0, so our feasible paths live in the half plane H+ = H+(0). Also, let us rescale—or, if the reader so prefers, "change our choice of physical units"—so that 2g = 1. Then our problem is exactly equivalent to that of determining the light rays—i.e., the minimum-time paths—in a plane medium where the speed of light c varies continuously as a function of position according to the formula c[x,y) = y/y. It is then reasonable to expect that, if we use a suitable discretization of our problem, then the optimal paths of the discretized problem will approach those of the original problem as the discretization parameter 5 goes to 0. Johann Bernoulli did this as follows. Let us define ysk=kS

for

S > 0 and it = 0 , 1 , . . . ,

and then divide the half-plane H+ into horizontal strips Sskd={(x,y)eR2:yi
for

k = 0,1,...,

of height S. Johann's discretization is then obtained by replacing c by a constant csk in each strip Sk. The constants c\ are defined by

4 = yvi+i

for

each

k

•

He then computed the light rays for the original problem by taking the limit of the light rays for the discretized problem as 6 j . 0. The light rays of the discretized problem can be studied using Snell's law. Clearly, the paths will be straight-line segments within each individual strip, and all that needs to be done is to determine how these rays bend as they cross the boundary between two strips. The answer is provided by the laws of optics as developed by Snell, Descartes, Fermat, Leibniz and Huygens. Snell had observed that, if the speeds of light on the two sides Hi, Hi of a line L in the plane are different constants «i, i>2, and a light ray R traverses Hi and then enters H2 after being refracted at L, then the ratio fj^V" °^ ^ e s m e s of the incidence and refraction angles is a constant, independent of 61. (By definition, 9i is, for i = 1,2, the angle formed by the part Ri of R that lies in Hi with the line perpendicular to L. It is clear that the Ri are straight lines.) Fermat subsequently showed that this is precisely what happens when light

127

is assumed to follow a minimum-time path. The law relating the incidence angles to the velocities of propagation is due to Leibniz and Huygens, and implies the law of Snell. It says that sin 0i vi ——— = — sin0 2 v2

. or, equivalently,

sin 0i sin0 2 = , vi v2

,,,,>

v(3.3.3)

Johann Bernoulli used (3.3.3) to study the light rays of the discretized problem. If R is such a ray, and we let Rk be the part of R that lies in the strip S|, and use 8sk to denote the angle of Rk with a vertical line, then (3.3.3) enabled him to conclude that the quantity

sin0f

t1/in is a constant, since in each strip S% the speed of our light ray is t/yjfe+1. Passing to the limit as 5 \. 0, we conclude that the sine of the angle 0 between the tangent to the brachistochrone and the vertical axis must be proportional to ^/y. Since dx

sin 0 = sjdx

1

+ dy2

,

we find that dx2 dx + dy2 ~ 2

Vy

'

where v is a constant. Then dx2 4- dy2 _ 1 dx2 vy I.

e., 1 + y'(x)2

C 1 - — , where C = - .

So the curve described by expressing the y-coordinate of the brachistochrone as a function of its x-coordinate will satisfy the differential equation

y'(x) = J^f^, V y(x) for some constant C.

(3.3.4)

128

The parametrized curves tp i-> (x((p),y(
C y(ip) = —(1-cos


0<
satisfy (3.3.4). It is easily seen that these equations specify the cycloid generated by a point P on a circle of diameter C that rolls without slipping on the horizontal axis, in such a way that P is at {XQ, 0) when ip = 0. Moreover, it is also easy to check that (*) given two points A and B in H+ there is exactly one curve passing through A and B and belonging to the family of curves (3.3.5). (The parameter count for (*) is obviously right: (3.3.5) describes a 2parameter family of curves; requiring that the curve pass through A fixes one of the parameters, and then asking that it pass through B fixes the other. Proving (*) requires a bit more work, but is still quite easy.) 4 4-1

Spurious solutions and the calculus of variations approach Remarks on Johann Bernoulli's

argument

The argument that we have presented is the one of Johann Bernoulli, and Equation (3.3.4) is the one that he wrote in his paper, followed by the statement "from which I conclude that the Brachistochrone is the ordinary Cycloid." (He actually wrote dy = dx./-^^, but he was using x for the vertical coordinate and y for the horizontal one. Cf. 34 , p. 394.) In present day notation, it is customary to adopt the convention that the symbol y/r stands for the nonnegative square root of r, but it is obvious that Johann Bernoulli did not have this in mind. What he meant was, clearly, what we would write today as

or, equivalently, y(x)(l + y'(x)2) = constant.

(4.4.2)

In particular, the solution curves should be allowed to have a negative slope, contrary to what a "modern" interpretation of (3.3.4) might suggest. But y' should stay continuous, so that a switching from a + to a — solution of (4.4.1) is not permitted.

129 Even with the more accurate rewriting (4.4.2), the differential equation derived by Johann Bernoulli also has spurious solutions, not given by (3.3.5). Indeed, for any y > 0, the constant function y(x) = y is a solution, corresponding to C = y. More generally, one can take an ordinary cycloid given by (3.3.5), follow it up to y>=7r—so that dy/dx=Q—then follow the constant solution y(x)=C for an arbitrary time T, and then continue with a cycloid given by (3.3.5). Such paths are, indeed, compatible with Huygens' law of refraction. Remark 4.1. The existence of the spurious solutions was pointed out by Brook Taylor (1685-1731) in 1715. (Taylor is known to mathematicians as the discoverer of "Taylor's expansion," the inventor of integration by parts, and the creator of the calculus of finite differences. Like many other mathematicians of his time, he was involved in a priority dispute with Johann Bernoulli, involving Taylor's 1708 solution of the problem of the center of oscillation, that went unpublished until 1714.) <£ 4-2

The calculus of variations approach

The spurious solutions, and all the other problems, such as the apparent arbitrariness of the requirement that y' be continuous, can be eliminated in a number of ways. For example, one can prove directly that the spurious trajectories are not optimal. Or one can use, as an alternative to Johann Bernoulli's method, the calculus of variations approach, based on writing the Euler-Lagrange equation. This equation gives a necessary condition for a function y* to minimize the integral

J= /

L(y(x),y'(x),x)dx

Ja

within the class y of all functions [a,b] B xt-^ y(x) e R that satisfy some appropriate technical conditions (for example, the requirement that y(-) is Lipschitz continuous) and are such that y(a) and y(b) have given values a, j3. Here, the "Lagrangian" 0 x R x [a, b] 3 {y,u,x) >->• L(y,u,x)

GR

is a given function, and Q is an open interval in R. Under suitable technical

130

conditions on L, yt, and the set y, if j/* is a minimizer then the condition

dx \ du X

V* (x) ,y'.(x),z))

=-Q- (v* (x), y', (x), x)

(4.4.3)

must be satisfied for almost all x. It is easy to see that the above result can be applied to the brachistochrone problem, provided that we postulateb that it suffices to consider curves in the x, y plane that are graphs of functions x t-> y(x) denned on the interval [a, b]. Then Constraint (iii) can be written (taking E = 0 and 2g = 1 as before) as dx2 +dy2 = y dt2 , which gives dt =

v/cte2 + dy2 — = L(y, y )dx.

where

L(y,u) = ^ ± Z .

(4.4.4)

Vy So Johann Bernoulli's problem becomes that of minimizing the integral J = /

L{y(x),y'{x))dx,

J a

subject to y(a) = a and y(b) = /?, where L is given by (4.4.4). This gives the equation l + y'(x)2+2y(x)y"(x)

= 0,

(4.4.5)

which is stronger than (4.4.2), since (4.4.2) is equivalent to y' + y' +2yy'y" = 0, i.e., to j/'(l + y' + 2yy") = 0, whose solutions are those of (4.4.5) plus the spurious solutions found earlier. It is easy to verify that the solutions of the Euler-Lagrange equation (4.4.5) are exactly the curves given by (3.3.5), without any extra spurious solutions, showing that, for the brachistochrone problem, the Euler-Lagrange method gives better results than Johann Bernoulli's approach. We now show that optimal control is even better. ''With optimal control, this "postulate" becomes a provable conclusion, as will be shown in 5 below.

131

5

The optimal control approach

Having shown that the Euler-Lagrange equation gives better results than Johann Bernoulli's method, we now look at another important step in the evolution of the theory of minimizing curves, namely, optimal control theory. We show that optimal control provides even better results in two ways: 1. With optimal control it is no longer necessary to postulate that the optimal curve is the graph of a function x i-» y(x). 2. In the optimal control setting, the problem of the existence of an optimum becomes trivial. 5.1

The brachistochrone as an optimal control problem

We can formulate Johann Bernoulli's question as an optimal control problem, in which the motion takes place in the x, y plane, and the dynamical behavior is given by x = u^/\y\,

y = vy/\y~\.

(5.5.1)

Here the control is a 2-dimensional vector (u, v) taking values in the set U = {(u,v):u2+v2

= 1}.

(5.5.2)

(Actually, for the "true" brachistochrone problem the motion is restricted to the upper half plane, so we could have written y/y rather than y/\y\. We choose to use the more general expression y\y\ because later, in the study of the reflected brachistochrone, we will want to work in the whole plane.) Let us use p, q, p0 to denote, respectively, the momentum variables conjugate to x and y, and the abnormal multiplier. Then the control theory Hamiltonian H(x,y,u,v,p,q,po,t) is given (using a = sgny) by the formula H(x,y,u,v,p,q,p0,t)

= (pu + qv)y/ay - p 0 .

The Pontryagin Maximum Principle then tells us that if a curve [0,T] 9 i .->£(*) = (*(*),»(*)) is optimal then there exist absolutely continuous functions t »->• p(t) and 11-> q{t), and a nonnegative constant po, such that, if we let u = x, v — y, and write p to denote the momentum vector (p, q), and \\p\\ to denote its Euclidean norm y/p2 + q2, then the Hamiltonian maximization conditions

• ( 0 -iwoii' "<,, = »

(5 5 3)

'-

132

as well as the "adjoint system" of differential equations

«„=„,

W-_*'*M±%M«)__-4ai 2

2 vow)

,

(5.5.4)

v<w)

are satisfied for almost all t. (Notice that \\plt)\\ ^ 0. Indeed, the Maximum Principle also gives the condition that H = 0. So ||p|| = 0 would imply p0 = 0, contradicting the nontriviality of the triple (p(t),q(t),p0).) If the constant p vanishes, then x = 0, so we get a vertical line. Otherwise, x is always ^ 0, showing that we can use x to parametrize our solution. Since y>{x)

=

!k ax

=

i = l = lt x u p

(5.5.5)

we have l+y'(x)2

= ^ -

(5.5.6)

and y

„{x)

=

l.*L p dx

i.. px

=

(s.5.7)

But (5.5.1) and (5.5.3) imply that a; =

*

Ibll

,

(5.5.8)

and then (5.5.4) and (5.5.7) yield »"(*) = -

2 ML & • 2

<5-5-9)

2t/P So 2yy" = - ^

= - ( 1 + y'2), and then l + y'2+2yy"

= 0,

(5.5.10)

which is exactly Equation (4.4.5). As before, this leads to the cycloids, with no "spurious solutions." Notice that this argument does not involve any discretization or any use of Snell's law of refraction. Notice also that in our control argument we have not postulated that the solution curves could be represented as graphs of functions y(x). We have proved it! This should be contrasted with the analysis based on the calculus of variations, where this fact had to be postulated.) This is one example showing that, for the brachistochrone problem, the optimal control method gives better results than the classical calculus of variations.

133

5.2

Optimality proofs and the existence problem

For a second example showing the advantages of the optimal control method over the classical calculus of variations for the specific case of the brachistochrone problem, we turn to the question of the rigorous proof of the optimality of Bernoulli's cycloids. Clearly, no argument based solely on necessary conditions for optimality will ever prove that a trajectory is optimal, because it could happen, for example, that there are no optimal trajectories, in which case the statement that "every optimal trajectory is a cycloid" would be vacuously true. If we really want to prove the optimality of Bernoulli's cycloids, an extra step is needed. To give a rigorous proof of the optimality of the cycloids, valid for all choices of A and B in the closed half-plane H+ = {x,y) : y > 0}, one can proceed in two ways. One possibility (cf. Bliss 7 ) is to use Hamilton-Jacobi theory. Another, much simpler, approach, is to prove first the existence of an optimal trajectory. Indeed, once this existence result is known for all A, B 6 H+, then it is clear that any optimal trajectory £» : [0, T] t-> H+ going from A to B must be such that its restriction £, \[a, /?] to every subinterval [a, /?] of [0, T] is also optimal for the problem P(a, /?) with endpoints £»(«), £*(/?). (This is a rather obvious point, known in optimal control theory as the "principle of optimality," and noticed in the case of the brachistochrone by Jakob Bernoulli.) If one assumes that £» (a) and £* (/?) are interior points of H+ for 0 < a < /3 < T, then the necessary conditions for optimality give a unique candidate for the solution of P(a,/3), namely, the unique curve in the family given by (3.3.5) that goes through £*(a) and £»(/?). So £* is such that ^»[[a,/?] is given by (3.3.5) whenever 0 < a < /? < T, and this easily implies that £* itself is a cycloid given by (3.3.5). To complete the argument, one has to exclude the possibility of a solution of the problem with endpoints A, B which intersects the x axis at some other point. This can be done by a simple qualitative argument, using the following lemma. L e m m a 5.1. Suppose L is a horizontal line contained in the interior of the upper half-plane H+. Let P, Q be distinct points of L. Then the straight-line segment joining P to Q is strictly faster than any trajectory C, from P to Q such that £ lies entirely in the lower closed half-plane determined by L and at least one point of C, lies strictly below L. Proof. Let T^, T^ be the corresponding times. Let y be the y coordinate of the points of L, so y > 0. Write P = (p,y), Q = (q,y). Let £ : [0,T] >-> H+

134 be the horizontal segment from P to Q. Then Te = Now let [a,b] 3 t i-> ((t) = (x(t),y(t)) in our statement. Then

\q-p\

Vy ' G H+ be a trajectory from P to Q as

m< warn = Vv(d
< ( 6 - o ) v ^ = T c V».

JO

Therefore T, > i| g^ ~- ^Pil = r € , and our proof is complete.

0

d

The following is an immediate consequence of the lemma. Corollary 5.2. Lei A, B be points of H+ such that A^ B, and assume that £ : [a, b] i-+ if+ is h'me optimal. Then £(t) is an interior point of H+ whenever a •

135

So everything hinges upon the existence result. It turns out that this is not at all trivial if the brachistochrone problem is reformulated in terms of the classical calculus of variations 0 , because (a) the Lagrangian L given by L(y, u) =

t^

has a singularity at y = 0,

and, even more importantly, (b L is not sufficiently coercive for the usual existence theorems to apply. (The precise meaning of (b) is that, as a function of u, L just grows like a constant times |u|, whereas for the usual existence theorems to apply L would have to grow like a constant times |u| r for some r such that r > 1. More sophisticated existence theorems for classical calculus of variations problems that include the brachistochrone are stated in Cesari n , 14.3 and 14.4, but these conditions are not at all simple, and in addition they depend on other results by Cesari that were to appear in a book that was never published.) In the control theory setting, however, the existence question is easily settled by a completely trivial application of the Ascoli-Arzela theorem: given A, B £ H+, let T be the infimum of the times of all trajectories going from A to B; let {£,-} be a sequence of trajectories from A to B, defined on intervals [0, Tj], such that Tj I T. Then the £y are all contained in a fixed compact subset of H+, as can easily be shown by applying Gronwall's inequality and using the fact that the right-hand side of (5.5.1) has less than linear growth as a function of x and y. Then (5.5.1) implies that the derivatives £, are uniformly bounded as well. So we can use the Ascoli-Arzela theorem to extract a subsequence that converges uniformly to a curve £ : [0,T] \-¥ H+. In principle, £ is just a trajectory of the "convexified" system, in which the evolution equations are those of (5.5.1) but the control constraint is u2 + v2 < 1 rather than u2 + v2 = 1. However, it is easy to see that £ is a true trajectory of the nonconvexified system because, if the inequality u(t)2 + v(t)2 < 1 was strict on a set of positive measure, then it would be possible to find an even faster trajectory from A to B by just reparametrizing £. So we see, once again, that in the control setting a question that is hard from the point of view of the calculus of variations can become easy or even trivial. The key point in the above argument is that there is at least one trajectory going from A to B in finite time, as can easily be verified. (If one of the points J4, B belongs to dH+, and A ^ B, then the argument would not work if the speed of light was y rather than ^/y, for in that case there would be no path joining A and B in finite time.) c

We thank F. H. Clarke for bringing this point to our attention.

136

6 6.1

The differential-geometric connection Is the standard formulation of the brachistochrone problem satisfactory?

The mathematical formulation of the brachistochrone problem, as presented in 3.1, is not completely natural, because it takes it for granted that we know that energy is conserved. More precisely, "energy conservation" is "assumed" in at least two conceptually different ways. First, the fact of energy conservation is a prerequisite for even stating the problem. This is clearly not intellectually satisfactory, because it would be much better to state the problem first, and then let the mathematical analysis discover energy conservation for us, after which we could use this new fact to find the solution. Second, the problem of "finding the minimum time curve from A to B" does not make sense for general A and B, unless the value of the energy E is specified. The problem usually studied in textbooks is another one, namely, that of "finding the minimum-time curve from A to B subject to an extra constraint E = E, for a given E." If this constraint is not given, then we really have, for each given pair (^4, B) of endpoints, a one-parameter family of problems, one for each value of E. (Alternatively, we may want to look at the unconstrained problem, in which we are also minimizing over all values of E. But this is clearly not what Johann Bernoulli or anyone else who studied the problem had in mind, and in addition is easy to solve: the infimum of the times is zero, and there is no minimizer.)

6.2

Hamilton's principal function

At this point, we digress to discuss an important idea of Hamilton, namely, the "principal" or "characteristic" function. This idea was not truly appreciated for more than century, until optimal control theory emerged to provide the right setting that would do it justice. So we will only describe it here in broad outline, following the 1944 article 4 5 by J. L. Synge. For the specific case of the brachistochrone, we will show below that the search for Hamilton's principal function leads directly to the discovery of the differential-geometric structure underlying the problem. Synge wrote that To Hamilton, optics and dynamics were merely two aspects of the calculus of variations. He was not interested in experiments. To him, optics was the investigation of the mathematical properties of of curves giving

137

stationary values to an integral of the type

v= v x y z

j ( > ' '%%d£)du-

(6 6 1}

--

Hamilton was, in fact, a great contributor—probably the gratest single contributor of all time—to the calculus of variations. Consider two points, A' with coordinates x',y',z' and A with coordinates x,y,z. Consider all possible curves connecting A' and A. To each curve corresponds a value of the above integral. [Note by HJS&JCW: Hamilton was working with Lagrangians that are homogeneous of degree one with respect to the velocities. In that case, the choice of the interval of definition of the curves is immaterial, since every curve can be reparametrized to be defined on [0,1] without changing the value of the cost integral.] Now compare these values. Is there one curve that gives a smaller value to the integral than all others? It would seem that there must be, but it would be a rash conclusion. Suffice it to say that "in general" there is a curve giving a smallest value to the integral. Then such a curve is a ray in optics. In the language of the calculus of variations, it is an extremal. Now here is Hamilton's great central idea. Regard the minimum (or stationary) value of the integral as a function of the six coordinates x',y',z',x,y,z of A' and A. This function is Hamilton's characteristic or principal function. There is nothing hard to understand about that. What is hard to follow is Hamilton's plan to develop all the properties of the extremals from the characteristic function. In fact, it is so hard that few mathematicians in the ensuing century have given serious consideration to this plan. With the rise of optimal control, "Hamilton's plan to develop all the properties of the extremals from the characteristic function" has become one of the best and most widely used tools in the modern analysis of optimal control problems, under the name of the "dynamic programming" approach. Synge's complaint that "few mathematicians in the ensuing century have given serious consideration to this plan" may have been true in 1944 but is certainly not true today. 6.3

What is Hamilton's principal function for the brachistochrone?

Let us look at the brachistochrone problem from the modern dynamic programming point of view. The natural questions to ask are

138

1. what is the state space for the brachistochrone problem, and 2. what is the problem's "principal function" in Hamilton's sense? More precisely, we would like to know 3. what, exactly, is the controlled dynamical system q=f(q,u),

q&Q,

uGU

(6.6.2)

that corresponds to the brachistochrone problem ? (and, in particular, what is the system's state space Q ?) 4. what is the cost functional

J?

and 5. what is the value function V, regarded as a function QxQ3

fe,

qt) •->• V(qi,

? i

) € l U {-00, +00}

of the initial and terminal points, as in Hamilton's

definition?

Remark 6.1. Naturally, Hamilton's definition and Synge's version of it have to be modified to adjust them to contemporary standards of mathematical precision. Synge's text quoted above seems to require the existence of an optimum for the principal function to be well defined. We take the point of view that V(qi,qt) is always well defined as an extended real number—i.e., a member o / I U {—oo, +oo}. Precisely, V(qi, qt) is the infimum of the costs of all the trajectories going from qt to qt. In particular, V(qi,qt) = +oo if and only if there exist no trajectories from qi to qt with cost < +oo. 0 We want to address these questions taking seriously the facts that energy conservation need not be known for it to be possible to state the problem mathematically, and that, consequently, there should not be a different brachistochrone problem for each value E of the energy, but only one brachistochrone problem. It is clear that J should be time. But what exactly should we choose for Q, U, and / ? Can we, for example, take Q = E 2 ? This would mean that one can specify arbitrary points qi and qt in R2 and define V(qi,qt) unambiguously. But we know that this is not so. Given qt and qt, the minimization problem as formulated in 3.1 only makes sense for a fixed value E of the energy, and only if both qt and qt belong to the E-dependent closed half-plane H+l — I.

139

Therefore there is a well defined value VE(qi,qt) for each value of E such that qt € H+ (jj and qt € H+ (—), but these values are in general different, and the infimum of all of them is zero. A common way to get around this difficulty is by stipulating that we are only interested in dropping the particle with zero initial kinetic energy. (One can even interpret Johann Bernoulli's statement of the problem, quoted in 2.1, as suggesting this possibility.) This, however, will not do, because it would have the following unnatural consequence: if we pick two points q, q' in an optimal arc T going from qi to qt, and use T(q,q') to denote the piece of T from q to q', then T(q,q') will not in general be an optimal arc from q to q' (because the particle that started falling with zero velocity at qt will not in general have zero velocity at q). In modern terminology, the problem with zero initial kinetic energy does not satisfy the principle of optimality.d Perhaps the state space is bigger than the plane? In other words, it might be the case that one can specify the position and something else at the starting time and a similar object at the terminal time. This will not do either. We cannot, for example, specify the initial and terminal positions and velocity vectors, or even just the positions and the lengths of the velocities. What can be specified is qi, qt, and the energy—or something yielding equivalent information, e.g. the length of the initial velocity vector—and from this there is no way to derive a state space S such that V is defined on S. The key problem is that Hamilton's principal function must be a function of an even number of parameters, since it is a function of the initial and terminal states, but the natural number of variables for the value function of the brachistochrone problem seems to be five, namely, two coordinates for the initial position, two for the terminal position, and one for the energy. 6.4

Lie brackets, integrals of motion, and the differential-geometric perspective

A much better way to pose the problem is to write down the equations of motion that truly correspond to Johann Bernoulli's problem as he stated it in June 1696. One should then let the mathematical analysis lead us to the discovery of energy conservation and the simplification resulting from it. This will lead us naturally to a four-dimensional problem whose analysis is best carried out from the differential-geometric perspective, by means of Lied

J a k o b Bernoulli's solution of the brachistochrone problem makes heavy use of the fact that a portion of a minimizer is itself a minimizer. So Jakob cannot possibly have been thinking that one should only consider the case of zero initial kinetic energy.

140

algebraic calculations. The true equations of motion for the brachistochrone are X =

v, w, V = —uw w = uv y=

(6.6.3)

Here x, y, v and w are the state variables, and u is the control. The values of u are allowed to be arbitrary real numbers. The variables x and y are the horizontal and vertical coordinates of our moving point, and v, w are the components of the velocity vector. The requirement that the point is freely falling "under the influence of its own weight" means that the force effectively acting on the point should be equal to a vector proportional to [0, — g] plus a "virtual force" that does no work, i.e., is perpendicular to the velocity. Equation (6.6.3) captures these requirements, by introducing a virtual force vector of the form [uw, — uv], where u is an arbitrary "control," taking values in E. Using "t" to denote "transpose," and writing q = [x, y, v, w]^, we get a familiar equation, namely, q = F(q) + uG(q),

(6.6.4)

where F = [v,w,0, —gtf, and G = [0,0, — w,v]^. This is a 4-dimensional system, with state space Q equal to M4, and control space U equal to E. 6.5

The diamond Lie algebra

Having expressed our equations of motion in the form (6.6.4), the first thing that one should do, from the differential-geometric perspective, is to compute the iterated Lie brackets of F and G, and determine the Lie algebra structure of our problem. Precisely, we use L to denote the "accessibility Lie algebra" of our system, that is, the Lie algebra of vector fields generated by F and G. Our first task will then be to compute L. It turns out that the four vector fields F, G, [F, G] and [F, [F, G}} are linearly independent over M. (that is, there exists no nontrivial linear combination of them with constant coefficients that vanishes identically), and in addition the relations

141

[G,[F,G]]=F,

(6.6.5)

[F, [F, [F, G}}} = [G, [F, [F, G}]\ = 0

(6.6.6)

hold. It then follows easily that L is four-dimensional. As an abstract Lie algebra, L is isomorphic to the diamond Lie algebra G4, a solvable Lie algebra that plays an important role in representation theory (cf. 4 , p. 59). This Lie algebra has a basis H,P,Q, E, with the commutation relations [H,P}= [H,Q] =

Q, -P,

[P,Q]= [HtE\=

E, [P,E} = [Q,E} = 0.

(To construct an isomorphism $ from G4 to L, define * ( P ) = [F,G], *(Q) = F, *(ff) = G,*(E)

= ~[F, F,G\] •

A concrete realization of G4 as an algebra of differential operators—acting on smooth functions on the real line—can be defined by "quantizing the harmonic oscillator," i.e., by taking P Q E H

dx ' multiplication by x, identity,

i<-PW) = i(-^ + * ! ),

so that H is the Hermite operator. 6.6

Energy conservation

We now compute the accessibility distribution A generated by L. Formally, q i-> A(q) is the map that assigns to each state q the linear space A(q) = {X(q)

:XEL}.

To find A we first observe that, even though the vector fields F, G, [F, G] and [F, [F, G]} are linearly independent over R, a simple computation shows that G = <;1F + <;2[F,G} +

<;3[F,[F,G}},

(6.6.7)

142

where the Q are smooth functions of q, given by v

Ci=—,

9

w

C2 =

9

, Cs =

v2 +w2 5-2-. V

6.6.8

From these facts one can easily show that every iterated Lie bracket of F and G is a linear combination of F, [F,G], and [F,[F,G]] with smooth coefficients. Since F(q), [F,G](q), and [F, [F,G]](q) are linearly independent at each point q E E 4 , we can conclude that A is 3-dimensional at each point. This means—using, for example, Probenius' theorem on the existence of integral manifolds—that, at least locally, there is a nontrivial integral of motion, i.e., a function with nonzero gradient which is constant along all integral curves of F and G, and then also along all solutions of (6.6.4). This integral of motion can be easily computed and turns out to be the energy E, given by E{x,y,v,w)

v2 + w2 = — - — + gy.

(6.6.9)

Notice that E is a smooth function on E 4 , with nowhere vanishing gradient. Therefore the level hypersurfaces of E—i.e., the sets SE^

{q : E(q) = E] ,

for E € E—are smooth 3-dimensional manifolds. (The set SE is given by the equation V=

2E - v2 - w2 Tg '

which shows that each SE is the graph of a smooth function (x,v,w) 6.7

Naive application of the Maximum

H-> y.)

Principle

The fact that our 4-dimensional state space E 4 is foliated by 3-dimensional "leaves" SE has profound implications for the optimization problem. Indeed, if we try to find the optimal trajectories by applying the Maximum Principle, we get nothing. This is a consequence of more general result, namely, that, ( # ) whenever a control system satisfies a nontrivial "holonomic constraint" such as E = constant, then the Maximum Principle is uninformative, because every trajectory is an extremal. (The word "nontrivial" here means that VE never vanishes.)

143

Remark 6.2. The reason for ( # ) is as follows. The necessary condition given by the Maximum Principle for a trajectory-control pair (£, n) to be optimal is that (£, n) be either a "normal extremal" or an "abnormal extremal." Moreover, being an abnormal extremal is a necessary condition for the pair (£, n) to be such that the terminal point of £ does not belong to the interior of the reachable set from the initial point. If a nontrivial constraint such as E = constant is satisfied, then the reachable set from any initial point q is contained in a submanifold of the state space of positive codimension. Hence the reachable set has empty interior. Therefore all trajectory-control pairs are abnormal extremals. ^ 6.8

Controllability

We now investigate the controllability properties of our system. First of all, it is clear that two points qi, qt cannot be joined by a trajectory of (6.6.3) unless they belong to the same leaf SE- We now show that the converse is true, that is, any two points qi, qt belonging to the same leaf of (6.6.3) can be joined by a trajectory. In other words, the restriction of (6.6.3) to each leaf is completely controllable. To prove our statement, we use the technique of "Lie saturates" introduced by Jurdjevic and Kupka. Fix E and let (6.6.3.E) denote the restriction of (6.6.3) to SE- By construction, the system (6.6.3..E) has the accessibility property. So to prove complete controllability it suffices to establish complete controllability of the Lie saturate family of vector fields C. It is clear that F, G and — G belong to C. So the desired conclusion will follow if we show that — F G C. But this follows easily, because the identity [G, [G, F]\ = — F implies e*GFe-*G

6.9

=

_

p

The existence problem

A precise definition of Hamilton's principal function V was given in Remark 6.1. It is clear that, if qi and qt belong to R 4 , then V(qt, qt) < +00 if and only if qi and qt belong to the same leaf SEIt is then reasonable to ask whether an optimal control steering qi to qt exists whenever V(qt,qt) < +00. Equivalently, we want to know if the infimum that occurs in the definition of V is in fact a minimum whenever it is finite. The usual way to prove existence of an optimum is to pick a sequence {??j}j£N of controls steering qi to qt with costs Cj that converge to the infi-

144

mum, look at the corresponding trajectories £,, and then try to produce a subsequence {r)j(e)}e€T$ of {^}jgN that converges in some sense to a limiting control TJOO and is such that the £,•(£) converge to a trajectory £00 of rjco and the costs Cj(^) converge to the cost CQQ of (£oo,»?oo)This method may fail for various reasons. The simplest one is that the £j may "go off to infinity." That is, there may not exist a subsequence {£j(e)}teN that stays in a fixed compact set. In the following definition, we give a name to the property that this particular obstruction to the existence of optimal controls does not occur. Definition 6.3. We say that an optimal control problem with dynamics (6.6.2) and Lagrangian cost functional with Lagrangian L satisfies the properness condition if for every pair qi,qt of states such that V(qi,qt) < +00 there exist a number c € l and a compact subset K of Q such that (1) V(quqt)


(2) whenever rj is a control steering qi to qt with cost < c, and £ is the corresponding trajectory, it follows that £ is entirely contained on K. 0 When a problem satisfies the properness condition, it still does not follow that optimal controls exist, because the minimizing sequences {T]J}J€N need not have subsequences that converge in any reasonable sense. On the other hand, it is commonly agreed that in this case optimal controls really exist if one enlarges the original class U of admissible controls by adding to it some "generalized controls." We will show that this is indeed what happens for our brachistochrone problem, and that in this case the generalized controls can be easily described in concrete terms, and turn out to be none other than "impulse controls." To begin with, we prove the following. Proposition 6.4. The minimum time problem for the brachistochrone system (6.6.3) satisfies the properness condition. Proof. Let qi, qt be two points of M4 having the same energy E. Fix an arbitrary c € E such that c > V(qi,qt). We have to find a constant A such that ||£(i)|| < A for all t € [0,r] whenever r < c and £ : [0,r] (->• K4 is a trajectory of (6.6.3) for some control t 1-4 r)(t) such that £(0) = qi and &T) = qt. Write £(*) = [x{t),y(t), v(t),w(t)]*. Let

9(t) = [v(t)Mt)V, at) = Kt)M*)V-

145

Then (6.6.3) implies that dt

|| C (t)|| 2 = - 2

9

•«,(*).

Therefore dt

2 2 IIC(*)IIS = 2g\w(t)\<9 +w(t)
+\\at)\\

and then, if we write

V>(*) = |llC(*)ll 2 -IK(o)|| 2 |, we find that < [\g2 + \K(s)\\2) ds < c(g2 + ||C(0)||2) + / V(*) ds. Jo Jo Then Gronwall's inequality implies that m

W)

< c(g2 + | | C ( 0 ) | | V < c{g2 + ||C(0)|| 2 )e c .

It then follows that

IICWII 2
0 6.10

• Application of the Maximum Principle on manifolds

Using the integral of motion E, we can regard E 4 as foliated by the 3dimensional level hypersurfaces of E, and conclude that every trajectory is contained in one of the leaves Sg. If E G E, then the Maximum Principle on manifolds (cf. 3T ) can be applied to the problem restricted to Sg. The conclusion turns out to be exactly the same as that for the unrestricted problem, except that the nontriviality condition for the momentum is slightly stronger. This extra strength is actually fundamental for, as explained in 6.7, without it the conclusion would be completely uninformative. If [a, b] 3 t H-> £„(£) € E 4 is an optimal trajectory for our problem, corresponding to a control [a, b] 3 t i-> n* (t) € E, then the usual version of the Maximum Principle yields the existence of a nontrivial momentum vector field [a, b] 3 t !->• 7r(t) 6 E 4

146

(often called an "adjoint vector") that satisfies the adjoint equation, the Hamiltonian maximization condition, and the additional condition that the maximized value of the Hamiltonian is a nonnegative constant 6 . The principle applied to the restriction to the leaves yields the stronger conclusion that n(t) cannot be orthogonal to the leaf Sg that contains £*. (In differential-geometric terms, -?v(t) is really a covectoron Sg at £*(£), and has to be nontrivial as such. Equivalently, if one insists on regarding 7r(i) as a vector in R 4 , then it should not be orthogonal to the tangent space to Sg at £*(£).) (Naturally, one could also avoid using the Maximum Principle on manifolds by invoking instead the fact that Sg is identified with M3 in an obvious way, since it is the graph of a smooth function (x,v,w) i-> y(x,v,w). This would amount to eliminating y from the equations by substituting for it its expression in terms of the other variables, for each fixed E.) The Hamiltonian is the function R4 x E x M4 3 (q,u,p) >-> R defined by H(q,u,p)

=
(6.6.10)

where the functions ip, ip are defined by 1>{q,p) =

•


Since u is completely unrestricted, H cannot have a maximum as a function of u unless tp(q,p) = 0. So V"(6(*).T(*)) = 0

for

a11

*•

(6.6.11)

t,

(6.6.12)

Differentiation of (6.6.12) yields p(^(t),Tr{t))

= 0 for all

where p{q,p)d=
147

A further differentiation—together with the fact that [G, [F, G]] = F—yields 0(M*)Mt))

+u(t)
= 0,

(6.6.13)

where e(q,p)^{p,[F,[F,G]](q)). This, together with the equalities ip = p = 0, determine u as a function of q, i.e., provide an optimal control in the form of a state space feedback law u = u(q), as we now show. First of all, (6.6.7) implies that *l> =
(6.6.14)

along our trajectory. Next, we use the crucial fact that ¥>(*(«),&(t)) 5* 0,

(6.6.15)

which follows from the stronger version of the Maximum Principle. (The precise argument is as follows. The covector n(t) must be nontrivial as a linear functional on the tangent space at £*(<) of the leaf containing £*(t). Since this space is spanned by the three vectors F(^(t)), [F,G](£*(£)) and [F,[F,G]](Ut)), one of the numbers ¥>(*(*),&(*)). p{*(t),t.{t)), 0(TT (*),£,(*)) must be nonzero. We know that p(ir(t),t;*(t)) = 0. If <^(7r(t), £*(*)) = 0, then (6.6.13) would imply that 6{n(t), £» (£)) = 0, and we would get a contradiction. Hence (6.6.15) holds.) Therefore (6.6.13) implies u{t)

(6 6 16)

—
--

that is U{t)

~

(«(t),F(Ut)))

(6 6 17)

•

- -

Moreover, it follows from (6.6.14), together with the fact that I/J = p = 0 along our trajectory, that Ci(Ut)M^(t),^(t)) Then (6.6.16) implies

+ C3(Ut))0(At),Ut))

= 0.

(6.6.18)

148

as long as C3 (£*(*)) ^ 0. Therefore u{t) = u(Ut)),

(6.6.20)

where , def

Ivg

uj(q) = u{x, y, v, w) =

(6.6.21) 2 2 . v2 + w'z It is then clear that u is a smooth function on the complement of the set S = {(x,y,v,w)

:v2+w2

= 0} .

(6.6.22)

The "singular" set £ corresponds, naturally, to the points where the kinetic energy vanishes, which as we know from our other analyses is where some singular behavior occurs. If we define Z(q)=F(q)+Lj(q)G(q), we see that Z is a smooth vector field extremals in the sense of the Maximum The resulting trajectories are easily use the feedback u = u(q), the dynamic

(6.6.23) 4

on M \E whose trajectories are the Principle. computed. First, observe that, if we equations become

x = v, Vw, v = —2g(v2 + w2)~1vw , 2 1 2 L ...2w..2 w = 2g(v22_+w )~ v —„ g 2 2 1 2 = {v + w )' {2gv - g{v2 + w2)) = g{v2 +w2)~1(v2 -w2).

(6-6-24)

If we introduce a complex variable z denned by z = v + iw, then the last two equations of (6.6.24) say that

i = ^ 1

that is, z=9—.

(6.6.25)

z

\\ '* The formulas of (6.6.25) define a well known dynamical system of the complex plane C. The right-hand side of the equations is a smooth vector field on C\{0}. The integral curves are circles passing through the origin and having center on the real axis (plus the imaginary axis, parametrized by E 3 t —> (0, — gt), which is also a solution since along it (6.6.25) just says that i = — gi). The parametric equation of the circles is z(t) = r{l + a e^), where a £ { + 1 , - 1 } and r is a positive real number.

(6.6.26)

149

Once it has been shown that the velocity vector {v,w) evolves according to (6.6.26), a trivial integration shows that the trajectories described by the position vector (x, y) are cycloids. 6.11

The complete synthesis and Hamilton's principal function for the 4-dimensional brachistochrone problem

We haven't yet completely answered all our earlier questions and determined Hamilton's principal function V. Let qi, qt be two points of R4. It is clear that V(qi, qt) = +oo unless both <7i and qt belong to the same energy hypersurface, because if the energies are different then the points cannot be joined by a trajectory, and the infimum of the empty set is +oo. On the other hand, if the points qi and qt belong to the same level hypersurface Sg, then we know that they can be joined by a trajectory, so V(qi, qt) will be finite. Moreover, it is reasonable to expect that such a curve will exist, and if it does not then we ought to be able to explain why. Write qi = (Ai,a,i), qt = (At,at), where Ai} at, At, at, belong to R 2 . The classical analysis carried out earlier seems to suggest that, once Ai, At and E are specified, there will be only one optimal arc joining them. So, if we specify Ai and At but do not fix the energy, then there should be a 1parameter family of such arcs (one arc for each value of E). Yet we now seem to need a 3-parameter family, because we want to accommodate all possible pairs (oj, at) (i.e., four parameters) as long as they give rise to the same energy (one constraint, and 4 — 1 = 3 ) . Equivalently, given E, we ought to be able to choose the directions 0i, Ot, of the velocity vectors ai and at in a completely arbitrary way. (Their magnitudes are determined by the fact that the energy at qi and qt must equal E.) But given Ai, At and E there is a unique cycloid from Ai to At corresponding to the energy E, and in particular the directions 6i, 6t at Ai, At of this cycloid are completely determined by Ai, At and E. In other words, we ought to be able to choose freely the two angles 9i, 6t, but it appears that we cannot do so. The solution of this apparent paradox is as follows. Suppose we fix an energy level E. Given any Ai, the set C(Ai, E) of those a's such that (Ai, a) € Sg is either empty or a circle, except for the "borderline" case when (Ai,0) £ Sg and C(Ai,E) consists of a single point. If C(Ai,E) is a circle, then through each point of {Ai} x C(Ai,E) there passes an integral curve of Z. The set of all points lying on such curves is 2-dimensional, and intersects the 1-dimensional set {At} x C(At,E) at just one point. This means that there is a unique pair (a.i,at) such that (Ai,a,i) and (At,at) lie on the same integral

150

curve of Z. The portion of that curve joining (At, en) to (At,at) is, of course, our famous cycloid. What is then the optimal trajectory joining (Ai, at) to (At, at) for general ai, at"? What is the corresponding value of V(Ai,ai,At,at)'? The answer is quite simple: the "optimal trajectory" is obtained by 1. applying an impulse control to rotate a; to a; instantaneously, 2. then following the cycloid from (Ai,cn) to

(At,at),

3. finally applying another impulse control to rotate at to at. (These are not true trajectories, of course, but it is easy to translate the above into a statement about a minimizing sequence of true trajectories.) The value V(Ai,cii, At,at) is therefore equal to V(Ai,a,i,At,dt). 7 7.1

Five modern variations on the theme of the brachistochrone Hestenes' "variation of the brachistochrone problem"

In 1956, M. Hestenes wrote (in

18

, p. 64):

... it is interesting to speculate on what type of brachistochrone problem the Bernoulli brothers might have formulated had they lived in modern times. An interesting variation of the brachistochrone problem can be described as follows: Consider an idealized rocket ship moving in a vertical plane. The ship is to be considered as a particle acted upon by a gravitational force g and a thrust of constant magnitude F and variable inclination (p ... There are no other forces acting on this ship. The pilot steers the ship by controlling the inclination
y = y(t)

0
subject to the constraints (with unit mass) x — F cos tp

y = F sin (p — g

and with x, y, x, y, prescribed at t = 0 and t — T, one which minimizes the time of flight T.

151

It can be shown that the path of least time has the following interesting property: there is a line L with direction fixed in space, and with position fixed relative to the ship, such that if a particle M is moving along L with constant speed, the ship will traverse a path of least time by keeping the thrust aimed at the particle M. Hestenes later on proceeds to solve the problem. Let us present a solution based on optimal control. The dynamical equations can be rewritten as x =z

z = F cos ip

y =w

w = F sin (p — g.

If [0,T] 9 ( 4 (x*(t), z*(t),yt(t),w*(t)) e I 2 is an optimal trajectory, corresponding to a control function [0, T] 3 t t-> ip* (t) € M, then the Maximum Principle yields a nonnegative constant -KQ and and a vector-valued function [0,T]3t^n(t)

=

(nx(t),irz(t),Try(t),Trw(t)).

The Hamiltonian is given by H(x,z,y,w,ip,px,pz,py,pw,po)

= px z+pz Fcosip+py w+pw (Fsmtp-g)-p0

.

The adjoint equation becomes •kx = 0

7rz = — 7rx

Then, if we write u)(t) — {nz(t),Ttw(t)), u(t) =

-ky = 0

nw = — •Ky .

we have tui,

(JQ +

where u>o and u>i are constant two-dimensional vectors. The Hamiltonian maximization condition then says that the vector u(i) = (cos ip(t), sin ip(t)) is given by

""» = »

•

<"•»

The Hamiltonian must vanish along our trajectory. Therefore, ir(t) can never vanish, for if 7r(t) = 0 it would follow that TT0 = 0, contradicting the nontriviality condition. Hence u>(t) cannot vanish identically. So u is a linear function which is not identically equal to zero. Therefore there is at most one value of t such that uj(t) = 0. It follows that the function u of (7.7.1) is well denned except possibly at one point i. Moreover, if such a i exists, then cj(t) = (t — i)cji, so if t < t u(t) = {—wi ui\ if t > i.

152

The "line L" of Hestenes' "interesting property" is the line Lt with direction <jj\, with a point At having a "position fixed relative to the ship" given by At = (x*{t),y*(t))

+u}0.

So in fact Lt is moving along with the ship. If a point qt on Lt moves with constant velocity UJ\, starting at Ao at time 0, then the point's position at time t is A* + toj\. Therefore the vector from the ship's position to qt is precisely u)(t). We have shown that the direction of the thrust is uj(t), from which Hestenes' "geometric property" follows. 1.2

The reflected brachistochrone problem

In 3, our analysis of the brachistochrone problem was carried out in a halfplane, by looking only for optimal trajectories that are entirely above the x axis, as in Johann Bernoulli's original problem. Let us pursue Hestenes' idea that "it is interesting to speculate on what type of brachistochrone problem the Bernoulli brothers might have formulated had they lived in modern times." An obvious question for the brothers to ask would have been that of finding out what happens if we work in the whole plane. This is in fact a much more natural mathematical setting for the brachistochrone problem. We shall refer to this new question as the reflected brachistochrone problem. (It is because we were thinking all along of this extension that we wrote y/\y~\ rather that ^Jy in (5.5.1)). For a recent study of the reflected brachistochrone problem from the dynamic programming perspective, see Malisoff 2 8 ' 2 9 . Solving this more general problem amounts to finding the light rays when the medium is the whole plane, and the speed of light is ^/\y\. In other words, we want to find the minimum time paths for the control system (5.5.1) with control set U given by (5.5.2) and state space equal to the whole plane. Notice that this problem is "completely controllable," in the sense that (C) any two points A, B of R2 can be joined by a feasible path, even if they lie on opposite sides of the x axis. To prove (C), observe first that the desired conclusion is trivial if A and B both lie in the open upper half plane or in the open lower half plane. So it clearly suffices to prove the conclusion if A lies in the open upper half plane and B belongs to the x axis. Moreover, we can assume that A lies on the same vertical line as B, for otherwise we can pick A' on the same vertical as B but strictly above B, and join A to A' first in finite time. So let us suppose

153

that A - (a,a) and B{a,0). Then the curve [0,a] 9 s 4 ( a , a - s ) certainly goes from A to B. Along this curve, dy = -^/ydt, so dy

ds

Therefore the total time T required to move from A to B along this curve is given by fa J0

ds

n

r-

yfa-s

so T < +oo, and (C) is proved. The crucial point here is that the right-hand side of (5.5.1) vanishes along the x axis, but this does not prevent the existence of feasible paths crossing the x-axis in finite time, because the "speed of light function" (x,y) <-+c(x,y) = y/\y] is not Lipschitz near the x axis. Remark 7.1. If the function c was Lipschitz-continuous, then the well known uniqueness theorem for ordinary differential equations would imply that every solution passing through a point on the x axis is a constant curve. This would obviously render it impossible for feasible curves starting at a point not on the x axis to reach the x axis in finite time. An important example of this situation is the case when the speed o light is \y\ rather than y/\y\. That case was also studied by Johann Bernoulli, as is clearly seen in the text we quoted at the end of 2.2, where he says that if, for example, the velocities were as the altitudes, then both curves [HJS & JCW: that is, the brachistochrone and the tautochrone] would be algebraic, the one a circle, the other one a straight line. It turns out that the "brachistochrones" for the case when the speed of light is \y\ are half circles whose center lies on the x axis. Moreover, these circles take infinite time to reach the x axis, precisely because the function J/ — ' >• |y| is Lipschitz. ^ We have seen that the complete controllability of (5.5.2) depends essentially on the non-Lipschitzian character of its speed of light function. However, this same non-Lipschitzian character also renders the Maximum Principle inapplicable, in its classical and nonsmooth versions (including the so-called "Lojasiewicz version," cf. Sussmann 35), since all these results require a Lipschitz reference vector field. Can we still use necessary conditions to find the solution, perhaps by invoking a more general version of the Maximum Principle that would be applicable to this nonsmooth, non-Lipschitzian case?

154

Suppose, for example, that we want to find an optimal trajectory £ going from A to B, where A lies in the upper half-plane and B is in the lower halfplane. Then one can easily show, first of all, that an optimal trajectory £ exists, using the Ascoli-Arzela theorem as before (cf. 5.2). Next, using the usual necessary conditions for optimality, e.g. the Euler-Lagrange equation or the classical version of the Maximum Principle, one shows that any portion of an optimal curve which is entirely contained in the closed upper half plane or in the closed lower half plane is a cycloid given by (3.3.5), or a reflection or such a cycloid with respect to the x axis. Next, one sees that £ cannot traverse the x axis more than once. (This is a very straightforward consequence of Lemma 5.1.) Therefore £ must consist of a cycloid T\ going from A to a point X in the x axis, followed by a second cycloid 1^ going from X to B. It remains to find X. Naturally, X is the only point on £ in a neighborhood of which the usual smooth and nonsmooth versions of the Maximum Principle are inapplicable. On the other hand, it is easy to produce a nonrigorous formal argument yielding the missing condition, as we now show. Let us apply the Maximum Principle formally, even though the technical conditions required for the validity of the usual versions are not verified. The maximized value of the Hamiltonian is clearly equal to ||p||\/[y| — po, and we know that this number is in fact equal to zero. So \\p\t)\\V\y\=Po

for all

t.

(7.7.2)

If we multiply (5.5.6) by \y(t)\, and use (7.7.2), we find

|y(*)l ( l + ^ ) » ) = M * ^ K = ^ .

(7.7.3)

The adjoint system (5.5.4) tells us that p is constant on each time interval that does not contain the point r such that £(r) = X. If we make the Ansatz that p is actually constant on the whole time interval, i.e., that the constant values of p on both sides of T are equal, we find that the cycloids Ti, T2 must be such that the quantity C=\y(x)\(l+y'(x)2).

(7.7.4)

which is constant along each Tj, actually has the same value for both curves. This extra condition determines X. The above argument is, of course, not rigorous. It turns out, however, that the recent "very nonsmooth" versions of the Maximum Principle developed by one of us—cf. Sussmann 39>4o,4i,42,43,44—apply to this problem and provide a

155

rigorous justification for our formal argument. The reason for this is that the new results do not require Lipschitz continuity—or even continuity—of the reference vector field. All that is needed is that the flow maps generated by the reference flow be differentiable in an appropriate sensed Remarkably, our "extra condition" has a simple geometric meaning. Recall that the cycloids of Johann Bernoulli's problem are generated by a circle rolling along the x axis. The extra condition then says that

(C) The rolling circles that generate the upper and lower cycloids must have equall radii. 7.3 A modest claim If we are going to "speculate" as Hestenes did, it is natural to ask not only what questions the Bernoulli brothers would have asked if they had lived in modern times, but also what kinds of answers they would have liked. This is of course a hard question to answer, and anything we say will have to be highly subjective. Granted this, we are firmly convinced that 1. The Bernoulli brothers would have liked the "interesting property" of the solution of Hestenes' problem (which is almost certainly the reason why Hestenes chose to emphasize this property) and also that 2. they would have found elegant the geometric condition of our solution of the reflected brachistochrone problem. This seems clear to us but, since de gustibus non est disputandum, we shall not pursue this particular argument further. 7.4

The brachistochrone with friction

In the problem studied in 6, we assumed that there was no friction. This implied that energy was conserved and that the four-dimensional state space was foliated by three-dimensional orbits of the system's accessibility distribution. ' F o r the specific case of the reflected brachistochrone, classical differentiability suffices. For more general problems, one needs other theories of "generalized differentiation," as e x p l a i n e d in 39,40,41,42,43,44

156 It is then natural to consider the more general case of the brachistochrone with friction, and to expect that for this problem there will be no nontrivial conserved functions of the state, so the accessibility distribution will be fourdimensional, and the system will have the accessibility property. We now outline some of the results, without carrying out a complete analysis. The dynamical behavior of the system is given by the equations x—

v, w, v = —uw — pv, w = uv — pw — g, y =

(7.7.5)

where, as before, u is an unbounded scalar control. The parameter p is a real number, the friction coefficient. The problem with friction corresponds to p > 0. When p = 0 we get the frictionless case considered earlier. If we write

F(q) =

v w -pv -pw

G(q) =

0 0 -w v

then (7.7.5) takes the form q=

F(q)+uG(q).

(7.7.6)

From now on, we assume that p ^ 0, since the case when p — 0 has already been discussed. Let L be the Lie algebra of vector fields generated by F and G. In order to analyze the structure of L, it will be convenient to introduce five new vector fields Z, V, W, X, and Y. The following eight formulas express F, G, and [G, F] in terms of the usual differential operator notation for vector fields, and define Z, V, W, X, and V:

F = vdx + wdv — pvdv - {pw + g)du vd„ G= - wdv + [G,F] = -wdx + vdy — gdv wdu Z= vdv + V = dv

w = X = Y =

dx 8y

157

Let S be the linear span of the eight vector fields F, G, [G,F], Z, V, W, X, and Y. It then turns out that S C I , because Z = a(F + [G,[G,F}}), W = 7 ([F, Z] + F + PZ), V = [G,W], X = pV-[F,V], Y = PW- [F, W], where 1 , 1 a = - and 7 = — . P 29 On the other hand, the formulas [F, [G, F]] = -p[G, F] + 2gX - 2pgV, [G,[G,F]] = -F-pZ, [F,Z] = [F,V} [F,W] [F,X] [G,Z] [G,V] [G,W]

= = = = = =

-F-pZ-2gW, -X + pV, -Y + pW, [F,Y] = 0, [G,X] = [G,Y] = 0, -W, V,

show that [F, S] C S and [G, S]CS. So L = S. A straightforward calculation shows that F, G, [G,F], Z, V, W, X, and Y are linearly independent over E. So L is an S-dimensional Lie algebra. Let A be the accessibility distribution of our system, so A(q) = {h(q) : h 6 L} if q 6 K4. Since the vector fields X, Y, V and W belong to L, it is clear that A(q) = M4 for every q. So our system has the accessibility property. An analysis similar to the one carried out in the previous section for the frictionless problem yields an optimal "feedback control" 2gv _Cy2 ~ vw u = — - — 5 - + 2 np — 25 - — r2> v + w involving an additional parameter £, given by _ TVy_

TTx

{7.7.7)

158

(Here irx, ny are the X and Y components of the momentum, which are easily seen to be constant along extremals.) In other words, through every point q € E 4 there passes a one-parameter family of optimal trajectories, depending on the parameter C,. 7.5

The classical brachistochrone as a degenerate problem

The family of brachistochrone problems with friction coefficient p constitutes an "unfolding" of the classical frictionless problem. For p ^ 0, the situation is generic, in the sense that the accessibility distribution is four-dimensional. The generic behavior of the time-optimal trajectories for such problems is precisely the one described here: the control is given in "feedback" form, depending on the state and an extra parameter. Formula (7.7.7) shows that the extra parameter only occurs as part of a term that is multiplied by p. It follows that when p = 0 the extra parameter disappears, and we get a state space feedback as determined earlier. 7.6

Deriving Snell's law from the Maximum Principle

Our third "modern variation" will involve going back to ancient times, long before 1696, and considering Fermat's minimum time principle and Snell's law of refraction. We have already explained how Snell's law is intimately related to the brachistochrone problem, since it played a crucial role in Johann Bernoulli's solution. So, if we pursue Hestenes' line of inquiry further, and speculate on other questions that the Bernoulli brothers might have asked "had they lived in modern times," it is easy to imagine them asking the following. (Q) Given that in these modern times powerful general theories have been developed, going far beyond the particular examples that were dealt with by ad hoc methods up to the 17th and early 18th centuries, do these general theories actually cover all these special examples whose study preceded and begat them? As an important special case of this question, the brothers would certainly want to know whether the necessary conditions of the calculus of variations and optimal control theory apply, in particular, to the derivation of Snell's law. So let us consider a medium such as the one we described when discussing Snell's law, in which the speed of light is constant on each of the two sides H-, H+ of a hyperplane H in K n , but the constant values c_, c+, are different.

159

Let us ask the obvious question, namely, whether Snell's law be derived from the Maximum Principle. It is easy to see that there is no way to formulate this question as a minimum time problem with a controlled dynamics q = f(q,u,t) in which / is continuous with respect to q, since | | / | | has to jump as we cross H. In the usual versions of the Maximum Principle, it is not required that / be differentiate, or even continuous, with respect to the control u. But here the situation is different, since we need to be able to deal with an / which is discontinuous with respect to the state variable q. Once again, it turns out that the recent versions the Maximum Principle developed in Sussmann 39>4o,4i,42,43,44 apply to this problem as well, and lead to Snell's law. The reason is that in these version all that is needed is that the reference trajectory £* arise from a flow {$t',t} consisting of differentiable maps, and there is no need for this flow to be generated by a continuous vector field. 7.7

Was Johann Bernoulli close to discovering non-Euclidean geometry?

Given a "speed of light function" (x,y) •-* c{x,y) on some plane region, the time along a curve is given by _ y/dx2 + dy2 c(x,y) since y ^f + -^ = c(x,y). This means that, if we adopt a slightly different point of view, and think of time as "length," then the function c defines a Riemannian metric on the set fi c = {{x,y) : c(x,y) > 0}, provided that c is sufficiently smooth. Let us assume that c(x,y) > 0 whenever y > 0, as was the case in our two examples. Then c gives rise to a Riemannian metric 2

_ dx2 + dy2 c(x,y)2

on the upper half plane. The "brachistochrones" are then the minimum length curves, that is, the geodesies. In the case of the "true brachistochrone," when c(x,y) = y/\y~\, these geodesies are cycloids (plus vertical segments) but maximal geodesies have finite length. On the other hand, in the case when c{x, y) = \y\, so that the Riemannian metric on the upper half plane is given by ds2=y-2(dx2+dy2), the geodesies are, as we have seen, half circles of infinite length.

(7.7.8)

So, using modern terminology, the circles that occur as solutions of this modified version of Johann Bernoulli's brachistochrone problem are the geodesies of a Riemannian metric on the upper half plane given by (7.7.8). We now know that this Riemannian metric defines the so called Poincare half plane, which is the simplest model of non-Euclidean geometry. This raises an obvious question: how close did Johann Bernoulli get to solving the famous problem of the provability of Euclid's fifth postulate? What renders the question particularly interesting is that 1. Johann Bernoulli lived at a time when the problem of Euclid's parallel postulate was very much at the center of mathematical research. For example, G. Saccheri published in 1733 his book 3 2 , entitled Euclides ab omni naevo vindicatus (that is, "Euclid vindicated from every flaw"), where he pushed very far the development of non-Euclidean geometry, only to end up dismissing it by claiming that a particular conclusion he had reached had to be rejected because it was "repugnant to the nature of a straight line." 2. The cycloids that occur in the original brachistochrone problem have many of the incidence properties of Euclidean lines, and therefore come close to satisfying all Euclid's postulates other than the fifth. But they do not yet provide a good model for non-Euclidean geometry because they "go to infinity" (i.e., approach the x axis) with finite length. 3. The half-circles of the Poincare model have all the desired incidence properties and also have infinite length, so they furnish a perfect model where all the Euclidean axioms other than the fifth are true. 4. Johann Bernoulli did study these half circles, as we pointed out in Remark 7.1. So in a sense he did discover the Poincare model. 5. Johann Bernoulli was definitely interested in the question, since he made significant contributions to the early stages of the development of the idea of geodesies on curved surfaces, by studying the curves of minimum length on a surface, which later 9 came to be known as "geodesies." 6. The solution of the problem of the independence of Euclid's postulate is extremely simple by today's standards, even though mathematicians took about 2200 years to find it. Actually, the solution is so simple that it can be explained to any undergraduate with a moderate understanding of plane analytic geometry. All that is required is to take any 9

T h e word "geodesic" was coined much later by Liouville (1809-1892).

161

reasonable way to compute "time" along curves, rename "time" by calling it "length," and then call the length-minimizing curves (that is, the "brachistochrones") "lines." For example, the case when c = \y\ leads us to taking the half circles with center on the x axis (plus the vertical half lines) and calling them "lines." Once this trivial renaming has been carried out, it is easy to verify that all of Euclid's postulates other than the fifth hold, but the fifth postulate fails. So Johann Bernoulli had essentially all the tools he needed to solve the problem and prove that Saccheri was wrong, by showing that Euclid's fifth postulate does not follows from the other ones. Why didn't he do it? One can surmise that he lacked the understanding of the geometric significance of the cycloids and half circles that he had found as solutions of his minimization problems. He certainly saw these curves as "geometric," but failed to see that they could serve as the "lines" of a different geometry. In modern terms, this is a trivial remark, namely, that one can turn around the well known fact that the segment is the shortest path between two points, and make it the definition of the notion of segment, while at the same time using a more general prescription to measure "length." But the idea that minimization problems can be the source of new geometries evolved very slowly over two millennia, and took a long time to crystallize, reaching final form in the nineteenth century, in the era of Gauss and Riemann, when it revolutionized mathematics and then moved on, at the beginning of the twentieth century, to bring a a fundamental change in our understanding of the physical universe. One can already find the idea in a rudimentary form in Fermat's least time principle'1, which can be interpreted as saying that light rays follow the shortest paths, with the proviso that "length" must now be defined in terms of a new and more refined geometry, in which the physical properties of space influence the geometry, and shortest paths can be curved or bent. This interpretation is suggested, for example, by G. Lochak, in 27 , p. 57-58 [our translation]: [Fermat's principle] anticipated the greatest revolution to be undergone by geometry in the future, namely, the discovery of non-Euclidean geometries, because it showed for the first time that the physical properties of a medium might cause the shortest paths not to be straight lines. ''which, as we have seen, was a direct ancestor of Johann Bernoulli's solution of the brachistochrone problem

162

In fact, Leibniz definitely uses the word "shortest" in his analysis of the refraction problem in his 1684 Nova methodus paper, showing that he was at least entertaining the possibility that light rays in a medium with variable speed of light truly were "shortest paths" in some new geometry. But it would appear that, in spite of these earlier glimpses of what would become one of the greatest scientific ideas of all times, the era of Johann Bernoulli was not yet ready to take the trivial step of turning around the lenght minimization property of segments. Johann Bernoulli passionately wanted to succeed and be famous, and was obsessed by the desire to surpass his contemporaries, especially his brother Jacob and his son Daniel*. It is thus sadly ironic that he, of all people, got so close to solving an important 2000-year old problem but missed because he failed to take a small step, a step so simple that today it can easily be explained to beginners. Had Johann Bernoulli "lived in modern times," he probably would have been torn between the curiosity to find out about non-Euclidean geometry and the desire not to hear about the tragedy of his missed opportunity to reach eternal fame and glory. Acknowledgements The writing of this paper began while the first author was Johann Bernoulli Visiting Professor at the University of Groningen, in May, June and July 1999, and continued while he was back at Rutgers University, partially supported by the National Science Foundation under NSF Grant DMS 98-03411. It was concluded while he was Weston Visiting Professor at the Weizmann Institute of Science in Rehovot, Israel, from October 2000 to January 2001. He is grateful to the University of Groningen, the Weizmann Institute, and NSF, for their help and support. References 1. Bell, E. T., Men of Mathematics. (Simon and Schuster, New York, 1937). 2. Beltrami, E., Saggio di interpretazione della geometria noneuclidea.Giornale di Matematiche VI, 284-312 (1868) (in Italian). "Johann was very jealous of Daniel's success. He once threw Daniel out of the house for having won a French Academy of Sciences prize for which Johannn had also been a candidate (cf. Bell l , p. 134). Probably with the help of his loyal disciple Euler (a student of Johann in Basel and a colleague of Daniel in Saint Petersburg), Johann plagiarized Daniel's pathbreaking work on hydrodynamics (cf. Guillen 1 7 ) .

163 (Eng. transl. in Stillwell, 33 , 7-34). 3. Berkovitz, L. D., Optimal Control Theory. (Springer-Verlag, New York, 1974). 4. Berna, P., N. Conze, M. Duflo, M. Levy-Nahas, M. Rais, P. Renouard, M. Vergne, Representations des groupes resolubles. (Dunod, Paris, 1972). 5. Bernoulli, Johann, Curvatura radii in diaphanis nonuniformibus . . . [The curvature of a ray in nonuniform media . . . ]. Acta Eruditorum, May, 206-211 (1697). {Opera omnia I, 187-193.) 6. Bernoulli, Jakob, Solutio problematum fraternorum . . . una cum propositione reciproca aliorum [Solution of problems of my brother . . . together with the proposition of others in turn]. Acta Eruditorum, May, 211-217 (1697). (Opera I, 768-778.) 7. Bliss, G. A., Calculus of Variations. (Open Court Publishing Co., Chicago, 1925). 8. Bonnard, B., V. Jurdjevic, I. A. Kupka and G. Sallet, Transitivity of families of invariant vector fields on the semidirect products of Lie groups. Trans. Amer. Math. Soc. 271, no. 2, 525-535 (1982). 9. Bonnard, B., V. Jurdjevic, I. A. Kupka and G. Sallet, Systemes de champs de vecteurs transitifs sur les groupes de Lie semi-simples et leurs espaces homogenes (French). In Systems Analysis Conf. (Bordeaux, 1978), Asterisque, Soc. Math. France, Paris, 75-76 (1980). 10. Bonola, R., Non-Euclidean Geometry. (Italian original: La Geometria non-Euclidea, The Open Court Publishing Co., 1912.) Eng. transl. by H. S. Carslaw, (Dover, New York, 1955). 11. Cesari, L., Optimization—Theory and Applications. (Springer-Verlag, New York, 1983). 12. Clarke, F. H., The maximum principle under minimal hypotheses. SI AM J. Control Optim. 14, 1078-1091 (1976). 13. Clarke, F. H., Optimization and Nonsmooth Analysis. (Wiley Interscience, New York, 1983). 14. Clarke, F. H., Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory. Grad. Texts in Math. 178, (Springer-Verlag, Berlin, 1998). 15. Euler, L., Methodus inveniendi . . . accepti. Lausanne, Geneva, 1744; (Opera omnia, ser. I, vol. 25, 1952). 16. Goldstine, H. H., A History of the Calculus of Variations from the 17th to the 19th Century. (Springer-Verlag, New York, 1981). 17. Guillen, M., Five equations that changed the world: the power and poetry of mathematics. (New York, Hyperion, 1995).

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18. Hestenes, M. R., Elements of the calculus of variations. In Modern Mathematics for the Engineer, (E. F. Beckenbach Ed., McGraw-Hill, New York, 1956). 19. Isidori, A., Nonlinear Control Systems. (Springer-Verlag, Berlin, 1995). 20. Jakubczyk, B., and W. Respondek eds., Geometry of Feedback and Optimal Control. (Marcel Dekker, New York, 1997). 21. Jurdjevic, V., and I. A. K. Kupka, Control systems on semisimple Lie groups and their homogeneous spaces. Ann. Inst. Fourier (Grenoble) 31, no. 4, vi, 151-179. 22. Jurdjevic, V., and I. A. K. Kupka, Control systems subordinated to a group action: accessibility. J. Differential Equations 39, no. 2, 186-211 (1981). 23. Jurdjevic, V., and I. A. K. Kupka, Linear systems with degenerate quadratic costs. Compt. Rend. Math. Rep. Acad. Sci. Canada 10, no. 2, 101-106 (1988). 24. Jurdjevic, V. Geometric Control Theory. (Cambridge U. Press, New York, 1997). 25. Kupka, I. A. K., Degenerate linear systems with quadratic cost under finiteness assumptions. In New trends in nonlinear control theory (Nantes, 1988), Lecture Notes in Control and Inform. Sci. 122, (Springer, Berlin, 231-239, 1989). 26. Lee, E. B., and L. Markus, Foundations of Optimal Control Theory. (Wiley, New York, 1967). 27. Lochak, G., La Geometrisation de la Physique. (Flammarion, Paris, 1994). 28. Malisoff, M., On the Bellman equation for optimal control problems with exit times and unbounded cost functionals. In Proc. 38th IEEE Conference on Decision and Control, Phoenix, AZ, Dec. 1999, (IEEE Publications, New York, 23-28, 1999). 29. Malisoff, M., A remark on the Bellman equation for optimal control problems with exit times and noncoercing dynamics. In Proc. 38th IEEE Conference on Decision and Control, Phoenix, AZ, Dec. 1999, (IEEE Publications, New York, 877-881, 1999). 30. Nijmeijer, H., and A. van der Schaft, Nonlinear Dynamical Control Systems. (Springer-Verlag, New York, 1990). 31. Pontryagin, L. S., V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes. (Wiley, New York, 1962). 32. Saccheri, G., Euclides ab omni naevo vindicatus: sive conatus geometricus quo stabiliuntur prima ipsa universale Geometriae Principia. (Milan,

165 Italy, 1733). 33. Stillwell, J., Sources of Hyperbolic Geometry. History of Mathematics, Vol. 10 (Amer. Math. Society, Providence, RI, 1996). 34. Struik, D. J., Ed., A Source Book in Mathematics, 1200-1800. (Harvard U. Press, Cambridge, Mass., 1969). 35. Sussmann, H. J., A strong version of the Lojasiewicz maximum principle. In Optimal Control of Differential Equations, N. H. Pavel Ed., (Marcel Dekker, New York, 293-309, 1994). 36. Sussmann, H. J., A strong version of the Maximum Principle under weak hypotheses. In Proc. 33rd IEEE Conference on Decision and Control, Orlando, FL, 1994, (IEEE publications, New York, 1950-1956, 1994). 37. Sussmann, H. J., An introduction to the coordinate-free maximum principle. In Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek Eds., (Marcel Dekker, New York, 463-557, 1997). 38. Sussmann, H. J., and J. C. Willems, 300 years of optimal control: from the brachystochrone to the maximum principle. (IEEE Control Systems Magazine, 32-44, June 1997). 39. Sussmann, H. J., Multidifferential calculus: chain rule, open mapping and transversal intersection theorems. In Optimal Control: Theory, Algorithms, and Applications, W. W. Hager and P. M. Pardalos Editors, (Kluwer Academic Publishers, 436-487, 1998). 40. Sussmann, H. J., Geometry and optimal control. In Mathematical Control Theory, J. Baillieul and J. C. Willems Eds., (Springer-Verlag, New York, 140-198, 1998). 41. Sussmann, H. J., Transversality conditions and a strong maximum principle for systems of differential inclusions. In Proc. 37th IEEE Conference on Decision and Control, Tampa, FL, Dec. 1998. (IEEE publications, New York, 1-6, 1998). 42. Sussmann, H. J., Resultats recents sur les courbes optimales. In Quelques aspects de la theorie du controle, Journee annuelle de la Societe Mathematique de France, (Paris, 1-50, June 2000). 43. Sussmann, H. J., A nonsmooth hybrid maximum principle. In Stability and Stabilization of Nonlinear Systems (Proceedings of the First European Community Nonlinear Control Network Workshop on Stability and Stabilization held in Ghent, Belgium, on March 15-16, 1999), D. Aeyels, F. Lamnabhi-Lagarrigue and A. J. van der Schaft Eds. Lect. Notes Control and Information Sciences 246, (SpringerVerlag, 325-354, 1999). 44. Sussmann, H. J., New theories of set-valued differentials and new versions of the maximum principle of optimal control theory. To appear in

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Part II Contributed Chapters

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169

SYMPLECTIC M E T H O D S FOR S T R O N G LOCAL OPTIMALITY I N T H E B A N G - B A N G CASE ANDREI A. AGRACHEV Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina ul.8 - 117966 Moscow, Russia and SISSA, Via Beirut 4 - 34014 Trieste, Italy E-mail: [email protected] GIANNA STEFANI Dip. di Matematica Applicata, G. Sansone, Via S. Marta 3 -50139 Firenze, Italy E-mail: [email protected] PIERLUIGI ZEZZA Dip. di Matematica per le Decisioni Via C. Lombroso 6/17 - 50134 Firenze, Italy E-mail:[email protected], URL: http://www.dmd.unifi.it/zezza/ We announce a new sufficient condition for a bang-bang extremal to be a strong local optimum for a control problem in the Mayer form. The controls appear linearly and take values in a polyhedron and the state space and the constraints are smooth, finite dimensional manifolds.

1

Introduction

We state a sufficient condition for a bang-bang extremal to be a strong local optimum for a control problem in the Mayer form. The controls appear linearly and they take values in a polyhedron and the state space and the constraints are smooth, finite dimensional manifolds; here we give only a sketch of the proof, a complete version of the result will be published elsewhere, see 7

The sufficient optimality condition contains two different sets of assumptions: the first (Assumption 1 and 2) concerns the regularity of the Hamiltonian flow while the second is the positivity of a suitable second variation. In the case of bang-bang extremals the control values belong to the boundary of the control set so that the kernel of the first variation does not contain any subspace and hence the usual second variation is not defined. To overcome this problem we consider the finite dimensional subproblem generated by perturbing the switching times. We prove that the sufficient second order

170

optimality conditions for this finite dimensional subproblem yield local strong optimality. We follow the Hamiltonian approach to strong optimality consisting in constructing a field of state extremals covering a neighborhood of the reference trajectory. This field of extremals is obtained by projecting, on the base manifold M, the flow of the maximized Hamiltonian emanating from the Lagrangian submanifold of the initial transversality conditions, which belongs to T*M; if this projection admits a Lipschitz continuous inverse, then we can estimate the variation of the cost at a neighboring trajectory by a function which depends only on the final point and it is hence independent on the control differential equation, the Lipschitz continuity of the inverse is guaranteed by the surjectivity of the tangent map. Let us point out that if the final point is fixed then the manifold of the admissible final points reduces to a singleton and hence, to obtain optimality, it is sufficient the surjectivity of the projection which corresponds to the classical non self-intersecting property of the state extremals, when the final point is not fixed, the optimality will also be guaranteed by the positivity of the second variation. The relations existing between the second variation and the symplectic properties of the Hamiltonian flow can be used to show that if the second variation is positive then the tangent map of the flow is surjective and the sufficient optimality conditions for the above mentioned function of the final point are satisfied. A crucial idea underlying the general approach is that the linear Hamiltonian flow of this linear quadratic problem has to be the tangent map of the flow of the maximized Hamiltonian. Some issues has already been addressed. In 5 we stated sufficient conditions for strong local optimality for an optimal control problem in Rn with unbounded controls, while in 4 we gave an intrinsic expression of the accessory problem and we studied the relations between the Hamiltonian flow and the index of the second variation. The geometric properties of the field of extremals needed to prove sufficient conditions for strong optimality has been studied in 6 . Similar conditions for time-optimality of a bang-bang trajectory have been obtained by A. Sarychev in 9 and 8 .

171

2

Main results

Let Xi, i = 1 , 2 , . . . , m be m distinct C°° vector fields defined on the C°° manifold M and let us define the multivalued function X:M^>TM,

X:x*+co{X1(x),...,Xm[x)},

where co denotes the convex hull. We consider an optimal control problem stated in the Mayer form Minimize

J(£) := co(£(0)) + cT{Z{T))

(2.1)

subject to i{t) e x{i(t)) ttO) 6 N0 , £(T) 6 JVr

(2-2) (2.3)

where N0, NT are C°° submanifolds of M. We are given, as a candidate optimal solution, a reference bang-bang Pontryagin extremal £; that is an absolutely continuous function £ : [0, T] - • M which is a solution of system (2.2)-(2.3), satisfies the Pontryagin maximum principle (PMP) and it is such that we can find a partition of the interval [0, T], {0 = t0 < tx < t2 < • • • < tr < U+i = T} with the property that, in each subinterval, the function £ is a solution of i(t) = XjMt)),

te[ti-i,n]

(2.4)

for some ji 6 { 1 , 2 , . . . , m } ; the values ti, i = 1,2,... ,r will be called switching times. We assume that Xj{ ^ Xji+1, for i = 1,2,..., r, for not introducing unnecessary switching times. Corresponding t o t h i s reference trajectory we define the time-dependent reference vector field h : [0, T] x M -> TM and we set hi ~ % 4 . _ l j t . ) : x >->•

Xj^x).

The time dependent reference vector field ht defines, by lifting to the cotangent bundle, the reference time dependent Hamiltonian H : [0,T] x T*M - • E, (t,£) ^

(£,ht(n£)),

where n : T*M -> M is the canonical projection and we set Hi := The maximized Hamiltonian H :T*M-*R,

l^

max

(l,v),

H^t._lit.y

172

is well defined and Lipschitz. For simpler notations we set x0 := |(0), xT := £(T). Recalling that any Hamiltonian Ht : T* M ->• R defines a Hamiltonian vector field, denoted by Ht, we can express the PMP by saying that there exist po £ {0,1} and a lift A of £ which is a solution of X(t) = Ht(X(t))

(2.5)

A(0) - po dco(x0) = 0 on TX0N0 \{T)+podcT(xT)=0 on TXTNT

(2.6) (2.7)

such that bo| + | | A | | ^ 0 Ht{X(t)) =

max (X(t),v) = H(X(t)). vEX(i(t))

The functions po CQ and po ex are denned on N0 and Nx, respectively, but they can be extended to the whole manifold M in such a way that the transversality conditions (2.6) and (2.7) hold on the whole tangent space. Let us denote by a, j3 : M —> E two functions such that a -p0c0 /3 — po ex

on N0, on Nx

A(0) = da(xo), X(T) =

-dP(xT).

In the normal case (po = 1) Q, /3 are cost functions equivalent to the original ones while in the abnormal case (po = 0) they are extensions of the zero function. When po = 0 all the costs disappear from our conditions and indeed we will study a problem with a zero cost. Proving that £ is a strict strong minimizer will imply that it is isolated among the admissible trajectories. The transversality conditions of the PMP can be stated saying that the end points of the adjoint covector belong to some specific Lagrangian manifolds, if we define A0 := {da(x) + T^No | x 6 N0} , A T := {-d/3(x) + T^Nx | x € NT} , then the transversality conditions (2.6)-(2.7) can be expressed as A(0) 6 A 0 ,

X(T) G A T .

173 The sufficient conditions will be stated analyzing the problem Minimize

J(fl := a(£(0)) + /?(£(T))

(P)

subject to (2.2) and (2.3). The points ti:=\(U),

i=0,l,...,r + l

will be called the switching points of the adjoint covector A. From the PMP we can deduce the following consequences which represent necessary optimality conditions for i — 1,2,..., r Hi(li) = Hi+i(£i) (d(Hi+1-Hi),Hi+1)(ei)>0. To state sufficient condition for f to be a strong local minimizer we need to strengthen these two conditions. We will assume that the following hold Assumption 1. (d(Hi+1 - Hi),Sw){ti)

> 0, * = 1,2,... ,r.

Assumption 2. The maximum max (\(t),v), t€[0,T] v€Xtf(t)) is attained at a vertex of X(£(t)) for all t ^ ti,t2,- • -,tr or along an onedimensional edge for t = t\, t2, • • •, tr. These conditions are strictly related to the properties of the flow of the maximized Hamiltonian H and they can be interpreted as regularity conditions on the maximized Hamiltonian near the adjoint covector. In fact the PMP implies that each switching point Hi belongs to the level set i?j+i — Hi = 0 and Assumption 1 yields that this level set is an hypersurface called switching surface. Moreover we can prove Lemma 2 . 1 . Under Assumption 1 there exists a neighborhood U of IQ such that we can define recursively, for i = 0 , 1 , . . . , r, the C°°-maps Ti : U - • M, and fa : U -> T*M

in the following way, set To : = 0,


the T{ 's are implicitly defined by (Hi - Hi+1) (expnWHiifa-iit))) Ti("i) = ti,

=0

174

while the fa 's are C°° symplectic diffeomorphism defined as <j>i{t) := exp(-Ti(e)Hi+1)

expn(e)Hi{4>i-i{l)).

Under Assumptions 1 and 2, by Lemma 2.1, we can say that the map "K : [0,T] x U - > [ 0 , r ] x T*M given by •K{t,t) = (t,exptHi+1(i(e))),

t £ [Ti(0,r i + i(/)],

is the unique solution of the Hamiltonian system \(t) = H(X(t)) A(0) = L Moreover the flow "K is C°° on each 2 n + 1 dimensional C°° manifold with boundary Oj, given by Oi := { ( M ) | / € U, Ti_i(0 < * < Ti(l)} C [0,T] x T*M. Let us remark that every solution of this system has the same number of switches as the adjoint covector A. Remark 2.2. Assumption 1 plays a role analogous to the one of the strengthened Legendre condition in the case of unbounded controls; in fact Lemma 2.1 says that we can define locally, in a tube around the adjoint covector, a timedependent Hamiltonian as {t,l)t-¥Hi(t,e),

for

M~1(t,t)£Oi.

Assumption 2, which concerns the uniqueness of the maximum point, ensures that with this choice we obtain the maximized Hamiltonian, for analogous conditions in the case of unbounded controls see 5 . In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial and hence the usual second variation, which is defined on the kernel of the first one, does not exist. To define an appropriate second variation we proceed starting from the following trivial fact: any choice of a subset of admissible variations for problem ( P ) would lead to a subproblem and hence to necessary conditions. Let us choose an appropriate r—dimensional family of variations corresponding to bang-bang trajectories, they are constructed by taking variations of the switching times. The optimality with respect to these variations will be sufficient (under the above stated assumptions) to prove strong local optimality of the reference trajectory. For a given a > 0 such that min i=l,...,r+l

(U — tj_i) > 2a,

175

let e G B(0, a) C W, set e 0 = £V+i = 0 and consider the time-dependent vector field obtained from the reference one in the following way (e,x) t-> ht(e,x) = hi(x) , t 6 (£,-_i + £•,•-!, *i +£i),

i = 1,2,.. .,r + 1.

In other words we have moved the switching time U by £j. Remark 2.3. Notice that a small e corresponds to a control variation which is small in the L1 norm but not in the L°° norm. We denote the flow of £(t) = ht(e,£(t)) by St : M x B ( 0 , a ) - » M and we consider the following finite dimensional subproblem of problem (P) Minimize

j(x,e)

:= a(x) + P(Sr(x,e))

(sub-P)

subject to x £ N0 , ST(x,e)

e NT.

Note that St : x ^

St(x,0)

is the flow of the reference vector field and hence St(xo, 0) = £(£)• Thanks to the choice of a,0 we have that the PMP implies that (xo,0) is a critical point for 7, that is dj(xo,0)=0, so that the second derivative of 7 at (xo,0) is well defined and J":=^27(xo,0) is a quadratic form on TXo M x E r which gives the second order approximation of 7. Thus the second variation of (sub-P) is the restriction of J " to the linearization of the constraints, namely if we set Ji = {{6x,e) e TXON0 x W : ST*{5x,e)

e