Control Theory of Systems Governed by Partial Differential Equations EDITORS:
A.K. AZIZ University of Maryland Baltimore County Baltimore, Maryland
J.W. WINGATE Naval Surface Weapons Center
White Oak, Silver Spring, Maryland
M.J. BALAS C. S. Draper Laboratory, Inc. Cambridge, Massachusetts
ACADEMIC PRESSIN
1977
New York San Francisco L?*d n A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1977, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR *TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT
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Library of Congress Cataloging in Publication Data
Conference on Control Theory of Systems Governed by Partial Differential Equations, Naval Surface Weapons Center (White Oak), 1976. Control theory of systems governed by partial differential equations. Includes bibliographies and index. 2. Differential Control theory-Congresses. 1. I. Aziz, Abdul equations, Partial-Congresses. Balas, III. II. Wingate, John Walter. Kadir. Mark John. IV. Title. QA402.3.C576
629.8'312'015 15353
ISBN 0-12-068640-6 PRINTED IN THE UNITED STATES OF AMERICA
76-55305
Contents List of Contributors Preface
REMARKS ON THE THEORY OF OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS J. L. Lions
STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS: A GENERAL THEORY
1
105
A. V. Balakrishnan
DIFFERENTIAL DELAY EQUATIONS AS CANONICAL FORMS FOR CONTROLLED HYPERBOLIC SYSTEMS WITH APPLICATIONS TO SPECTRAL ASSIGNMENT David L. Russell THE TIME OPTIMAL PROBLEM FOR DISTRIBUTED CONTROL OF SYSTEMS DESCRIBED BY THE WAVE EQUATION H. O. Fattorini SOME MAX-MIN PROBLEMS ARISING IN OPTIMAL DESIGN STUDIES
119
151
177
Earl R. Barnes
VARIATIONAL METHODS FOR THE NUMERICAL SOLUTIONS OF FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS 0. Pironneau
209
SOME APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY TO DISTRIBUTED PARAMETER SYSTEMS W. H. Ray
NUMERICAL SOLUTION OF THE TRANSONIC EQUATION BY THE FINITE ELEMENT METHOD VIA OPTIMAL CONTROL M. O. Bristeau, R. Glowinski, and O. Pironneau
231
265
List of Contributors A. V. BALAKRISHNAN, University of California Los Angeles, California 90024 EARL R. BARNES, IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598
M.O. BRISTEAU, IRIA/LABORIA, Domaine de Voluceau, 78 Rocquencourt, France
H. O. FATTORINI, Departments of Mathematics and Systems Science, University of California, Los Angeles, California 90024 R. GLOWINSKI, IRIA/LABORIA, Domaine de Vol uceau, 78 Rocquencourt, France
J. L. LIONS, IRIA/LABORIA, Domaine de Voluceau, 78 Rocquencourt, France
0. PIRONNEAU, IRIA/LABORIA, Domaine de Voluceau, 78 Rocquencourt, France W. H. RAY, Department of Chemical Engineering, State University of New York, Buffalo, New York 14214 DAVID L. RUSSELL, Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Preface These proceedings contain lectures given at the Conference on Control Theory of Systems Governed by Partial Differential Equations held at the Naval Surface Weapons Center (White Oak), Silver Spring, Maryland on May 3-7, 1976. Most physical systems are intrinsically spatially distributed, and for many systems
this distributed nature can be described by partial differential equations. In these distributed parameter systems, control forces are applied in the interior or on the boundary of the controlled region to bring the system to a desired state. In systems where the spatial energy distribution is sufficiently concentrated, it is sometimes possible to approximate the actual distributed system by a lumped parameter (ordinary differential equation) model. However, in many physical systems, the energy distributions are widely dispersed and it is impossible to gain insight into the system behavior without dealing directly with the partial differential equation description. The purpose of this conference was to examine the control theory of partial differential equations and its application. The main focus of the conference was provided by Professor Lions' tutorial lecture series-Theory of Optimal Control of Distributed Systems-with the many manifestations of the theory and its applications appearing in the presentations of the other invited speakers: Professors Russell, Pironneau, Barnes, Fattorini, Ray, and Balakrishnan.
We wish to thank the invited speakers for their excellent lectures and written summaries. All who were present expressed their satisfaction with the range and depth of the topics covered. There was strong interaction among the participants, and we hope these published proceedings reflect some of the coherence achieved. We appreciate the contributions of all the attendees and the patience shown with any fault of organization of which we may have been guilty. We thank the Office of Naval Research for their financial support of this conference. Finally, special thanks are due Mrs. Nancy King on whom the burden of typing this manuscript fell.
ix
"REMARKS ON THE THEORY OF OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS" J. L. Lions
Introduction
These notes correspond to a set of lectures given at the Naval Surface Weapons Center, White Oak Laboratory, White Oak, Maryland 20910, May 3 through May 7, 1976.
In these notes we present a partial survey of some of the trends and problems in the theory of optimal control of distributed systems. In Chapter 1 we present some more or less standard material, to
fix notations and ideas; some of the examples presented there can be thought of as simple exercises. In Chapter 2 we recall some known facts about duality methods,
together with the connection between duality, regularization and penalty (we show this in an example); we also give in this chapter a recent result of H. Brezis and I. Ekeland (actually a particular use of it) giving a variational principle for, say, the heat equation (a
seemingly long standing open question, which admits a very simple answer).
Chapter 3 gives an introduction to some asymptotic methods which can be useful in control theory; we give an example of the connection
between "cheap control" and singular perturbations; we show next how the "homogeneization" procedure, in composite materials, can be used in optimal control.
In Chapter 4 we study the systems which are non-linear or whose state equation is an eigenvalue or an eigenfunction; we present two
examples of this situation; we consider then an example where the control variable is a function which appears in the coefficients of the highest derivatives and next we consider an example where these two
1
J. L. LIONS
2
properties (control in the highest derivatives and state = eigenfunction) arise simultaneously.
We study then briefly the control of free
surfaces and problems where the control variable is a geometrical argument (such as in optimum design).
We end this chapter with several
open questions.
In Chapter 5 we give a rather concise presentation of the use of mixed finite elements for the numerical computation of optimal For further details we refer to Bercovier [1].
controls.
All the examples presented here are related to, or motivated by, specific applications, some of them being referred to in the Bibliography.
We do not cover here, among other things: the controllability problems (cf. Fattorini [1], Russell [1] in these proceedings), the stability questions, such as Feedback Stabilization (let us mention in'this respect Kwan and K. N. Wang [l], J. Sung and C. Y. Yii [1], and Sakawa and Matsushita [1]; cf. also Saint Jean Paulin [1]);
the identification problems for distributed systems, which can be put in the framework of optimal control theory, and for which we refer to G. Chavent [1], G. Chavent and P. Lemonnier [1] (for applications to geological problems), to G. I. Marchuk [1] (for applications in meteorology
and oceanography), to Begis and Crepon [1]
(for applications to oceanography), to J. Blum (for applications to plasma physics); cf. also the surveys Polis and Goodson [1] and Lions [11];
problems with delays, for which we refer to Delfour and Mitter [1] and to the bibliography therein; multicriteria problems, and stochastic problems. For other applications than those indicated here, let us refer to the recent books Butkovsky [1], Lurie [1], Ray and Lainiotis. P.
K. C. Wang [1].
The detailed plan is as follows:
Chapter 1. Optimality conditions for linear-quadratic systems. 1.
A model example.
1.1 Orientation
l.],
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
1.2 The state equation 1.3 The cost function.
The optimal control problem
1.4 Standard results 1.5 Particular cases 2.
A noninvertible state operator.
2.1.Statement of the problem 2.2 The optimality system 2.3 Particular cases 2.4 Another example
2.5 An example of "parabolic-elliptic" nature 3.
An evolution problem 3.1 Setting of the problem 3.2 Optimality system 3.3 The "no constraints" case 3.4 The case when
Uad = {vlv > 0
a.e. on
E}
3.5 Various remarks 4.
A remark on sensitivity reduction 4.1 Setting of the problem 4.2 The optimality system
5.
Non well set problems as control problems 5.1 Orientation
5.2 Formulation as a control problem 5.3 Regularization method Chapter 2. 1.
Duality methods.
General considerations
1.1 Setting of the problem 1.2 A formal computation 2.
A problem with constraints on the state 2.1 Orientation
2.2 Setting of-the problem 2.3 Transformation by duality 2.4 Regularized dual problem and generalized problem 3.
Variational principle for the heat equation 3.1 Direct method 3.2 Use of duality
3
J. L. LIONS
4
Chapter 3.
Asymptotic methods.
1.
Orientation
2.
Cheap control.
An example
2.1 Setting of the problem 2.2 A convergence theorem 2.3 Connection with singular perturbations 3.
Homcgeneization
e
3.1 A model problem 3.2 The homogeneized operator 3.3 A convergence theorem Chapter 4. 1.
Systems which are not of the linear quadratic type.
State given by eigenvalues or eigenfunctions 1.1 Setting of the problem 1.2 Optimality conditions 1.3 An example
2.
Another example of a system whose state is given by eigenvalues or eigenfunctions 2.1 Orientation
2.2 Statement of the problem 2.3 Optimality conditions 3.
Control in the coefficients 3.1 General remarks 3.2 An example
4.
A problem where the state is given by an eigenvalue with control in the highest order coefficients 4.1 Setting of the problem 4.2 Optimality conditions
5.
Control of free surfaces
5.1 Variational inequalities and free surfaces 5.2 Optimal control of variational inequalities 5.3 Open questions 6.
Geometrical control variables 6.1 General remarks 6.2 Open questions
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
Chapter 5.
Remarks on the numerical approximation of problems of optimal control
1.
General Remarks
2.
Mixed finite elements and optimal control
2.1 Mixed variational problems 2.2 Regularization of mixed variational problems 2.3 Optimal control of mixed variational systems 2.4 Approximation of the optimal control of mixed variational systems
5
Chapter 1
Optimality Conditions for Linear-Quadratic Systems 1.
A Model Example Orientation
1.1
We give here a very simple example, which allows us to introduce a number of notations we shall use in all that follows. 1.2
The state equation
Let
S2
be a bounded open set in
Let
A
be a second order elliptic operator, given by n
n
(1.1)
A
i
J
, with smooth boundary
Rn
=l
where the functions
z
i
Ju
(a.-(x) 13
ai
,
a
ai3
a
(x)
L-() ; we introduce
belong to
0
+ ao
J
the Sobolev space H1(S2)
(1.2)
i
ll
J
provided with the norm
(1.3)
- jaj, X2)1/2
1
Ilmll =
where
H _
(1.4)
02dx)1/2 = norm in
L2(S2)
(all functions are assumed to be real valued); provided with (1.3), H1(S2)
is a Hilbert space; for o, y ( D2aii Ni
(1.5) We assume
(1.6)
A
to be
as dx
H1(Q)
we set
+L J9ai aax ,y dx + f9a04'dx
H1(q) - elliptic, i.e. 2
, a >0 , vo E
7
H1 (c )
I
J. L. LIONS
8
The state equation in its variational form is now:
a(y,*) = (f,*) + fr v*dr
(1.7) where
(f,*) = f f*dx
f given in
,
"control variable" v is given in We recall that of
,y
space
on
r
V4rcHI(52)
L2(52)
L2 (r)
, and where in (1.7) the
.
one can uniquely define the "trace"
it is an element of
;
H112(r))
VyeH1(s2)
L2(r)
(actually of a smaller
and the mapping
y --- * I r
is continuous from
-
H1(s2)
L2(r)
.
Therefore the right hand side in (1.7) defines a continuous linear form on
H1(52)
so that, by virtue of (1.6):
,
Equation (1.7) admits a unique solution, denoted (1.8)
by
y(v); y(v) eH1(s2)
is affine continuous from
y(v)
v
and the mapping L2(r)
-
1(2)
.
The interpretation of (1.7) is as follows: Ay(v) = f
(1.9)
ay(v)
(1.10)
= v
in
sZ
on
r
,
A
where
as =
a
cos(v,xi) , v = unit normal to
r
directed
J
A
toward the exterior of aid eLW(52),
52
;
of course, under only the hypothesis that
(1.10) is formal; in case
aid eWl-(,Q)
as (i.e.
(1)
ax
A-0) V
k)
,
then one can show that
k
H2(Q)
00 a
2
' axi ax &L2 (2) J
y(v)eH2 (S2)
1) (
and
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
(1.10) becomes precise.
In the general case one says that
solution of (1.7) is a weak solution of (1.9)(1.10)
.
9
y = y(v)
p
We shall call (1.7) (or (1.9)(1.10)) the state equation, y(v) being the state of the system. The cost function.
1.3
The optimal control problem.
To each control v we associate a cost J(v) defined by (1.11)
where
J(v) = fr Iy(v)-zdl2dr + N fr v2dr
zd
Let
is given in v
L2(r)
and where
belong to a subset
Uad
N
of
,
is a given positive number.
L2(r) (the set of admissable
controls); we assume (1.12)
Uad
is a closed non-empty convex subset of
We shall refer to the case
L2 W)
Uad = L2(r) as the "no constraint"
case.
The problem of optimal control is now (1.13)
1.4
find inf
J(v)
Standard results.
,
vE Uad
(cf. Lions [1])
Problem (1.13) admits a unique solution u (the optimal control). This optimal control u is characterized by (J'(u), v-u)
>_ 0
VV&Uad
(1.14)
Up Uad
where
(J'(u), v) = d
J (u4 v)Ir=0 (this derivative exists).
The condition (1.14) which gives the necessary and sufficient condition for
u
to, minimize
J
a Variational Inequality (V.I.).
over
Uad
is (a particular case of
J. L. LIONS
10
An explicit (and trivial) computation of
J'(u)
gives (after
dividing by 2)
Ir (Y(u)-zd) (y(v)-y(u)) dr + N fr u(v-u) dr (1.15)
Vve Uad, uc Uad.
#
Transformation of (1.15) by using the adjoint state. In order to transform (1.15) in a more convenient form, we introduce the adjoint state p defined by
A* p = 0
in 2
,
(1.16)
aA*=y - zd where we set
on
r
y(u) = y, A* = adjoint of
A
The variational form of (1.16) is
(1.17)
a* (P,l,) = Ir (Y-zd)* dr
V*eH1 (52)
where we define
(1.18)
a* (,,4r) =
Let us set X = fr (Y-zd) (y(v)-Y)dr
by taking
y(v)-y
in (1.17) we obtain
X = a* (p,y(v)-y) = a(y(v)-y,p) = (by using (1.7)) = Ir (v-u)p dr
and (1.15) becomes
fr (p+Nu) (v-u) dr > 0 (1.19)
ue Uad'
ve Uad'
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
We can summarize as follows the results obtained so far: control
the optimal
of (1.13) is characterized through the unique solution
u
{y, p, u}
of the optimality system given by:
=
(1_2nt
,
-u
=y- zd
av
on
r,
Ir (p+Nu)(v-u)dr > 0 VvE Uad uE Uad
1.5
Particular cases. 1.5.1
Uad = L2(r)
If
(1.21)
The case without constraints.
p + Nu = 0
, the last condition in (1.20) reduces to
.
Then one solves the system of elliptic equations:
Ay = f, A*p = 0
in
0
(1.22)
aA+Np=O,ap=y-zd on and
u
is given by (1.21). 1.5.2
In case
Uad = {vi v >_ 0 a.e. on
Uad
r)
is given by 1.5.2, the last condition (1.20) is
equivalent to (1.23)
u
0, p + Nu >_ 0, u(p+Nu) = 0
i.e.
(1.24)
.
u = sup (0, N) =
N p
11
J. L. LIONS
12
Then one solves the system of non-linear elliptic equations:
Ay=f,A*p=0 in St, (1.25)
a Np =0, and
y
a
zd
on
r
is given by (1.24).
u
Remark 1.1
By virtue of the way we found (1.25), this system admits a unique solution
{y,p}
.
Remark 1.2
We have two parts on
r
r- = {xI xer, p(x) s 0}, r+ = {xl xer, p(x) > 0} (these regions are defined up to a set of.measure u = 0
r+
on
.
The interface between
and
r
as a free surface or as a commutation line.
0
r+
on
r ) and
can be thought of
#
Remark 1.3
For interesting examples related to the above techniques, we refer to Boujot, Morera and Temam [l]. 2.
#
A non invertible state operator. 2.1
Statement of the problem
In order to simplify the exposition we shall assume that A = - o
(2.1)
but what we are going to say readily extends to the case when
A
self-adjoint elliptic operator of any order (or to a self-adjoint system).
We suppose that the state y
qy=f
- v
in
(2.2) on
r
.
2
,
is given by
is any
OPTIMAL CONTROL OF DISTRI6UTED SYSTEMS
But now if
A
denotes the unbounded operator
-A
13
with domain
= 0 on r), 0 e spectrum of A so that A is not
jpe Hl(52), A eL2(S2),
invertible; but a necessary and sufficient condition for (2.2) to admit a solution is (2.3)
(f-v,l) = 0
and then (2.2) admits an infinite number of solutions; we uniquely define
y(v)
by adding, for instance, the condition
(2.4)
M(Y(v)) = 0
where
M(') = TiT f9 dx,
,
js2j
= measure of
2
Summing up: we consider control functions v which satisfy (2.3); then the state y(v) of the system is given as the solution of (2.2) (2.4).
#
The cost function is given by
(2.5)
3(v) = frly(v)-zdl2 dr + N f2v2dx
We consider EUad = closed convex subset of L2(r) and of the (linear) (2.6)
set defined by (2.3)
and we want again to solve (2.7)
2.2
inf J(v), ve Uad
The optimality system.
One easily checks that problem (2.7) admits a unique solution which is characterized by (we set
y(u) = y ):
fr(y-zd)(Y(v)-Y)dr + N(u,v-u) > 0 (2.8)
ucUad
#
yve Uad'
u,
14
J. L. LIONS
We introduce now the adjoint state p as the solution of
Ir(y-zd)dr
-op =
a = y-zd M(p) = 0
on
in
s
r
.
We remark that (2.9) admits a unique solution. If we take the scalar product of the first equation in (2.9) with
y(v) - y , we obtain
(-op, y(v)-y) = Inl 1r(y-zd)dr (1, y(v)-y) = 0
(by virtue of (2.4)) = - Jr- (y(v)-y)dr +
+ (P, -6(Y(v)-Y)) _ - 1r(Y-zd)(Y(v)-Y)dr + (p,-(v-u))
(the use we make here of Green's formula is justified; one has just to think of the variational formulation of these equations). Then (2.8) reduces to (2.10)
(-p+Nu,v-u) ?. 0
Vvc Uad, uE Uad
Summarizing, we have: the optimal control u , unique solution of (2.7), is characterized by the solution {y, p, u} optimality system:
I -ny 1
= f-u, -AP =
a = 0, M(y) = 0,
tr(Y-zd)dr
= y-zd
on
in
r
M(p) = 0,
(-p+Nu, v-u)
0
Vve
Uad
Ut Uad
2
of the
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS 2.3
Particular cases
Let us suppose that (2.12)
Uad =
{v
I
(v,1) _ (f,l)}
i.e. the biggest possible choice of
Uad
Then (2.10) is equivalent to
(2.13)
p + Nu = c = constant and
up- Uad
i.e.
-(p,l) + N(u,l) = c (2.14)
c = N M(f)
1521
= N(f,l)
i.e.
.
Then one solves first the system:
-Ay + N = f - Mf ep =
1r(y-zd)dr
(2.15) av
= 0,aV= y - z d
on
r
,
M(y) = M(p) = 0 and then
(2.16)
u=Mf+N'
#
Let us now suppose that (2.17)
Uad = {vJ v >_ 0 a.e. in g,
under hypothesis (2.18)
Mf > 0
(v,1) = (f,1)}
15
J. L. LIONS
16
which implies that
Uad
is not empty (case
Mf < 0) .
does not reduce to
{0}
(case
Mf = 0
)
or
Then the solution of (2.10) is given by
u=N+Mf+r-Mr
(2.19) where
r= (N+Mf xFR, x being a solution of
(2.20)
x = M(r)
fr(y-zd)dr
(actually unique if k =
+ Mf
0)
admits a solution
x = M(r)
Indeed, let us check first that
at least assuming that
,
L-(2)
N if we set p(x) = 0
p(x) = M((x-x) ) then for
p(x) is an increasing function, for
- c, p(X) = x - M(x) = x - Mf
x
x
enough, hence the result follows; let us notice that (2.19) does not depend on the choice of now check that
(2.21)
u
satisfies (2.10).
x
M(u) = M(f)
,
large
defined by
satisfying (2.20); let us
We can write
u = N + Mf - x + r = (N + Mf We have
u
0
and
+ u, v-u) = (Mf - x + r, v-u) = (r,v-u)
(-
N but (r, u) = 0
hence
+ u, v-u) = (r,v)
(-
N hence the result follows.
0
since
r
and
v
are > 0
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
The optimality system is given by
-AY =f - (N+Mf-x)+, -op = T2T 'r(y-zd)dr (2.22)
a = 0,
= y-zd
on r
X = M {(N + Mf - a) Remark 2.1
)
,
M(y) = M(p) = 0 ,.
.
#
Regularity of the optimal control.
It follows from (2.20) or (2.21) that (2.23)
ue H1(Q)
since
,
+ Mf - x eHl(S2)
.
N
Let us also remark that if does not improve (2.23)
zd a H1/2(r)
then
p e H2(12)
but this
#
.
Remark 2.2
One can find (2.19) (2.20) by a duality argument.
(cf.
Chapter 2 for the duality method). 2.4
Another example.
As an exercise, let us consider the state equation -Ay = f
2
in
,
(2.24)
1
a=v
on
r
which admits a set of solutions {y + constant} iff (2.25)
- fry dr = fnf dx
We define the state (2.26)
M(y) = 0
y(v) .
.
as the solution of (2.24) which satisfies
17
18
J. L. LIONS
If we consider the cost function
(2.27)
J(v) = frIY(v)-zdl2 dr + N frv2 dr
then the optimality system is given by
-AY = f, -AP = n fr(y-zd)dr
av
= u,
= y-z
av
on
d
in
s2
r
(2.28)
M(y) = M(p) = 0
fr(p+Nu)(v-u)dr
where
Uad
0
Vve Uad, ue Uad
is a (non-empty) closed convex subset of the set of v's in
L2(Q) which satisfy (2.25). 2.5
An example of "parabolic-elliptic" nature Let us consider now an evolution equation
(2.29)
at - Ay = f-v
fr veL2(Q)
Q = St x JO, T[
in
,
with boundary condition
(2.30)
a = 0
on
z = r x ]0, T[
and (2.31)
y(0) = y(T)
on
2
(where y(t) denotes the function x-y(x,t))
.
The equations (2.29) (2.30) (2.31) admit a solution (and actually a set of solutions y + constant) iff (2.32)
fQ v dx dt = fQ f dx dt
.
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
19
Let us then define the state of the system as the solution
y(v)
of (2.29) (2.30) (2.31) such that (2.33)
fQ y(v) dx dt = 0
.
If the cost function is given by J(v) = fQly(v)-zdl2 dx dt + N f v 2 dx dt, N > 0, zdeL2(Q)
(2.34)
Q
and if
Uad
is a (non empty) closed convex subset of the v's in
L2(Q)
such that (2.32) holds true, the optimality system is given by
a - Qy=f - u -
k - op = Y-Zd at
a (2.35)
- TQ7fQ(Y-zd) dx dt
on
z
y(0) = y(T), p(0) = P(T)
,
fQydxdt=fQpdxdt=0 fQ (-p+Nu) (v-u) dx dt > 0
Vve Uad
ue Uad
3.
An evolution problem. 3.1
Setting of the problem.
We consider now an- operator A as in Section 1
(cf. (1.1)); we
use the notation (1.5) and we shall assume there exists
a
and
a>O
such that
(3.1) a(o,m) + X14,12
--
allmll2 V4e Hl(S2)
(this condition is satisfied if that
a0, a3 eLm(2)
and
aij&L (St) such
J. L. LIONS
20
n
i j - a1E i2 , a1A
a..(x)
E
i,j=1
We consider the state equation:
(3.2)
at + Ay = f
(3.3)
a = v
(3.4)
Y(0) = yj
Q = S2x]o,T], feL2(Q)
in
vcL2(Z),
on
E
(1)
on
,
2, y0cL2(2)
-
This problem admits a unique solution which satisfies y cL2(O,T;H1(S2))
(3.5)
.
(cf. Lions [1] [2] for instance, or Lions-Magenes [1])
The variational formulation of this problem is
(L, *) + a(y,i,) = (f,*) + fr v$dr VyeH1(s2)
(3.6)
with the initial condition (3.4). Let the cost function
J(v) 2
(3.7)
J(v) = fE Iy(v)-zdl
and let
Uad
be given by
dE + N fE v2dz, zdeL2(2), N > 0
be a (non empty) colsed convex subset of
L2(z)
consider the problem of minimization: (3.8)
inf J(v), ve Uad '
3.2
Optimality system.
Problem (3.8) admits a unique solution, say u, which is characterized by
(1)
We write
a
av
instead of a
ev A
,
.
We
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
21
(J'(u), v-u) > 0 Vve Uad, ue Uad
(3.9)
i.e. (where we set
y(u) = y):
f (y-zd) (y(v)-y)dZ + N fz u(v-u)dz > 0 (3.10)
Vve U
ad'
uc Uad
In order to simplify (3.10) we introduce as in previous sections the adjoint state p given by
(3.11)
aP* = y - zd
[P(T) = 0
on
(7)
z
on
Then
fE (y-Zd) (y(v)-y)dz = jr, p(v-u)dE
so that (3.10) becomes (3.12)
j
yve Uad, ue Uad
(p+Nu) (v-u)dz > 0
The optimality system is given by
at I
(3.13)
1
+Ay=f,-
at + A* p = 0
av,d = u
=
y
-z
on
on
y(0) = y0, p(T) = 0
fF (p+Nu) (v-u)dz > 0
(1)
We write av* instead of as A
;
in Q,
,
2 ,
dve Uad, uc U d . a
J. L. LIONS
22
The "no constraints" case.
3.3
If we suppose that (3.14)
Uad = L2 (E)
then (3.12) reduces to p + Nu = 0
(3.15)
.
Then one solves first the system in {y, p}
+ Ay = f, (3.16)
a Y(
and then
u
aP a
+ A* p = 0
+ U'p = 0, ap* = y - zd av
in on
Q
E
0) = y0, p(T) = 0
is given by (3.15).
Remark 3.1
We obtain a regularity result for of
u = - N p
L2(0,T; H1/2(r)) (and one has more, since
M eL2(Q) 3.4
,
if we assume more on
Tha case when
zd ).
;
u is an element
peL2(O,T; H2(12)) and
#
Uad = {vfv > 0 a.e. on Z}
In the case when (3.17)
Uad = {v1 veL2(E), v >_ 0 a.e. on E}
then (3.12) is equivalent to (3.18)
u
0, p + Nu > 0,
u(J+Nu) = 0
on Z
i.e.
(3.19)
u = N p-
lnen the optimality system can be solved by solving first the non-linear system in {y, p} given by
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
a+Ay=f, -at+A*p=0 in
a
(3.20)
p- = 0, a2 = y -zd
-
on
23
Q
z
N
y(0) = y0, p(T) = 0
on
2
and by using next (3.19). Remark 3.2 We obtain
(as in Remark 3.1) the regularity result on the
optimal control: ue L2(0,T;H112(r))
(3.21)
3.5
.
Various remarks.
Remark 3.3
For the "decoupling" of (3.16) and "reduction" of the "two point" boundary value problem in time (3.16) to Cauchy problems for non linear equations (of the Riccati-integro-differential type) we refer to Lions [1] [3] and to recent works of Casti and Ljung [1], Casti [1] Baras and Lainiotis [1](where one will find other references) for the decomposition of the Riccati equation.
We refer also to Yebra [1],
Curtain and Pritchard [1], Tartar [1]. Remark 3.6
We also refer to Lions, loc. cit, for similar problems for higher order operators A, or operators A with coefficients depending on x and on t ; also for operators of hyperbolic type, cf. Russell [2], Vinter [1], Vinter and Johnson [1]. 4.
A remark on sensitivity reduction
4.1
Setting of the problem Let us consider a system whose state equation is again (3.2),
(3.3), (3.4) but with a "partly known" operator A let us consider a family
A(t)
of operators:
.
More precisely,
24
J. L. LIONS
(4.1)
A(r)m = - E aal
where
e R
3O) + E
axe + a0(x,K) 6
we suppose that
;
Iaid,
a0 c Lm(2xR)
ail
(4.2)
Then for every
(4.3)
a z 1i 2
r} n.J
z ai
the state
,
at + A(4)y = f
in
. ate- = v
F.
AN = y0
on
on
a>O,tlr, eR is the solution of
Q
s2
The cost function is now (4.4)
fE ly(v,?.)-zd12 dE + N fz v2 dE
We know that
A(,Y)
is "close" to
A(r0)
, and we would like to obtain
an optimal control of "robust" type, i.e. "stable" with respect to
changes of A(d "around" A(40)
.
p
A natural idea is to introduce a function
(4.5)
f
such that
P is > 0, continuous, with compact support around (C)dr. = 1
(of course the choice of about the system). (4.6)
p
will depend on the information we have
We now define the cost function 2
J(v) = fR
fEty(v,e-)-zdl
The problem we want now to solve is (4.7)
inf J(v), ve Uad
dE + N fE v2 dE
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
denotes a (non empty) closed convex subset
where, as usual, Uad of
25
L2(z)
4.2
The optimality system.
Problem (4.7) admits a unique solution u , which is characterized by
dz + N f u(v-u)dE
fP(d& fz (y(u,t.)-zd) (y(v,r) (4.8)
vv e
Uad
ue Uad
be the solution of
Let
-
p=0
+ A*
ate- _ y()
(4.9)
p(T) = 0
on
-z
2
d
in
Q
on
E,
.
Then multiplying the first equation (4.9) by y(v,s) 0 = - Iz
so that (4.8)
(v-u)dz
dz + Iz
educes to Nu) (v-u) dz
(4.10)
0 Vve Uad, ut Uad
Summarizing, the optimality system is given by
+ A(n)y = f, - a +
(4.11)
aV
= u, A(Y,)
a v( ) A* c
y(0) = y0, p(T) = 0
and (4.10).
we obtain
in
0
= y - z d on z
,
Q
0
J. L. LIONS
26
Remark 4.1
For numerical applications of the preceding remark, as well as for other methods of reduction of sensitivity in the present context, we refer to Abu El Ata [1] and to the bibliography therein. Remark 4.2
y
If
mass + 1 at K,0) in the weak star topology
&,)d
of measures (i.e.
support) and if we denote by
v
uP
continuous with compact
the solution of (4.7), then one can
show that (4.12)
-+ u in L2(E) weakly
u
P
where (4.13)
5.
u
solves inf
ve Uad
Non well set problems as control problems 5.1 Orientation.
Let us consider the following (non-well-set) problem (this problem arises from a question in medicine, in heart disease; cf. Colli-Franzone, Taccardi and Viganotti [1]): in an open set
with boundary
Q
r0Ur 1, a function
satisfies an
u
elliptic equation (5.1)
Ay = 0
,
and we know that
(5.2)
a
-
=O
rl
on
A
and we can measure (5.3)
y = g
on
S
Figure 1
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
If
g
is precisely known, this uniquely defines
27
but, as it
y
is well known, in an unstable manner.
The problem is to find
y
on
ro
Formulation as a control problem
5.2
Ay(v) = 0
(5.4)
in
y(v) = v
on
of our system as the solution of
y(v)
Let us define the state 2 ro ,
aA(v)=0 on rl (we assume that the coefficients of
A
are such that this problem
admits a unique solution).
We introduce
Uad
as the smallest possible closed convex subset
of
L2(ro) which "contains the information" we may have on the values
of
y (the "real one") on
ro
;
in general it will be of the form
Uad = {vI veL2(r0), mo(x) < v(x) < ml(x)
on r o
,
(5.5)
mo
and
ml
given in
L'-(r o)}
We introduce the cost function (5.6)
J(v) = IS ly(v)-gl2 dS
and we want to solve the problem (5.7)
If
Uad
(5.8)
inf J(v), ve Uad
has been properly chosen (i.e. not too small)
inf J(v) = 0
which is attained for
v = the value of y
on
r0
.
But. of course, this is again an unstable problem and following the idea of Colli-Franzone, Taccardi and Viganotti, loc. cit. we are
28
J. L. LIONS
now going to regularize the above problem of optimal control. Remark 5.1
Another approach to the problem stated in 5.1 Is given in Lattes-Lions [1] via the Quasi Reversibility method. 5.2
Regularization Method
There are a number of methods available to "stabilize" (5.7). Following Colli-Franzone, et al., we introduce the Sobolev space and the Laplace-Beltrami operator
on
r0
Uad = {vI ve H2(r0), m0 <_ V <_ ml
on
er
.
H2(r0)
We now consider
0 (5.9)
(we assume that
Uad
is not empty) and we define, for
JC(v) = J(v) + e Ir0 hero
(5.10)
r0}
v12
dr 0
e > 0
,
.
The new problem we are considering now is inf Je(v),
(5.11)
ve Uad '
Let
ue
be the unique solution of this problem.
The optimality
system is given by
Ay= 0,A*p=0 in
2
ay=0 onrl A
(5.12)
aP-
= y - g
on
S, = 0
y = u,.p = 0
on
ro ,
fr
0
(- a3 + e Ar A*
0
on
rl/S
u) (v-u) dro >_ 0 vve Uad' ue Uad
29
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
We refer to the paper of Colli-Franzone et al for the choice of and the acutal numerical implementation of this method. regularization methods, cf. A. N. Tikhonov [1].
e
For other
Chapter 2 Duality Methods 1.
General considerations. 1.1
Setting of the problem.
Let
V
functions
and and
F
F
Q G
and
be two real Hilbert spaces we consider two from
G
V
and
Q
such that
R
are lower semicontinuousand convex on
V
and
Q respectively, such that
- m
(one says that
and
G
F
and
+m
are not identically + G
are proper convex functions).
Let be given an operator (1.2)
A e L(V;q)
and let us consider J(v) = F(v) + G(AV)
(1.3)
We consider the minimization problem inf J(v), veV .
(1.4)
Remark 1.1
Problem (1.4) will be called the primal problem (P). introduce its dual (P*).
We want to
#
Remark 1.2
By virtue of the fact that
F
and
G
can take the value +
the formulation (4.3) (4.6) contains - as we will see in the example given in Section 2 - cases with constraints on the control variable v Remark 1.3 In the applications what is given is
equation); one can generally choose
F, G, A
J(v) (and the state in several different
manners - which lead to several different dual problems.
31
#
f
32
J. L. LIONS
1.2
A formal computation
We are going to obtain in a formal manner the dual problem (P*) For a justification (under suitable hypotheses on
F, G, A) cf.
T. R. Rockefeller [1], cf. also I. Ekeland and R. Temam [1].
We recall that if
is a proper convex function on a Hilbert
4'
space H , its conjugate function 4'* is defined on the dual H* of H
(which can be identified with 4'* (h*) = sup
H
or not) as:
[
- 4'(h)]
h
(1.5)
h* a H*, = scalar product in the duality H*, H
We introduce (1.6)
4'(v,q) = F(v) + G(q)
and we remark that inf J(v) = inf inf sup [4'(v,AV-q)- ] v q. q*
(1.7)
indeed the sup with respect to
q*
is
+ -
unless q = 0 .
If we now commute - in a formal manner - the inf and the sup in (1.7), we obtain inf J(v) = sup inf [4'(v, Av-q] - ] q* v,q = sup [-sup [
q* Let us set
(1.9)
Av-q=4.
v,q
.
.
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
33
Then sup [ - P(v,Av-q)] = sup [ - F(v) + < -q*. V,z
= F* (A* q*) + G* (-q*) and we obtain (1.10)
inf J(v) - sup [-F*(A* q*) - G*(-q*)] q*
The problem (P*) is (1.11)
(P*):
sup [-F*(A* q*) - G*(-q*)] q*
.
Remark 1.4
Even if (P) admits a unique solution, say u, (P*) does not necessarily admit a (unique) solution.
(cf. Example below in Section 2;
other examples are given in Lions [4].) *
In case (P*) admits a solution
qo
, one has
(1.12)
F(u) + F* (A* q0*) = < u,A* q0*>
(1.13)
G(Au) + G*(-q0*) =
.
A problem with constraints on the state.
2.
2.1
Orientation.
We are going to consider a problem whose state equation is a linear parabolic equation and where constraints on the control variable are given through constraints on the state: We show, following an idea of J. Mossino [1], that proper use of duality (as in Section 1) "suppresses" the state constraints at the cost of losing existence of
(1)
A*
is the adjoint of
A ; A* &L(Q*; V*).
J. L. LIONS
34
a solution of the dual problem.
But this procedure gives useful
tools for the numerical solution of such problems. 2.2
Setting of the problem.
We consider the state equation (as in Chapter 1, Section 3) given by
(2.1)
at + Ay = 0
in S2 -10,T[
(2.2)
a = v
z = r x]O,T[
on
,
A y(0) = 0
(2.3)
,
(We assume the right hand sides of (2.1) (2.3) to be zero, which does not restrict the generality.) Let
y(v)
Let
yl
(2.4)
be the solution of (2.1) (2.2) (2.3). be given in
L2(c2)
.
We define
Uad = {v) vEL2(Z), y(T;v) = yl} ,
and we assume that
yl
is such that
a closed convex subset of
Uad
is not empty.
L?(1)
Let the cost function be given by J(v) _ !Z Iy(v)-zdl2 dz + N !E v2dz
(2.5)
We consider the problem inf J(v), v6 Uad
(2.6)
2.3
Transformation by duality
We define, with the notations of Section 1: V = L2(E)
,
Q = L2(E) X L2(2)
Then
Uad
is
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
AV = {y(v) 1E , y(T,v) } e Q
F(v) = - jE v2 dE
G(q) = G1(gj) + G2(g2), q = {qj, q21
'
,
with
12
G,(o,) = j I
E Iq 1 -z d
0 if g 2=
yl
G2(g2) = + - otherwise.
Then (2.6) coincides with (1.4) (with a factor).
One checks easily
that
F*(v) _
(2.7)
IE v2 dE
,
V* = V = L2(B)
!E q I dE + !E gJ Zd dg , Q*
G1 (qj) _
=Q,
* G2 (q2)
For
j.9 q2 yj dx
{ql, q2} e Q , let us define (q)
as the solution of
in Qx]O,T[ , - aL + A* p = 0 (2.8)
a4l
avA*
s(T) = q2 Then one checks that (2.9)
A* q = 4+(q){71
on
52
.
35
36
J. L. LIONS
Indeed, taking the scalar product of the first equation (2.8) with
y(v) , one obtains
0 = fQX]0,T[ (- at + A*,P) Y(v) dx dt =
fEgl y(v) dE + IE $ v dE - fQ W) Y(T,v) dx
i.e.
= fE 4, v dE , hence (2.9) follows.
Then according to (1.10)
(2.10)
inf J(v)
inf [2M fE 4'(q)2 dE + I fE q dE q
- fE q1 zd dE - fQ q2 y1 dx] ;
We see that the dual problem (the inf) is a problem without constraints q
on the "control variable" q ; but it is not coercive in
q2 , so that
we do not have necessarily existence of a solution of the dual problem; but we have existence of a solution of the regularized dual problem:
(2.11)
innf[
fE +(q)2 dE + Y fE qi dS + . IQ q2 dx
- fE ql zd dE - fQ q2 Y1 dx]..
#
Remark 2.1
Optimality system for the regularized dual problem. Let
q0 _ [q0E, q2
4,(q°)
4E
it is characterized by
be the solution of (2.11).
If we set
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
Iz 4 (q)dz + Iz (gof-zd) ql dE + f
(2.12)
(E q0 -yl) q2 dx = 0
vq
L by
We define the "adjoint state" z (=z
+ Az = 0
at (2.13)
in s?x]0,T[
az =on
2
,
P Then
0 = IQx]O,T[ (
+Iz
+ Az) 4,(q) dx dt
N IE 4,e 1,(q) dz
z ql dz + I2 z(T) q2 dx ;
therefore (2.12) becomes
zd) q1 dz + IW (z(T) + qZE-y1) q2 dx = 0
Iz (2.14) Vq
and we finally obtain the optimality system
A* E = 0,
a + A z
zd - ze,
(2.15)
a
=
A
(T) _
1
(y1-z E (T)),
. ,E
z (0) = 0 e
in
= 0
9x]O,T[
on z. on
S ,
,
37
J. L. LIONS
38
with the approximate optimal control
given by
u E
uE +
(2.16)
2.4
tE
on
E
.
f
Regularized dual problem and penalized problem
We are going to show, in the setting of the preceding Section, the close connections (actually the identity in this case) which exist between the method of duality and the penalty method. We consider again the problem (2.5) (2.6) and we define the penalized problem as follows: for
(2.17)
Let
uE
(2.18)
r - 0 , we define
J&(v) - J(v) + E Jy(T;v) - Y112
be the unique solution of the penalized problem:
J(u) - inf J(v) ,
veL2(z)
Then one shows easily that
(2.19)
uE - u in
(2.20)
JE(uE) -. J(u )
L2(z) weakly as
E -+ 0
.
The optimality system is given as follows; we set
yE = Y(uE)
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
and we define
pe
39
as the solution of
ap
- ate + A* pe = 0
(2.21)
ap ape
= ye - zd
in
S2-]0,T[
on
2
P(T) = e (y (T)-y1)
The optimal control
,
on
2
is characterized by
u e
E(ye-z d) y(v) d2 + N I2 uvd2 + '(y (T)-yl, y(T;v)) = 0 (2.22)
V veL2(2)
;
using (2.21), (2.22) reduces to
pe + Nue = 0
on 2 ; hence the
optimality system; ap ay atE + Aye = 0 , - ate + A* p' = 0
aye (2 .23)
1
in
52x]0,7(
ap e
= ye - zd av + N pe - 0 , ao A* A
y(0) = 0 , Pe(T) = e (ye(T)-yl)
on
on
2
2
40
J. L. LIONS
This system is actually identical to (2.15) with
41E _ - pe, zC = YE
3.
.
Variational principle for the heat equation. Direct method
3.1
We present here a particular case of a recent result of H. Brdzis and I. Ekeland [1].
We want to show that, under suitable hypotheses given below, the solution
of the "heat equation":
u
I lat+Au=f in u = 0
(3.1)
E ,
on
u(x,0)
S2-10,T[ ,
u0
in
2
L realizes the minimum of a quadratic functional. We assume that
A* = A
(3.2)
and that, if
a((k,*)
A
alimll2 vmEHO (Q), a > 0
a(m,o)
(3.3)
is the linear form associated to
Remark 3.1
The result below readily extends to higher order elliptic operators A .
M
We assume that feL2(Q) (actually one could take (3.4)
where
H-1 (2)
u0eL2(Q)
.
= dual of
feL2(0,T,H-1(Q)
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
41
We define
(3.5)
U = (kf 4sH1(2x] 0,T[), 4, = 0
on
z,
By virtue of (3.3), A is an isomorphism from whose inverse is denoted by
(3.6)
A-1
J(0 = JO [1 a($) +
+
.
(x,0) = u0(x)
Ho (2)
onto
on
H-1(2)
We now set
a (A 1(f
-
)) - (f,o)] dt +
Im(T)12
where we have used the notation (3.7)
a(.*) - a(.k,,o)
.
We are going to check that inf J(o) = J(u), u
(3.8)
solution of (3.1)
,
qscU
the inf in (3.8) being attained at the unique element
u
Proof: we set
where
*
spans the set of functions in
H1(Qx)0,T()
such that
*=0 on E,*(0)=0. We have
(3.9)
J(.) = J(u) + K(*) + X(u,$)
(3.10)
K(V)
fO [- a(p) +
a
(A-1 at)] dt +
au X(u,*) = JT [a(u,*)-a(A 1(f - at)' 0
+ (u(T), 4'(T))
A-1
-
I41(T)I2
a )-(f,*)] dt +
s2}
42
J. L. LIONS
But from the first equation (3.1) we have
A-1
and
a(u, A-1
(f
at) = u _ (u, A A-1
at
X(u,*) =
)
= (u,
)
, so that
at)-(f,$)]dt +(u(T) , *(T))
JT 0
But taking the scalar product of the first equation (3.1) by
it
X (u,$) = 0 so that J(o) = J(u) + K(ey)
(3.11)
since
K(*) > 0 3.2
and
iff
K(ey) = 0
,p = 0 , we obtain (3.8)
Use of duality
Let us define
(3.12)
F(¢) =
a(,k)
on
HO(B)
r Then the conjugate function
(3.13)
a(A-1
F*(m*) =
F*
*)
of
F
is given on
H-1 (2)
by
,
and
JW = IT 0
(3.14)
4EU
CF(O)+F*(f - -al) - <4,f - >] dt +
Iu0I2
-
gives
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
It follows that Iu012
Iu012
and that JW =
(= J(u)) iff
2
F(,k) + F*(f - at
_< 0, f ->
i.e.
= derivative of
i.e.
=
i.e.
A-1 (f
- 1)
F*
at
f - at
43
Chapter 3
Asymptotic Methods 1.
Orientation
The aim of asymptotic methods in optimal control is to "simplify" the situation by asymptotic expansions of some sort. This can be achieved by one of the following methods: (1) simplification of the cost function - this is, for instance the case when the control is "cheap", cf. Section 2; (ii) simplification of the state equation, by one of the available asymptotic methods: (j) the most classical one is the use of asymptotic expansions in terms of "small" parameter that may enter the state equation,
i.e. the method of perturbations, in particular the method of singular perturbations; we refer for a number of applications in Biochemistry or in Plasma Physics to J. P. Kernevez [1], Brauner and Penel [1], J. Blum [1] and to the bibliography therein; cf. also Lions [7].
(jj) the homogenization method for operators with highly oscillating coefficients;cf. Section 3; (JJJ) the averaging method of the type of Bogoliubov-Mitropolski [1]; we refer to Bensoussan, Lions and Papanicolaou [1]; (iii) simplification of the "synthesis" operator by the choice of a particular feedback operator (in general on physical grounds) ;
We do not consider this aspect here; we refer to Lions [4], Bermudez [1], Bermudez, Sorine and Yvon [1];.it would be apparently of some interest to consider this question in the framework of perturbation methods. 2.
Cheap control.
An example.
2.1 Setting of the problem.
With the notations of Chapter 1, Section 3.1, we consider the state equation given by
45
46
J. L. LIONS
(2.1)
at + Ay = f
in
a = v
z
on
Q = 2x]O,T[
A
y(0) = y0
2
on
We consider the cost function
(2.2)
Je(v) = IE Iy(v)-zdl
where
c > 0
is "small"
2
dz + e IE v2 dE
.
This amounts to considering the control v as "cheap" - a situation which does arise in practical situations, where one often meets the case where acutally Let let
ue
Uad
e = 0
.
be a (non-empty) closed convex subset of
L2(z) , and
be the solution of
Je(uE) = inf
J&(v), ve Uad
(2.3)
ueeUad We want to. study the behavior of
ue
as
0
We shall see that this question is related to problems in singular perturbations. 2.2
A convergence theorem.
Let us set (2.4)
uc
y(ue) = ye
is characterized by
IE (Y,-ad)(y(v)-y e)dE + e IE ue(v-ue)dz >_ 0, Vve Uad (2.5)
u
e
U
ad
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
We define
0(v) = y(v) - y(O) (where here y(0) denotes the solution y(v) of (2.1) for v = 0) ; we have
{v) + Ac(v) = 0, in Q (2.6) avA A
0(v)It=O = 0 on 2
If we set (2.7)
0(uE) = 0e
(2.5) can be written
f
0E(0(v)-0E)dz + e fZ uE(v-u6)dz >_
(2.8)
'- fZ (zd-y(0))(0(v)-0E)dE
Let us consider the case when (2.9)
r = a2
is a
and let us write the Set which are zero for
(2.10)
t.;:: 0
- + A0 = 0
variety ,
C"
in
I
of all distributions
m
in
9xJO,T[
and which satisfy
52x3-m
T[
One can show (cf. Lions-Magenes [1], Vol. 3) that one can define, in a unique manner
{di (2.11)
- 1E } e D' (E) x D' (E) a J
D'(E) = space of distributions on
z ,
47
48
J. L. LIONS
the mapping
0 -
with the topology of
,
{O1z
at- iE } being continuous from a
D' (2x]-',T[)) -
f
(provided
D'(E) x D' (z)
We then define
(2.12)
K = 10 I ocj , 41z &L2(E), a
2=
I+ib
eL2(E)I
A
which is a Hilbert space for the norm (f2[[2+(b)2]dL)1/2
(2.13)
We define next (2.14)
Kad
Kad = (ol oeK, M,e Uad
is a closed convex subset of
K
With these notations, (2.8) is equivalent to
eKad
4. FE
(2.15)
(de9 0-06) 2 + L (E)
e)) 2 L (E)
(I"l0e)
(zd-y(0), 0-0 e) 2 L
Vme Kad
.
(E)
We can now use general results about singular perturbations in Variational Inequalities; using a result of D. Huet [1], we have:
me y 0o
where mD
in
L2(2)
as
e - 0 ,
is the solution of
(2.16)
(40,0--00) > (zd-y(D),m-0D) VOe Tad ,
#0FKad where
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
Kad = closure of
Kad
49
K
in
(2.17)
K - {ml 001.01Z
eL2{z)}
But if Proj_ = projection operator in
K
on %d , we have
Kad
(2.18)
00 = Proj_
(zd-y(0))
Kad
and going back to (2.19)
yE
one has:
yE -. Y(0) + Proj
_
(zd-y(0))
in
L2(z)
Kad
Remark 2.1
One deduces from (2.19) the convergence of
UE
in a very weak
topology. 2.3
Connection with singular perturbations
Consider now the "no-constraints" case. (2.20)
Then (2.18) reduces to
00 = zd-y(O)
so that
(2.21)
yE -. zd
in
L2(z)
which was easy to obtain directly. But since in general, considering not have
zd
to be smooth, one does
the convergence (2.21) cannot be improved
zdIt=O = yo'r , (no matter how smooth are the data) in the neighborhood of z
.
There is a singular layer around
t = 0
on
t - 0
on
z .
The computation (in a justified manner) of this type of singular layer is, in general, an open problem. We refer to Lions .[8] for a computation of a surface layer of
similar nature, in a simpler situtation, and for other considerations along these lines.
50
J. L. LIONS
3.
Homogeneization
A model problem
3.1
Notation: We consider in
Rn functions
y - ai .(y)
with the
following properties:
(3.1)
Jaij &L-(R n)
,
aij
is Y-per iodic, i.e. Y = ]O,y0 (x... X]O, Yo
a1
is of period
yo
in the variable
and
yk
E aij(y) riej ' a z qZ , a > 0 ,.a.e. in y ; for
e>O , we define the operator
Ac
by
n
A&o
(3.2)
az (aij(E)
)
.
J
Remark 3.1
The operator
is a simple case of operators arising in the
AE
modelization of composite materials; operators of this type have been the object of study of several recent publications; let us refer to de Giorgi-Spagnolo [1], I. Babuska [1] [2], Bakhbalov [11, BensoussanLions-Papanicolaou [2] and to the bibliography therein. The state equation We assume that the state
(3.3)
in
+ + At)yc = f
(3.4)
ay = v avAt
(3.5)
ye,t=0 = y0
on
z
on
,
2
The cost function is given by
y&(v)
is given by
Q = 2x]O,T[
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
(3.6)
Let
Je(v) = jE lye(v)-zdl2 dE + N jE v2dz, N>O,zde L2(s)
Uad
be a closed convex subset of
L2(E)
51
.
.
By using Chapter 1, we know that there exists a unique optimal control
ue
, solution of
Je(ue) = inf
(3.7)
Je(v), ve Uad, ue Wad
The problem we want to study is the behavior of 3.2
as
e -+ 0
The homogeneized operator
Let us consider first the case when
then that, .when
(3.8)
ay
= v
f
is fixed.
One proves
in
on
E
,
Q
,
A
ylt=O = yO A
v
0 ,
e
at+Ay=
and where
u
on
SZ
,
is given by the following construction.
One defines firstly the operator
(3.9)
for every
Al = - Day- (aij(y) aa)
j
one defines
constant, of
Al(X3-yj) = 0 (3.10)
X3 Y-periodic
and one defines next
XJ
on
Y
;
as the unique solution, up to an additive
J. L. LIONS
52
aij
al
(Xj-yj, X3-yj),
JYJ = measure of
Y
(3.11) a a-'-y- dy
ayi
Then n 2
A
(3.12)
i j=1
aij axa axj
which defines an elliptic operator with constant coefficients; called the homogenized operator associated to 3.3
be defined by (3.8); we define
J(v) = fE Iy(v)-zdl2 dz + N fE v2 dz
and let
u
be the unique solution of J(u) = inf
(3.14)
J(v), veUad' ue Uad
We have:
(3.15)
ue -+ u
in L2(E)
as
a-0.
Proof:
Let us set (3.16)
Since
(3.17)
is
A convergence theorem
Let us consider the "homogeneized control problem": let
(3.13)
A
A`
ye(ue) = ye, y(u) = y
J (v) >_ N fZ v2dz
Plle L2(E)
we have
<_ constant
and by virtue of the uniform ellipticity in (3.1), we have
y(v)
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
(3.18)
53
IlyEII
L2(O,T;H0(Q)) and also ay
(3.19)
Iat
sC
V2 L (o,T;H
1
(9))
It follows from (3.18) (3.19) that (3.20)
y6I
E compact set of
L2(E)
and we can extract a subsequence, still denoted by
(3.21)
(3.122)
uE - u
yE- y
in
L2(E)
a e Uad
weakly,
L2(0,T;H0(s2)) weakly,
in
ayE
uE, yE, such that
in
yEi2 -.yI,
L2(O,T;H- 1(2)) weakly,
in
L2(E)
Therefore (3.23)
lim inf JE(U) > JE Iy-zdl
2
dE + N JE (u)2dz = X
E-+0 But for every
v e Uad' we know that (cf. (3.8)) y&(v) . y(v) ay
L2(0,T;H1(s)) weakly and also that
(tv)
a
- at y(v)
weakly; therefore (3.24)
ye(v)l,: - y(v)l,
so that
(3.25)
JE(v) - J(v)
in
L2(E)
in
strongly
in
L2(0,T;H-1(2))
54
J. L. LIONS
Then the inequality
Je(ue)
X < J(v), vs
(3.26)
Je(v)Vve Uad
gives
Uad
But one can show that y
(3.27)
so that
X = J(u), hence (3.26) proves that Since
(3.28) Since
lim sup Je(ue)
J6(ue) -+ J(u) jE dye-zdl
2
u = u
J(v) Vv, we have
.
dE - fE Iy-zd(2 dE
(cf. (3.23)) it follows from
(3.28) that
N fE u2 dz - N jE U2 dE
(3.29)
Since
u
ue-+u
in
-t u
in
L2(E)
weakly, it follows from (3.29) that
L2(E)
strongly.
Remark 3.2
Let us consider the optimality system:
ate +As ye = f,- ate + (As)
a
a
ave As
pe = 0
= us,
= ye- zd
av
on
(Ac)*
y6(O) = y 0, p E{ T) = 0, o n S t together with
jE (p+Nu (3.31)
I
ue s Uad
(v-ue) dE
0 Vve Uad
in
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
Then, as
e - 0
55
,
L2(O,T;HO(8)) weakly,
in
(3.32)
in L2(O,T;Ho(Q)) weakly, (3.33)
where
ue
u
{y,p,u}
in
L2(z)
,
is.the solution of the "homogeneized optimality system"
Q
(3.34)
avA
= u, -'3p
avA*
-
y - zd
y(O) = y0, p(T) = 0
on
on
.
z
2
with Jfz (p+Nu)(v-u) dz >_ 0 Vve Uad' (3.35)
ue Uad
Remark 3.3
In the "no constraint" case, (3.31) and (3.35) reduce to
pe + Nue
0, p + Nu = 0
on
z
The optimality system can then be "uncoupled" by the use of a non linear partial differential equation of the Riccati type.
The
above result leads in this case to an homogeneization result for these non-linear evolution equations.
Chapter 4
Systems Which Are Not of the Linear Quadratic Type 1.
State given by eigenvalues or eigenfuncitons. 1.1
Setting of the problem.
Let
2
this is not indispensable) boundary,
Let functions
Rn , with a smooth (although
be a bounded open set in
a;1
be given in
r ;
R is supposed to be connected.
, satisfying
St
ai3 = aji &L"(Q), i.J = 1,...,n (1.1)
E aid{x} i:j
a > 0
>_ a E
a.e. in
s2
.
Let us consider, as space of controls:* (1.2)
U = L-(SZ)
and let us consider (1.3)
Uad
such-that
Uad = bounded closed convex subset of
L'(Q)
.
We then consider the ei envalue problem:
JAY+vYAY
in S2,
(1.4)
-Y = 0 -on r ; it is known (Chicco (1]) that the smallest eigenvalue in (1.4) is simple and that in the corresponding one-dimensional eigen-space there is an eigenfunciton
0
We therefore define the state of our system by (1.5)
.where
(y(v), a(v)}
X(v) = smallest (or first) eigenvalue in (1.4), and Ay(v) + vy(v) = a(v) y(v)
(1.6)
y(v) >_ 0
in
Sz
Iy(v)l = 1 (1-1 = L2norm)
57
in
s?
, y(v) = 0
on
r
58
J. L. LIONS
The cost function is given by (1.7)
J(v) = I.9 ly(v)-zdl2 dx
and the optimization problem we consider consists in finding (1.8)
inf J(v), ve Uad '
1.2
Optimality conditions.
It is a simple matter to see that
v - {y(v), x(v) } is continuous from
U
weak star
(1.9)
into H1(2)
weakly x R
.
Indeed
a(0) + I2vo (1.10)
x(v) = inf
2
dx
2 Im
where
aij(x)
a-
a
Therefore if (1.10) that
x(vn)
vn
v
in
dx
L°°(s) weak star, it follows from
is bounded, hence y(vn)
is bounded in
H,(Q)
can then extract a subsequence, still denoted by y(vn), x(vn)
that y(vn) - y in L2(2)
weakly and x(vn) .+ x .
strongly, and we have
Ay+vy=xy,y=0 on r,
Y'0, IYI=1 so that
y = y(v), x = x(v)
But
; we
such
y(vn) -P y in
59
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
it immediately follows from (1.9) that
r there
exists
ue Uad (not necessarily unique) such that
(1.11)
J(v) = inf J(v), vc Uad
We are now looking for optimality conditions. course to study the differentiability of make first a formal computation.
(1.12)
The main question is of
v - {y(v), x(v)}
.
We set
y(v0 + v)Ie=o = y, d (v0 + rv)1t=0
assuming, for the time being, these quantities to exist.
(1.6) v by v0+ v and taking the Ay + v 0
S
Replacing in
derivative at the origin, we find
y + v y(v0) = x(v0) Y + .Y(v0)
i.e.
(1.13)
AY + v0Y - x(v0)Y = -vy(v0) + xy(v0)
Of course (1.14)
Since (1.15)
on
,y = 0
Iy(v)I =
1
r
we have
(y, y(v0)) = 0 .
Formula (1.10) gives (1.16)
X(v) = a(y(v)) + f2 v y(v)2 dx
hence (1.17)
Let us
i = 2a(y(v0),.Y) + 2 f. v0 y y(v0)dx + f. v y(v0)2 dx
J. L. LIONS
60
But from the first equation (1.6) with the scalar product with
v = v0 we deduce, by taking
y
a(y(v0).y + I2 v0y(v0)Y dx = a(v0) I9 y(vo)y dx =
= (by (1.15)) = 0 so that (1.17) gives (1.18)
a = f9 v y(v0)2 dx
The derivative
.
6,a} is given by (1.13) (1.14) (1.15) (1.18)
Remark 1.1 Since
a(v0)
is an eigenvalue of
A + v 01
solution iff (-vy(v0) + a y(v0), y(v0)) = 0
(1.13) admits a
,
which is (1.18).
We can now justify the above calculation: {y(v), k(v)}
(1.19)
with values in
is Frechet differentiable in
L"(9)
D(A)x R
V
where (1.20)
D(A) = {oI 4e HO(ST), Ale L2(2))
This is an application of the implicit function theorem (cf. Mignot Cl)); we consider the mapping
(1.21)
1b,a,V _._L.
+ v4 - 4
D(A) x Rx U - L2(2)
.
This mapping, which is a 2d degree polynomial, derivative of F with respect to o,A at
(1.22)
q,?
(A
is C"
oo, a0, v0
The partial
.
is given by
+v0-X0)O -40
We consider S1
= unit sphere of
(D(A)nS1)x Rx U
.
L2(SZ)
If we take in (1.22)
and we restrict
F
to
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
00 = y(v0), x0 = x(v0)
(1.23)
then (1.22) is an isomorphism; therefore by applying the implicit.
function theorem, there exists a neighborhood in (D(A)f)S1)x Rx U
and there exists a
of y(v0),",v,
yxAxU
v,
function
C"
v - {K1(v), K2(v)} (1.24)
U -ii Y x A such that F(K1(v), K2(v), v) = 0, ve U , (1.25)
K1(v0) = y(v0), K2(v0) = x(v0)
We have (y(v0),x(v0,v0) y + ax (y(v0),x(v0),v0)i.
+ av (y(v0),x(v0), v0) = 0
which gives (1.13), hence (1.18) follows and (1.16) (1.15) are immediate.
N u
is
instead of
v0
We are now ready to write the optimality conditions: if an optimal control then necessarily (1.26)
(J'(u), v-u) >_ 0 We Uad
We introduce
with
v-u
instead of
v
,
anti
i.e.
Ay + uy - x(u)
'
= - (v-u) y(u) + x y(u)
(y, y(u)) = 0 , (1.27)
j2 (v-u) y(u)2 dx
y=0 in
r.
u
62
J. L. LIONS
Then (1.26) becomes (after dividing by 2), if y(u) = y (1.28)
f (Y-zd) ., dx >_ 0
Vve Uad
In order to transform (1.28) we introduce an adjoint state {p,µ} such that
Ap + u p - X(u)p = y-zd + µy (1.29)
p = 0 on r
;
(1.29) admits a solution iff (1.30)
(1+0 1Yl2 = (Y,zd).
We uniquely define (1.31)
by adding the condition
p
(p,y) = 0
Then taking the scalar product of (1.29) with
j'
, and since
(y,y) = 0,
we obtain
IQ (Y-zd) Y dx = Ap+up-a(u)p,Y) _
_ (p,AY+uY-a(u).Y) _ (p,(v-u)Y) + i(p,y) _ - (p,(v-u)Y)
so that we finally obtain the optimality system: in order for
u
an optimal control it is necessary that it satisfies the following system, where
y(u) = y
Ay + uy = X(u)Y, y '- d, IYI = 1 (1.32)
Ap + up - x(u)p = Y(Y,zd) - zd, (p,Y) = 0,
y, p = 0 and
on
r
to be
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
;fQ py (v-u) dx >_ 0
63
Vve Uad'
(1.33)
ue Uad
I
'
Let us also remark that the system (1.32) (1.33) admits a solution. 1.3
An example.
The following result is due to Van de Wiele [1].
We consider the
case: (1.34)
Uad= {vI k0
<_ kl a.e.}, kieR
.
Then (1.33) is equivaleant to:
py0 if (1.35)
(the sets
xe521<=>u(x)=k1,
py
0
if
xc SZO <_> u(x) = ko
py
0
in
52\(S20 U S21 )
O, 521
are defined up to a set of measur:: 0
But it is known that - for a 2d order elliptic operator
(1))
-y(x)>0
a.e. so that (1.36) actually reduces to
(1.36)
p >_ 0
on
521,
ps0
on
520
p=0
on 2
(go U 21)
We are going to conclude from this result that
f if (1.37)
zd is not an eigenfunction for A+uI, and if u is any
optimal control, then necessarily ess sup u = k1, ess inf u = k0 .
(1) One can define more precisely these sets up to a set of capacity 0
J. L. LIONS
Suppose on the contrary that, for instance, ess sup u < kl Then one can find
such that
k0 < u + k < kl
,?.38)
E.,t
k > 0
y(u+k) = y(u), X(u+k) = x(u)+k
and
u+k
is again an optimal
control; we have therefore similar conditions to (1.36), but now, by virtue of (1.38), the analogs of ;(,,+k) = 0
-:1uded.
in
and
20
s21
are empty and therefore
i.e. (cf. (1.32) y(y,zd) = zd , a case which is
2 ,
Therefore ess sup u = kl
.
Another example of a system whose state is given by eigenvalues or eigenfunctions. 2.1
Orientation
We give now another example, arising in the operation of a reactor.
For a more complete study of the example to follow,
together with numerical computations, we refer to F. Mignot, C. Saguez and Van de Wiele [1]. 2.2
Statement of the problem
The operator.A is given as in Section 1.
v&L-(52), 0 < k0 < v(x) < kl a.e.
Uad = {vI
(2.1)
The state
{y(v), X(v))
is defined by
Ay(v) = a(v) v y(v)
in
2
(2.2)
y(v) = 0
on
r
X(v) = smallest eigenvalue, (2.3)
Y(v) and
y(v)
(2.4)
0
in
2 ,
is normalized by (y(v),g) = 1, g given in L2(St)
We set (2.5)
My(v) _
We consider
!g y(v) dx
.
in
i2)
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
65
and we define the cost function by (2.6)
J(v) = f2ly(v) - My(v)1
2
dx
We are looking for (2.7)
inf J(v), vc Uad
.
Remark 2.1
In (2.4) one can take more generally (2.8)
geH-1(2)
In particular if the dimension equals 1, we can take
g = z Dirac measures (cf. Saguez [1])
(2.9)
Remark 2.2
The above problem is a very simplified version of the operation of a nuclear plant where
y(v)
corresponds to the flux of neutrons
and where the goal is to obtain as smooth a flux as possible, which explains why the cost function is given by (2.6). 2.3
Optimality conditions
As in Section 1 we have existence of an optimal control, say- u, in general not unique.
We prove, by a similar argument to the one in Section 1, that-
is Frechet differentiable from Vad
v - y(v), a(v) set (2.10)
y(u) = y, a(u) =
I
y = °
d
we obtain from (2.2)
A ,
D(A) x R .
If we
J. L. LIONS
66
(A-xu) y = (au+x(v-u))Y,
r,
y=0 on (2.12)
(Y,g) = 0, i j) uy2 dx + x j., (v-u) y2 dx = 0
The optimality.condition is (2.13)
(y-My, Y-M(Y)) > 0
But
(y-My, My) _ MY-My),y) = 0
Vve Uad
so that (2.13) reduces to (y-My,Y)
(2,.14)
0
Vve Uad
We define the adjoint state
{p,µ}
(A-xu)p = y - My + pg,
, by
p = 0
on
r
,
(2.15)
(p,g) = 0
where
µ
is such that (2.15) admits a solution, i.e. (y-My,y) + F+(g,y) = 0
(2.16)
i.e.
µ = - (y-MY,Y)
Taking the scalar product of (2.15) with that
,y
and using the fact
(g,,') = 0 , we have
(y-M(y,Y) = ((A-xu)P,Y) = P,(A-xu)Y)
= ((iu+x(v-u))Y,P) replacing
x
by its value deduced from the last equation in (2.12), we
finally obtain j.9 LP -
y,uy Y] y (v-u) dx > 0
(2.17)
ue Uad
Vve Uad,
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
Therefore, if
67
is an optimal control, then one has
u
(A-),u)y = 0,
(A-au)P = y - My - (Y-My,Y)9, (2.18)
(9,Y) = 1, (9,P) = 0,
y=p=o and (2.17).
on
r
#
We go one step further, by using the structure of (2.1).
We introduce, as in Section, 1.3,
si =
( Ix (
u(x) = ki }, i = 0, 1 ,
St\(QoUS2I}
and we observe that (2.17) is equivalent to
iMY Y) s 0 y(p - Y,u
on
Q,,
0
on
5Z0,
Y(P - y;uy Y)
Y(P- Y,uy y) =0 on 2\(Q But since
(2.19)
y > 0
a.e. this is equivalent to
p - y,uy u y<0
on
stl
0
on
909
p -
Y,uy Y
lY.uY1
y = 0 on 9\(20U21)
We deduce from this remark that
OU21).
Uad
given by
68
J. L. LIONS
if
g e H-1(2)
and
g J L2(2) (and even if g j HI(g),
(2.20)
g = constant on r) then ess sup u = kl, ess inf u = k0
Proof : Suppose for instance that
ess sup u < kl
k0 < cu(x) < kl
(2.21)
a.e.
in
Y(au) = y(u), a(vu) = oa(u)
But
.
Then we can find
such that
a > 1
2 .
so that
ou
is again an optimal
control and therefore one has the analog of (2.19) but this time with 20
and
empty; i.e.
2l
(2.22)
y
p -
0
a.e.
in
y.Uy
Q
From the first two equations in (2.18), we deduce from (2.22) that (2.23)
y - M(y) = (y-My,y)g
a.e.
in
52
hence the result follows, since (2.23) is impossible under the conditions stated on
g
in (2.20).
Remark 2.3
All what has been said in Sections 1 and 2 readily extend to other boundary conditions of the self-adjoint type. 3.
Control in the coefficients 3.1
General remarks
We suppose that the state of the system is given by
(3.1)
x(3.2)
where
(3.3)
- E aai (v(x) aY-)- f in 2, feL2(2), y = 0
on
r
ve Uad
Uad =
vsL (a), 0 < k0 <_ v(x) _< kl
a.e. in
Q}
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
69
Of course (3.1) (3.2) admits a unique solution y(v)6HI(2) It is generally not true that L`(s) weak star
HO(R)
v + y(v)
is continuous from
weakly and there are indeed counter-examples
(cf. Murat [1] [2]) showing that, for cost functions of the type (3.4)
J(v) = !y(v)-zdl2
there does not exist'an optimal control. Remark 3.1
The control appears in the coefficients of highest order in the operator; when the control appears in lower order terms, the situation is much easier.
Remark 3.2
Problems of optimal control where the control appears in the highest order derivatives are important in many questions; we refer to the book Lurie [1].
Orientation
In what follows we consider a situation (cf. Cea-Malanowski [1]) when
J(v)
is of a special form, implying continuity of
J
for the
weak star topology.
3,2 An example We suppose that the cost function is given by J(v) _ (f,y(v))
(3.5)
.
Remark 3.3
One can add constraints to (3.3), of the type (3.6)
Is va(x)dx In case
f = 1
given,
a given > 0
or < 0 integer'.
, the problem is to find the composition of
materials such that the rigidity of the plate is minimum, or maximum if one looks for the sup of
J(v) , where
v
is subject to (3.3), and
also possibly to a condition of the type (3.6).
J. L. LIONS
70
According to Remark 3.3 it is of interest to consider the two problems, respectively studied by Cea-Malanowski [1] and by KlosowiczLurie [1]: (3.7)
inf J(v), ve uad
(3.8)
sup J(v), ve Uad
The main point in solving (3.7) is the following: J(v)
v -
(3.9)
is lower semi continuous from
Uad
provided with
the weak-star topology -. R . Proof. vn
Let (3.10)
v
y(vn) = Yn
in the weak star topology of
C(2)
.
We set
.
We have
(3.11)
IlYn11
l
5 C.
HO(St)
therefore we can extract a subsequence, still denoted by yn such that
(3.12)
yn
but in general
y in Hp(st) y ¢ y(v)
special structure of
.
J(J)
J(vn) - J(v) = f
weakly, We observe now (and this is where the very comes in) that vn
)grad(yn-y(v))12 dx
(3.13)
- tQ (vn-v) (grad y(v)l 2 dx
Indeed, if we compute the right hand side, we obtain
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
71
vn grad yn grad y(v) dx
f.9
(3.14)
+ f2 vigrad y(v)I2 dx
but from (3.1) with
;
v = vn, y = yn, we obtain
f.9 vn grad yn grad y(v) dx = f,Q f y(v) dx
= jr. vIgrad y(v)j2 dx = J(v)
so that (3.14) actually equals Since (3.15)
J(vn) - J(v)
vn > 0 (since vn, v c Dad) ; it follows from (3.13) that
J(vn) >_ J(v) - f2 (vn-v) Igrad y(v)l 2 dx
and since
vn-v - 0
in
L"(SZ) weak star and since
fixed L1 function, fS2 (vn-v)Igrad y(v)l 2dx -. 0
lim inf J(vn)
(3.16)
i.e. (3.9).
grad y(v)l2
is a
and (3.15) implies
J(v)
#
It immediately follows from (3.9) that (3.17)
problem (3.7) admits a solution.
#
Remark 3.4
We refer to Cea-Malanowski, loc. cit, for further study of problem (3.7), in particular for numerical algorithms. Remark 3.5
The existence of an optimal solution in problem (3.8) seems to be open; the proof presented in Klosowitz-Lurie loc. cit. does not seem to be complete, but this paper contains very interesting remarks on the necessary conditions satisfied by an optimal control, assumi.'tg it exists.
#
72
J. L. LIONS
Remark 3.6 cf. also Barnes [1] (these Proceedings). 4.
A problem where the state is given by an eigenvalue with control in the highest order coefficients.
Setting of the problem
4.1 In
(4.1)
L«(2)
we consider the open set
U = {vJ veL-(2), v >_ c(v) > 0
depends on For every (4.2)
Let
v e U Av
k
defined by
a.e. in
2 , where
c(v)
v}
we define the elliptic operator
_ - E az
be given
U
> 0
.
Mx) a
)
.
We define the state
y(v)
as the first
eigenfunciton of the problem Av y(v) = x(v) 1v+k) y(v)
in
2 ,
(4.3)
y(v) = 0
where
on
r ,
(we can normalize y
ly(v)l = 1)
by
x(v) = smallest eigenvalue, i.e.
f2vIgrad012 dx (4.4)
0(v) =
inf f2(v+k)
qseH1(2)
2
dx
We consider the cost function (4.5)
J(v) = f. v dx
and we want to minimize constraint (4.6)
x(v) = x(1)
J(v)
over the set of v's in
U
subject to the
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
73
Remark 4.1
This problem has been considered in Armand [1], Jouron [1].
In
the applic4tion to structural mechanics, n=2 , v corresponds to the width of the structure, and we want to minimize the weight for a first eigenvalue fixed, equal to the eigenvalues of the structure with uniform width equal to 1
.
Remark 4.2
One will find in Jouron, loc. cit, the study of the analogous problem under the added constraint v(x) >_ c > 0, c fixed.
(4.7)
4.2
Optimality conditions
We see, as in Section 1, that in the open set
U
the functions
is Frechet differentiable with values in
v - (y(v), ),(v)}
Hl(2) x R
If we set
JY
dr.
(4.8)
a =
1Y(v) = y, X(u) = A (u arbitrarily fixed for the time being), we obtain:
Au y + Av y = A(u+k) y+ A v y+ i(u+k) y i.e.
(Au - X(u+k))y = av y - Avy + i(u+k)y (4.9)
y=0 on t This is possible iff the right hand side is orthogonal in
L2(2)
to
hence (4.10)
a f
(u+k)y2 dx = !Q v[Igrad
yj 2
If we assume that there exists the set (4.6), then there exists that
e R
- a y2] dx
u c U
which minimizes (4.5) on
(Lagrange multiplier) such
y,
74
J. L. LIONS
(4.11)
(J'(u),v) + r, a = 0 Vv
i.e., using (4.10):
(4.12)
1
=
1
[Igrad yj
2
- X y2] = 0
fQ(u+k)y dx in (4.12) (4.13)
a=a(u)=a(1) (grad
yI2 -
=
Since
,
so that (4.12) can be written
X(1)Iy(2 = constant = cl
.
fQu)grad yJ2 dx
x(l)
we easily find that
fQ(u+k) y2 dx
(4.14)
N
cl = 12uldx > 0
2
We are going to check that, reciprocally;
if
u e U , with
y = y(u)
is such that
.(u)
satisfies
(4.15)
1 Igrad yJ2 - a(l)lyJ2 = cl = positive constant then
u
is an optimal control
Proof:
Let us multiply the equality in (6.15) by (v-u) and integrate over
Q ; we obtain cl[J(v)-J(u)] = JQvlgrad yJ2 dx - a(1) f2(v+k)lyl2 dx - [fgu Igrad yI2 dx - a(1)fQ (u+k)
IyI2 dxj
= IQvlgrad yI2 dx - a(1) f2(v+k) Iy12 dx >_ 0 (by (4.4))
5.
Control of free surfaces. 5.1
Variational Inequalities and free surfaces
Let
Q
be a bounded open set in
a(o,*) be given on
Rn
HQ(Q) (to fix ideas) by
and let a bilinear form
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
a(q,*) = E f a..(x)
as dx +
ax
f2a
i
(5.1)
a0, aiJ eL"(Q), ai eL"(g)
75
a
'
.
We assume that
a(A,4)
(5.2)
a > 0, oc
aI1kl12,
Hi
0(S2)
where (5.3)
Let
IIqs11
K
in
= norm of k
H1(S2)
be given such that
(5.4)
K
is a (non-empty) closed convex subset of
Then it is known (cf. Lions-Stampacchia [1]) that if H-I(s2)
, there exists a unique
y
f
is given in
such that
(5.5)
a(Y,0-Y) > (f ,4-Y)
VocK ;
(5.5) is what is called a Variational. Inequality (V.I.). Remark 5.1 If we get
(5.6)
y = y(f)
IIy(fl)-y(f2)II
, we have
c
11f1-f211 -1 H
(52)
Remark 5.2
In the particular case when
(5.7)
a,
is symmetric:
V4i,4reHl(2)
then finding (5.8)
y
satisfying (5.5) is equivalent to minimizing
a(,k,o) - (f,1)
over
K
then the existence and uniqueness of y
in (5.5) is immediate.
dx:
J. L. LIONS
76
Example 5.1
Let us suppose that g
1K
a.e. in
21 , g given such that
(5.9) K
is not empty
Then one can, at least formally, interpret (5.5) as follows; if we set in general (5.10)
then
Ao _ - z aai
y
(aid
as )
+ E ai
ax
- + a0 0
should satisfy
AY-f>0 , y - g
(5.11)
0
(Ay - f) (y-g) - 0
in
Q
with
y = 0
(5.12)
on
r
We can think of. this problem as "a Dirichlet problem with an
obstacle", the "obstacle" being represented by
g
.
The contact region is the set where y(x) - g(x) = 0, xes
(5;13)
;
outside the contact region we have the usual equation Ay = f
(5.14)
where
represents, for instance, the forces.
f
The boundary of the contact region is a free surface. one has
y = g
and
axi
Formally
= N- on this surface. axi
Remark 5.3
For the study of the regularity of the free surface, we refer to Kinderlehrer [1] and to the bibliography therein.
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
77
Remark 5.4
For a systematic approach to the transformation of free boundary problems into Y.I. of stationary or of evolution type, we refer to C. Baiocchi [1] and to the bibliography therein. Remark 5.5
Actually it has been observed by Baiocchi [2] [3] that one can transform the boundary problems arising in infiltration theory into uq
asi Variational Inequalities (a notion introduced in Bensoussan-Lions
[1] [2] for the solution of impulse control problems).
There are many interesting papers solving free boundary problems by these techniques; cf. Brezis-Stampacchia [1], Duvaut [1], Friedman [1], Torelli [1], Conmincioli [1] and the bibliographies of these works. 5.2
Optimal control of Variational Inequalities
We define the state
of our system as the solution of the
y(v)
V.I. (with the notions of Section 5.1): y(v)eK, (5.15)
a(y(v), d-y(v)) '- (f+v, o-y(v))
VoeV
where (5.16)
v = control function.
ve U = L2(2),
The cost function is given by (5.17)
(where
2
J(v) = ly(v)-zdl
Jm = norm of
o
+
N1vi2
in
L2(2)).
The optimization problem is then: (5.18)
inf J(v), ve Uad = closed convex subset of
U
It is a simple matter to check that (5.19)
there exists
us Uad
such that
J(u) = inf J(v)
J. L. LIONS
78
Remark 5.6
For cases where we have uniqueness of the solution of problems of this type, cf. Lions [6].
#
Remark 5.7
One can think of prob;em (5.18) as an optimal control related to the control of free surfaces.
would be to try to find surface (in case
K
In this respect a more realistic problem minimizing the "distance" of the free
ve Uad
is given by (5.9)); cf. Example 5.1, Section 5.1)' This type of question is still largely open.
to a given surface. cf. also Section 6.
We assume from now on that
K
is given by (5.9).
It follows
from (5.6) that 5.20)
IIY(v1) - Y(v2)Il
c
Iv1-v21
so that, by a result of N. Aronszajn [1] and F. Mignot [2], the function
is "almost everywhere" differentiable (an extension
y(v)
v
of a result of Rademacher, established for
Rn ).
We set formally (5.21)
and we set
Y =
Y(u+(v-u))1t=0
y(v) = y ; a necessary condition (but this is formal since
we do not know if
u
is a point where
y
is differentiable; for
precise statements, cf. F. Mignot [2)) of optimality is (5.22)
(y-zd,y) + N(u,v-u) ? 0
The main point is now to see what
vve Uad
jy
satisfies:
f Ay - (f+u) >_ 0, (5.23)
y - g >_ 0, l (Ay - (f+u)) (Y-g) = 0
.
looks like.
The optimal state
y
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
79
Let us introduce
Z = set of x's in 4 such that (5.24)
y(x) - g(x) = 0
(Z is defined up to a set of measure 0
Then one can show that, at least "essentially":
rq=0 (5.25)
on 'Z
Aq = v-u
y=0
on
,
on 2\Z r
.
This leads to the introduction of the adjoint state by
p=0 (5.26)
on
Z
,
A* p = y - zd
p-=0
on
On
4 \Z
r.
Then
(Y-zd, y) _ (p, v-u)
so that (5.22) becomes (5.27)
(p+Nu, v-u)
0
the optimality system is'(formally) given by (5.23) (5.26)
Conclusion:
N
(5.27).
Example 5.2 Let ug assume that (5.28)
Uad = U .
Then (5.27) reduces to (5.29)
vve Uad, ue Uad
p + Nu = 0
80
J. L. LIONS
so that the optimality system becomes:
Ay + N p -. f >_ 0,
y-9'-0, (Ay +
p - f) (y-g) = 0
2
in
(5.30)
= 0
on
Z
(defined in (5.24))
A*p=y - zd
R\Z,
on
We introduce a bilinear
Let us give another form to (5.30). form on
0 = H0(2) - HO(Q)
by
A(y,p;,O,*) = a(y,o) + N a*(p,*) + . (p,,) - -
(5.31)
(y,4y)
where
(5.32)
a* (0,+y) =
We observe that
A(y,p;y,p) = a(y,y) +
..(5.33)
a*(p,p)
c[IIYII2 + IIPII2]
N
Given (5.34)
4
in
H1(2)
we set *(x) - g(x) = 0
Z(4s-g) = set of x's in 2 such that
Then (5.30) can be formulated as: (.
A(y,p;o-y,*-p)
z
(5.35)
d0,4 c
y=0
such that
Z(y -g)
on
p41
(5.36)
y,pEk, y ? g,
0
on
Z(y-g)
This is a quasi-variational inequality.
#
,
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
5.3
81
Open questions
Due to Remark 5.5, it could be of some interest to study
5.3.1
the optimal control of systems governed by quasi-variational inequalities.
Even after the interesting results of Mignot [2] for the
5.3.2
optimal control of stationary V.I., many questions remain to be solved for the control of Y.I. of evolution. Let us give now an interpretation (cf. Bensoussan-Lions
5.3.3
[2], [3]) of
y(v)
when
K = {k14 < 0 on
(5.37)
22}
and, to simplify the exposition,
fn grad 0 grad ,ydx + E f2 g,(x)
a(m,*) _
dx
xj
(5.38)
+f2addx, where the gj's are , say, in C1(sy) (in order to avoid here any technical Then
difficulty).
, the solution of the corresponding V.I.
y(x;v)
(5.15), can be given the following interpretation, as the optimal cost of a stopping time problem.
We define the state of a system, say
zx(t)
, as the solution of
the stochastic differential equation: dzx(t) = g(zx(t))dt + dw(t), (5.39)
zx(0) = x, x652 where
g(x) = {gj(x)}, and where w(t) is a normal Wiener process in Rn
In (5.39) we restrict Let
A
t
to be a.s. <_ Tx = exit time of 2
be any stopping time, < Tx
We define the cost function (5.40)
.
Vx(e) - E fe eat [f(yx(t)) + v(yx(t))]dt 0
Then (Bensoussan-Lions, loc. cit.)
82
J. L. LIONS
(5.41)
inf yx(e) = y(x;v) e1--r
x
nuestion:
is it possible to obtain a result of the type (5.25)
by using (5.41)? 6.
#
Geometrical control variables. 6.1
General remarks
Geometrical control variables can appear in several different ways.
Among them: (i)
the state can be given by a state equation
which contains
Dirac masses at points which are somewhat at our disposal; (ii)
the control variable can be the domain itself where we
compute the state.
#
In the direction (i) we confine ourselves here to refer to Lions [1] [3], Saguez [1] for linear systems cf. also Vallee [1].
Another type of problems containing Dirac masses (all these questions are interesting also for practical applications) is considered in Amouroux [1] and Amouroux and Babary [1]. For non-linear systems, this leads to very interesting questions, also about Partial Differential Equations! mentioned to the author by Komura [1].
Problems of this type were
We refer to Bamberger [1],
Benilan and H. Brezis [1].
In the direction (ii) the first (and actually the main!) difficulty lies in a convenient definition of the domains.
If one
parametrizes smoothly the boundaries of the admissible domains then at least as far as existence is concerned there are no great difficulties; cf. Lions [9].
The most general results for the largest possible
classes of domains seem to be at present those of
Chenais [1] [2].
An interesting idea for representing the domain is due to J. Cea [1] [2], with an explicit application in J. P. Zolesio [1]. Assuming this done, or assuming everything is smooth, the next step is to obtain necessary or necessary and sufficient conditions of optimality.
(Let us remark that the approach of Cea, loc. cit.,
simultaneously gives conditions of optimality).
A systematic account
of this type of problem, with 'several interesting applications, is
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
given in Pironneau [1]; cf. also Pironneau [2], these the bibliography therein.
83
proceedings, and
For extensions of the Hadamard's formula
(variation of Green's formula in terms of the variation of the domain), cf. Murat-Simon [1], Dervieux-Palmerio [1]. 6.2
Open questions
6.2.1
It would be interesting to extend Hadamard's formula to
variational inequalities.
A (very partial) attempt toward this goal is
made in Lions [10]. The following question (which seems very difficult) is
6.2.2
motivated by a paper of Nilson and Tsuei [1] (which presents a much more complicated situation.
Let us consider a family of surfaces
parametrized in some way, where Let us define surface
rl
2(v)
r(v)
lies between NO and
as the open set between
r(v)
r(l)
and a fixed
.
In the domain
2(v)
r(v)
we consider the free boundary problems
84
J. L. LIONS
Ay(v) - f >_ 0, (6.1)
y(v) - g '- 0,
I where
f , g
(Ay(v) - f) (y(v) - g) = 0 are given in
operator given in
2(0)
;
2(0)
and
in (6.1)
conditions that we do not specify. (cf. Section 5.1), denoted by
S(v)
in
A
y(v)
s(v)
is a second order elliptic Is subject to some boundary
This V.I. defines a free surface .
The general questions is: what are the surfaces can approximate by allowing and
r(1) ?
r(v)
S(v)
to be "any" surface between
that one r(0)
(Notice the analogy between this problem and a problem of
controllability).
Chapter 5
Remarks on the Numerical Approximation of Problems of Optimal Control 1.
General remarks.
Methods for solving numerically problems of optimal control of distributed systems depend on three major possible choices: (i)
choice of the discretization of the state equation (and
the adjoint state equation), both in linear and non-linear systems; (ii)
choice of the method to take into account the constraints;
(iii) Choice of the optimization algorithm. Remark 1.1
If the state is given (as in Chapter 4, Section 1) by the first
eigenvalue of course (i) should be replaced by the choice of a method to approximate this first eigenvalue.
#
The two main choices for (i) are of course (il)
finite differences;
(i2)
finite elements.
The main trend is now for (i2) and we present below in Section 2 a mixed finite element method which can be used in optimal control.
There are many ways to take into account the constraints, in particular, (iil)
by duality or Lagrange multipliers;
(ii2)
by penalty methods.
Remark 1.2
An interesting method (cf. Glowinski-Marocco [1]) consists in using simultaneously Lagrange multipliers and penalty arguments. Remark 1.3
One can also consider the state equation, or part of it (such as the boundary conditions) as constraints and use a penalty term for them (cf. Lions [1], Balakrishnan [1], Yvon (3]). The algorithms used so far for (iii) are: (iiil)
gradient methods in particular in connection with (il);
85
J. L. LIONS
86
(iii2)
conjugate gradient methods in particular in connection
with (i2); (iii3)
algorithms for finding saddle points such as the Uzawa
algorithm.
Remark 1.4
All this i; also related to the numerical solution of Variational Inequalities for which we refer to Glowinski, Lions,
Trfmolieres in. Mixed finite elements and optimal control.
2.
2.1
Mixed variational problems.
We first recall a result of Brezzi [1], which extends a result of Babuska [3].
(cf. also Aziz-Babufka [1].)
Let
be real
4,1, 412
Hilbert spaces, provided with the scalar product denoted by ( (and the corresponding norm being denoted by ll and
b
Ili
,
i=1,2)
.
,
)i
Let
a
be given bilinear forms:
(2.1)
is continuous on
41''1 - a(41,*1)
t1 "
1 b(41'''2) is continuous on 4l x 42
(2.2)
We shall assume throughout this section that the following hypothesis hold true: we define
B e L'(41;4'2)
by
(2.3)
= b(41,,2)
we assume (2.4)
a(41,41)
0 4 1e41
(2.5)
a(41,41)
c,111.1111
(2.6)
sup 01
lb(41'*2) Ih41 1
1
, a>O, V41e
KFr B
--c11*2112,c>0
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
87
Remark'2.1
If we introduce
(2.6)'
B*cL(02;01) , then (2.6) is equivalent to
IIB*4r2114,1> c 11*2112vy2 E 42
.
We now set (2.7)
n(o;tr) = a(01.*1) + b(+r1.02) - b(61.$2)
4) _ t1 x 4,2
where
on
4' x 4,
an
a , we look for
.
Problem: given a continuous linear form
L(*)
e' such that
(2.8)
n(,O;4) = L(4V) V*&$
This is what we call a mixed variational problem. refer to Brezzi, loc cit. and to Bercovier [1].
For example, we The result of Brezzi
is now:
under the hypothesis (2.4) (2.5) (2.6) problem (2.8) admits a unique solution and
(2.9)
11,0114)
<_ cl IILII,
The idea of the proof is as follows: we observe first that by virtue of (2.6)
find
B
is an isomorphism form
ScL(4'2;4'1)
If we define AFL(4'1;4'1') (2.10)
onto
4'1/KerB
identity.
such that
by
= a(ol.*1)
and if we write (2.11)
L(i<) = L1(y1) + L2(*2). Lie*'i
9
t2'
, so that we can.
J. L. LIONS
88
then (2.8) is equivalent to (2.12)
A41 + B*02 = L1
(2.13)
-B41 = L2
If we set
D1 = -SL2 , we have
-B(oD1) = 0
i.e. z=ml-Dle Ker B
and (2.12) is now equivalent to Az + B*b2 = L1 - AD1
(2.14)
= gl
But by virtue of (2.5) A is an isomorphism from (Ker B) - (Ker B)' B
is an ismorphism for
from
p1/Ker B But
t'2 ~ (('1/Ker B)'
B*
so that
is an ismorphism
_ (Ker B)' + ('l"Ker B)'
; and then
(B*)-1
z = A Ihl, 02 =
2.2
-t'
'P2
k1
Regularization of mixed variational problems
We now follow Bercovier [1].
We define a regularized form
nP,e
of
n
by
(.k;4f) = n(O;4V) = n(o;V') + p(o1 *1)l + e(02,4r2)2 (2.15)
P,e>0. We remark that, from (2.4) and (2.7) we have (2.16)
0 V&
so that
(2.17)
p11011112.+ e11021122
Therefore there exists a unique element (2.18)
0p ,e = 3 6 'p
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
89
such that
ff(3;4) = L(ay) V* e 4.
(2.19)
We are now going to prove: assuming that one has (2.4) (2.5) (2.6), and for and
(2.20)
p
small enough, one has
e/p
c(p+ ) IILII,,' The proof of (2.20) is in two steps.
(2.21)
We introduce
y) = n(4;*) +
nP(4;*) =
We remark that this amounts to replacing P(Ik1'*l)l
elenent
(2.22)
and leaving
0P _ ¢e$
b
invariant.
such that
f($;*) = L(+) vs
4
We are going to show (2.23) (2.24)
Cp IILII,p
II$-mll4, 5 C
IILII
from which (2.20) follows.
Proof of (2.23). We have
(2.25)
n($-m;Vl) + p($1
so that $1-4,1 a Ker B
.
0
a(41.9y1)
by
a(41.,*1) +
Therefore there exists a unique
J. L. LIONS
90
Therefore if we take
tt(4;$) > aII*I II
= 4 -
in (2.25) we have (since
d'
if I a Ker B)
allml -111, < P
!ImI III
ul-01111
hence
(2.26)
a,1 III
41 -01II1 ` a
-
Therefore s
!!41111
c IILII,, , (we denote by
T-I
Therefore (2.26) implies
constant).
II41-41111 < Cp IILII,4,.
We take now in (2.25)
(2.28)
Q 1141111 + Ilml II1
small enough,
hence, for
(2.27)
Ilml II1
Sup Yl
-
* = {9'1,0} ; we notice that
Itt(0;V+1,O)l C 1102112 - C'
1101111
and that (2.25) gives
(2.29)
IT(4- ;*1,°)I
Sup --
ll -- -
Ilmllll
so that (2.18) (2.29) give
(2.30)
C
1142_+2112
< C' Ilml-mllll + X1141111
(2.27) and (2.30), with
1141111
:5 CIILII,
,
imply (2.23).
c
various
OPTIMAL CON T ;UL C.
[-Sl RIBUTED SYSTEMS
91
Proof of (2.26)
We have now (2.31)
0
Taking
( 2 .3 2 )
and using (2.16), we ch!,i,,
P1I;I- m lII
s "'(P212
into (2.31) we obtain
Taking 4 _ {*1,O}
In (m-0;* Sup
-
"P22 2
fib 1
,0)1
= P'
II
1
-
so that by using (2.28) we obtain
(2.33)
II02-;21125(c'+P)II ;l
- $1111
From (2.32) (2.33) we deduce that (2.34)
Il;l -ml Ill < C P 1102112
.
But with (2.34), (2.33) implies
(2.35)
II;2-m2II2 < C
and therefore
1102112
1102112 < C P 1102112 + 11;2112
small enough 11;2112 < C IIm2112
and since by (2.23) we have 11;2(12 < C IILII, we have finally
(2.36)
11;2112 n C
11 LII,,
hence it follows that for
E
P
92
J. L. LIONS
But (2.34) (2.35) (2.36) imply (2.24). Remark 2.1
If we make a stronger hypothesis than (2.4) (2.5), namely
(2.37)
a (m1,o1) >
a11o1112 vm1E,151
then if we introduce (2.38)
nE(d;is) = n(. ;*) + E(02'*2)2
we have
(2.39)
nC(m;o) >_ allolllI +
E1102112
and therefore there exists unique
(2.40)
4,
e,6
such that
t1 (4E;y) = L(ay) V*ei .
One has then (cf. Eercovier, loc. cit.)
(2.41)
III-4Elilt < C e
1IL114>
Remark 2.2
Let us define the adjoint form
(2.42)
by
tt*(O;y) = n(`V;O)
If we define
(2.43)
tt*
a*
by
a*(01,*1) = a(*1,,O1)
then
(2.44)'
tt*(4;*) = a*(o1,*1) - b(p1,m2) + b(41,*2)
This amounts to replacing
a
not affect (2.4) (2.5) (2.6).
by
a*
and
b
by
-b .
These changes do
We have therefore similar results to the
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
93
above ones for the adjoint mixed variational problem. Remark 2.3
Usual variational elliptic problems can be formulated in the preceding setting (cf. Bercovier, loc. cit.); then the approximation results (2.20) or (2.41) lead in a natural way to mixed finite element methods.
Optimal control of mixed variational systems.
2.3
Orientation We now introduce the standard problems of optimal control for elliptic systems in the mixed variational formulation. Let operators
and
U
and
K
(2.45)
K e L(U;4')
(2.46)
C & L(4';H)
Let
be real Hilbert spaces; we are given two
H C
:
be given by (2.7) and we assume that (2.4) (2.5) (2.6) hold
n
y(v) e p
Then there exists a unique element
true.
(2.47)
n(y(v);'V) _ V*e(P
such that
.
This is the state of our system.
The cost function is given by
(2.48)
J(v) = IICy(v)-zdIIH + Let
Uad
NIIvIIU ,
N>0,
Zd&H
be a (non-empty) closed convex subset of
U
Ype optimization problem we want to consider is now inf
(2.49)
Since U -, 4,
(2.50)
v - y(v) ,
J(v), ve Uad
is an affine continuous (cf. (2.41)) mapping from
(2.49) admits a unique solution y(u) = y
,
u
; if we set
J. L. LIONS
94
it is characterized by (CY-zd, C(y(v)-y))H + N(u,v-u)U > 0 Vve Uad (2.51)
ue Uad
The adjoint state Using Remark 2.2, one sees that there exists a unique element
pe4
such that
(2.52)
n*(P:V') _ (Cy-zd,C*)H V>res
we call
p
the adjoint state.
Transformation of (2.51). By taking
i' = y(v)-y
in (2.52) we obtain
(Cy-zd, C(Y(v)-Y))H = x*(P; Y(v)-Y) = n(Y(v)-Y;P) _ (2.53) =
We define (2.54)
K*
K(v-u),p>
.
by
(K* p, v) U =
By virtue of (2.53) (2.54), (2.51) reduces to (K*p+Nu,v-u) U >_ 0
Vve Uad
(2.55)
ue Uad
The optimality system is finally:
n*(P;*) _ (CY-zd,Ctir)H V *e4, together with (2.55).
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
95
Remark 2.4
Since (2.56) (2.55) is equivalent to the initial problem (2.49) which admits a unique solution, the system (2.56) (2.55) admits a unique solution. 2.4
Approximation of the optimal control of mixed variational
systems
We now consider another bilinear form
n(o;*) = a(46l,*l) + b(4l"'2) - b('01,*2)
(2.57)
with hypothesis on
a
and
6
similar to (2.4) (2.5) (2.6).
Therefore,
with the solutions of Section 2.3, there exists a unique element
y(v)
such that
tt(Y(v);tl') _ V4re4
(2.58)
We shall assume that there exists
(2.59)
.
p > 0
"small" such that
IIY(v)-y(v)II,, <_ C pIIvIIU
Remark 2.5 If we take
n
by (2.15) then, by virtue of (2.20), we have
(2.59) with
(2.60)
p
+ 6
.
Remark 2.6
if we assume (2.37) and (2.6) and if we choosen = ,re by (2.38), then, by virtue of (2.41) we have (2.59) with (2.61)
p = e
(2.62)
(Cy-zd, C(y(v)-y))H + N(u,v-6)U i- 0
,
Vve dad
given
96
J. L. LIONS
We have now: if we assume that
it
and
n
satisfy (2.2) (2.3) (2.4) then
11u-upU s Cp112
Remark 2.7
The special hypothesis (2.26) (or an hypothesis on
n*
) is
needed only for defining the adjoint state; (2.56) is valid without this hypothesis. Proof of (2.56). Since
N > 0 , it is enough to consider what happens for a
bounded set of v's in U (2.64)
.
Y(v) = y(v) + r,
By virtue of (2.49) we can write
IIr1I4, ` Cp
therefore
so that
(2.65)
IIJ(v) - J(v)II
We now take
v=u
Cp
in (2.43) and
v=u
in (2.55).
We obtain
(CY-zd,C(Y(u))H + (Cy-zd,C(Y(v)-Y))H - Nllu-ull
We now define
(2.66)
J(v) = IiCY(v)-zdIIH + Nllvll
and we denote by (2.67)
u
J(u) = inf
the "approximate" optimal control .1(v), ve Uad, ue Uad
0
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
It is characterized, if we set
y(u) = y, by
(Cy-zd, C(y(v)-y))H + N(-u.v-6)H > 0 Vv& Uad (2.68)
ue Uad
We now prove the following result:
(2.69)
jju-uIIU <_ Cp1"2
Proof.
We choose (2.70)
v = u (resp. v=u) in (2.51) (resp 2.64)).
(Cy-zd. C(y(u)-Y))H + (Cy-zd, C(y(U)-;))H - NIIu-ulj > 0
But (2.66) is equivalent to
IIC(Y-y)IIH
(2.71) (Cy-zd,
C(y(u)-y)) + (Cy-zd, C(Y(u)-Y))
Using (2.59) we have
IIYO)4II0 = 11y0)46)114'
CpI iJIN.
6(u)-Y114' = 11;(u)-y(u)114'
CPIiuIIU
so that (2.67) implies (2.72)
We obtain
Iju-ull
cp(IiuIIu + 011U)
But if we choose a fixed
NII;II
v0E Uad
i(u) < i(v0)
so that (2.68) implies (2.65).
we have
constant
97
98
J. L. LIONS
Remark 2.8
We can extend all ti;i i tr;eoor, to the case of evolution equations. Remark 2.9 For some extens;cr.
o n.:c-1irear problems, we refer to
Bercovier, loc. cit. Remark 2.10
By using the methods of finite elements for standard elliptic problems (as in Aziz ed: [1], 8abus`ka [1], Brezzi [1], Ciarlet-
Raviart [1], Raviart-Thomas [1], Oden [1]) and the above remarks, one obtains in a systematic manner mixed finite element methods for the optimality systems; cf. Bercovier [1]. Remark 2.11
For other approaches, cf. A. Bossavit [1], R. S. Falk [1]. Remark 2.12
We also point out the method of Glowinski-Pironneau [1] who transform non-linear problems in P.D.E. into problems of optimal control, this transformation being very useful from the numerical viewpoint.
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49.
50. 51. 52.
53. 54.
101
R. S. FALK [1] Approximation of a class of optimal control problems with order of convergence estimates. J.M.A.A. 44, (197-3), 28-47. H. FATTORINI [1] These proceedings. A. FRIEDMAN [1]
E. de GIORGI and S. SPAGNOLO [1] Sulla convergenza degli integrali dell' energia per operatori ellittici del 2° ordine. Boll. U.M.I. 8 (1973), 391-411. R. GLOWINSKI [1] Lagrange and penalty. R. (LOWINSKI, J. L. LIONS and R. TREMOLIERES [1] Analyse Numerique des Ine uations Variationelles. Paris, Dunod, 197 R. GLOWINSKI and I. MAROCCO 1] Sur 1' approximation... R.A.I.R.O. (1975), 41-76. R. GLOWINSKI and 0. PIRONNEAU [1] Calcul d'ecoulements transoniques. Colloque IRIA-Laboria, Versailles, December 1975. D. HUET [1] Perturbations singuliPres d'Inegalit6s Variationnelles. C.R.A.S. 267 (1968), 932-946. C. JOURON [1] Etude des conditions necessaires k'optimalit6 pour un probleme d'optimisation non convexe. C.R.A.S. Paris 281 (1975). 1031-1034. J. P. KERNEVEZ [1] Control of the flux of substrate entering an enzymatic membrane by an inhibitor at the boundary. J. Optimization Theory and Appl. 1973. [2] Book to appear. D. KINDERLEHRER [1] Lecture at the I.C.M. Vancouver, 1974. B. KLOSOWICZ and K. A. LURIE [1] On the optimal nonhomogeneity of a torsional elastic bar. Archives of Mechanics 24 (1971), 239.
55.
56. 57. 58.
59.
60. 61.
249. 62. 63.
64.
65.
KOMURA [1] Personal Communication, Tokyo, 1975. C. C. KWAN and K. N.WANG [1] Sur is stabilisation de la vibration elastique. Scientia Sinica, VXII (1974), 446-467. asi Reversibilit6 et R. LATTES and J. L. LIONS (1] La Methode de Elsevier, English translation, Applications. Paris, Dunod, 1967. by R. Bellman, 1970). J. L. Lions [1] Sur le contrAle optimal des syst6mes gouvern6s par des equations aux deriv6es particles. Paris, Dunod-Gauthier Villars, 1968. (English translation by S. K. Mitter, Springer, uations diff&rentielles o rationnelles et probl4mes 1971.) [2] 3 Some aspects of the optimal aux limites. Springer, 1961. control of distributed ftEemeter systems. Reg. Conf. S. in Appl. Math., SIAM, G, 1972. L4J Various topics in the theory of optimal control of distributed -systems, in Lecture Notes in Economics and Math. Systems, Springer, Vol. 105, 1976 (B. J. Kirby, ed.); 166303. [5] Sur le contr8le optimal de systbmes distribu6s. Enseignement Mathematique, XIX (1973), 125-166. [6] On variational inequalities (in Russian), Uspekhi Mat. Nauk, XXVI (158), (1971), 206-261. [7] Perturbations singulibres dans les prob1tmes aux limites et en contr8le optimal. Lecture Notes in Math., Springer, distribu6s: propridt6s contr8le optimal de 8 323, 1973. de comparaison et perturbations singulibres. Lectures at the Congress :Metodi Valutativi nella Fisica - Mathematics:, Rome, December 1972. Accad. Naz. Lincei, 1975, 1T-'?. (9] On the optimal control of distributed parameter systems, in Techniques of optimi10 Lecture zation, ed. by A. V. Balakrishnan, Acad. Press, 1972.
102
66.
67.
68. 69. 70.
71.
J. L. LIONS
in Holland. J. L. LIONS and E. MAGENES [1] ProblLmes aux limites non homo Lnes et applications. Paris, Dunod, Vol. 1, 2, 1968; Vol. 3, 1970. English translation by P. Kenneth, Springer, 1972, 1973. J. L. LIONS and G. STAMPACCHIA [1] Variational Inequalities. C.P.A.M. xx (1967), 493-519. K. A. LURIE [1] Optimal control in problems of Mathematical Physics. Moscow, 1975. G. I. MARCHUK [1] Conference IFIP Symp. Optimization, Nice, September 1975. F. MIGNOT (1] Contr8le de fonction propre. C.R.A.S. Paris, 280 (1975), 333-335. [2] Contr6le dens les Inequations Elliptiques. J. Functional Analysis. 1976. F. MIGNOT, C. SAGUEZ and J. P. VAN DE WIELE [1] Contr8le Optimal de systPmes gouvern6s par des problAmes aux valeurs propres.
Report Laboria, 72.
73.
74. 75.
76.
77.
78. 79. 80. 81. 82.
83. 84. 85.
86.
J. MOSSINO [1] An application of duality to distributed optimal control problems...J.M.A.A. (1975). 50, p. 223-242. [2] A numerical approach for optimal control problems...Calcolo (1976). F. MURAT [1] Un contre exemple pour le probleme du contrSle dans lea coefficients. C.R.A.S. 273 (1971), 708-711. [2] Contre exemplea pour divers problPmes ou le contr9le intervient dans les coefficients. Annali M. P. ed. Appl. 1976. F. MURAT and J. SIMON [1] To appear. R. H. NILSON and Y. G. TSUEI [1] Free boundary problem of ECM by alternating field technique on inverted plane. Computer Methods in Applied Mech. and Eng. 6 (1975), 265-282. J. T. ODEN [1] Generalized conjugate functions for mixed finite element approximations..., in The Mathematical Foundations of the Finite Element Method, A. K. Aziz, ed., 629-670, Acad. Press, New York, 1973. 0. PIRONNEAU [1] Sur les problPmes d'optimisation de structure en M6canique des fluides. Thesis, Paris, 1976. [2] These proceedings. M. P. POLIS and R. E. GOODSON [1] Proc. I.E.E.E., 64(1976), 45-61. P. A. RAVIART and J. M. THOMAS [1] Mixed finite elements for 2nd order elliptic problems. Conf. Rome, 1975. W. H. RAY and D. G. LAINIOTIS, ed. [1] Identification, Estimation and Control of Distributed Parameter Systems. R. T. ROCKAFELLAR 1 Conjugate duality and optimization. Reg. Conf. Series in Applied Math. SIAM. 16, 1974. D. L. RUSSELL [1] These proceedings. [2] Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory. J.M.A.A. 40 (1972), 336-368. C. SAGUEZ [1] Integer programming applied to optimal control. Int. Conf. Op. Research, Eger. Hungary, August 1974. J. SAINT JEAN PAULIN [1] Contr8le en cascade daps ua problZme de transmission. To appear. Y. SAKAWA and T. MATSUSHITA (1) Feedback stabilization of a class of distributed systems and construction of a state estimator. IEEE Transactions on Automatic Control, AC-20, 1975, 748-753. J. SUNG and C. Y. YU [1] On the theory of distributed parameter systems with ordinary feedback control. Scientia Sinica, SVIII, (1975), 281-310.
OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS
87.
88.
89. 90. 91.
92.
103
L. TARTAR [1] Sur 1'6tude directe d'equations non lineaires intervenant en th6orie du contr8le optimal. J. Funct. Analysis 17 (1974),1-47. [2] To appear. A. N. TIKHONOV [1] The regularization of incorrectly posed problems. Doklady Akad. Nauk SSSR,153 (1963), 51-52, (Soviet Math. 4, 1963, 1624-1625). G. TORELLI [1] On a free boundary value problem connected with a nonsteady filtration phenomenon. To appear. A. VALLEE [1] Un problpme de contr8le optimum dans certains problcmes d'evolution. Ann. Sc. Norm Sup. Pisa, 20 (1966), 25-30. J. P. VAN DE WIELE [1] REsolution numerique d'un probl6me de contr8le optimal de valeurs propres et vecteurs propres. Thesis 3rd Cycle. Paris 1976. R. B. VINTER [1) Optimal control of non-symmetric hyperbolic systems in n-variables on the half space. Imperial College Rep. 1974.
93. 94. 95. 96.
97.
R. B. VINTER and T. L. JOHNSON [1] Optimal control of non-symmetric hyperbolic systems in n variables on the half-space. To appear. P. K. C. WANG [1]. J. L. A. YEBRA (1). To appear. J. P. YVON (1] Some optimal control problems for distributed systems and their numerical solutions. [2] Contr8le optimal d'un probleme de fusion. Calcolo. [3] Etude de la methode de boucle ouverte adaptee pour le contr8le de systbmes distribu6s. Lecture Notes in Economics and Math. Systems, 107, (1974), 427-439. [4] Optimal control of systems governed by V.I. Lecture Notes in Computer Sciences, Springer, 3 (1973), 265-275. J. P. ZOLESIO (1) Univ. of Nice Report, 1976.
We also refer to: Report of the Laboratoire d'Automatique, E.N.S. Mecanique, Nantes: Calculateur hybride et Syst1'mes a parambtres r6partis, 1975
"STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS: A GENERAL THEORY" A. V. Balakrishnan* A large class of filtering and control problems for linear systems can be described as follows. y(t)
(say, an
0 < t < T < m
m x .
1
We have an observed (stochastic) process
vector),
t
representing continuous time,
This process has the structure:
y(t) = v(t) + n0(t) where
n0(t)
is the unavoidable measurement error modelled as a white
Gaussian noise process of known spectral density matrix, taken as the Identity matrix for simplicity of notation. composed of two parts: random 'disturbance'
The output
the response to the control input nL(t)
v(t) u(t)
is
and a
(sometimes referred to as 'load distur-
bance' or 'stale noise') also modelled as stationary Gaussian; we also assume the system responding to the control is linear and time-invariant so that we have: t
B(t-s) u(s)ds + nL(t)
v(t) = J
O
where
is always assumed to be locally square integrable, and
where
is a 'rectangular' matrix function and
IIB(t)II2dt < -
.
FO
* Research supported in part under grant no. 73-2492, Applied Mathematics Division, AFOSR, USAF
105
A. V. BALAKRISHNAN
106
We assume further more that the random disturbance is `physically realizable' so that we can exploit the representation: t
nL(t) = I0 F(t-p) N(p) do
where
is a rectangular matrix such that
F(p)
f o IIF(s)II2ds < where, in the usual notation, AA*
IIAII2 = Tr.
We assume that the process noise process
n0(t)
nL(t)
is independent of the observation
.
It is more convenient now to rewrite the total representation as: y(t,w) = v(t,w) + Gw(t)
t
t
v(t,w) = J0 B(t-s) u(s)ds + J0 F(t-s) w(s)ds
where GG* =
I
.F(t)G* = 0 w(-)
is white noise process in the. appropriate product Euclidean space,
and
IIF(t)II2dt
<
JO 00
We hasten to point out that we may replace the white noise formalism by a 'Wiener process' formalism for the above as: t
Y(t,w) =
v(s,w)ds + G W(t,w) 0 t
v(t,w) = I0 B(t-s)u(s)ds +
ft0 I
F(t-s)dW(s,w)
STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS It makes no difference to the theory that follows as to which formalism is used.
The optimization problem we shall consider is a stochastic
control ("regulator") problem in which the filtering problem is implicit:
to minimize the effect of the disturbance on the output (or some
components of it).
More specifically, we wish to minimize:
ft
[Qv(t,w), Qv(t,w)]dt
E J
0
(1.2) t
fo E
denoting expectation, where for each
t
,
u(t,w)
only upon the available observation up to time t
.
must 'depend'
We can show [1]
that under the representation (1.1), (1.2), the optimal control may be sought in the class of 'linear' controls of the form: t0
K(t,s)dY(s,w)
u(t,w) = J
in the Wiener process formalism, or rt
K(t,s) y(s,w)ds 0
in the white noise formalism.
This problem embraces already all the stochastic control problems for systems governed by ordinary differential equations by taking the special case where the Laplace transforms of rational.
But it also includes a wide variety of problems involving
partial differential equations where the observation process each
t
are
and
Y(t)
for
has its range in a finite dimensional Euclidean space (measure-
ments at a finite number of points in the domain or on the boundary for example).
One may argue that any physi- " measurement must be finite
dimensional; in any case, the extension to the infinite dimensional case brings little that is new, and we shall not go into it here.
As a simple example of a non-rational case we may mention: F(t) =
t-3/2
e-
1/t
(1.4)
A. V. BALAKRISHNAN
108
arising
from boundary input in a half-infinite rod [5].
associated process
nL(t)
Note that the
is not 'Markovian' even in the extended
sense [2].
To solve our problem, our basic technique is to create an 'artificial' state space representation for (1.1).
It is artificial in the
sense that it has nothing to do with the actual state space that originates with the problems. example belgw.
We shall illustrate this with a specific
Without going into the system theoretic aspects in-
volved, let us simply note that the controllable part of the original state space can be put in one-to-one correspondence with the controllable part of the artificial state space. Let
denote
H
observation process.
L2[0,o;Rm] Let
D(A) = E6cH
m
is the dimension of the
denote the operator with domain
A
in
H:
is absolutely continuous with derivative
also]
e H
fl
where
and
Af =fl Let
B
denote the operator mapping the Euclidean space in which the
controls range, into
H
by:
8 u(t) ti B(c)u(t)
0 < c <
,
and similarly Fw(t)
Assume now that 0 < t < m
.
0 < G <
, F(r)w(t) F(t)
and
B(t)
are 'locally' continuous, in
Then we claim that (1.1) is representable as (a partial
differential equation!)
3c(t) = A x(t) + Bu(.t) + Fw(t)
; x(0) = 0 . (1.5)
y(t) = C x(t) + Gw(t) (or appropriate 'Wiener-process' version), where
C
is the operator
defined by:
Domain of [or,
C = [fcH
I
f(t)
is continuous in
is 'locally' continuous] and
0 < t <
STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS
C6 = b(0)
[value at the origin of the 'continuous function' representativ' of
f(.)]
.
We can readily show that
is in the domain of
x(t)
assumption of local continuity.
C
because of t.rc
On the other hand we do not need to
make the 'exponential rate of growth' assumptions as in the earlier version of the representation [3]. that (1.5) has the solution:
To see this we have only to note
(assuming that
is locally squor-
integrable):
t
t S(t-a)Bu(a)dc +
x(t) =
J0 where
J0
S(t-a) Fw(a)da
is the semigroup generated by
S(t)
A .
Now
t
h(t) = f0 S(t-o) Bu(a)do
is the function:
t
h(t,c) = J0B(c+t-o) u(a)da
and
is locally continuous in
h(t,c)
continuity of t
.
Hence
h(t)
0 < c <
0 < c < m , because of the
is in the domain of
C
, for
Moreover r0 t
B(t-a) u(a)da
C h(t) = 1
Similarly
t
t F(t-a)w(a)do
S(t-a) Fw(a)do =
C
J0
J0
which suffices to prove the representation.
Of course to complete th'
representation we have that the cost functional (1.2) can be written:
f0t E
ft
[QCx(t), QCx(t)]dt + E
Cu(t), u(t), u(t)]dt
(1.71
0
In this form we have a stochastic control problem in a Hilbert space,
and we may apply the techniques of [4]; except for the complication
A. V. BALAKRISHNAN
110
that
is now unbounded, uncloseable. The 'operators'
C
and
B
F
are
Hilbert-Schmidt and in this sense there is a simplification. Even though
C
is uncloseable, let us note that
Cx(t) = Jt B(t-a) u(a)da + I F(t-a)w(a)da 0 0t
and hence is actually locally continuous in
g(P) = Jp C'S(p-a) J
0 < t , and
0 < p < t
F6(a)da
O
defines a linear bounded transformation on Wn(t) = L2 ((O1t)1Rn) where
.
is the Euclidean space in which
Rn
w(t)
ranges, into
WO(t) = L2((O,t),Rn) 0 < t
for each
We shall only consider
.
u(t)
such that
t
L (t,s) y(s)ds
0 < t < T
g(t) = JO L (t,s) f(s)ds
0 < t < T
u(t) =
(1.8)
J0
where t
defines a Hilbert-Schmidt operator mapping
WC(T) = L21(0,T); R
W0(T)
WC(T)
into
where
I
p
where
RP
every
t
is the real Euclidean space in which The Hilbert-Schmidtness implies that
.
Schmidt also, a.e., and that T f0
t
JO IIL(t,s)II2.S dt <
.
u(t)
ranges for
L(t,s)
is Hilbert-
STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS
ill
It is not difficult to see that t
u(t) = J0 L(t,s)y(s)ds
r0t
t
x(t) =
S(t-o) B u(a)da + 1
fo
S(t-o) Fw(a)da
y(t) = C x(t) + G w(t) defines 2.
uniquely, for each
The Filtering Problem.
Let us first consider the filtering problem for (T.1) taking to be identically zero.
We shall see that this is an essential step in
solving the control problem.
Thus let, in the notation of Section 1,
(t
x(t,w) =
S(t-a) Fw(a)da
I
(2.1)
0
y(t,w) = Cx(t,w) + Gw(t)
As we have noted earlier, the only difference from the standard problem treated in [4] is that
is uncloseable.
C
Nevertheless since
t
Cx(t,w) =
F(t-a) w(a)da 10
and is continuous in the element in y(s,w)
we see that
w , we note that, denoting by
0 < S < t
,
is a weak Gaussian random variable with finite WO(t)
for each
t
.
Moreover yt
has the covariance
operator:
I + L(t) L(t)*
where
L(t)
yt(w)
defined by
W0(t)
yt(w)
second moment in
for each
t
is defined by
L(t)f = g
;
g(p) = fo p F(p-a) f(a)ds
0 < p < t
I
A. V. BALAKRISHNAN
1%
did is linear bounded on operator on
W0(t)
.
WO(t)
I
for each
t
is the identity
yt(w)]
(
belongs to the domain of
x(t,w)
and
;
Let
x(t,w) = E [x(t,w) Then
into
Wn(t)
C
and each
am l further C x(t,w) = E [Cx(t,w)
I
yt(w)]
(2.2)
the novelty in this relation arising from the fact that unbounded.
This can be seen readily as follows.
is
C
We note that (see
[4])
x(t,w) = E [X(t,w) Yt(w)*] [I + L(t) L(t)*]-1 Yt(w)
(2.3)
where
t K(t,s)f(s)ds
E [x(t,w) yt(w)*]6 = 10
,here P
K(t,p) = S(t-p) f p S(p-o)F
:n;i
the corresponding element in It
fp 0
0 ,'lid
F(t-p+) F(p-o)*do
is locally continuous in
it follows that
H
x(t,w)
F(p-o)* do
is given by f(s)ds
0 < t , for any
is in the domain of
C
0 <
<
6(-)
in
for each
W0(t)
.
t
and
Hence w
jnd further a simple verification establishes (2.2) since the right side :f (2.2) is given by L(t)*]-
E [Cx(t,w) yt(w)*] ind for any
f
in
WO(t)
[I + L(t)
:
E [Cx(t,w) yt(w)*]6 = C E [x(t,w) yt(w)*]6 Relation (2.2) enables us to extend the arguments in [4] to show that
STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS
z(t,w) = y(t,w) - Cx(t,w) is again white noise.
P6(t)
Let
0 < t < T
denote
E [(X(t,w) - x(t,w)) (x(t,w} - X(t,w))*] Then that
.
P(t) = E [x(t,w) x(t,w)*] - E [z(t,w) z(t,w)*]
P(t)
maps into the domain of
as an element of
where
R
WO(T)
113
C
.
and it follows
The covariance operator of
has the form
is Hilbert-Schmidt and hence the Krein factorization theorem
(the Kernels being strongly continuous) as in [4] yields (I+R)-1 = (I-L)* (I-L)
where
L
is Volterra and (I-L)
Moreover =
(I-L)-1
where
M
I + M
is Hilbert-Schmidt also.
Hence we can write
where
t If = g
;
J(t,a) z(a,w)da
g(t) = l
0
and following [4] we must have that J(t,a) = S(t-a) (C P6(a))* so that t
P6(t)x = 10 S(o)F F*S(a)*xda
S(t-a)(C P(a))*(C 6 P(a))S*(t-o)da 6
ft0
(2.4)
A. V. BALAKRISHNAN
114
and in turn we have that, for
x
and
y
in the domain of
A*
[P6(t)x,Y] = [Pf(t)x,A*y] + [Pf(t)y,A*x] + [Fx, FY] - [C P6(t)x, C Pf(t)y)
Pf(0) = 0
;
(2.5)
.
Further we have: rt
x(t,w) =
5(t-o) (C Pf(a))* (y(a,w) - Cx"(a,w))da 1
0 rt
S(t-a) (C Pf(a))* C x(a,w)da
1
0
+ I0t S(t-a) (C Pf(a))*Y(o,w)do
This is an 'integral equation' that has a unique solution. 2(t,w)
.
z(t,w)
(2.6)
satisfies.
Moreover (2.6)
For suppose there were two solutions
The difference, say
h(t)
, (fixing the
w )
z1(t,w)
t
I0 S(t-a) (C Pf(a))*C h(a)da
h(t)
and hence we can deduce that:
C h(t) = -
C S(t-a) (C Pf(a))*(C h(a))da
f
'0
But
is an element of
C
L2(O,T)
and the right-side defines a
Hilbert-Schmidt Volterra transformation which is then quasinilpotent. Hence
must be zero.
C
C Xl(t,:o)
Hence
z(t,w)
=
,
, would satisfy
Hence
C z2(t,w)
remains the same:
z(t,.,) = y(t,.u) - C Xl(t,w) = y(t,w) - C R2(t,w)
STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS
115
But t
x(t,w) =
J(t,a) 2(o,w)do
J
0
proving the uniqueness of solution of (2.6).
We could also have deduced
this from the uniqueness of the Krein factorization.
We can also re-
write (2.6) in the differential form in the usual sense (see [41): x(t,w) = Az(t,w) + (C Pf(t))*(y(t,w) - Cx(t,w)) z(0,w) = 0 Let us
yielding thus a generalization of the Kalman filter.equations. note in passing here that A - (C Pf(t))*C is closed on the domain of
for t- 0
and the resolvent set includes the open
A
It does not however generate a contraction semigroup
right half plane. .
The proof of uniqueness of solution to (2.5) can be given by invoking the dual control problem analogous to the case where
C
is
bounded, as in [4] but will be omitted here because of limitation of From this it will also follow that
space.
is monotone in
[Pf(t)x,x]
t. Let
Cn
be defined on
by:
H
r10 /n
Cnf = g
;
g(t) = n
I
b(s)ds
111
Then
is bounded.
Cn
Hence it follows that
E (Cn x(t,w)) (Cn X(t,w)*) t
1{0 (Cn 5(a)F) (Cn S(a)F)*dcr
and as
n
goes to infinite, the left side converges strongly and the
right side yields C (C R(t,f))* ;
R(t,t) = E [x(t,(o) x(t,w)*]
.
A. V. BALAKRISHNAN
116
In a similar manner we can show that E [(C X(t,w) (C x(t,w))*] = C (C R(t,t))* E [X(t,w) X(t,w)*] _
(t,t)
E [(C x(t,w) - x(t,w)) (C x(t,w) - C z(t,w))*] = C (C P6(t))*
We are of course most interested in the case seen that
[Pf(t) x,x]
is monotone.
T
We have
.
Also t
[Pf(t)x,x] < [R (t,t)x,x] =
[S(o)F F*S(o)*x,x]da 10
Let us assume now that
IIF*S(o)*xll2da = [R x,x] < - .
(2.7)
TO
(This is clearly satisfied in our example (1.4).) Then
Pf(t)
also converges strongly, to
into the domain of
P. , say; further
P. maps
and satisfies
C
P. = Rm - i.0) (C p)*(C P.) S(a)*da
and hence also the algebraic equation:
0 = [Pm ,A*y] + [Py,A*x] + (F*s,F*y] - [C P x,C Py)
(2.8)
which has a unique solution. 3.
The Control Problem.
Because of space limitations, we shall have to limit the presentation to, the main results, emphasing only the differences arising due to
the unboundedness of
C
.
Thus, defining as in [4, Chapter 6), and
STOCHASTIC FILTERING AND CONTROL OF LINEAR SYSTEMS
117
confining ourselves to controls defined by (1.7); x(t,w) - xu(t,w) = z(t,w)
C z(t,w) + Gw(t) _ 3'(t,w)
where
Xu(t,w) = A x(t,w) + B u(t,w)
we can invoke the results of section 2 to obtain that z(t,w) = y(t,w) - C x(t,w) where
x(t,w) = E [X(t,w) yields white noise.
0 < P < t]
We can then also proceed as in [4] to show that we
can also express any
u(t,w)
satisfying (1.7), also as
(t
u(tw) = j
m(t,p) z(P,w)dp
Y
where the operator is Hilbert-Schmidt.
The separation theorem follows
easily from this, and-we can show that the optimal control is given by T
u0(t,w)
ft (Q C S(P-t)B)* x(P,w)dp
where
x(P.w) = x(P,w) + xu(P.w)
and hence as in section 2, is the unique solution of
(3.1)
A. V_ BALAKRISHNAN
118
x(p,w) = A X(p,w) + B u0(p,w)
+ (C Pi(p))*(Y(p,w) - C x(p,w))
z(O,w) = 0
.
Further we can follow [4], making appropriate modifications of the unboundedness of
C
,
to deduce from (3.1) that
u0(t,w) = - (PC(t)B)*x(t,w)
where
Pc(t)
(3.2)
is the solution of
[Pc(t)x,Y] = [PC(t)x,Ay] + [PC(t)Ax,Y]
+ [QCx, QCY] - [(PC(t)B)*x, (Pc(t)B)*Y]
PC(T) = 0
for
x,y
in the domain of
REFERENCES 1.
;
(3.3)
A
References
A. V. Balakrishnan: "A Note on the Structural of Optimal Stochastic Controls", Journal of Applied Mathematics and Opptimization, Vol. 1,
No. 1, 1971+. 2.
Y. Okabe:"Stationary Gaussian Processes with Markovian Property and M. Sato's Hyperfunctions", Japanese Journal of Mathematics, Vol. 41,
3.
A. V. Balakrishnan: "System Theory and Stochastic Optimization", Proceedings of the NATO Advanced Institute on Network and Signal Theory, September 1972, Peter Peregrinns Its., London. A. V. Balakrishnan: Applied Functional Analysis, Springer-Verlag,
1973, pp. 69-122.
4.
1976. 5.
A. V. Balakrishnan: "Semigroup Theory and Control Theory".
"DIFFERENTIAL DELAY EQUATIONS AS CANONICAL FORMS FOR CONTROLLED HYPERBOLIC SYSTEMS WITH APPLICATIONS TO SPECTRAL ASSIGNMENT" David L. Russell* 1.
Introduction
This article is part of a continuing program of research aimed at the development of control canonical forms for certain distributed parameter control systems.
This, in turn, is part of a larger effort
being undertaken by a number of research workers, to arrive at a fuller understanding of the relationships between controllability of such systems and the ability to stabilize, or otherwise modify the behavior of, these systems by means of linear state feedback. [15], [11].)
(See [9], [10],
The present article is largely expository and will rely
on the paper [12] for certain details.
Nevertheless, we do present some
results which go beyond those already presented in that paper.
Let us recall the control canonical form in the context of the discrete finite dimensional control system.
Wk+l = Awk + guk, w ( En, u E E1
If one starts with
w
0
= 0 , the control sequence
uo, ul,
..., un_1
produces the state
* Supported in part by the Office of Naval Research under Contract No. 041-404. Reproduction in whole or in part is permitted for any purpose of the United States Government.
119
DAVID L. RUSSELL
120
An-lguo
wn =
+ An-2gul +...+ Agun-2
+ gun-1
uo 1
(An-lg,
_
An-2g,
..., Ag,g
U(u}
=
un-2 u n- 1
)
=_
The system is controllable just in case this "control to state" map is nonsingular, i.e., just in case
U
is a nonsingular
nxn
matrix.
We
shall assume"this to be the case.
It is possible then to use the matrix U to "carry" the system (1.1) from the space
En
control sequences
w over into the space
of state vectors {u}
n
of
by means of the transformation (1.3)
The transformed system is
'k+l = U-1ALk + U
lguk
(1.4)
is the last column of the nxn identity matrix and
The vector en
A= I al
1
0
0
a2
0
1
0
0
0
1
0
0
an-1
an
where the
ai
...
0
,
are the components of the vector
U-1Ang
ently, the unique scalars for which n A g =
a1An-1g + a2An-2g +...+ an-lAg
+ ang
or, equival-
DIFFERENTIAL DELAY EQUATIONS
121
We refer to (1.4) as the control normal form of the system (1.1).
To pass to the control canonical form one employs the "convolution type" transformation 1
0
0
...
0
-al
1
0
...
0
a -2
-al
1
...
0
_an-3...
an-2
-an-1
(1.6)
C'V
1
the result of which is to produce C-lenuk
k+l
C+IA (1.7)
k + enuk 0-
0 with
en
and now
, as before, equal to 0 1
0
1
0
...
0
0
0
1
...
0
0
0
0
...
1
...
al
A=
(1.8)
a
an-2
an-1
The system (1.7) is the control canonical form for (1.1).
It is
significant because it enables one to see immediately the effect of linear state feedback 1
2 ,...,kn-
u = (k l ,k 2
l
kl
,kn)
n-1
DAVID L. RUSSELL
122
The closed loop system is
k+l = (A + Since
0
1
0
...
0
0
0
1
...
0
0
0
0
an-1+k2
an-2+k3
A + enk =
.. . an+kl
1
...
al+kn
the coefficients of the characteristic polynomial of the closed loop system matrix
A + e k
, and hence its eigenvalues, can be determined
n
at will by appropriate selection of
kl,k2,...,kn
.
The canonical form (1.7) is equivalent to the scalar n-th order system U
k+l =
.
k
In the work to follow we will see that certain infinite dimensional control systems can be reduced to a canonical form comparable to this,
namely, 2
e-Y (t,0) + jr
p(2-T)r.(t,w) d; + u(t) 0
by an entirely analogous procedure, likewise involving a "control to state" map followed by a transformation of convolution type comparable to (1.6). 2.
Control Problems for Hyperbolic Systems Let us consider the scalar hyperbolic equation 2
2
a
+ y at -
ax
r(x)w = g(x)u(t),
0.5 x 5 1, t>_ 0
DIFFERENTIAL DELAY EQUATIONS
where
is a constant, the real function
y
r E C[O,1]
and
123
g E L2[0,1].
We shall suppose further that boundary conditions (2.2)
a0w(O,t) + bo
ax
(O,t) = 0, alw(l,t) + b1 ' (l,t) = 0
are imposed at the endpoints
ax
article we shall suppose that
bl t 0
the article we also assume that on the case
b
0
¢ 0
and
x = 0
.
x =
1
.
Throughout this
Throughout the main body of
bo = 0, a
0
t 0 , but we will comment
in the last section of the paper.
The Strum-Liouville operator 2
(2.3) x
with boundary conditions of the form (2.2), bl # 0, bo = 0, ao ¢ 0
has distinct real eigenvalues Al < X2 < , " ' < Xk < Xk+l ` with (cf. [4])
ak = (k - j)2 I12 + 0(1), k and corresponding eigenfunctions L4[O,1]
.
Ok
(2.4)
forming an orthononnal basis for
Taking the inner product of (2.1) with
Ok
we have
wk'(t) + ywk (t) + Xkwk(t) = gku(t), (2.5)
k = 1, 2, 3, ... where
w(x,t) , the presumed solution of (2.1), has the expansion,
convergent in
L2[0,1]
w(x,t) =
E
,
Wk(t)4k(x)
k=1
Letting
,
vk(t) = wk(t)
and setting
DAVID L. RUSSELL
124
Wk(t)
wkl
`;'k
1
1
Yk(t) (2.6)
vk(t)
zk(t)
where
wk = 2 (- Y +
Y2-4ak
wk=(-Y -
2-4ak
(2.7)
(2.8)
(2.5) is transformed to Yk(t) = wk Yk(t) + hk u(t)
zk(t) = wk zk(t) + hk u(t)
(2.9)
(2.10)
.
In (2.7), (2.8) we shall use the convention that
lies either
on the non-negative real axis or the non-negative imaginary axis. numbers
hk, hk
The
in (2.9), (2.10) are
-T--T ,
hk = "'k
-")k
hk - --wk
(2.11)
-wk
and have the property
limhk= limhk =T k-..
k-w
A slightly different transformation is used if some
y = 0
k (so that wk = wk = 0) or if y2 = 4ak (so that
ak = 0
for
wk = wk).
For
and
brevity of treatment we do not discuss these special cases here but they can be brought within the same framework.
DIFFERENTIAL DELAY EQUATIONS
125
From (2.4), (2.7) and (2.8) we see that
2 + i(k -
O'k
w
k
If we let
2)n
+ 0(k), k
= - 2 - i(k - 2) n +
0(j),
w,
k -+ ° °
(2.12)
(2.13)
.
w-k = wk+1' y-k = Zk+l , h-k = hk+l, k = 0,1,2,...
we can
replace (2.9), (2.10), (2.12), (2.13) by
yk = ``'kyk + hku(t), - = < k <
wk
= -
Because the
(2.14)
+ i(k - 2)n + 0(i), - m < k <
wk
(2.15)
take the form (2.15) itis known (see, e.g. [6], kt
[5], [14], [8], [13]) that the functions
e
form a Riesz basis
(image of an orthonormal basis under a bounded and boundedly linear transformation) in
L2[0,2]
.
invertible
There exists also a dual Riesz
fk#E .
basis consisting of functions
pk' - m < k < m , for which
roewkt r2
()
k
pE(2-t) dt = (ewk ,pl)L2[O,2] = 6E
=
0,
(2.16)
The biorthogonality property (2.16) enables us to study the controllability of the system (2.14) (equivalently (2.1), (2.2)) quite readily.
An arbitrary control
u E L2[0,2]
has the expansion
0o
u(t) =
E
k=-.
µk
pk(t),
E
I"kI2 <
k=-m (2.17)
r2
µk=
f.e
'k(2-t)
u(t) dt, -o'
If we begin with
yk(0) = 0,
DAVID L. RUSSELL
126
and apply the control
in (2.14), the variation of parameters
u(t)
formula gives
ewk(2-t)u(t)
j2
yk(2) = hk
(2.18)
dt = hk µk
The first equality in (2.18) defines the control to state map for this system, i.e. the map
Uu = {hk
U
L2[0,2] - t2
:
(2 ewk(2-t)u(t)
dt
I
-
described by
< k < m ..
Jo
It should be compared with the analogous matrix (cf. (1.2)).
(2.19)
JJJ
We want
U
in Section 1
to be one to one, hence invertible on its
U
range, which we accomplish with the Approximate Controllability Assumption
hk t 0, - W < k <
The states reachable at time t = 2 consist of sequences
{hkµk
I --
1µk12<m,
E
k= -m
a dense subspace, which we denote by
R
, of
t2
.
In terms of the
original system (2.1), (2.2) this means that we can reach states Z
Wk "k
k=1
v(',2)
'k
('.2) = k!1 Vk
with the
wk, vk
of the form
Wk = r wk'
2< 1
I; kl
=
vk
2 I wk k=1
9k vk'
E k=1
2
DIFFERENTIAL DELAY EQUATIONS
127
We remark that it is known from [13] that the time interval of length 2 is minimal in order that
should have dense range and that the range
U
is not altered if we take an interval of length greater than 2.
Hence
the choice of the interval [0,2] and the control space 12(0,2]. Our plan now is to proceed just as in Section 1. the solution of (2.14) with initial state in always lies in
with
R
But
.
U
:
For
U E Lloc
R , as defined above,
R and U-1 is defined on onna Hence this map can be used to
R
L2[0,2]
U-1 R c t2 L2[0,2] ontotransform (2.14) from a system in :
.
R c t2
L2[0,2]
to a system in
In order to carry this out successfully, however, we need a certain ewkt, - - < k <
property of the functions
where the
u
have the
asymptotic property (2.15)., We have 2
+ e-y = e
e
+ e`Y (2.20)
in+o(k)
e-y[e
=
where clearly
Ek = O(
and hence
)
< k< -
+11 = Ek, 'skl2 E
<
k=--
If we now define
p 2-t where the
(2.21)
sk
Z
=
k=--
are the "biorthogonal functions" defined in (2.16), we
pk
clearly have
y
wk2 e
+e
=f
2
mkT
(2.22)
dT, - w< k <
e
0
t Multiplying by
e
ec)k
we have
(t+2) +e
yeat
2e'ok
= J of
(t+t)
Q('F7'r
-
dT
128
DAVID L. RUSSELL
This functional differential equation is then satisfied by any linear wkt combination Z Ck e . It serves as the analog of the homogeneous k=-m version of (1.7).
The first step in the derivation of the control normal form for (2.14), which is analogous to (1.7), is to use the map
U
defined in
(2.19), to effect the change of variable 2
yk{t) = hk
wk(2-T)
(t,T) dT =
e 1
0
(2.24)
t>_0, -- < k < -
.
The inverse map is clearly
yk(t) E k=
1
k(.) = U
h
(2.25)
{yk(t)}
k
A formal derivation of the functional equation satisfied by proceeds as follows.
Et7 fdyk(t)
a" t T at
=
hk
k=
k
`wkyk(t);hku(t) }
_ `E k=
hk
k
2
_ (using (2.24))
+ u(t)
Z k=-
kE
PkT
k
I 02
``
DIFFERENTIAL DELAY EQUATIONS
129
Integrating by parts we now have _
t
(1r2 emk(2-s) a"
r
k=-
at
+
o
t s
as
(_t2.'2 (t,0)+u(t)
E
PTT
-
k=
Now using
(2.22) we obtain
at
(
E
k=
0
+ k= -
Since the sequences in
L2[0,2]
`\
a(t,s as
l
) + PTT Z(t.0)/ ds PITT
(_(t2)_e Y G(t.0)+u(t)) PTT
{e``'k(2-'r) I
and
{pk(,r))
.
(2.26)
are dual to each other
, the first sum (at least formally - in general
is not actually in
pTT, of
mk(2-s) e
2
t, T ) =
a
a (t,s L2[0,2]) is the expansion, in terms of the functions
a"latTZ + pT,7(t,0)
The second sum can be written
(t,0)+u(t)) E ka-m
and again formally,
PTT
p; FT can be viewed as the expansion of the
E
k=-o. distribution 6(-r-2), since 2 e
f0 To avoid a multiple of
6(-r-2)
0
= 1, -
.
appearing on the right hand side of
(2.26) we set j(t,2) + e -Y j(t,0) = u(t)
(2.27)
DAVID L. RUSSELL
130
and, from our earlier remarks, we now have T at
=
a'T
(2.28)
.+ TTT (t,0)
The equations (2.27), (2.28) constitute the control normal form for (2.14) (equivalently (2.1), (2.2)) and should be compared with (1.4) in Section 1.
The above formal derivation is justified in a rigorous manner in [12].
We now proceed to the control canonical form which, if (1.6) is to be paralleled, should be obtained with a "convolution type" transformation.
The transformation which we use is, in fact, Y-(t,') =
(2.29)
defined by
2(t,v) =
da
- IT p.T-a
(2.30)
0
With this, substitution of (2.30) into (2.27) yields 2
(t,2) = (t,2) + I
p7'£-aj (t,a) da
0
2 e-Y G(t,0) + u(t) + I p(2-,) (t,a) da 0
and, since (2.30) clearly gives
(t,0) _ (t,0) , we have 2
PTY:TT dt,T) dT
e-ydt,0) - u(t) + I
(2.31)
0
Now substituting (2.30) into (2.28) we have, again using the fact that
(t,0) = Z(t,0)
,
DIFFERENTIAL DELAY EQUATIONS
0
P T-a i(t,a) dal
-
et ( (t.t
131
ro
tT
at
-
to
PT,-a
Fo
i;(t,a) da - pTTT i;(t,0)
+ TaT (t,T) +
Noting that
a
a p T-a
0
aT
aT
atT aT
-
da
at
.
and
integrating by parts in the second integral above we have
tT at
a
tT aT
--P(- a T_a
=
FD
(
to at
tQ -
as
da
This integral equation has only the solution
tT- at,T
at
ati
=0
(2.32)
The equations (2.31) and (2.32) together consitute the control canonical form of (2.14) (equivalently (2.1), (2.2)) and should be compared with (1.7) in Section 1.
Equation (2.32) simply amounts to left translation,
hence (2.31) is a neutral functional equation for Again the above passage from the control normal form to the control p T-a
canonical form has only been carried out formally, since
(t,T)
do not, in general, have derivatives in
L2[0,2]
.
and
We again
refer the reader to [12] for a more rigorous argument. 3.
Spectral Determination For Hyperbolic Systems We have noted in Section 1 that for finite dimensional systems the
control canonical form is useful in establishing that the eigenvalues of the closed loop system can be placed at will with appropriate choice of
DAVID L. RUSSELL
132
the feedback row vector k
Our purpose now is to show that the
.
canonical form developed in Section 2 can be employed to the same end with reference to the system (2.1), (2.2). The "natural" space for study of the system (2.1) is the '°finite energy" space
consisting of function pairs (w,v) in
HE
x L[0,2] 2H[0,2] with
1
w(O) = 0
.
Supplied with an inner product (3.1)
= 10 [v('x)v(x)+w-(x)w'(x)+w(x)(r(x)+r0)(x)]dx, o
E
(with
r
0
chosen so that
r(x) + ro > 0 , x ( [0,1]) and associated
norm
II(w,v)IIHE _ [((w,v),(w,v))HE]l/2
HE
becomes a Hilbert space.
Let us consider the situation wherein the control
u(t)
is
determined by the feedback relation (3.2)
u(t) = ((w,v),(k,t))H
with
(k,t) E HE
functions
.
We expand
w, v, k, t with respect to the eigen-
of the Sturm-Liouville operator (2.3) with boundary
.0i
conditions (2.2) : a W=
E i=1
a
wi q'i ,
v=
a
a
k.
E
11E=1
ki
i=1
and compute, from (3.1),(3.2)
i=1
vi 41i -
DIFFERENTIAL DELAY EQUATIONS
u(t) = f
l [v(x) t x
+
0
ax
133
(x) ax (x) + w(x)(r(x)+ro)c x J dx
fl
[v(x) t x
+ (Lw+row)(x)c x ] dx
0
=
[vi ei + (xi+ro)wi V ] E i=1
.
Now taking the transformation (2.6) into account we have
u(t) =
E i=1
E [yi(ti+(ai+ro)wilki )+zi(ti+(ai+ro)wilki )]
(3.3)
i=1
E [yiai + zi i=1
From the fact that
it can be shown quite readily (see [4],
(k,t) E HE
for example) that
E
itil2
<
E
ailkil2 < -
(3.4)
i=1
i=1
and the conditions (3.4) are also sufficient in order that From this it is easy to see. that
and
ai
ai
(k,t) E HE .
in (3.3) can be chosen to
be arbitrary sequences with E i=1
if
(k,t) E HE
ail2
< -,
E i=1
lail2
<
is chosen appropriately.
system (2.14) (again letting
Thus in the context of the
a_k = ak+l' k = 0,1,2,...) we may assume
u(t) generated by the feedback law
DAVID L. RUSSELL
134
u(t) -
Z
ak Yk(t)
k-{a,,}
with
an arbitrary sequence satisfying
a
E
kl2
< r
(3.5)
k--m
Arguing in reverse, each such {ak} corresponds to some (k,t)EHE.
To
pass to the expression of u in terms of the variable i we use (2.24), whence cok(2-r)
E ak hk 12 e k=-m o
u(t)
v(t,t) dT
(3.6)
.
But for use in the canonical form (2.31), (2.32) we need u(t) in terms of t
0;(t,-) so that
From (2.29), (2.30) we have
.
2 f2 u(t) -
E akhk k=--
0k(2-T)
2
E
c (t,T)dr
e o
(2--r)
akhk j e wk
k=»
Wt-'r) -
(t,a)da ] dT
p('2-
ro
0
This result can be simplified as a result of the following proposition. Proposition 3.1
There are non-zero complex numbers
Pk
with
0
and
p
being independent of 2
wk(2-T) r
k
r I
fo
o
2
_ Pk
where the functions 2.16
.
(3.7)
, such that 1
p(.(t,a) daj dT (3.8)
dt , - m< k <
0 Pk
are the biorthogonal functions defined by
DIFFERENTIAL DELAY EQUATIONS
135
aW ;T
Proof Since the
form a basis for
EM
Cue
L2[0,2]
we can write
, C = Wt.-). pi)L2[0,2]
and it is enough to establish the result (3.8) for the special cases
!il T
<j <
e J Since (cf. (2.16))
/2
eW3.T
dti = b
Jr
0
all we have to show is that 2
jo
wk
-
Ie
e
For
Re(s) > Re(w3) 2
63
(3.9)
a-s1 pcp-a)
a
d&
0
-
1-e
s-wj
+
r e-5p
rp
p pT7 e m' a
da d t
(3.10)
fo
pp a
a
dc, &
10
(letting
T = r+2
e-sp
(
o
2 1
denote the Laplace transform, putting
and using the fact that p(s) - 0, s > 2)
(-s+w3)2 =
w a
p
2
dp - Jo
(-5+co')2
=
dT =
J
consider the expression
e'p
f 0e-sp
dol
a
P
o
LLL
l-e
- 1(p)(s) L(e
s -
a-sr (
+ e-2s Fo
a )(s) a
r+2
p r+-a a r
da dr (cont.)
DAVID L. RUSSELL
136
1e
_
(-S+w )2 s-Ca
j
wa
w3 a
[e-2s
da- L(p) (s )] L(e
p2 - a e U
j)(s)
=(using (2.22)) +
-e(-s+wj)2
2s (ej2+e)
e-
1
-
S
(3.11)
Since
p
has support in [0,2], (3.11) is entire in s , except for a s = wj
(possible) pole of first order at -2wk
1
11+e
wk-wj
e
-Y
For
s = wk, k ' j, we have
-wka
2
p(a) daJ
e
-
.
o
(3.12) e-2wk
2wk
I o p(2-t) e
[e
`''k-`''j
wki
2
+e-y-
d.r]
where we have again used (2.22).
Byt if we take
the right hand side of (3.10) by
e
2
Io e
wk (
2-T[e"J"-
,
s = wk
and multiply
(3.12) clearly shows
wa
Io.s
0
J
j
p(2-a) e j da d, = 0,
k 3'
j
What we have shown, in effect, is that (cf. (2.29))
wjt
wk(2-T)
2
e
j
(C(e
))(T) dT
= 0,
k # j
0
wk(2-r)
Since the
e
2 ,
pj(ti) form dual bases for
L [0,2]
w.t C(e
for some scalar
(3.13)
)= jpj pj
and, since
C
is bounded and can readily be seen
to be boundedly invertible, we conclude that the bounded away from zero.
pj
are bounded and
The result (3.8), and with it the proof of the
proposition, then follows.
DIFFERENTIAL DELAY EQUATIONS
137
A more operator-theoretic proof and interpretation of Proposition One can also see from (2.22) and (3.11) that
3.1 appears in [12].
Pj _ d .
{e2S+e_1_e25L()(s)] I
(3.14)
s=wj
a result which is in agreement with the manner in which the biorthogonal functions
are constructed in"[6], [5], [14] and which will be
pj
useful in Section 4.
Now we are in a position to prove our major result. Let distinct complex numbers
Theorem 3.2
vj, - - <
< W
,
be selected with the property
2
hJZ
<m.
(3.15)
J- mI
Then there is a
pair,
such that the feedback relation (3.2)
(k,t) E HE
yields a closed loop system 2
+Y at -
at
2 -7
(3.16)
HE
have been replaced by the eigenvalues
wj
for which the eigenvalues vj
= 0
+ r(x)w - g(x) ((w,at).(k,t))
ax
.
Proof
Consider a system (cf. (2.14))
i=vjyj, having the
vj
<j<m
as eigenvalues.
(2.24), (2.29) but with the system (cf. (2.31), (2.32))
wj
Carrying through the transformations replaced by the
vj , we arrive at the
DAVID L. RUSSELL
138
4(t"')
-r-
a'r
at
= 0
(3.17)
2
e(t,2) + e-11(t,0) ` J T e(t,T)d-r o,
(3.18)
wherein (cf. (2.21), (2.16))
v2
q2-T
=
_
(ej+e ')
E
q
j=-m (2
e
J
T.
(3.19)
) dc = bj
q
(3.20)
0
Now the earlier work of the present section shows that if the feedback relation (3.2) is used in (2.31), (2.32) we obtain
t'T at
-
t'T
aT
=0
(3.21)
2
) .(t,T)d'r
(t,2) + e Yr'(t,0) = J 0
(3.22) 2
+
E ajpjhj J ;=-0D 0
p(2-) T .(t,tr)dT
To prove our theorem then we need to show the existence of a sequence {aj}
with
E
1aj12<
(3.23)
for which (3.21), (3.22) agrees with (3.17), (3.18), i.e.,
E ajpjhj pj (2--r) = q(2-TT j=
(3.24)
DIFFERENTIAL DELAY EQUATIONS
139
6o that (3.21), (3.22) will agree precisely with (3.17), (3.18). Let
q 2-T
E
=
d. pj 2-T
(3.25)
j=
Then by the biorthogonality relation (2.16),
r2 dj = 1
wjw q(2-T) dT
e 0
v32
r2 =
e
1
v.T
4).T
r2
q2-TdT+J (e3-e3 ) qT'f-Jdt 0
0
v.2 (cf. (3.19), (3.20))
+
vJ
(e
+e-Y)
(fcoj
f 0
da
e°C dT) q 2-T
dt
j
v .2
r2
7 dT
(i:J rear dT
(e ' fe 'Y) +
(3.26)
fo Now
If IJ 0
wJ.
J vj
Te°Ld`}
q
7dr
`- 2v/2lwj-vj) sup le°`I
2
< 2J
0 I1
,. vjJ eat(
I
CI dT
11gIIL2[0,21
where the "sup" is taken over the straight line segment joining and
Wj
in the complex plane.
this quantity is uniformly bounded, independent of (3.26) we see that we have
vj
From (2.15), (3.15) it is clear that j
.
Returning to
DAVID L. RUSSELL
140
v.2 dj = (e 3 +e-Y) + YjIWj-vjl (3.27)
w.2 = (e Y +e-Y) +
wherein the
Yj
,
Yj
Yjl.j-vjl
are uniformly bounded complex numbers.
Then
(3.24) becomes (cf. (3.25), (3.27))
j
yj lcaj-vj I
E
) =
P
j=
E aAhj P
) ,
whence
a3
=
Yj_W_-yjI ijhj
Then from (3.15) and (3.7) we have (3.5) and the proof is complete. In [12] we show that with boundary control, where becomes
0
g
in (2.1)
and the second equation in (2.2) becomes
alw(l,t) + bl
(l ,t) = u(t)
(3.28)
the condition (3.15) is replaced by
E
Iwj-v.1
(3.29)
< m .
j=
We also show there that the asymptotic relationship (2.15), i.e.,
=-2+i(k-k)n+O(k)
,
can be replaced by
with
Y
+ i ( k - k)n +0(k)
an arbitrary complex number, by taking
DIFFERENTIAL DELAY EQUATIONS u(t) = u(t) + u(t) u(t) = a2w(l,t)
,
(3.30)
+ b2 az 0,t) + C2 at (1,t)
al b1
D
141
f 0
.
I
a2 b2 /
After this "boundary feedback", the resulting system with the second equation of (2.2) replaced by
(1,t) = u(t)
(al-a2)w(l,t) + (b1-b2) aax (l,t) - C2 aawt
can be further modified by feedback similar to (3.2). that a combination
The result is
of feedbacks (3.30), (3.2) for boundary control
(3.28) applied to (2.1) (with g=0) can produce any desired eigenvalues
vk = -
+ i(k
Z)n + bk
2 with
z
Ibkl2
Thus the "asymptotic line"
<
be preke-rved with distributed control
g(x)u(t)
must
Re(w)
or boundary control
(3.28) with distributed feedback, but can be altered to
y arbitrary, if we allow boundary feedback as well.
Re(v)
Then, within the
established "asymptotic line" eigenvalues can be selected at will, provided the relevant condition (3.15) or (3.29) is maintained, with distributed feedback similar to (3.2).
This provides a very nearly
complete spectral determination theory for control systems (2.1), (2.2) (or (3.28).). 4.
Spectral Determination for Certain One-Dimensional Diffusion
Processes.
Let us now consider a diffusion process related to the system (2.1), namely,
at - a?2
+ r(x)w = g(x)u
(4.1)
ax
and with precisely the same boundary conditions as before, which we repeat for convenience:
DAVID L. RUSSELL
142
aow(O,t) + bo
at
(O,t) = 0, a1w(l,t) + b1
at
(l,t) = 0.
(4.2)
For a system of this type it is natural to use a feedback relation of the form 1
(w'(x)k' x
u(t) = J
+ w(x)(r(x)+r0)c x )dx
(4.3)
0
t(x) = 0
which corresponds to
With use of such a feedback
in (3.2).
law the closed loop system becomes
at+Llw=0
where
Ll
,
(4.4)
is the operator 2
+ r(x)w
(Llw)(x) = ax
1
- g(x) f(w'(x)k'(x)+w(x)(r(x)+r0)vcrdx 0
with boundary conditions again of the form (4.2).
(4.5)
Now the eigenvalues
of the operator (4.5) are precisely the squares of the eigenvalues which would be obtained for the system (2.1) with
and with the
= 0
.
feedback control (3.2), i.e., with a special (in the context of (3.2)) feedback law for which the dependence on
v = at is zero.
return, therefore, to the system (2.1) with
y = 0
Let us
and explore the
effect of the control law (4.3). Since we are taking
y = 0
in (2.1) now, (2.7), (2.8) becomes
'Icivcm-k+l
-
k=1,2,3, ...
This, together with the fact that we are taking place of (3.3) u(t) =
E i=1
[yiai-ziai3
t(x)
(4.6) 0 , gives, in
DIFFERENTIAL DELAY EQUATIONS
143
and then, in (2.14), we have (4.7)
u(t) =
akyk(t)
E
k=-.. With
y = 0
k = 1,2,3,...
ak = - a-k+l '
we have (cf. (2.11))
hk = hk = gk/2
and, therefore, in (2.14) we have hk = h_k+l
All of this means that the function
of (2.21) can be rewritten in
p
the "symmetric form" 2
PI =
B (e
+1)Pk + E
(e-
2
+1)pk(2-t)
k=1
k=1
(4.8) (e<,)`
E
k=1
+1) P:Ty + PTA
,
since 2
pk(2-t) = P-k+l 2-t
PPJU
- e
Using (4.7) and (4.8) we see that the feedback relationship (4.7) now becomes
u(t) =
E akPk hk Pk : -
E 4k hk pk(2-t) k=1
k=1
(4.9) k=
=1
akPk hk Pk('F
-
E
k=1
akik
What we need next is a relationship between shall obtain from the formula (3.14).
hk
Pk
e 2wk
and
We have, since
p-k f
ik
, which we
y = 0 ,
DAVID L. RUSSELL
144
e2s + e-1 - e2s L(5) (s)
=
e2s + 1
e2s L(5) (s)
-
= es [es+e-s-es L(p)(s)] Now, since
p(t) = o, t > 2
,
e..st
es L(P)(s) = es f
-pTtTdt
0 2
(setting T=2-t) = es
)
e-s(2-i
f
0
2
= e-s fo e
Hence
es + e-s
es L(p)(s)
-
-(-s1T
1'\T/dT = e-S L(P)(-s)
is an even function vanishing at
and from this we conclude that
±r,,l, ±w2, tw3,
pk = d5'
{es[es+e-s-e sL(P)(s)]}s=-wk
[es+e-s-esL(p)(s)]s=-w
e-`Ak Td
=
s
e
.k
-wk dds [es+e-s-esL(P)(s)] S
k
2wkPk
{e-s[es+e-s-esL(P)(s)]}s=w
e-2{`'k
ds
= - e k
Then (4.9) also assumes the symmetric form
u(t) =
= k=1
aOk hk (pk 2-t
+ PP-T))
(4.10)
It is now an easy matter to see that, with this type of feedback, we can realize eigenvalues provided (3.15) holds and
vk
for the closed loop system (4.4)
DIFFERENTIAL DELAY EQUATIONS
145
vk = - v-k+l
(i.e., A is moved to the eigenvalues of
vk
,
-wk
is moved to
(cf. (2.3)) are
L
-vk).
Then, whereas
Xk = wk, those of
L1
(cf. (4.5)) are
µk=vk , k=1, 2, 3, Now
vk = wk + hkek
,
{ek}
square summable, otherwise arbitrary.
Hence 2
µk = ak + 2wkhkek + hk ek
If
{*k}
=
+ wkhk C 2ek +
wkk
is a square summable sequence the equations 2
k
2ek +
1, 2, 3, ...
k
have the square summable solution sequence
-2wk+ ek 2hk (choosing the branch of the square root which reduces to hk - 0).
2wk
as
Hence we have
Let distinct complex numbers
Theorem 4.1
µj, j = 1, 2, 3, ...
be
selected with 2
E j=1
wjhj
Then there is a function
(4.11)
k E H1[0,1], k(0) = 0, such that the feedback
law (4.3) yields a closed loop system (4.4) wherein the eigenvalues of
DAVID L. RUSSELL
146
the operator
wj
,
5.
and
are the numbers
L1
Thus the
µj,
j = 1, 2, 3, ...
.
can be moved "wj times as far" as one can move the
aj
at will", subject only to (4.11).
Remarks on Canonical Equations of Higher Order Almost forgotten by now is the restriction
imposed at the beginning of Section 2. restriction; for if we take
b
0
bo = 0, ao
0
Nevertheless, it is a crucial
t 0 (whether
a
0
$ 0
or not will then
be immaterial) the canonical form undergoes a very decided change.
To
see that this is the case, consider the simple situation wherein y = 0, r(x) = 0, ao = a1 = 0, bo = bl = 1 (cf.
.
The eigenvalues of
L
(2.3)) are then
k = 1, 2, 3, ...
k2n2,
o,
with corresponding orthonormal eigenfunctions
fo(x)
1,
mk(x) = v2 cos k,ix, k = 1, 2, 3,
Passing to formulae analogous to (2.12), (2.13) we have
wo = 0,
wk = krri,
coo =0, wk=-krri,
k =1,2,3,
We have an eigenvalue of multiplicaity two at zero corresponding to the equation (cf. (2.5))
wo(t) = gou(t)
(5.1)
The corresponding exponential functions (cf. (2.16) and foregoing discussion) eUt,
1
=
eikrrt,
k =
(5.2)
±1, ±2, ...
form the familiar Fourier basis for
L2[0,21
.
But, corresponding to
the multiple eigenvalue at zero, the homogeneous counterpart of (5.1), i.e.,
w;-(t) = 0 , has independent solutions 1, t and the totality of
DIFFERENTIAL DELAY EQUATIONS
147
functions which must be taken into account includes those in (5.2) and also the function t L2[0,2] t
.
Since those in (5.2) already form a basis for
.
, we no longer have an independent set when we add the.function
The moment problem associated with the problem of controllability
(cf. (2.17)) now becomes 2
y0(2) = h0
u(t)dt
(5.3)
(2-t)u(t)dt
(5.4)
0
2
yo(2) = ho
= hlk4
(2 1
e
ikr(2-t) u(t)dt,
Whereas the equations (2.17) can be solved µk12 E
km
< m
,
k = ±1, ±2, ...
.
(5.5)
0
i f yk = hk Nk
,
this is not possible for (5.3)-(5.5) because once
y0(2), yk(2)' k = ±l, ±2,
...
are selected, u is determined and thus
y0(2) is determined - it cannot be selected arbitrarily.
Thus the
system
dy0
= yo
(5.6)
(5.7)
dt0 = h0 u(t)
dtk = ikn yk + hk u(t),
is not controllable using controls
k = ±l, ±2, ...
u E L2[0,2)
.
(5.8)
In general (see [131)
this situation can be repaired by allowing controls u ( L2[0,2+e] , Another approach is e > 0 . But then uniqueness of control is lost. required.
What is needed is a space of controls whose dimension is "one more To make a than the dimension of L2[0,2]" - in some appropriate sense.
DAVID L. RUSSELL
148
long story (which eventually will appear elsewhere) short, the appropriate space turns out to be H1[0,2] carry
relative to H-1[0,2]
L2[0,2]
H-1[0,2] , the dual space to
The control to state map, U, will
.
t2 and the normal form, obtained from (5.6),
into
(5.7), (5.8) by use of
U
, will "live" in
H-1[0,2]
.
A convolution
type transformation is then used to produce the control canonical form,
which in this case turns out to be
'(t+2)="(t) + u(t) "living" in H-1[0,2]
L2[0,2] L2[0,2]
or
H1[O,2]
or ,
(5.9)
,
, according to whether
respectively.
is obtained by simply applying
to
u
lies in
The "principal part" of (5.9),
t(t+2) = t(t)
,
the principal part
of
(t+2) = fi(t) = u(t) the operator (5.8).
d/dt
.
(5.10) "
,
Of course (5.10) is the canonical form for (5.7),
The added equation (5.6) is accounted for by the differentiation
in (5.9).
This is entirely analogous to what happens in the finite
dimensional case.
More generally, we might suppose that we have a system which,
after modal analysis, takes the form
µtZt + ftu(t),
dtt
dF=wkyk+hk with the
µt
,
<,
t = 1, 2,
(5.11)
u(t), -m
(5.12)
all distinct and
wk = (k+a)tti + 0(k),
Iki
Such a system would be obtained, for example, if one considered an electronic device involving essentially no delays (such as a conventional circuit with very short wiring paths between components) attached to a long antenna in which the distributed character of the state and finite speed of signal propagation could not be ignored.
If only the
DIFFERENTIAL DELAY EQUATIONS
149
system (5.12) were taken into account the canonical form, obtained just as in the earlier sections of this paper, would assume the form 2
(t+2) = e2anifi(t) +
p(2-w)
u(t)
fo
(see [12] for details).
When we adjoin the system (5.11) the "natural"
space of controls becomes H-n[0,2] and the control to state map n[0,2] . U Hl2 The final canonical form becomes :
.
k (t+2) = e
with
D
2omi
2
k(t) +
f0
p 2-T DK(t+ti)dr + u(t)
(5.13)
denoting the differential operator n
n (t e
e=1
If some of the
- µe)
caL
,
µt are repeated eigenvalues, the form of some of
the equations (5.11), (5.12) would have to be different if we are to have controllability but (5.13) would still be the canonical form.
As
in Section 3, eigenvalue assignment theorems can be obtained with the use of the form (5.13).
Finally, although we have discussed the problem of eigenvalue assignment for diffusion, or "heat", equations in Section 4, there remains the question of what the control cannonical form for such systems will eventually turn out to be.
Our conjecture is that it will
take the form of a "differential equation of infinite order"
ak
k=0
Ad
=
dt
k \
n k=O
(1
-
ddt) X
) = u (t)
k
Comparable canonical forms may also be formed for the Euler-Bernoulli
beam equations and other systems having no minimal controllability interval [2], [3], [7]).
DAVID L. RUSSELL
150
References 1.
2.
3.
4. 5.
6.
7.
Courant, R. and D. Hilbert:"Methods of Mathematical Physics, Vol. II - Partial Differential Equations", Interscience Pub. Co., New York, 1962. Fattorini, H. 0. and D. L. Russell: "Exact controllability theorems for linear parabolic equations in one space dimension", Arch. Rat. Mech. Anal., Vol. 43 (1971), pp. 272-292. "Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations", Quart. Appl. Math., Vol. 32 (1974), pp. 45-69. Graham, K. D. and D. L. Russell: "Boundary value controllability of the wave equation in a spherical region", SIAM J. Control. Levinson, N.: "Gap and Density Theorems", Amer. Math. Soc. Colloq. Publ., Vol. 26 (1940), Providence, R.I. Paley, R. E. A. C. and N. Wiener: "The Fourier Transform in the Complex Domain", Amer. Math. Soc. Colloq. Publ., Vol. 19 (1934), Providence, R.I. Quinn, J. P.: "Time optimal control of linear distributed parameter systems", Thesis, University of Wisconsin- Madison, August :
1969. 8. 9.
10.
11.
12.
13.
14.
Riesz, F. and B. Sz.-Nagy: "Functional Analysis", F. Ungar Pub. Co., New York, 1955. Russell, D. L.: "Linear stabilitzation of the linear oscillator in Hilbert space", J. Math. Anal. Appl., Vol. 25 (1969), pp. 663-675. "Control theory of hyperbolic equations related to : certain questions in harmonic analysis and spectral theory", Ibid., Vol. 40 (1972), pp. 336-368. "Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods", J. Diff. Eqns., Vol. 19 (1975), pp. 344-370. "Canonical forms and spectral determination for a class : of hyperbolic distributed parameter control systems", Technical Summary Report #1614, Mathematics Research Center, University of Wisconsin - Madison, February 1976. (Submitted to J. Math. Anal. Appl.) "Nonharmonic Fourier series in the control theory of distributed parameter systems", J. Math. Anal. Appl., Vol. 18 (1967), pp, Schwartz, L.: "Lt.ide des sommes d'exponentielles", Hermann, Paris, :
:
1959. 15.
Slemrod, M.: "A note on complete controllability and stabilizability for linear control systems in Hilbert space", SIAM J. Control, Vol. 12 (1974), pp. 500-508.
"THE TIME OPTIMAL PROBLEM FOR DISTRIBUTED CONTROL
OF SYSTEMS DESCRIBED BY THE WAVE EQUATION" H. 0. Fattorini
Introduction.
1.
A prototype of the problems considered here is
that of stabilizing a vibrating system by means of the aplication of suitable forces during a certain time interval.
To be specific,
consider a uniform taut membrane clamped at the boundary region
.
of a plane
the membrane vibrates freely (no
t = 0
Up to*the time
thus its deflection
external forces are applied):
r
u(x,y,t)
satisfies
the wave equation
a2u -
at
c2
a22 +
ax
2 a 2
((x,y)
ay
E 2
,
t -5 0)
and the boundary condition u(x,y,t) = 0 where
2 c
((x,y)
f(x,y,t)
,
t s 0)
the modulus of elasticity (resp. the
= p/p , p (resp. p)
density) of the membrane. (1)
( r
At time
t = 0
an external force
begins to be applied; the deflection of the membrane then
satisfies the inhomogeneous wave equation
a
at
*
=
c2 ( a22 + a22 I + f ax ay /
((x,y) E 2
,
t > 0)
This work was supported in part by the National Science Foundation under grant MPS71-02656 A04.
151
H. O. FATTORINI
152
and the same boundary condition as for (1.2)
u(x,y,t) = 0
t < 0
((x,y) E r, t > 0)
.
We assume that the magnitude of the force is restricted by the constraint (1.3)
(C
ff
If(x,y,t)l2 dx dy < C
(t > 0)
a positive constant fixed in advance) while its objective is that
of bringing the energy of the membrane
E(t)
to zero in a time
7
7
2
{
T > 0
U) at
2
aU
+
c2
[ ax)
as short'as possible.
21
2
+
ay
dx dy
J
In other words, we
want to bring the membrane to a standstill as soon as practicable within the limitations imposed on the use of force by the constraint (1.3).
applied
here E(t)
=0
Figure 1
153
THE TIME OPTIMAL PROBLEM
Three questions arise naturally in connection with this problem. (a)
Is it at all possible to reduce the energy by means of a force
T
a finite time
E(t)
to zero in
subject to the constraint
f
(1.3)? (b)
exist a (c)
Assuming the answer to (a) is in the affirmative, does there f0 If
that does the transfer in minimum time? exists, is it unique?
f0
What additional properties
(say, smoothness) does it have?
Problem (a) is a typical controllability problem, and we show Section 3 that it has a solution.
This is scarcely surprising in view
of the extremely lavish class of controls at our disposition.
(A more
realistic situation would be that in which we can only use a finite number of control parameters, for instance
m f(x,Y,t) =
where the functions fl, ..., fm
E k=l
fk(t)bk(x,Y)
b1, ..., bm
are fixed in advance and we can vary
subject to constraints of the form
Ifk(t)1 < C
(1
<_ k <_ m,t > 0)
.
This case, however, is much more complicated; the controllability problem (a) may not have a solution at all, even if we replace the final condition
E(t) = 0
by
E(T) < e
for a given
e > 0).
Problem (b) refers to the existence of optimal controls and it is well known that, at least in the linear case considered here, its ....solution follows from the solution to (a) via a simple weak compactness
argument as in [2], [3], [1].
This is done in Section 4.
We examine in Section 5 problem (c).
There we prove an analog of
the celebrated PONTRYAGIN maximum principle in the form obtained by
BELLMAN, GLICKSBERG and GROSS for linear systems in [2] and generalized
H. 0. FATTORINI
154
to infinite-dimensional situations by BALAKRISHNAN [1] and the author The basic technique here is that of "separating hyperplanes" used
[6].
in [2] for the solution of a similar control problem in finite dimensional space.
It turns out that treating the present problem directly would
involve us with some of its special features (say, finite velocity of propagation of disturbances) that play no significant role on it.
It
is then convenient to cast it into the formalism of second order differential equations in Hilbert spaces.
This is done in Section 2
and the results obtained in the following sections are then seen to be applicable to many different situations.
We examine in Section 6 some variants of the original problem obtained by replacing the constraint (1.3) on the control (which is not necessarily the only physically significant one) by other types and we show that versions of the maximum principle also hold in these cases. Second-order equations in Hilbert space.
2.
We begin our quest for
generality by considering the problem in Section 1 number of dimensions. space
RP(p >_ 1)
in an arbitrary
be a bounded domain in Euclidean
Let then s
with sufficiently smooth boundary
r
and consider
the operator
(2.1)
aku
(Au)(x) = c'au(x) = c2 k=l Z
is defined in the customary way as the
(ak = a/axk).
The domain of
set of all
in the Sobolev space
u
A
that satisfy the Dirichlet
H1(E)
boundary condition (2.2)
u(x) = 0
(x ( r)
and such that. ou , understood in the sense of distributions, belongs to
L2(c)
.
(Recall that
distributional derivatives
consists of all
H1(E))
alu, ..., a u p
belong to
P
Pull 2l H
=
(lull2L2
(c)
laku1122 k=l
L (E)
)
u E L2(g)
whose
L2(g)
with norm
THE TIME OPTIMAL PROBLEM
It is well known ([7]) that operator in
L2(S)
is a negative definite self adjoint
A
u(x,t)
If
.
155
is a solution of the inhomogeneous
wave equation 2
(2.3)
a u =
c2ou + f
(x
0)
that satisfies the boundary condition u(x,t) = 0
(2.4)
and we denote by
(x E r,t _ 0)
u(t)
the function in
t -_ 0
with values in
L2(g)
given by
u(t)(x) = u(x,t) and define
similarly, then
f(t)
is (at least formally) a
u(-)
solution of the abstract differential equation u"(t) = Au(t) + f(t)
(2.5)
(t > 0)
.
We are then naturally led to the following abstract formulation of the time-optimal problem considered in Section l: Let space and
A
(Au,u)
for some
w > 0
<_
-
w11u112
(2) (u E D(A))
, where
indicates the scalar product in
be given elements of
u0, ul, Vol V
H
.
1
(a')
(2.7)
Does there exist a control
f(-)
Ilf(t)II < C
such that the corresponding solution of (2.5) with (2.8)
u(0) = u0
u'(0) = ul
u(T) = v0
u'(T) = vl
satisfies (2.9)
be a Hilbert
a self adjoint operator such that
(2.6)
Let
H
H
156
H. 0. FATTORINI
for some
T > 0 ?
(b')
Assuming there exists a control
requirements in (a), does there exist a to
(v0,v1) (c')
f
satisfying the that transfers
f0
(u0,u1)
in minimum time T ? What additional properties does
f0
have? (3)
In order to put the problem in a somewhat more precise footing, we must examine the equation (2.5) with some care.
We start with the
homogeneous equation (2.10)
u"(t) = Au(t)
(t > 0)
A solution of (2.10) is, by definition, a twice continuously differen-
is satisfied everywhere.
C(t) = c(t,A)
for all
t
and (2.10)
Solutions of (2.10) exist for "sufficiently To make this precise, define
smooth" initial data (2.8). (2.11)
u(t) E D(A)
such that
tiable function
S(t) = s(t,A)
where 1
1
c(t,a) = cos(-X) Yt
(2.12)
C(t), S(t)
1 2sin(-a)'Z
s(t,a) _
,
t
.
(4)
computed through the functional calculus for self adjoint
operators ([9], Chapter XII).
In view of (2.6) the spectrum of
A
is
contained in the negative real axis, so that
11C(t)II 5 1
Let
K
,
be the domain of
definite square root of
11S(t)II 5 1
(-A)7 -A
.
,
(t >0)
.
the unique self adjoint, positive
Then it is not difficult to deduce from
standard functional calculus arguments that if
(2.13)
u0 E D(A), ul E K
,
u(t) = C(t)u0 + S(t)uI
is a solution of (2.10) with initial data (2.8) and that, moreover, it is the unique such solution.
As for the nonhomogeneous equation (2.5),
THE TIME OPTIMAL PROBLEM
if
is, say, continuously differentiable in
f
t > 0
157
the (only)
solution of (2.5) with null initial data is given by the familiar formula
t
(2.14)
S(t-s)f(s)ds
u(t) = 0
is of
(the solution with arbitrary initial data
u0 E D(A), u1 E K
course obtained adding (2.13) to (2.14)).
However, the nature of our
control problem is such that the definition of solution introduced
above is too restrictive (for instance, we will be fprced to consider controls
f
that are much less than continuously differentiable).
is continuous (as a H-valued
C(t)u
functional calculus that
t
function) for any
and continuously differentiable for
u E H
with
(C(t)u)' = AS(t)u ; note that
into
D(A))
S(t)u
t
tive
and
AS(t)u
S(t)
maps
is continuous for any
.
into
H
u ( K).
is continuously differentiable for any
(S(t)u)' = C(t)u
In
It is again a consequence of the
view of this, we proceed as follows.
u E H
K
u
E K
(thus
K
Also,
with deriva-
Making use of all these facts we extend the
previous notion of solution in a way customary in control theory, namely we define rt
S(t-s)f(s)ds
u(t) = C(t)u0 + S(t)uI +
(2.15)
1
0
to be the (weak) solution of (2.5), (2.8) whenever f
u0 E K, ul E H
and
is a strongly measurable, locally integrable function with values in
H. (5)
It is not difficult to see, on the basis of the previous
observations, that
is continuously differentiable (with
derivative t
(2.16)
u'(t) = AS(t)u0 + C(t)ul + J
C(t-s)f(s)ds
0
and that the initial conditions (2.8) are satisfied.
It is not in
H. O. FATTORINI
158
general true that
u
can be differentiated further, so that it may not
be a solution of (2.5) in the original sense. 2.1 Remark.
In the case where
A
is defined by (2.1), (2.2) the
functional calculus definitions of
(-A)1"2, C(t), S(t)
can be
explicited as follows.
(0 < a0 < a1 5 ...)
be the
eigenvalues of functions.
A ,
Let
{cpn}
Then 1
(2.17)
(-an}
a corresponding orthonormal set of eigen-
1
(-A)2u
=
n(u,con)wn
E k=l
(-A)1"2
the domain of
consisting of all
u
E
E
such that the series
on the right-hand side of (2.17) converges, or, equivalently, such that
We also have
E>
1
C(t)u =
(cos Xnt)(u,(pn)`Dn n=0 1
1
S(t)u = E (a nsin ant)(u,con)`Dn n=0 for all
u
E
L2(9)
2.2 Remark.
.
Some of the assumptions in this section (as, for
example, (2.6) or the restriction of
A
to the class of self adjoint
operators) can be weakened without modifying many of the conclusions in the next sections. 3.
We comment on this in §6.
Solution of the controllability problem. We look now to problem (a)
in §1
in its abstract form (a').
(3.1)
(t '- 0)
Ilf(t)II < C
and such that, for some initial data
u(0)
Its solution involves finding an
satisfying
H-valued function
T > 0
,
the solution of (2.5) with preassigned
u0 , u'(0) = u0
other words, such that
satisfies
u(T) = u'(T) = 0
;
in
159
THE TIME OPTIMAL PROBLEM
(3.2)
J S(T-t)f(t)dt = - C(T)u0 - S(T)u1
(3.3)
Jo C(T-t)f(t)dt = - AS(T)u0 - C(T)u1
Existence of a solution to (3.2), (3.3) for
T
large enough will
follow from some simple manipulations with
C(-)
by introducing some useful notations.
K = K x H
Let
and
S(-)
.
We begin
endowed with the
norm 2
2
II(u,v)IIK = IIuIIK +
where the norm in immediate that
K
K
IIvIIH
is defined by
IIuji
is a Hilbert space.
=
II(-A)1/2uIIH
Elements of K
.
will be denoted
by row vectors or column vectors as convenience indicates.
S(t)
the operator from
H
into K
It is We denote by
defined by
S(t)u =
and observe that, in this notation, the two equations (3.2), (3.3) can be condensed into the single equation C(T)u0 + S(T)u1
f S(T-t)f(t)dt = -
(3.4)
AS(T)u
Let now 0
cp,,y
t 5 T
E D(A)
both
differentiable and S'(O)u = u
1
such that
p(T) = -1
,*(0) - 0 yr(T) = 0 u
+ C(T)u
be twice continuously differentiable scalar functions in
p(0) = 0
If
0
;
t
S(t)u
(p'(0) = 0 gyp' (T) - 0
*'(0) = 0 *'(T) _ -1 and
t - C(t)u
.
are twice continuously
S'(t)u = C(t)u, S"(t)u = AS(t)u, S(0)u = 0,
C'(t)u = AS(t)u, C"(t)u = AC(t)u, C(0)u = u, C'(O)u = 0
H. O. FATTORINI
160
(see the comments preceding (2.15). that, if
u, v E D(A)
Then integration by parts shows
and
f(t) = cp(t)Au - cp"(t)u + *(t)Av - it"(t)v we have
(3.5)
IT S(T-t)f(t)dt
=
(:)
and it is easy to see that we can.choose gyp, y in such a way that
(3.6) M(T)
lif(t)II
<_ M(T)(Ilull +
I(Aull + 11Y11 + IIAvII)
a nonincreasing function in
T > 0
which does not depend on and
We perform now some computations with follows directly from its definition that
S(.)
It
.
satisfies the "cosine
functional equation"
(3.7) for all C
C(C)C(TI) = 2' C(C+TI) + - C(C-TI) ,
Tl
.
to
,
rl
and integrating with respect to
1:'
in
that
0 < ' <_ C
(3.8)
or by
We obtain also from the definition of
Writing (3.7) for
S(C)C(T1) = 7 S(C-#,t) + 2 S(E-1)
We apply next both sides of (3.7) to an element of Au, u E D(A)
H
of the form
, obtain4ng
S"s()C(T1) Au = T S"(g+rl)u + -
C'(C+n)u +
and, upon replacing
71
by
T1'
(p-u C'(C-n)u
u,v.
~
and integrating in the interval
THE TIME OPTIMAL PROBLEM
(3.9)
S(C)S(TI)Au = 7 C(C+l)u - -- C(g-0)u We replace now
C
(3.9) and integrate in
by
T - t
by
,1
,
t
in (3.7), (3.8) and
0 r t < T , obtaining
fo C(T-t)C(t)u dt = T C(T)u + -T S(T)u
I S(T-t)C(t)u dt = J0 C(T-t)S(t)u dt =
S(T)u
I S(T-t)S(t)Au dt = T C(T)u - -T S(T)u
or, taking (3.10)
u EH
v
,
E K,
JD S(T-t)(C(t)u + AS(tv) dt
2 S(T)u + 2 C(T)v T 2
C(T)u +
1
S(T)u +
S(T)v T
AS(T) v
We go back to the controllability problem assuming for the moment that u0, ul E K
We look for a solution of the problem of the form
.
C(t)u + AS(t)v + fl(t)
f(t)
where
u E H
,
v ( K
and
are as yet unspecified.
fl
In view of
(3.10), equations (3.2) and (3.3) become
(3.11)
(3.12)
C(T)v - . S(T)v + 4 S(T-t)fl(t) dt
S(T)u +
- C(T)u + . S(T)u +
7
7
AS(T)v + 4 C(T-t)fl(t) dt
AS(T)u0 - C(T)ul
.
161
H. 0. FATTORINI
162
Clearly both equations will be satisfied if we take
v2
u = -T u1
(3.13) and choose
u0
in such a way that
f1
S(T)u T f0 S(T-t)f1(t) dt =
(3.14)
T
0
S(T)u1
which clearly can be done in view of (3.5) and comments preceding it and the fact that
belong to
u0 , u1
K
Now, it is not difficult to
.
show that 1
(3.15) for
u
IIAS (t)u1i < I1(-A) 2u11 E K
,
so that 1
Ilf(t)II
TIIul11 + III(-A)2u011 1
1
+ T M(T)(IIu011 + II(-A) u011 + Ilu111 + 11(-A)2u111) and it is then clear that
11f(t)II < C
for
large enough.
T
We have
then proved 3.1 THEOREM.
Let
u0 E K
,
u1
E H
.
Then the controllability
problem has a solution for sufficiently large under the added assumption that follows.
Let
u1
( K
.
and define, for every
e =
T
.
We get rid of it next as u
E H
3
e(u) = {t
; IIC(t)ull < ellull ; 0 <_ t < T}
.
In view of the cosine functional equation (3.7) for have
2C(t/2)2 = C(t) + I
.
Accordingly, if
t E e(u)
211C(t/2)uli ' 211C(t/2)2u11 > 1 - IIC(t)uli
=.q =
t/2
we
THE TIME OPTIMAL PROBLEM
which makes it clear that if words, that measure of
(3.16) for all
e(u) e(u)
and
E E
.
then
t/2 fe(u)
are disjoint.
'lf e(u)
cannot exceed
fo IIC(t)u11 2 dt u
E e(u)
t
or, in other
This means that the
2T/3 ; hence
Ilull2
27
(This argument is due to GIUSTI.)
Define now
N(T)u = fT C(t)2u dt
Clearly
N(T)
is a self adjoint operator and we can write (3.15) as
follows:
(N(T)u,u) ' which shows that
27 Ilull2
N(T)
is invertible and that
IN(T)-lll `5 T' We examine now (3.14) again in the light of the preceding comments on N
.
Write
f1 = f2 + f3
f2(t) =
where
C(T-t)N(T)-1S(T)ul
T
Then it is clear that fp C(T-s)f2(s) ds = T S(T)ul
Call now
v(T) = f 0 S(T-t)f2(t) dt
preceding (3.5) construct an f3
and, making use of the comments
such that S(T)u0 + Tv(T)
fn S(T-t)f3(t) dt = - T 0
To prove that this is possible, and that small norm for
T
f2
will have sufficiently
large enough we only have to show that
163
S(T)u0 -
H. 0. FATTORINI
164
Tv(T)
and that
E D(A)
The statement for
IIA(S(T)u0 - Tv(T))II
S(T)u0
preceding observations; as for help of (3.9).
v0 = VI = 0
remains bounded as
T
is a direct consequence of (3.15) and Tv(T)
it can be easily proved with the
This ends the proof of Theorem 3.1 for the case The general case can be easily deduced from the one just
.
solved using the invariance of equation (2.5) with respect to time reversal.
u0, v0 E K, ul, vl E H
In fact, let
that there exists a solution
.
Take
T
so large
(resp. f2) of the controllability
fl
problem with (u0,u1) (resp. (v0,vl)) as initial data and zero final data in
0
t_ T with
Ilfl (t)II
(rasp. IIf2(t)II < T)
<_
(0 <_ t .<_ T)
Then
f(t) = fl(t) + f2(T-t)
solves the general controllability problem. 4.
Existence of optimal controls.
(u0,u1), (v0,v1) ( K
Given
shall call any strongly measurable function satisfying (3.1) and driving an admissible control.
(u0,u1)
to
f
H
with values in
(v0,v1)
we
in some time
T > 0
We have established in the previous section
that continuous admissible controls always exist: we show next that, giving up continuity in favor of measurability time optimal controls exist as well. 4.1
Let
THEOREM.
optimal control
f0
(u0,ul), (vo,vl) E K .
driving
(u0,ul)
to
Then there exists an
(v0,v1)
in minimum time TO
The proof is an infinite dimensional analogue of that in [2]. Since the extension has already been carried out ([3], [1], [6]) in varying degrees of generality, we only sketch it here. infimum of all
drives
(u0,u1)
T
to
Let
be the
TO
f
for which there exists an admissible control (v0,v1)
of admissible controls driving T1 > T2 > ...
,
Tn
in time (u0,u1)
TO
T
.
to
that
Choose now a sequence
{fn}
Tn
with
(v0,v1)
in time
THE TIME OPTIMAL PROBLEM
and consider
{fn} as elements of the space
Chapter III) extending {fn}
Since the sequence
is uniformly bounded in
exists a subsequence (which we still denote to an
;
H) (see [8],
fn = 0 L2(0,T1
there.
there
H)
;
{fn}) that converges weakly
which, as easily seen, must vanish in ta TO
f0
satisfy (3.1) almost everywhere. to
L2(0,T1
by setting
(Tn,T1)
to
fn
165
f0
The fact that
and must
drives
(u0,u1)
follows from taking limits in the sequence of equalities
(v0,v1)
C(Tn)u0 + S(Tn)ul - v,
-
IT
n S(T -t)f (t) dt = 0
n
AS(Tn)u0 + C(Tn)u1 -
°
v1
which can be easily justified on the basis of the weak convergence of {fn}
(see [3] for further details)). Let
The maximum principle.
5.
(u0,u1)
and
f0
be a control joining two points TO
in minimum time
(v0,v1)
isochronal set (of f0) to be the set of all
=
f0 0
v
(u,v) E H x H
,
the
of the form
S(T0-s)f(s) ds
f
(that is, for some strongly measurable
f
for some admissible control f
S2(=2(T0))
T
(U) (5.1)
and define
that satisfies IJf(t)II
a:e. in
<_ C
t >_ 0)
.
We assume in the sequel (as we plainly may) that
is convex(6).
that
2
from the definition of
2 c K
.
C = 1
.
It is clear
It is also immediate that
S2
Two crucial properties of the isochronal set are:
(1) The interior of
2
(in
K)
is non void.
(ii) (w0,w1)=(v0,v1)-(C(t)u0+S(t)u1,AS(t)u0+C(t)ul) is a boundary
point of
9
.
The proof of (I) follows essentially form that of Theorem 3.1. (u,u') E K .
Let
By "running backwards" equation (2.10) we can assume that
(u,u') = (u(T0),u'(T0))
for a solution
(u(0),u'(0)) E K ; precisely,
of (2.10) with
H. O. FATTORINI
166
C(TO)u(0) + S(TO)u'(0)
u
(5.2)
AS(TO)u(0) + C(TO)u'(0)
u'
where u(0)
C(TO)u - S(TO)u'
u'(0)
-AS(TO)u + C(TO)u'
(5.3)
(the justification of (5.2) and (5.3) is an easy consequence of formulas (3.7), (3.8) and (3.9)). find a control
According to Theorem 3.1 we can now
such that
f
TO
S(TO-t)f(t) dt =
f
10
(:)
with
!If(t)II < MII(u(0), u' (0))IIK M
a constant independent of
(0 5 t s TO)
(u(0),u'(0))
.
,
But, on the other hand, it
follows from (5.3) that
II(u(0),u' (0)HK = II(u,u' )IIK so that if
is sufficiently small the control
II(u,u')IIK
admissible.
f
will be
This shnws that the origin is an interior point of
The proof of (i!) follows from (i). not a boundary point of
SZ
.
Sd
In fact, assume (w0,w1)
is
Taking into account that the function
t -+ C(t)u0 + S(t)u1
is continuous in
is continuous in
it is not difficult to deduce the existence of a
T1 < TO
and a
H
v1
t -+ AS(t)u0 + C(t)u1
Ct)u0 + S(t)u1 E 9
-
r
and that
such that
r < 1
vO (5.4)
K
AS)uO + C(t)u1
(Tl
t <_ TO)
THE TIME OPTIMAL PROBLEM
But this clearly means that C(t)up + S(t)ul
vp
fT
(5.5)
-
AS(t)u0 + C(t)ul
vl
where
0 S(Tp-s)f(s;t)ds
=
o
is an admissible control with
(5.6)
Ilf(s;t)II < r
(0 < s s TO)
We observe next that
TO-t lim t-. TO
in
K
S(10-s)f(s;t)dt = 0 10
, so that making use of the remark at the beginning of this
section we can construct a (5.7)
IIg(s;t)II
1
- r
g(s;t)
,
(1 s s 5 t)
such that
rT -t
rt S(t-s)g(s;t) =
(5.8) I
if
t
0
0 I
is sufficiently near
S(Tp-s)f(s;t)dt
0 T0
.
Hence
fTO S(TO-s )f(s;t)ds 0
It S(t-s)(f(s+T0-t;t)+g(s;t))ds 0
which shows, in view of (5.5), that in time
t < TO
(u0,ul)
can be driven to (v0,vl)
which contradicts the optimality of
TO
.
This
proves (ii).
We can now apply one of the standard separation theorems of functional analysis ([8], Chapter V) and deduce the existence of a
167
H. O. FATTORINI
168
nonzero continuous linear functional (5.9)
y in
such that
,c
Y(u,u') : Y((w0,w1))
in the isochronal set
for all (u,u')
any linear functional in
K
st
.
It is easy to show that
must be of the form 1
Y(u,u') _ (u*,(-A) u) + (u*l,u')
for some u*,u E H
But then (5.9) can be written in the following
.
form:
T
(u,(-A)S(T0-t)f(t)dt) + 100
100 C(T0-t)f(t)dt)
1
T T (u*0,(-A)2 f 0 S(T0-t)f0(t)dt) + (U*,, 100 C(T0-t)f0(t)dt)
O for all admissible controls 1
f
.
Since we can write T
T 5(T-t)f(t)dt) = 1
(u*0,(-A) 2
0
1
((-A)
'Z
S(T-t)u**,f(t))dt
0
and T
rT
C(T-t)f(t)dt) J
5.1
THEOREM.
in minimum time such that
0
TO
Let .
f0
fo
(C(T-t)u*,f(t))dt
be a control driving -(u0,u1)
Then there exist
to.
(v0,v1)
u0, ut E H, IIu0*lI2 + IlujIJ2 > 0
THE TIME OPTIMAL PROBLEM
169
1
(5.10)
((-A) T S(T-t)u*0 +
f0(t))
suPllfll`1S(T-t)u0 + C(T-t)u1,f)
a.e. in 0 5 t < T Clearly, (5.10) does not provide us with information on
f0(t)
at
points where 1
(-A)2 S(T-t)uo + C(T-t)u = 0
However, there can be only a finite number of these points in the interval
0 s t s TO
In fact, assume this is not the case, and let
.
{tn}
be a sequence of zeros of
some
t
there.
It is clear that
'
in
0 s t < TO
converging to
u(t) =
is a genuine
solution of (2.10) with u(t)=0 and u'(t)=1im(tn-t)'1(u(tn)-u(t)) = 0 which, by uniqueness, shows that identically zero.
(hence
is
This is absurd it view of the fact that
0 u*
and u
cannot vanish simultaneously.
We obtain then the following 5.2
COROLLARY.
If
is an'optimal control,
f0
,P(uo,u'i,t)
(5.11)
f0(t)
(0 <- t 5 TO) II
except perhaps at a finite number of points
t0 <
< to
where
0
Since
'(u*,u*,t)
is a continuous function, we see that
be piecewise continuous (i.e. continuous in the intervals (tn,TO))
f0
must
(0,t), ...,
H. O. FATTORINI
170
f0(t)
FIGURE 2
Another immediate consequence of (3.11) is uniqueness: f0, fo Then
be two controls that transfer optimally g = 1/2(f0+f0)
(u0,u1)
in fact, let to
(vo,v1)
must be as well optimal and, because of (5.11) it
must satisfy
ilg(t)II =
1
(0 <_ t
<_ T)
except perhaps at a finite number of points. at every continuity point of
f0
But then
f0(t) =
or 0 , which shows that
f0
'0(t)
and
% must have the same discontinuity points and coincide everywhere else.
6.
Generalizations. The maximum principle in other geometries. We re-examine briefly the "abstract" time-optimal problem set up
in Section 2.
The assumptions on
A
there, while general enough to
include the optimal problem in Section 1 are too strong to yield results
THE TIME OPTIMAL PROBLEM
in other problems.
RP
For instance, the assumption that
w < 0
(2.6) for a
(in this case
should satisfy
does not hold for the Laplacian in the whole space w = 0 )
cant change in the theory.
This, however does not cause any signifi-
.
In fact, we only need to assume that
self adjoint and that (2.6) holds for some The operators
A
171
A
is
perhaps negative.
(-A)1'2, A-1, which make several appearances in the
treatment, are then replaced by (A-A)1/2
for
x > - w .
A more significant generalization is that OF abandoning the assumption that
A
is self adjoint and supposing only that the Cauchy
problem for the equation (2.10) is well posed in the sense of [4], [5]. The operator-valued functions
are then defined in terms of
the solutions of (2.10) and it is possible to show that (XI-A)-1
exist for
x
(xI-A)112,
large enough and that all the properties needed
of these entities in the treatment of the abstract time-optimal problem in Section 2 (in particular the cosine functional equation (3.7)) hold. The following caveat, however, is important: The solution of the controllability problem (a') in Section 2 that was given in Section 3 was based on uniform boundedness of
C(.)
and the result
and
does not necessarily hold without these conditions.
Nevertheless, the
theory can be carried forward in the following sense: if the controllability problem (b') has a solution then the result in Section 4 applies to show existence of an optimal control and the results in Section 5 apply as well with some modifications. tions:
The operator
N
We list some of these modificawhich was used in
defined in Section 3
Section 5 to show that the isochronal set
2
has interior points is
now defined by fT
N(T) _
C(t)C*(t)u dt 0
The proof that
N(T)
has a bounded inverse runs much like in the self
adjoint case, however now we take M(T) = sup{(IC(t)(!;
modification.
i?(u**,u?r,t)
0 < t
T}
.
e = (2M(T) +
l)-1
where
The existence theorem 4.1 needs no
Finally, in Theorem 5.1 we must replace the function
defined there by
(xI-A*)1"25*(T0-t)u*
+
C*(TO-t)ut, where X > - co as in the comments opening this section.
H. 0. FATTORINI
172
The preceding considerations are based on a desire to carry the results to what appears to be their natural level of generality; we mention here that there is no need of assuming
H
to be a Hilbert
space(a Banach space will do) although this extension does not seem to have a wide range of applicability outside of systems described by the wave equation in one space dimension.
We show that the techniques in the present paper can be success-
fully applied to variants ofthe problem in Section 1.
Assume the
constraint (1.3) is replaced by
f?x(O,T) If(x,t)l 2 dx dt f- C
(6.1)
.
The resulting time optimal problem can be cast into abstract form and treated much in the same way as the one discussed in Section 2 and 5.
The solution of the controllability problem in Section 3 applies of course to the estimate (6.1), as well as the existence Theorem 4.1, with minor modifications.
driving time from
(u0,o1) E K
f0
to
the optimal control, TO (v0,v1) E K
is defined in the same way as
9 = 9(10)
f E L2(O,TO;H)
controls
Q
satisfying (6.1).
.
the
The isochronal set
but using this time It can be proved just as
is a convex subset of K with nonvoid interior and that
well that (w0,w1)
Call
The separation theorem can be
is a boundary point of
u0, u E H , not
then applied to deduce the existence of two elements
both zero and such that
(6.2)
(00
(4,(u*,ut;t),f0(t))dt = sup
f00
where the supremum is taken with respect to all
f
in
L2(0,TO;H)
with
1T0
IIf(t)II2 dt.s 1
0
(we are of course assuming that Section 5,
C = 1
in (6.1))and
denotes, as in
THE TIME OPTIMAL PROBLEM
173
1
(-A)2 S(T0-t)ua + C(T0-t0)u}
.
It turns out then that
(Jb0
(6.3)
f0(t)
(6.4)
f0(t) =
(0 s t s FO)
111,
(0 < t
if necessary replacing 2u*,u should have norm 1 in L (0,T;H)
by .
T0)
7Xuo,au
in order that
The representation (6.11) implies
also uniqueness of the optimal control: as a matter of fact, the only information needed to prove uniqueness is that
TO
(6.5)
in fact, if if
IIf0(t)112 dt =
f0
f0, ?
1
are two optimal controls so is
is to satisfy (6.5) we must have
g
-(f0+f0) = g
f0(t) = f0(t)
(note that both
must be continuous in view of (6.3). The methods outlined in the preceding remarks apply equally well to constraints of the type
.r IIf(t)IIp dt s 1
(6-6) for
1
for
p = 1
<_ p : -
(although the existence theorem 4.1 has no counterpart
, unless H-valued measures are used as controls).
We comment finally, on two cases of some physical interest; namely, those in which the constraint (1.3) is replaced by (6.7)
or by
fgx(0,T) lf(x,t)I dx dt s C
H. 0. FATTORINI
174
If(x,t)I f C
(6.8)
a.e. in
HX(O,t)
.
The constraint (6.8) causes no problem as regards existence; however, the "natural" way of extending our results in Sections 3 and 5 fails,
as the wave equation does not give rise to a well-posed Cauchy problem in
L'(-T)
.
The same observation applies to the constraint (6.7), in
which case the "natural" setting will be the space
L1(.=)
.
In this
case, moreover, the question of existence of optimal control seems hopeless unless the class of controls is conveniently enlarged.
REFERENCES 1.
A. V. Balakrishnan, Optimal control problems in Banach spaces, SIAM
2.
R. Bellman, I. Glicksberg and 0. Gross, On the "bang-bang" control problem, Quart. Appl. Math. 14 (1956), 11-18. H. 0. Fattorini, Time- timal control of solutions of o rational differential equations, SIAM Journal on Control 2 (1964). 54-59. H. 0. Fattorini, Ordinary differential equations in linear topological spaces, I. Jour. Diff. Equations (1969), 72-105. H. 0. Fattorini, Ordinary differential equations in linear topological spaces, II, Jour. Diff. Equations (1969), 50-70. H. 0. Fattorini,_The time o timal control problem in Banach spaces, Appl. Math, and Optimization 1 (1974), 163-188. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1, Interscience, N.Y., 1962. N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, N.Y., 1957. N. Dunford and J. T. Schwartz, Linear Operators, Part II, Interscience, N.Y., 1963.
Journal on Control, 33
3.
4. 5.
6. 7.
8. 9.
1(965), 152-180.
FOOTNOTES
The past history of the membrane is obviously irrelevant and is only
1.
introduced here for the sake of picturesqueness; the relevant data are only 2.
u(x,y,O), ut(x,y,O)
In the case at hand, (2.6) is satisfied with w =
first eigenvalue of
A
.
110,
-110 < 0
the
We examine later cases where (2.6) is replaced
by a weaker inequality, with a view to other applications. 3.
We note that the, control problem
(a'), (b'), (c')
can be reduced
to a similar control problem for a first order equation in the usual way:
set
U = (u(t),u'(t))
and write (2.5) in the form
U'(t) = AU(t) + Bf(t)
175
THE TIME OPTIMAL PROBLEM
where
and
is the operator of "projection on the second coordinate".
B
However, the existing results on the time-optimal problem for firstorder equations ([3],[6]) pertain to the case where
B
is the identity
operator and thus cannot be applied here. 4.
The choice of the square root is obviously irrelevant;
c
and
s
are entire functions of both their arguments. 5.
The reader unfamiliar with Lebesgue-Bochner integration theory may
consider
f
to be piecewise continuous, in which case the integral in
(2.16) is an ordinary Riemann vector-valued integral.
After all, it
follows from the maximum principal in Section 5 that the optimal control f0
is piecewise continuous (although the proof uses Lebesgue-Bochner
integrals') 6.
2
is also bounded, and an argument like the one used in the proof
of Theorem 4.1 shows that it is closed.
These properties are not
significant in our analysis. 7.
The somewhat overexplicit treatment of the controllability problem
in Section 3 was designed in such a way that. its extension to the present situation is immediate.
"SOME MAX-MIN PROBLEMS ARISING IN OPTIMAL DESIGN STUDIES" Earl R. Barnes
1.
Introduction
In 1773 Lagrange [1] attempted unsuccessfully to determine the shape of the strongest column.
This is the shape of the column whose
buckling load is largest among all columns of given length and volume. For columns with circular cross sections the problem was solved by Clausen [2] in 1851.
It was solved for columns with general cross
sections in 1962 by Tadjbakhsh and Keller [3]. problems have been studied by many authors.
Since that time similar
Typical among these are
the problem of optimally designing vibrating beams studied in [4], and the problem of designing optimal circular arches studed in [5]. Several In
other interesting examples are included in [6], [7], [8] and [9].
most of these structural design problems the critical buckling load is the lowest eigenvalue of a self-adjoint diffferential equation.
The
critical buckling load is therefore the minimum of a certain Rayleigh quotient.
It is this minimum which has to be maximized in obtaining an
optimal design.
Structural design problems therefore lead to max-min
problems.
Traditionally,
these problems have been treated as problems in the
calculus of variations.
However, in many cases, it is desirable to
impose constraints on the structure to be designed which make the design problem more amenable to treatment by optimal control techniques than by classical variational techniques.
For example, in determining the
shape of the strongest column as in [3] and [9], one is led to, columns
which taper to a point at various places along the column. clearly
undesirable from a practical point of view.
This is
Therefore, in
formulating the strongest column problem, a positive lower bound should be imposed on the thickness of admissible columns.
177
The need to impose
EARL R. BARNES
178
similar constraints in several other structural design problems is pointed out in [6].
Our main purpose in this paper is to provide
techniques for solving optimal structural design problems in the presence of thickness constraints.
The techniques we develop have applications to a variety of other problems.
They can be used to determine stability conditions for solu-
tions of second order differential equations with periodic coefficients. Such problems arise in determining the stability or instability of an elastic structure subjected to a periodically varying force.
It is a
well-known fact that the stability of such a structure is dependent upon the frequency and amplitude of the applied force.
We shall
describe this dependence in Section 4 and give conditions on the frequency and amplitude that will guarantee stability.
It turns out that
the problem of determining these conditions leads to extremal eigenOf all Sturm-Liouville operators belonging to a certain
value problems.
class it is required to determine the one whose n-th largest, and the one whose n-th eigenvalue is smallest, n = 1,2,...
.
Since the eigenvalues are minima of Rayleigh quotients, determining the stability conditions require solving max-min and min-min problems. A third class of max-min problems that can be solved by the method These
we present here arises in the design of cooling fins and spines.
are extended surfaces attached to heated bodies such as engines and radiator tubes for the purpose of dissipating heat to a surrounding We shall study these problems in Section 5.
medium. 2.
Extremal Eigenvalue Problems Consider an untwisted column of length
similar cross sections with areas
a < A(x) < b
(2.1) where
a
and
b
, volume
V
0<x
We wish to determine the strongest admissible
This is the one whose critical buckling load is largest.
Let an admissible column be subjected to an axial load may cause it to buckle. plane.
, and
, 0 < x < R , satisfying
are given positive constants.
be termed admissible. column.
,
A(x)
z
Let
w(x)
P
which
In its buckled state the column will lie in a
denote its lateral deflection from the straight
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
position. (2.2)
Then
179
satisfies the equation
w
(EI(x)wxx)xx + Pwxx = 0, 0 < x < R,
together with some set of boundary conditions such as w(0) = wx(0) = 0, (2.3)
wxx(R) = Pwx(R) + (EI(t)wxx(a))x = 0,
or
w(0) = wx(0) = 0,
WW = wxx(e) = 0.
(2.4)
Conditions (2.3) correspond to a column clamped at x=R.
x=0
Conditions (2.4) correspond to a column clamped at
hinged at
and free at x=0
and
x=c..
In (2.2)
E
is Young's modulus of the column material and
is the moment of inertia of the cross section at
x
I(x)
about a line
through its centroid normal to the plane of the deflected column. all cross sections are similar
I(x)
is related to
A(x)
Since
by
1(x) = aA2(x)
where
a
is a proportionality constant determined by the shape of
cross sections. Let
y(x), 0(x)
A2(x)wxx(x), p(x)
=
and
be new variables defined by
a
A-2(x), x = « .
y(x)
This change of variables trans-
forms the differential equation (2.2) into (2.5)
y" + Ap(x)y = 0
.
The boundary conditions for y'(0) = 0 (2.6)
y(t) = 0
y
are obtained in [3].
They are
EARL R. BARNES
180
in the clamped-free case, and Y(0) + RY'(0) = 0 (2.7)
Y(R) = 0 in the clamped-hinged case.
The column will not buckle until the load A =
great that
Ea
P
is sufficiently
exceeds the lowest eigenvalue of (2.5) with appro-
priate boundary conditions.
Let
A1(p)
denote the lowest eigenvalue
of (2.5) together with one set of the boundary conditions (2.6), (2.7). The optimal column design problem may be formulated as: Maximize
(2.8)
A1(p)
subject to k
p-1/2(x)dx
f0
= V
h
b-2
where
and
h =
H =
.
These constraints are the fixed volume
constraint (2.9)
lZ0
A(x)dx = V
,
and the limited thickness constraint (2.10)
a < A(x) < b
imposed on admissible columns.
Problem (2.8) has been solved in [10]
for the case of columns 'clamped at
x = 0
A similar problem is solved in [9].
and hinged at
x = e
.
This is the problem of deter-
mining the optimum taper of a thin-walled tubular column of constant thickness.
In this case the cross sectional areas are annuli with
moment of inertia
I(x) = aA3(x)
For columns hinged at each end the
deflection is governed by the differential equation
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
181
Y" + XP(x)Y = 0 (2.11)
Y(0) = Y(t) = 0 The optimal design problem for such columns is maximize
(2.12)
a1(p)
subject to
1R0
0-1/3(x)dx = V
h < p(x) < H where
a1(p)
,
,
is the lowest eigenvalue of (2.11).
solved in [9] for
h = 0
arbitrary positive
h
and
and H
H = m .
This problem was
We shall give the solution for
in Section 3.
Problems (2.8) and (2.12) are special cases of a general class of extremal eigenvalue problems which we shall now study. Let
p(x)
be a measurable function defined on an interval [0,L)
and satisfying inequalities of the form H
are given bounds.
For each such
Al(P) < a2(P) < A3(p) <
p
h < p(x) < H , where let
..
denote the eigenvalues of one of the boundary-value problems (2.13)
y" + ap(x)y = 0, 0 < x < £, aly(0) + 81Y'(0). = 0,
a2Y(t) + 82y'(R) = 0;
(2.14)
y" + (a-P(x))y = 0, 0 < x < R, aly(o) + sly'(0) = 0 a2Y(R) + 82y'(R) = 0,
h
and
EARL R. BARNES
182
where the i
= 1,2
Let
and
ai
are real numbers satisfying
8i
Jail + 18i) # 0
In the case of problem (2.13) we shall assume that
.
f1(x,p),...,fm(x,p)
defined on the region
be
m
h > 0
.
given real-valued continuous functions
(0,Q) x (h,H)
, and let
c1,...,cm
be
m
given
constants.
As our notation indicates, the eigenvalues are functionals of
p
ai(p),
i
We shall be interested in the extremal eigen-
.
value problems: (2.15)
minimize
a(p), n = 1,2,...,
maximize
An(p), n = 1,2,...,
and
(2.16)
subject to e
(0 fi(x,p(x))dx = ci,
(2.17)
h < p(x) < H A measurable function
i
.
p
, defined on [0,t] and satisfying these condi-
tions will be termed admissible. denoted by
A
.
= 1,...,m
The class of admissible
We shall assume that
A
is not empty.
theorem gives necessary conditions for an admissible solution of (2.15). Theorem 2.1. where
An(p)
to be a
An analogous result holds for problem (2.16). For a fixed
n
,
let
p*
be solution of (2.15),
refers to the n-th eigenvalue of (2.13).
an eigenfunction corresponding to the optimum value there exist Lagrange multipliers that
p
will be
p's
The following
Let y*
an(p*)
.
denote Then
no > 0, nl'" 'nm, not all zero, such
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
183
2m
(2.18)
max
{n0y*(x)p +
h
E
nifi(x,p)}
i =1 2m
oy*(x)p*(x) +
=
nifi(x,p*(x))
E
i=l
for almost all
x
in
Similarly, if if
p*
[0,tj
an(p)
.
denotes the n-th eigenvalue of (2.14), and
is a solution of (2.15) and
y*
then there exist Lagrange multipliers
the corresponding eigenfunction, no > 0, nl,...,nm, not all zero,
such that
m
{noy* (x)p +
min
(2.19)
h < p
nifi(x,p)}
E
i=1 m
= noy*2 (x)p*(x) +
E
nifi(x,p*(x))
i=1
for almost all
in [0,a,]
x
.
The proof is based on a generalized multiplier rule found in [11, Chap. 4].
In order to state this rule we require a few definitions.
For a fixed
psA
let
R
z0 = an(p)
Let as
space
zi
= J0 fi(x,p(x))dx,
denote the totality of the vectors
Z p
and
ranges over Em+l
.
z*
Let
Definition.
of vectors
A
.
Z
z = (z0,zl,...,zm)
denote the point in Z
Z
obtained
at
corresponding to z*
with the following property:
any finite collection of vectors from the form
= 1,...,m
is a subset of Euclidean m+l-dimensional
By a derived set for
K c Em+l
i
p*
.
we shall mean a set If
K11...,kN
is
K , there exists a surface of
EARL R. BARNES
184
N
Z(el,...,eN) = z* +
in
Z
for
6 > 0
Z
j=}
kjej + o(e), 0 s e. 5 6, j
and sufficiently small.
o(e)
is a quantity satis-
fying
ow
lim f e I-.0 i d
Theorem 2.2.
derived set for
Z
= 0, where
IeI =
N z
j=1
(Generalized Lagrange Multiplier Rule) at
z*
.
Then there exist multipliers
Let
K
be a
t0 > 0
kl,...,Rm , not all zero, such that
m Z
tiki
> 0
i=0
for each vector
k = (k0,kl,...,km) E K
This theorem is given in [11, page 1783. Theorem 2.1 will become clear shortly.
Its connection with
First we shall prove a lemma.
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
Lemma 2.1. an(p)
Let
and
p
a
denote any two elements in
denote the n-th eigenvalue of (2.13), n = 1,2,...
185
A
and let
Then
.
rs
lim + t -+ 0
where
y
-xn(p)JOY2(x)(a(x)-o(x))dx
An(p+t(a-p))-an(P)
t
=
¢
py (x)dx fo
is the eigenfunction corresponding to For
Proof.
0 < t <
1
pt = p+t(o-p)
let
an(pt)
eigenfunction corresponding to
an(p)
.
and let
yt
denote the
, normalized so that
py2dx
to ptyt2dx = 0
Then
Yt + An(pt)pt(x)yt = 0
and
y + an(p)P(x)Y = 0
.
Upon multiplying the first of these equations by second by
yt
and the
and integrating by parts we obtain
R J0
y
rk
a n(pt)pt(x)ytydx = JO an(p)p(x)Yytdx
This implies that (jt -an(pt)JO(a-P)YtYdx
an(p+t(a-p))-an(p)
t
R
fo By letting
pytydx
t - 0 , and making use of the continuous dependence of the
EARL R. BARNES
186
eigenvalues and eigenfunctions of (2.13) on
, we obtain the conclu-
p
sion of the lemma.
Proof of Theorem 2.1.
Since eigenfunctions are unique only up to
a scalar factor we may assume that rR
p*y*2(x)dx = an(P*)
fl
f0
X E (0,A)
Let
and
p
E [h,H]
k = (k0,kl,...km) E Em+1
defined by
k0 = -(P-P*(x))Y*2(x)
(2.20)
= fi(x,P) - fi(x,p*(x)), We shall show that, except for vectors Let
kl,k2,...,kN
p1,...,pN E [h,H]
= 1,...,m .
i
forming a set of measure zero, the
x's
form a derived set for
k
vectors of type (2.20). and
Consider the vector
be fixed.
at
Z
z*
.
denote an arbitrary finite collection of Then there exist values
xl,x2,...,xN E (0,e)
such that
ko = -(Pj-p*(xj))y*2(xj) and
kj = fi xj,Pj) - fi(xj,P*(xj)), where j = 1,...,N
kj =
xl < x2 <
For simplicity we shall assume that xj
are points of continuity of
weaker measurability assumptions. xi + N6 < xj
j = 1,...,N
.
.
Let
... < xN
and that the
The proof is valid under much 6 > 0
xi < xj , and such that
if
E = (c1) ...,CN)
p*
.
be chosen such that
xN + N6 < R
.
be a vector of real parameters satisfying Let
Let
0 < ej < 6
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
X1 = xl I j = 2,...,N
.
Xj = xj + El + ... + Ej-1
Clearly, the intervals
187
,
Ij = [X..X.+E
are nonover-
lapping.
Define the admissible function
by
p
N
P*(x), x E
U j=l
x ( Ij,
,
i
I
= 1,...,N
.
R
Let
and
z0 (E) = an(PE)
fi(x,PE(x))dx,
zi(E) =
i
= 1,...,m.
I0
An easy consequence
of Lemma 2.1 is that k
y*2(x)(PE (x)-P*(x))dx + o(c)
z0(E) = an(P*) - I 0 N
(2.21)
an(P*) -
Ejy*2(x.)(Pj-P*(xj)) + 0(E)
E
j=l N
= z* +
cjk? + o(c)
E
j=1
Here we have used the continuity property of diff@rentiating the integral in (2.21).
P*
at the points
xj
in
Similarly, we have
N
2.
Zi(E) = I f.(x,P*(x))dx + E Ej[fi(xj,Pj)-fi(xj,P*(xj))1 + o(c) Q
j=l
(2.22) N
E.k1. + o(c), E i + j=1 J J
i
= 1,...,m
Combining (2.21) and (2.22) we see that the vector z(E) = (z0(E),zl(E),...,zm(E))
satisfies
N
Z(E) = Z* +
E
j=0
Ejkj + o(E) E Z
EARL R. BARNES
188
This completes the proof that vectors of the form (2.20), for almost all
x
,
form a derived set.
It now follows from Theorem 2.2 that there exist multipliers el,...' m
RO > 0,
such that
m -xay*2(x)(P-P*(x)) +
t1[fi(x,P)-fi(x,P*(x))] > 0
E
i=1
for all
p
E [h,H]
and for almost all
,
no = k0
i
, ni = -zi Theorem 2.1.
x E [0,e]
.
By taking
= 1,...,m , we obtain the conclusion of
,
In a similar manner the following theorem can be proved. For a fixed
Theorem 2.3.
where
let
n
P*
be a solution of (2.16)
Let y*
refers to the n-th eigenvalue of (2.13).
An (p)
the eigenfunction corresponding to the optimum value there exist Lagrange multipliers
min
h
no > 0,
nl,...,nm
an(P*)
.
denote Then
such that
m E nifi(x,P)}
{nay*2(x)P +
i=1
(2.23)
E nifi(x,P*(x))
= n0y*2(x)P*(x) +
i=1
for almost all
x E [0,9,]
Similarly, if if
p*
an(P)
.
denotes the n-th eigenvalue of (2.14), and
is a solution of (2.16) and
y*
the corresponding eigen-
function, then there exist Lagrange multipliers such that in
max
h < P< H
E nifi(x,P))
inaY*2(x)p +
i= l
(2.24) 2m
= nay*(x)P*(x) +
E nifi(x,P*(x)) i=1
for almost all
x
in
[0,e]
.
no >> 0'
nl'" ''nm
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
3.
The Shape of the Strongest Tubular Column Let
p*
189
denote a solution of problem (2.12).
p*
As we have seen,
determines the shape of the strongest thin-walled tubular column in
the class of columns hinged at
x = 0
and
and volume, and similar cross sections. of Theorem 2.3, there exist constants
ft
, and having fixed length
According to condition (2.23) no > 0
and
n
such that
min Ln0y*2(x)p + no-1/3] h < p < H (3.1)
= n0y*2(x)p*(x) + for almost all
[O,1]
in
x
no*-1/3(x)
.
We shall assume that the quantities satisfy no
and
a, b, V
in (2.9) and (2.10)
at < V < bt
.
When this is the case it is easy to show that
> 0
.
Without loss of generality, we take
n
are
For convenience we shall drop the
*
on
p*
and
y*
no = 1/3
Condition
.
(3.1) implies that if
h
(3.2)
n3/4(Y(x))-3/2
p(x) =
if
H
for almost all
n3/4(Y(x))-3/2 < h
x
if
h < n3/4(Y(x))-3/2
H
n3/4(Y(x))-3/2 > H
[0,t]
in
Since y(O) = 0 , for values of x have n3/4(y(x))-3/2 > H and p(x) = H
.
sufficiently close to
0
we
For these values of
x
the
differential equation (2.11) is simply (3.3)
y" + AHy + 0
.
It is instructive to view the solution in the phase, or In phase space, (3.3) implies that the point
y,y'
(y(x),y'(x))
plane.
is moving
along the ellipse (3.4)
y'2 + AHy2
=
y'2(0)
in a clockwise direction.
See Fig.
1.
The assumption
at < V < bt
190
EARL R. BARNES
.
implies that
is not identically equal to
p(x)
0 < x1 < R/2
will come a time
H
.
when the condition
Therefore, there n3/4(y(xl))-3/2 = H
is satisfied. Then for sufficiently small values of x >xl, we must have n3/4
=
p(x)
For these values of
(Y(x))-
the differential equation (2.11) becomes
x
An3/4y-1/2
y" + and the point
= 0
(y(x),y'(x))
y'2 + 4an3/4yl/2
(3.5)
3/2
is moving clockwise along the curve =
4anH-1/3
y'2(xl) +
If h is sufficiently small, all points on this curve will satisfy n3/4y-3/2 > h We shall assume this to be the case. This amounts to .
the assumption that the optimal column nowhere achieves the maximum allowable thickness determining
b
We leave to the reader the problem of
.
in the case where the maximum allowable thickness is
p*
achieved by the optimal column. In the case we are considering the point along the curve (3.5) on the interval (3.4) on the interval Fig.
1.
Clearly,
[s.-xl,,¢]
y'(z/2) = 0
,
(y(x),y'(x))
as is indicated by the arrows in Moreover, since eigenfunctions are
.
unique only up to a scalar multiple we may assume that scaled so that
y(2/2) =
1
.
Solving this equation for
y
has been
Equation (3.5) must then be given by
y'2 + 4an3/4y112 = 4X9
(3.6)
moves
(xl,x.-x1) , and along the curve
3/4
y'(xl)
.
and substituting into (3.4) gives
the equation
for
3anH-1/3
Y'2 + AHy2 = 4an3/4 _
(3.7)
(y(x),y'(x)),
0 < x < xl
.
This equation can be solved for
on this interval the solution shows that
terms of
x
(3.8)
xl -
arc sin 1XH
4n
3/4H _
nH-4/3 -3nH
y
in
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
191
I u,vi
,
`
s , y.
AH y
=4111
3A H
(Y(11. '4111
I
Figure 1. The path of the solution is indicated by arrows.
In order to obtain a second expression for function
e(x)
we introduce a new
xl
defined by the requirement xl < x < k/2
y(x) = sin4e(x),
.
Substituting this into (3.6) gives the differential equation
i.= 3de sino
an3/4 ,
e(x1) = arc sin
for
(3.9)
e
.
xl
< x < z/2
(nl/8H-l/6) ,
o(t/2) = ,r/2
This implies that
x1 = s,/2 -
2
an3/4
,r/2
2
sin ode
arc sin (nl/8H-1/6)
192
EARL R. BARNES
n1/8H-1/6
Z =
Let
(3.8) and (3.9) then imply the equation
Z
and
Z
a
3
sin ede
J
443Z
for the unknowns
("/2
2
2
arc sin
H
(3.10)
.
= 112
aH arc sin Z
ZZ
A second equation for these unknowns is
.
implied by the condition f£/2 0
We have x
rt/2 p-1/3(x)dx
= f
0
I H-1/3dx + 112 n-1/4sin2edx
0
H-1/3
A
+ 2Z 3 J
F72for
Z
.
5
sin ede = V/2 Z
n/2
5
sin ede arc s in
Z
arc sin
r/2
+
44
arc sin -Z2+ 2Z_15 f (3.11)
2H-1/3
Z2
arc sin
YrT
xl
Z Z
=
V/at
3 sin ede
Tr/2 arc c sin Z
It is easily shown that this equation has a unique solution
in the range
0 < Z <
1
Having determined Z , n can be obtained from the equation Z = n1/8H-1/6 If this value of n satisfies n3/4 > h , then the optimal column never achieves the maximum allowable thickness. case the appropriate value for
Given these values of
n
and
a a
can be obtained from equation (3.10). , the cross sections
A(x)
strongest thin-walled tubular column are obtained as follows. solve the initial value problem
Y"+as(Y)Y=0, 0<x<£. Y(0) = 0
,
Y,(0) =
In this
4an3/4_3xnH-1/3
of the First
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
193
where H
if
n3/4y-3/2
H
p(Y) =
n3/2y-3/2 A(x)
n3/4y -3/2 < H
if
.
is given by
a ,
nl/2y1/2(x)
A(x) =
la where
x1
0<x<x1
,
,
x1 < x < t - xl
k-X1 <x
is given by (3.8).
The case where the optimal column
achieves the maximum allowable thickness can be treated in an analogous manner. 4.
Lyapunov Zones of Stability for Hill's Equation We use the term "Hill's equation" to mean any homogeneous, linear,
second-order differential equation with real, periodic coefficients.
For the present discussion we restrict our attention to equations of the form
y" + (A-Bp(x))y - 0
(4.1)
where
A
and
s
- . < x <
are real parameters and
function satisfying and
,
for all
p(x+t) = p(x)
is a piecewise continuous
p
x
.
It is known that if
bounded.
On the other hand, an improper choice of
A
result in all nontrivial solutions being unbounded. B = 0
x
are properly chosen, then all solutions of this equation will be
a
then all solutions are bounded for
solutions are unbounded for
A > 0
constant, hence periodic, solution. to determine conditions on ness of all solutions.
A
and
For
.
and
8
will
For example, if
A > 0 and all nontrivial A = 0
the equation has a
Our main purpose in this section is B
that will guarantee the bounded-
We begin with an example to motivate these
considerations.
Consider a thin uniform rod of length L verse vibrations due to periodic and forces Let the line segment its undeflected state.
0 <
< L
, undergoing small trans-
p(t) = p(t+t), - - < t <
represent the position of the rod in
In its deflected state, let
w(E,t)
denote the
EARL R. BARNES
194
deflection at w
at time t
E
Under very mild assumptions (cf. [12])
.
satisfies the differential equation 4
where
E
2
2
a
at
is Young's modulus of the column material and
moment of inertia of cross sections. m the rod.
is the
I
is the mass per unit length of
We assume that the rod has hinged ends so that in addition to
(4.2) we have the boundary conditions 2
(4.3)
2
w(O,t) = a £ (O,t) = 0
,
a
w(L,t) = - (L,t) = 0 a6
Stability of the rod would require that all solutions of (4.2), (4.3) be bounded.
In studying the boundedness of solutions of this
boundary problem, it is customary to restrict attention to solutions of
the form N
w(g,t) =
(4.4)
E
yk(t)sin
k
k=1
that is, to solutions which have a finite Fourier sine series representation.
By substituting this solution into (4.1) it is found that yk
satisfies
(4.5)
(7)2P(t)) Yk(t) = 0
rgyk(t) +
This equation is of the form (4.1).
,
k = 1,...,N
.
The solutions (4.4) are bounded
for arbitrary initial conditions if and only if all solutions of (4.5)
We shall determine stability conditions for
are bounded. in (4.1). 6 =
If these conditions are satisfied for
a
and
6
A = EI(k)4 and
k = 1,...,N , then we can assert that solutions of (4.5) are
bounded for all
t
.
Numerous applications requiring a stability analysis of equation (4.1) are discussed in [13]. studied by many authors. In [14] and [15]. [16] in 1899.
The stability question itself has been
Good summaries of what has been done are given
The most fundamental results were given by Lyapunov
For a modern treatment of these ideas see [15, Chapters 2
195
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
We shall now describe the part of this theory that is required
and 3].
for our work.
For a fixed
.
a > 0
A1(T) < A2(T) <
let
,
...
denote the
successive eigenvalues of the boundary-value problem y" + (A-Bp(x+T))y = 0 (4.6)
Y(0) = y(&) = 0
.
Let An' = min An(T)
A" = Max An(T)
,
,
0 < T < L
and let
n = 1,2,...,
f R [y'2+Bp(x)y2]dx f0 A " = min 0
(4.7)
y2dx f0
where this latter minimization is taken over all absolutely continuous functions
satisfying
y
A" < A1 < A" < A' < A" < 2- 2
-
0
We shall sometimes use the notation
Theorem 4.1. if
x
A"
An ,
For a fixed
a
(A1(B), A"(B))
,
n = 1,2,...
A,(B) on
A"(B)
to indicate the
a
(AR(e), An'+1(B))
,
n = 0,1,...
.
lies in one of the intervals
a
If
.
Then
, all solutions of (4.1) are bounded
lies in one of the intervals
All solutions are unbounded if
.
A' A" < ... 3-< 3
1
dependence of the numbers
y'(0) = y1(t)
and
y(0) = y(L)
a
is the endpoint of one of these
intervals, then (4.1) has at least one nontrivial periodic, hence bounded, solution.
This theorem follows from Theorem 3.1.3 in [15]. (An, An+1)
The intervals
are called stability intervals, and the intervals
are called instability intervals. Theorem 4.2.
As
k
T = (2k+1)7r/A +
both
and
AZk+l
A2k+l
satisfy
(2k+1)-1a-1 /2 (l p(x)dx + o(k-1)
0
(An, An)
EARL R. BARNES
196
and both
A2k+2
and
satisfy
A2k+2
,R
p(x)dx + o(k-1)
T = 2(k+l),r/k + (k+l/4 0
This theorem appears as Theorem 4.2.3 in [15]. the length of the
It implies that
instability interval tends to zero as
n-th
n
.
Theorem 4.2 shows that the asymtotic
Consider problem (4.6).
behavior of the eigenvalues depend only on
f
p(x)dx
We remark,
.
however, that the asymtotic convergence may be very slow.
The asymto-
tic behavior of the eigenvalues of (4.6) suggests the consideration of the boundary-value problem y" +. (x-sp(x))y = 0
,
0 < x < a
(4.8)
Y(O) = y(t) = 0 and the associated extremal eigenvalues (4.9)
xn = min xn(p)
,
n = 1,2,...,
(4.10)
A" = max xn(p)
,
n = 1,2,...,
where the
min
(4.11)
max
and
ftn
10R
h = min p(x)
x" = OM .
Theorem 4.3.
values of
and
y
By taking
p(x+T)dx = M ,
interval
,
H = max p(x)
The interval ,
,
0 < x < R
.
constant in (4.7) we see that
h < p(x+T) < H
(All, An+1) n
h < p(x) < H
p(x)dx ,
p(x)dx = M =_
J0R where
are taken subject to
the interval
A" < A0
.
Since
the following Theorem is clear.
,
lies inside the stability
(All,
n = 0,1,...
We also define
Moreover, for sufficiently large
.
(x", xn+1)
is not. empty.
We shall now give a brief indication of how the results of Section 2 can be used to compute the values consider
xj
xl(p*) = x .
.
Let Let
p*
y*
An, x
n = 1,2,...
First,
be a measurable function satisfying (4.11) and be the corresponding eigenfunction of (4.8).
197
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
Then according to condition (2.24) of Theorem 2.3, there exist constants no > 0
max {n0y*2(x)p-np} = n0y*2(x)p*(x) - np*(x) h < p < H
(4.12)
If
such that
n
,
hi < M < Ht
,
then it is easy to show that
There are two possibilities for satisfying
p*
no > 0
and
n > 0
If the set of points
.
h
and
H
However, if
.
interval of positive measure, then we must have
y*2(x) = n
.
x
has measure zero, then (4.12) implies that
y*2(x) = n
achieves only the values
consequently
.
p*
on an
y*"(x) = 0 , and
p*(x) = B-1a1(p*) , on this interval.
Both possibilities
are taken care of by the formula
if y*2(x) < n
h
(4.13)
If
p*(x) =
min {8-la1(p*), H)
6-IX1(p*) < H
,
y*2(x) > n .
a simple phase plane analysis similar to that
used in Section 2 shows the existence of a point
0 < xl
< t/2
such
that
sin a gh x
p <x <x l
VX_ ON
T r
,
xl
<x
12 .
R
p*(x)dx = M
The condition
implies that
f0
a1M FA-07h
x 1
The condition
y*'(xl) = 0
holds for
y*'(x) = 0 implies that
xj = al(p*)
xl < x <
./2
, and the condition
is the smallest solution of the
equation
a g h A 9 M= n a-sh
in the range
sh < a < BH
.
On the other hand, if it turns out that
g-lal(p*) > H , a direct
EARL R. BARNES
198
substitution
of (4.13) into (4.8) and solving the resulting equation,
yields the equation
A- gh cot ash xl = T tan Vx_- jH (t/2-x1) for a = al(p*) , where RH-M 2 H-h
_ xl
To find
an
for
n >
divide the interval
1
[0,1]
into
n
equal parts and consider the boundary-value problem y" + (a-8p(x))y = 0
,
0 <'x < t/n
(4.14)
y(O) = y(t/n) = 0
,
and the constraints
p(x)dx = M/n
,
(4.15)
h < p (x) < H
.
Maximize the first eigenvalue of (4.14) subject to (4.15). denote the solution and let Extend
p*
p*(x+t/n) = p*(x) y*(x+a/2) = -y*(x)
y*
Extend
.
.
(2.24) of Theorem 2.3.
y*
p*
denote the corresponding eigenfunction.
to the entire interval
periodically
Let
[O,t]
by defining
semi-periodically by defining
Clearly, the extended functions satisfy condition Moreover, it can be shown that these functions
are in fact optimal.
In summary we have the following: Theorem 4.4.
For
n = 1,2,..., x"
, defined by (4.10), is the
smallest solution of the equation a£-BM
in the range
= n,r
8h < a < 8H
provided such exists, and is ctherwise the
smallest solution of the equation
s cot sh xl/n =
tan
(t/2-x1)/n
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
in the range
a
> OH , where
xl
199
2(H-h
A similar result holds regarding the numbers
an
defined in (4.9).
We shall simply state the result in the next theorem and leave the proof to the reader. Theorem 4.5.
n = 1,2,..., an
For
is the smallest solution of
the equation
gf-a coth
x1/n = a-gh tan Bh (L/2-x1)/n
B
Bh < a < BH
in the range
,
provided such exists, and is otherwise the
smallest solution of the equation
a BH cot a-BH x1/n = Bh tan a-Bh (z/2-x1)/n in the range
a
Example. (4.16)
> BH
.
Here
j
x1 = 2MHRh
All solutions of the equation
y" + (A-cos 2x)y = 0
are bounded if
lies in one of the intervals
a
(1.57577, 3.35029),
(0, .31541),
(4.62272, 8.35746),
15.36003),
(16.63322, 24.36123),
(25.63445, 35.36189),
48.36228),
(49.63551, 63.36254),
(64.63578,
99.36285),
(100.63607, 120.36294), ...
These are the intervals
(36.63511,
80.36271),
(81.63595,
.
(an, an+1),
applying Theorems 4.3, 4.4, 4.5, with
(9.63054,
n = 0,...,10
a = 1
,
,
obtained by
to equation 4.16.
The first application of optimal control theory to stability problems for Hill's equation was made by Brockett in [17]. 5.
A Variational Problem Arising in the Design of Cooling Fins When it is desired to increase the heat removal from a structure
to a surrounding medium, it is common practice to utilize extended surfaces attached to the primary surface.
Examples may be found in the
cooling fins of air-cooled engines, the fin extensions to the tubes of radiators, the pins or studs attached to boiler tubes, etc.
In this
section we shall study annular fins attached circumferentially to a cylindrical surface.
See Fig. 2.
The question we ask is this:
Given
200
EARL R. BARNES
a fin of fixed weight and length, and thickness
> h
and
< H
,
how
should it be tapered in order to maximize the rate of heat dissipation to the surrounding medium.
The answer was conjectured by Schmidt [18]
in 1926 for fins with no minimum or maximum thickness constraint imposed on them.
He proposed that the optimum fin should taper, narrowing in
the direction of heat flow, in such a way that the gradient of the temperature in the fin is constant. by Duffin [19] in 1959.
This conjecture was proved rigorously
Since that time a number of papers have
appeared treating various aspects of the optimal design problem. list [20], [21], [22], [23], to name a few.
We
In [23] the present author
obtained the optimum taper of a rectangular fin subject to thickness constraints.
We shall now show how to obtain analogous results for
annular fins.
Figure 2.
Annular Fin on a Cylinder
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES x, y, z
In
plane.
201
space we take the fin to be parallel to the x,y
We assume the fin is sufficiently thin that there is no appre-
ciable change in its temperature in the
z
direction.
If the tempera-
ture of the surrounding medium is taken to be zero, and if we assume that Newton's thermal conductivity is unity, then the flow of heat in the fin is governed by the steady state heat equation (5.1)
(p 2X )
ax
where
+ ay (p 2y) - q u = 0, (x,y)
ES
is the annular region of the fin in the
S
u = u(x,y)
is the temperature in the fin,
of the fin, and
q > 0
y
constant.
x,y
p = p(x,y)
plane.
is the thickness
is the cooling coefficient, here assumed
We shall assume that the outer edge of the fin is insulated
so that the appropriate boundary conditions are u = T (= steady state temperature)on
rl
(5.2)
where
of
S
and
rl .
v
r2
are, respectively, the inner and outer boundaries
is a unit outward normal to the boundary of
S
.
Newton's law of cooling implies that the heat dissipated per unit time by the fin is given by (5.3)
ff.q u dxdy S
The weight of the fin, which we assume to be fixed, is proportional to (5.4)
f p(x,y)dxdy = M
.
The thickness of the fin satisfies the constraints (5.5)
h < p(x,y) < H
for some positive numbers
and
h
H
satisfying
h f f dx dy < M< H f f dx dy S
S
Our problem is to determine
p
and
u
satisfying conditions
(5.1), (5.2), (5.4), and (5.5), in such a way that the integral (5.3) is maximized.
Ouffin and McLain [24] have given a max-min formulation
EARL R. BARNES
202
of this problem. (5.1) by
In order to obtain this formulation multiply equation
u
and integrate, using Green's theorem, to obtain
I
LP (aX)
2
2
+qu2] dxdy.
+ p (ay)
JJ [qu
ax (p ax)
ay (p
ax)]udxdy
S
(5.6) r
+ j
=T1
rl
+
up a do
I
lr2
p
rl
do
up av
avdo
On the other hand, by simply integrating equation (5.1) over the region S
, again making use of Green's theorem, we obtain f1 qudxdy = jI [ax (p ax) + ay (p ay)] dxdy S
S
(5.7)
=J
paw do
rl
Combining this with (5.6) we obtain 2
(5.8)
ff
qudxdy = 1J
T
S
(p ax)
2
+ p (ay)
+ qu2] dxdy
S
The differential equation (5.1), together with the boundary condition
u = T
on
is just the Euler equation for minimizing the
rl
integral on the right in (5.8).
We can therefore write 2
(5.9)
min ff Lp (ax)
If qudxdy = S
T
v
2
+ P (ay)
+ qv2] dxdy
S
where the minimization is taken subject to
v = T
on
rl
.
The problem
of tapering the fin to maximize the rate of heat dissipation is therefore equivalent to the max-min problem
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
2
(5.10)
max min p
2
+ p(ay)
JJ [p(aX)
u
203
+ qua] dxdy
S
where the minimization is taken over functions rl
satisfying
u
and the maximization is taken over functions
p
u = T
on
satisfying (5.4)
and (5.5).
The problem we have formulated is valid for fins on cylinders of arbitrary cross-sectional type.
However, we shall now restrict our
attention to circular cylinders of radius
R
In this case it is clear that the functions
and fins of length p'
and
R
which solve
u
(5.10) must depend only on the distance to the center of the cylinder. Let the center line of the cylinder be along the Fig. 2.
p(x,y)
Introduce the variables .
r =
+y
z
axis as in
(r) = u(x,y), p(r) =
In terms of these variables the problem (5.10) becomes
R +R
(5.11)
max min J p
[rp(r)o'2(r) + gro2(r)]dr R
m
where the minimization is taken over absolutely continuous functions
4,
satisfying (5.12)
(R) = T
and the maximization is taken over piecewise continuously differentiable functions
p(r)
satisfying
R+R
(5.13)
rp(r)dr = fR
h < p (r) < H
.
,
EARL R. BARNES
204
The boundary-value problem (5.1), (5.2) transforms into
dr (rp(r) dr) - grm(r) = 0 (5.14)
0(R) = T, 4'(R+R) = 0
.
The technique used to prove condition (3.14) in [23] can be used to prove the following theorem. Theorem 5.1.
Let
The pair
(5.14).
there exists a constant (5.15)
max h < p < H
for each
r
(r)-n] =
.
p*(r)[m*'2
be a solution of (5.11).
condition (5.15) implies that close to R+q
such that
n > 0 p[OI2
be functions satisfying (5.13) and
q*
is a solution of (5.11) if and only if
(r)-n]
[R, R+z]
in
(p*,m*)
Let
and
p*
(p*,p*)
p*(r) = h
For these values of
r
Since
*'(R+¢) = 0
for values of
r
sufficiently
, equation (5.14) is of Bessel
It therefore seems unreasonable to attempt an analytic solution
type.
of (5.11).
Instead we shall give an iterative procedure which can be
used to obtain numerical solutions.
h < p(r) < H
are no constraints of the form satisfies
0'2(r) =
n,
First we remark that in case there
R < r < R+.t
p, the optimal
on
o
Substituting this into (5.14)
.
gives a simple differential equation from which
p
can be determined.
This analysis is carried out in [19]. To facilitate the discussion of a numerical solution of (5.11) we introduce some notation.
All functions involved will be considered as
iembers of
L2[R, R+A], the space of square integrable functions on
[R, R+t]
We shall use the symbol
.
fR0o(r)p(r)dr
of two functions in
will be denoted as usual, by functional
g
11
g(p) = min
to denote the inner product
L2[R, R+,t]
The norm in this space
For convenience, we define a
.
on the class of nonnegative r
(5.16)
11
0-p
p's
by
R+z
[rp(r)o'2(r) + grm2(r)]dr
1
R
where the minimization is taken over functions satisfying (R) = T Problem (5.11) is then to maximize
g
subject to (5.13).
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES Let
(p2,02)
205
be pairs of functions satisfying (5.14).
Then by arguing as in the proof of Theorem 3.2 in [23], it can be shown that
(5.17)
1
.
rm'2(p2-p1)dr - 2 JO (p2-pl)2dr
9(p2) - g(pl) < I
ro'2(p2-pl)dr
0
where
IT (2R+L)L}2/h
K
.
This means that g is differentiable,
and its gradient is given by dg(p) =
where
r4'2
is the function which solves the minimization problem (5.16).
0
Let
be the function which maximizes
p*
sequence of functions converging to i)
ii)
Let If
pl
subject to (5.13). 'A
g
can be generated as follows.
p*
satisfying (5.13) be chosen arbitrarily.
p1,...,pk
have been chosen, take
pk+1
to be the solution
of R+A
maximize
[vg(pk)'(p-pk)
(
-
T (p-0k)
2]dr
R
subject to rR+R
rp(r)dr -
Z(5.18)
)R
n
h < p(r) < H pk+l
is the solution of a simple moment problem which can be easily
solved numerically.
It is clear from (5.17) that
We shall now show that the sequence converges to
g(pk+l) > g(pk)
g(pl), g(p2), ..., actually
g(p*)
By completing the square in the integrand in ii) above, we see that
pk+l
is the point in the convex constraint set defined by (5.18),
nearest the point
pk + K Vg(pk).
It follows that
EARL R. BARNES
206
(5.20)
(Pk + K vg(Pk) -
satisfying the constraints in (5.18), and in particular for
p
p = pk+1 For
k >
1
,
(5.17) implies that
"P W -PO
g(pk+l) - g(pk)
K(Pk +
vg(pk)-pk+l).(Pk+l-Pk)
K
+ "Pk+l-nk"
Ilpk+l-pkII
-
2 > 2 Upk+l-pkll2 Since
(by (5.20)).
[9(pk+l) - 9(pk)] << 9(p*) - 9(pl)
it follows that
k=1
liM Ilpk+l-pkfl = 0
(5.21)
k
(5.17) implies that 0
vg(Pk)-(P*-Pk)
g(p*) - g(Pk)
K(pk +
=
K K(pk-pk+l).(p*-pk+l) 0
< (vg(pk) as
by (5.21).
k -*
lim g(pk) = g(p*).
This completes the proof that
k- o
Thus for sufficiently large of
p*
k
,
the efficiency of
pk
is close to that
.
For a different treatment of problem (5.9) see [25]. ACKNOWLEDGEMENT
I acknowledge with pleasure the assistance of my colleague Matthew Halfant.
He drew the figures and assisted with some computer
programming required to test the validity of certain results.
MAX-MIN PROBLEMS IN OPTIMAL DESIGN STUDIES
207
REFERENCES 1. 2.
J. L. de Lagrange, "Sur la figure des colonnes, Miscellana Taurinensia," 1770-1773, p. 123. T. Clausen, "Uber die Form architektonischer Saulen," Bulletin physicomathematiques et Astronomiques, Tome 1, 1849-1853, pp. 279294.
3.
4.
5.
6. 7.
8.
9.
10.
11.
12.
13. 14.
15. 16. 17.
18. 19.
20.
21.
I. Tadjbakhsh and J. B. Keller, "Strongest columns and isoperimetric inequalities for eigenvalues," ASME Journal of Applied Mechanics, Vol. 29, Series E. 1962, pp. 159-164. F. I. Niordson, "On the optimal design of a vibrating beam," Quarterly of Applied Mathematics, Vol. 23, No. 1, 1965, pp. 47-53. B. Budiansky, J. C. Frauenthal, J. W. Hitchinson, "On optimal arches," Transactions of the ASME, Series E, December 1969, pp. 880-882. W. Prager and J. E. Taylor, "Problems of optimal structural design," J. Applied Mechanics, Vol. 35, 1968, pp. 102-106. J. E. Taylor, "The strongest column: An energy approach," J. Applied Mechanics, Vol. 34, No. 2, Transactions of the ASME, Vol. 89, Series E, 1967, p. 486. Journal of Optimization Theory and Applications, Vol. 15, No. 1, January 1975. M. Feigen, "Minimum weight of tapered round thin-walled columns," J. Applied Mechanics, Vol. 19, Transactions ASME, Vol. 74, 1952, pp. 375-380. E. R. Barnes, "The shape of the strongest column and some related extremal eigenvalue problems," to appear in Quarterly of Applied Mathematics. M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley and Sons, Inc., New York, 1966. S. Lubkin and J. J. Stoker, "Stability of columns and strings under periodically varying forces," Quarterly of Applied Mathematics, Vol. 1, No. 3, 1943, pp. 215-236. V. V. Bolotin, "The Dynamic Stability of Elastic Systems," Holden Day, Inc., San Francisco, 1964. W. Magnus and S. Winkler, "Hill's Equation," Interscience Publishers, New York, 1966. M. S. P. Eastham, "The Spectral Theory of Periodic Differential Equations," Hafner Press, New York, 1973. A. M. Lyapunov, "Sur une equation lineaire du second ordre, C. R. Acad. Sci. Paris 128, 1899, pp. 910-913. R. W. Brockett, "Variational methods for stability of periodic equations," in Differential Equations and Dynamical Systems, Hale and LaSalle, editors, Academic Press, New York, 1967. E. Schmidt, "Die warmeubertragung durch rippen," Zeit. d. Ver. Deutch Ing. 70, 1926, pp. 885-890. R. J. Duffin, "A variational problem relating to cooling fins," J. Mathematics and Mechanics, Vol. 8, 1959, pp. 47-56J. E. Wilkens, Jr., "Minimum mass thin fins and constant temperature gradients, SIAM J., Vol. 10, 1962, pp. 62-73. C. Y. Lin, "A variational problem with applications to cooling fins," SIAM J., Vol. 10, 1962, pp. 19-29.
208
22.
23.
24.
25.
EARL R. BARNES
C. Y. Lin, "A variational problem relating to cooling fins with heat generation," Quarterly of Applied Mathematics, Vol. 19, 1962, pp. 245-251. E. R. Barnes, "A variational problem arising in the design of cooling fins," Quarterly of Applied Mathematics, Vol. 34, No..l, April, 1976, pp. 1-17. R. J. Duffin and D. K. McLain, "optimum shape of a cooling fin on a convex cylinder," J. Mathematics and Mechanics, Vol. 17, 1968, pp. 769-784. J. Cea and K. Malanowski, "An example of a max-min problem in partial differential equations," SIAM J. Control, Vol. 8, No. 3, 1970, pp. 305-316.
"VARIATIONAL METHODS FOR THE NUMERICAL SOLUTIONS OF FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS" 0. Pironneau
ABSTRACT This paper is concerned with the resolution of partial differential equations with an additional boundary condition but an unknown boundary. Several methods are mentioned (variational inequalities, transformations of variables, relaxation methods, evolution methods) but the emphasis is put on the methods of optimum design and their numerical implementations.
PLAN
1.
Introduction
2.
Variational inequalities and transformations of variables
1.
3.
Relaxation methods and evolutions methods
4.
The methods of optimum design
5.
Conclusion
INTRODUCTION
Free boundary problems are boundary-value problems in which part of the boundary, the free boundary, is unknown and must be determined as a part of the problem.
Such problems arise in continuum mechanics
when two fluids are involved (ex. jets, waves...) or when a system with certain desired properties is to be designed (inverse problems in the design of airfoils, for example).
It is beyond the scope of this paper
to survey the work that has been done in this area.
For one thing the
task would be enormous since free boundary problems arise in many different fields.
The oldest work that I have seen is the one mentioned
in Lamb (1879) on the determination of the profile of minimum drag in
209
210
0. PIRONNEAU
Cryer (1970) mentions a paper by Trefftz (1916).
Stokes flow.
Let me
just say that a minimum references list on the subject should include the work of Hadamard (1910), Southwell (1946), Garabedian (1956), Lions (1972).
Among the various methods, that I know, for solving free boundary
problems one should distinguish between the general methods and those which are designed specifically for certain problems.
Among the
general methods one must also distinguish between those which have theoretical proofs of convergences and those which do not.
Thus we
have divided the paper in three paragraphs accordingly, but since they are rather new, we shall speak mostly of the general methods of optimum design, which are known to converge.
2.
VARIATIONAL INEQUALITIES AND TRANSFORMATIONS OF VARIABLES Variational inequalities
2.1.
Among the methods which are not general perhaps the best one is the method of variational inequalities. problem:
given
..., n, f E (2.1)
; aij(x), a0(x) E L-(2), aij = aji, i, j=l,
S2 C Rn
L2(52), g E
L2(1'), find
3- (aij a0
min {(
Let us consider the following
1
S2
a
i
such that:
k
+a0
Igodrl o > 0 a.e. in S2}
2)dx-J
52
52
whe re
H1.(2) _ {kEL2(2)Iaxi a- ( L2(2) If there exists a0 >_ a
,
a > 0
such that
i=l,...,n)
aijtitj >_ algl2 Vt E Rn
then (2.1) has an unique solution and it is such that there
exists 52+, 2° with
52+
fl 2° = 0, 9
U S2° = S2 .
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
(2.2)
as)
- az (aid
+ ao0 = f
211
2+
in
i
(2.3)
av
r+ = g
or k! r+ = 0
(T+
= asz+, the boundary of
s2+,
v = a1 -co-normal)
(2.5)
and a I5=0
5=0
(2.4)
o=0
in
20
(S=as2+nas2°)
.
Therefore the free boundary problem: find (4,2+) such that (2.2)(2.4) can be replaced by (2.1).
Problem (2.1) can be discretized by the method of finite elements and solved by numerical techniques related to those of variational equalities (see Glowinski-Lions-Tremolieres(1976)). Several free boundary problems have been solved successfully by this method: the determination of the plastic region for a Bingham fluid (Cea-Glowinski (1972)), the determination of the region of infiltration in a dam (Baiocchi-Comincolo-Guerri-Volpi (1973)), the determination of a cavitation bubble attached to an airfoil (BourgatDuvaut (1975)) and the determination of the surface of fusion of a two-phases problem (Bourgat-Duvaut (1976)). This method gives good results for the following reasons: 1.
The discretization need not follow the unknown boundary
2.
The accuracy on
0
does not depend on the accuracy of
Q+
(see
figure 1) 3.
It is not necessary to know a good estimate of
Q+
Unfortunately not all free boundary problems can be formulated as variational inequalities, or quasi-variational inequalities (BensoussanLions (1973)).
For a survey of the problems on which this method is
feasible see Baiocchi (1973).
2.2. Transformation of variable The essence of the method is to find a new statement of the problem by a change of variable and/or coordinates.
The new problem
0. PIRONNEAU
212
will be either a non-linear partial differential equation with nonlinear boundary conditions but fixed domain, or even a free boundary problem of the kind studied in §2.1 (Baiocchi (1973)).
Let us just
mention briefly a famous example: the inverse problem in incompressible flow, (see for example Stanitz (1952)) i.e. find the shape of the wall If the nozzle is symmetric
of a nozzle that gives a desired velocity.
and of infinite length, let o(x,y) be the potential, *(x,y) the stream function, given k0(s), o,,,k the problem is to find S = (S1,S2) such that (see figure 2)
am = 0
(2.6)
OIS = Oo(s)
,
n
an
= O
01-
mm
or
=0.*IS2=k,
(2.7)
a0
IS soky
an
Let e(x,y) be the angle of the stream lines with the Ox axis, then it is easy to show that in the (o,*) plane the nozzle becomes a stripe
(- m,+ o) x [o,k] in which AO = 0
(2.8)
The continuity equation implies ae =
(2.9)
where
v(0)
-
? av
v ao
any
ama V(O) = as o
(2.11)
O(s) = 0o(s)
(s) = s2
4.
0
,y = 0, * = k
is obtained from the equations
(2.10)
(Ex:
on the boundaries
V(,) = 21o)
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
213
Therefore the free boundary problem (2.6) has been replaced by a Neuman problem (2.7)-(2.8) plus a non-linear equation (2.9)-(2.10). Naturally this kind of technique is not general but when it works it is
quite remarkable; for instance some inverse problems in transonic flow are easier than the computation of the solution of transonic equation with fixed boundaries. 3.
RELAXATION METHODS AND EVOLUTION METHODS 3.1.
Relaxation methods
Relaxation methods (or fixed point methods) were studied by Southwell (1956) and are widely used by engineers.
The principle is
the following: suppose we must find (S,,h) such that (3.1)
ee = f
(3.2)
OIr = hIr , r = EUS = a2,E n s = 0
(3.3)
an is = 91s
where
f,
in
Qs
are given functions of
h,11g
one can choose an initial guess
S0
for
L2(Q) S
and
z
is given.
Then
and solve (3.1) with the
boundary conditions a
(3.4)
an
and compute
130 = 9130 1o1Z-hit S1
such that
olS1 = hlsl
This way it is possible to generate a sequence {S1 i> which will (hopefully) converge to the desired solution.
Refinements of this
method were given by Garabedian (1956), Cryer (1970), ... Relaxation methods are quite general and very easy to use; however, no proofs of convergence are available and in fact it is usually necessary to start with a good initial guess, otherwise the method diverges.
214
O. PIRONNEAU
Evolution methods
3.2.
It is a standard procedure in physics to replace a stationary
problem by a time dependent problem which has the stationary problem for equilibrium state.
For example, the interface between two viscuous fluids in equilibrium in a rotating cylinder, under gravity, can be determined by starting with the fluids at rest, and slowly accelerate the cylinder to the desired speed, while following the interface. interface at time
t + dt
The position of the
is determined from the position at time
t
and the knowledge of the speed of the fluid particles. In this example the time dependent problem is given by the physics
of the problem, whenever the free boundary problem is a straightforward modelization of a physical problem, the time dependent problem is easy to construct.
Note that it is not fair to put these methods in this paragraph since proofs of convergence are indeed available for them, if they are properly applied.
They are easy to use but they are rather slow and
not always numerically stable, because they correspond to an explicit discretization scheme for a parabolic problem.
A seepage problem in a
dam has been solved numerically by this method (Zienkiewicz (1971)). Cea (1974) showed that it is generally possible to formulate free boundary problems in this manner;however,the computation of the speed of the particles is not at all straightforward so that his method really belongs to the next paragraph. 4.
THE METHODS OF OPTIMUM DESIGN Consider problem (3.1)-(3.3) and let
(4.1)
and
(4.2)
o46 S = f
41S
in
oS
be the solution of
2s, oSIr = hIr
the solution of S =
S y17 =
h)z
Then any solution
S
'
an IS = 91S
of (3.1)-(3.3) is also a solution of
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
min
(4.3)
SES
S
where
Iv(,4 S-,yS)12dx} J.
S
is the set of admissible boundaries
such that
S
, and (4.1)-(4.2) have solutions in
aszS = E U S
215
2S
.
Problem (4.3) is an optimal control problem of a distributed parameter system where the (open loop) control is a part of the boundary; such a problem is a problem of optimum design.
Existence theorems for the solutions of (4.3) were given for special cases by Garabedian (1964) and Lions (1972). It was shown by Murat-Simon ('1974), Chesnay (1975), Pironneau (1976)
that problems of the type of (4.3) with convex functional and linear operator
have solutions if
a
contains uniformly Lipschitz boundaries
(boundaries of bounded variations in
is sufficient).
R2
The Lipschitz
condition can even be removed in certain cases (Pironneau (1976)).
Problems (4.3) can be solved by the techniques of the calculus of For notational convenience
variations and mathematical programming.
let us illustrate the methods on two model problems
(4.4),
min SES
(4.5)
-eq+4 - f
{1
{!2 I0-Odl2dxj
ek = f,
,
an IS = 0
,
S = as2}
S = a9}
'1s = 0
s
4.1.
Reduction to a fixed domain Following Begis-Glowinski (1973), (1976), Murat-Simon (1974),
Morice (1974), let
T
1
S
be a (smooth) mapping of
transforms all admissible domains
125, S ( S
Then problem (4.5), for instance, becomes
*
These assumptions can be relaxed.
R2
into
R2
into a fixed domain
which Q0
216
O. PIRONNEAU
(4.6)
{to T
min
1
fT w1det T'Id20}
=J
20
, Vw E HI(Q0)
S20
where mT -= m oT
of
T'
, and
T
Hl 0(2 0) _ { EHl
tT'-lvw>Idet T'Jd2
T'I d52oI
T ET
(o)IOIr0= 01, }, V SZ
'
=
' 'T E
1aTA ax3
H1(2o)
tT
'
= transpose P
is the set of all admissible mappings.
Problem (4.6) is a problem of optimal control where the control appears in the coefficient of the P.D.E.
This type of problem was
studied by Chavent (1971) (among others); they can be solved by the method of steepest descent (or conjugate gradient). equations are rather complex as one might guess.
the drawback of requiring the explicit knowledge of complicated domains is not a trivial matter.
However, the
This method has also r
which for
It has however the
advantage of being easy to discretize.
Begis-Glowinski (loc. cit.) have tried this method on a simple geometry where the upper boundary of an open rectangle is free. set
r
The
is then the homotheties which transform the free boundary into
a prescribed horizontal straight line.
Thus the method is the
simplest of the kind and it works quite well except perhaps when the unknown solution is too far off the prescribed horizontal; in this case the finite elements grid becomes rather peculiar so that the discretized problem is far from the continuous one (see figure 3).
Morice (1974) worked also in these lines but used
-r= (conformal mappings of
Sts
into
520}
(*)
The advantage of this method is that the formulation (4.6) is much simpler and there are ways to keep the discretization grid uniform
*
Morice later extended this idea to more general implicit mappings (quasi-conformal mappings).
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
during the iterations (see figure 4).
Thus at the cost of increasing
the number of state variables, the method is very good. it is not completely general.
217
Indeed if
S2S
Unfortunately
is not simply connected
it becomes very difficult to find a conformal mapping for it.
To conclude this paragraph, let me say that a compromise between the "easy to find" r's (best would be an implicit method) and the
-r's
that makes (4.6) simple is still to be found. 4.2. A descent method in the variable domain: the continuous case Suppose that
S = {x(s)JsE[o,l]}
S' = {x(s)+Xa(s)n(s)IsE[0,lJ} (i.e. X small) where
n(s)
is a solution of (4.4) and let
be an admissible boundary close to
is the outward normal of
S
at
x(s)
S .
By
definition if
(4.7)
E(S) =
2
then
loS-4dl2dx
f
S
E(S') - E(S) >_ 0 Let
6E
T
(with self explanatory notations)
VS'
E
e
be the left member of (4.7).
Suppose that a > 0, by (4.7)
6E = J [IOs-'kdl2-Ims'-mdf21dx + IQQIdJ s
it can be shown that 1I(oS-'k S2 -" 0
when
a -+ 0 ,
so that, by
L
differentiation and from the mean value theorem (since narrow strip around S ) 6E =
I25
1
w here
60 = 4
(4.8)
-4,S .
-d)6
S'-2S
+ o(a)
The P.D.E. in (4.4) in variational form is
(v4vw+4w)dx = 2S
+
S2
fwdx
I
2S
Yw ( H1(52s)
.
is a
218
O. PIRONNEAU
Therefore (4.9)
satisfies
6o,h
(vosvw+bow)dx = J xa(fw-Vovw-ow)dS
1
S
12S
Thus if we let
(4.10)
*
be the solution of
(v,yvw+yw)dx = f2s (0S-0d)wdx
Vw E Hi(2S)
J
2S P utting
in (4.9) and
w = ,y
f
w = 69s
aa(f*-v4v*-oir)dS
(,k S-0d)6odx =
2S
in (4.10) leads to
S
so that 6E = Is(1d2 + f-*-vy}dS + o(a)
(4.11)
A similar computation can be done for (4.5) if one notes that in
2
S
64,
(4.12)
where
oho = 0, 6415 = - ),a
ka(IAS-'bdI2
6E = 2 fs
*
(4.13)
is a solution of
- an an )dS
an
+ o(1t)
so that
+ o(X)
is the solution of
any =
0S-Od
in
9S, *Is = 0
.
It can be shown that (4.11) and (4.12) are valid for non positive
a' s . Therefore let us consider the-following algorithm for solving say (4.4) where
S = IS twice differentiable}
.
2t9
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
Algorithm 1
Step 0
(4.14)
Choose
S
0 0i
set i= 0
E S S
Step 1
Compute
Step 2
Compute *i by solving (4.10) with S = Si, 0S = 0i
Step 3
Set
Step 4
Compute
= 0
by solving (4.8) with
i
S = Si
ai = - (0 1 -md)2 - fyi + 01*i + V¢iV4t Xi
solution of
min E(S'(),)) with S'(x)={xi+1(s)=xi(s)+xai(s)ni(s)IsE[0,1]}
and set
Si+1
=
S'(xi) ,
i
= i + I
and go back to 1
.
.
THEOREM 1
Any C2-accumulation point of a sequence {Si}i>0 generated by algorithm 1
(4.15)
is a stationary point of (4.3) and satisfies 10S-4dl2
+ (f-,k)* - 707* = 0
Proof: Algorithm 1 is a method of steepest descent for (4.4) locally in the space of admissible
a's such that
S' E S
.
The convergence
proof proceeds just as in the case of an ordinary gradient method. From (4.11) with
a
as in step 3
Ei+l - Ei = -xi
,
iV*i]2dSi+o(xi)
[IBS S
So that each iteration decreases quantity in (4.15) is zero. 4.3.
E
of a non trivial quantity until the
For a rigorous proof see Pironneau (1976).
Implementation of algorithm 1
Thus algorithm 1 proceeds like the method of relaxation except that its convergence can be proved, at the cost of integrating a second P.D.E., the adjoint system.
O. PIRONNEAU
There are cases where algorithm 1 can be implemented directly: i) when the numerical integration of the PDE's can be done with a very good precision (see figure 5); ii) when the cost function of the optimum design problem is very sensitive to the positions of the free boundary. In all other cases a simple discretization of the equations of continuous problem would fail to enable us to solve the optimum design problem with a good precision because the numerical noise of the discretization makes it impossible to find a positive
a
solution of (4.14).
Therefore let us derive a formula similar to (4.11) for the
discrete case. For a given
S
let
0
zh
be a triangulation of
2h , a polygonal
approximation of 2So n
i e. T
2h =
f
= triangle where
Th ,
_I
h
Th f1 Th = one side or one node,
denotes the length of the largest side, let
H = {wow linear on Th}, i=l,...,n ; w continuous
If
is,the function in
wJ
'H
which equals 1 at the node j and
zero at all other nodes, then {w3}mj=1
is a basis for H (m - number of
nodes).
A first order finite element discretization (4.8) is (r = number of interior nodes) r
(4.16)
(vOvw4+.,w1)dx = fQ
f2
fwJdx
E J=1
h
h
o"w.
J=l,...,r ; m =
Equation (4.16) is a linear system in
0J
which can be solved by a
relaxation method, for example. Similarly (4.10) is approximated by m (4.17)
.r
(v*vw +*w )dx = t
21h
where
is the solution of (4.16)
('"d)wJdx, J=1,...,m ;
i< =
E *Jwj j=l
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
Let
S'(a)
be the boundary obtained from
S
221
by moving the nodes
0
along the lines of discretization "perpendicular" to
(see figure 6)
2h
then the discrete analogue of (4.11) can be obtained in a similar way: (1d-0d)2
(4.18)
1 6Eh = '2' -Q
+ f*-4*-VOV*)dx + oh(x)
h-2 h
In establishing (4.18) we have used the fact that continuously upon the coordinates of the nodes.
and
k
9'
depend
Now from the mean
value formula for integrals
(4.19)
oh(o)
1 6Eh = IS 0
Thus the discrete version of algorithm 1 is Algorithm 2: We assume that the user has an automatic triangulation subroutines which generate the interior nodes from the boundary nodes. For simplicity we assume that the first point of Step 0 - Choose
0
, set i=0 , choose
Step 1 - Compute
qs
from (4.16) with
Step 2 - Compute
,,
from (4.17) with
Step 3 - Let
z
tih
sh sth
remains fixed.
choose 0 E (0,1)
Qh Qh
sJ
is the curvilinear abcissa of the node
determined as follows: let
ai
J
and
al
where
6Eh3 = ISi (I-d12 +
a1J(s)
are
be the sines of the angle of the
discretization lines, let
(4.21)
S , set
be the number of boundary nodes on
s-s. i it i i a (s) - aJ + sJ+ j (aJ+l-aj) for s E (si ,sj+1), J=1,...,e
(4.20)
where
S
So
is given by (4.20) with
A a`) a'J(s) aids
as as
ak = 6Jk ,
k=l,...,e
0. PIRONNEAU
222
then take a
(4.22)
aEhj/(sj-sj-1)
Step 4 - Obtain. S(a)
j=2,...,m ; a = 0
by moving the nodes of
discretization lines of a quantity x aj
on the
Si
and compute the smallest
integer k such that (with self explanatory notations)
Eh(S(a)) - Eh(Si) f - pa E
(4.23)
a = ()k
j=1
and such that Si
,
to the next discretization line "parallele" to Step 5 - Set
Si+l
of the distance of
is not nearer than, say
S(2_k)
Si+1 = S(2-k)
Si
compute a 'new triangulation around
from its nodes computed in step 4, set
i = i+l
and go back to
step 1.
Conjecture - If the automatic triangulation subroutine is such that the position of the interior nodes depend continuously upon the position of the boundary nodes (and it is the case-of some such subroutines, for a given t
)
then the sequence {Eh(Si))i>O
will be a
decreasing sequence and {Si) will converge to an accumulation point of the discrete analogue of (4.4).
Justification - First of all (4.23) has always a non zero solution because t
i2
jZl (aP (s-sj-1) is the slope of x -o. Eh(S(a))
Therefore algorithm
at A = 0 (see (4.19)(4.21)(4.22)).
2 is the method of steepest descent applied to
(4.4) with 2 - 2h .and (4.16) instead of the continuous PDE; with one difference however, the triangulation is rebuilt at each iteration (so that each triangulation is well suited to the new geometry). since the cost function changes with the discretization, with the new triangulation might be greater than (4.23).
Eh(Sj)
Therefore,
Eh(Si+l) ,
despite
It should be noted that it might be possible to prove the
convergence of algorithm 2 by using.a model similar with the one described by Klessig-Polak (1972), by asking to increase the number
223
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
of nodes when the cost increases.
However, the following argument seems
to indicate that the cost is unlikely to be increasing. It is theoretically possible to consider
Eh
as a function of the
coordinates of all the nodes (instead of the boundary nodes only).
And
indeed in doing so one is very close to the concept of mappings of 2S into
20 , studied in the previous section. O'Carroll et al (1976) have
tested this method, by the way.
Now the change of cost due to a shift
of a boundary node tangentially to
S
is certainly smaller than the
change of cost due to a normal shift.
In the first case one changes
the definition of the continuous problem also while in the second case one changes the discretization of the problem.
It seems also that the
change of cost due to a shift of the interior nodes would be of the same order, if not smaller, than for the above tangential shift. Similarly, an implementable algorithm can be obtained for problem To derive an analogue of (4.12) one needs to use the following
(4.5).
formula (Murat-Simon (1974)): if 4 satisfies
then when
=J
J2
VoVwdx
a
is replaced by
fwdx
vw E H01 (2) ,4 E H01 02)
52'
, 64 E H1(2)
satisfies
J2vs0vwdx = - j,xvw.(f;+ dive" -(e'+te')v4)dx+o(x) vw E
HQ(S2)
(x,y) + xe(x,y) _ (x+), el(x,y), y+x e2(x,y)) are the coordinates of the,
transform of (x,y) by the map
I+xe
that transforms
st
into
ei
Then the analogue of (4.19) is aEh = -2j 9xv$[fe+divev4-(6'+t$')v4]dx + rsxa145`#d12dS+oh(x)
where
is the solution in
r
of
H,(9)
f2 v*vwdx = f2(4S-4d)wdx
and Ho 2h The reader will easily construct an implementation of
These equations remain valid when into
K
.
Vw E Ho(st)
$2
algorithm 1 in this case, by letting.
triangles that touches S , where
e
is changed into
0=0
except in the strip of
is taken linear and in tune with a.
O. PIRONNEAU
224
5.
CONCLUSION
An engineer faced with a free boundary problem may be in one of the two positions: either he knows very well his problem intuitively and he is able to find a good estimate of the solution; or he does not know much about the answer and he is not willing to develop a special subroutine to find it.
Then, if he is in the first case he will be
probably better off by using the relaxation method; but if he is in the second position he would prefer to call an "automatic" subprogram. Such efficient automatic scheme is still to be found; the last section of this paper contains one that looks reasonably promising on account of the fact that free boundary and optimum design problems are numerically usually quite stiff.
ACKNOWLEDGMENT I wish to thank J. R. Bourgat, A. Dervieux and P. Morice for their helpful suggestions. REFERENCES 1. 2.
3. 4.
5.
Baiocchi C. (1975) - Cours au Collage de France. Begis D., Glowinski R. (1975) - Application de la m6thode des 616ments finis A l'approximation d'un problCme de domain optimal. Applied Math. & Optimization, Vol. 2, No 2. Benssouasan A., Lions J. L. (1973) - CRAS 276, pp. 1189-1193, Paris. Bourgat J. F., Duvaut G. (1975) - Calcul num6rique avec ou. sans sillage autour d'un profil bidimensionnel sym6trique et sans incidence. Rapport Laboria No 145. Bourgat J. F., Duvaut G. (1976) - R6solution num6rique d'un problCme de Stephan A 2 phases par une in6quation variationnelle (a paraitre).
6.
Bourot J. (19714) - CRAS A.278.455.
7.
C6a J. (1975) - Une m6thode nur6rique pour Is recherche dun Romaine optimal. Proceedingal.F.I.P. Congrps Nice. C6a J., Glovinaki R. (1972) -.M6thodes num6riques pour 1'6coulement laminaire d'un fluide rigide visco-plastique incompressible. Intern. J. of Computer, Math. B. Vol. 3, pp. 225-255. Chavent G. (1971) - Th6se de Doctorat, Paris. Chesnais D. (1975) - On the existence of a solution in a domain identification problem. J. of Math. Anal. and Appli. 52,2. Cryer C. W. (1970) - On the approximate solution of free boundary problems using finite differences. J. of the Association for Comp. Machinery, Vol. IT, N° 3, pp. 397-4n. Garabedien P. R. (1956) - The mathematical theory of three dimensional cavities and jets.Bull. Amer. Math. Soc. 62, pp. 219-
8.
9. 10.
11.
12.
235.
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
13.
14. 15. 16.
17. 18. 19.
20. 21. 22. 23. 24. 25. 26.
27.
225
Garabedian P. R. (1964) - Partial Differential Equations. Wiley, New York. Glowinski R., Lions J. L., Tremoliere R. (1976) - Analyse num6rique des in6quations quasi-variationnelles. Dunod (to be, published). Hadamard J. (1910) - Legons sur le calcul des variations. Gauthiers-Villars. Klessig R.,Polak E., (1970) - A method of feasible directions using function approximation with applications to min-max problems. E.R.L.-M287, University of,California, Berkeley. Lamb H. (1879) - Hydrodynamics. Cambridge University Press. Lions J. L. (1972) - Some aspects of the optimal control of distributed parameter systems. RESAM, 6 SIAM. Murat R., Simon J. (19T4) - Quelques r6sultats our le contr8le par un domain g6om6trique. University de PARIS VI, Rapport interne Lab. Analyse Num6rique, R° 74003. Morice P. (19714) - Une m6thode d'optimisation de forma de domain e. Proc. CongrLs IFIP-IRIA, Paris. Springer Verlag. O'Carroll M. J. and H. T. Harrison (1976) - Proc. ICCAD Conf. Rappalo, Italy. Pirdnneau 0. (19T3) - On optimum profiles in Stokes flow, J. Fluid Mech., Vol. 59, pp. 117-128. Pironneau 0. (1976) - These de Doctorat, Paris. Southwell R. V. (1946) - Relaxation methods in theoretical physics, Vol. 1, Clarendon Press, Oxford. Stanitz J. (1952) - Design of two-dimensional channels, NACA technical note 2595. Trefftz B. (1916) - Uber the Kontraktion kreisf6rmige FlUssigkeitsstrahlen. Z. Math. Phys. 614, pp. 34-61. Zienkievicz 0. C. (1971) - The finite elements in Engineering Science. McGraw-Hill, London.
226
O. PIRONNEAU
Figure 1
- (From Bourgat-Duvaut (1976)).
Fusion of an ice cube when the bottom plane is heated.
Even though
tlye free boundary is not well approximated, the isothermal lines are computed. accurately.
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
227
11
im
am
III
Figure 2
Illl 111111 111111 111111 111111
Ilj 11j11IfII
111111 1
111
11 II I
11111
I11III
/111111
Ill
IN IN
IIIIIIII
lull VIII
11111111
III
11111111
11111
11111111
VIII
111 II111111
VIII 1101
1011111
III
1111101
VIII
IIIIIIII
0111
I II 11111111
VIII
Figure 3 - (From Begis-Glowinski (1975)).
Computation of a free boundary (upper line) by the method of BegisGlowinski and the corresponding finite element triangulation in a good case and in a bad case.
228
0. PIRONNEAU
Figure 4 - (From Morice (1974)).
Computation of a free boundary by Morice's method and the corresponding triangulation.
FREE BOUNDARY PROBLEMS AND OPTIMUM DESIGN PROBLEMS
Figure 5 - (From Bourot (1974)).
Computation of the minimum drag profile in Stokes flow by the method of Pironneau (1973).
Figure 6
Sl is obtained from So by movingrthe nodes of discretization of So of a suitable quantity: aaj . along the lines of discretization of
2so "perpendicular" to So
229
"SOME APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY TO DISTRIBUTED PARAMETER SYSTEMS" W. H. Ray
1.
Introduction In this chapter we shall discuss a large number of real and
potential applications of state estimation and control theory to distributed parameter systems.
Because of the inherent practical difficulties
of state measurement in distributed systems, the major problems in control implementation are often due to difficulties in obtaining reliable estimates of the state variables,
For this reason, the major emphasis
of this chapter shall be on distributed parameter state estimation.
The
reader is referred to other survey articles for an overview of parameter identification [1,2] and to the other chapters in this volume for a discussion of the control of distributed parameter systems.
A second
troublesome problem arising in applications is the development of efficient computational algorithms which will allow real time implemen. tation of these state estimation and feedback control schemes.
In what
follows, we shall give examples aF how these numerical problems can be solved in practice.
The next section gives an overview of practical considerations and the algorithms available for distributed parameter state estimation and presents a survey of reported applications.
Following this, a
detailed case study of the real time implementation of state estimation and feedback control to a laboratory model process will be discussed in some detail.
Finally, a general survey of remaining practical problems
will be presented along with some suggestions for future research
directions. II.
A Survey of State Estimation Algorithms and Applications The field of sequential state estimation has grown enormously in
scope and popularity since the original work of Kalman [3] and Kalman
231
232
W. H. RAY
and Bucy [4] in the early 1960's.
However, it was not until 1967 that
the first extensions of these ideas to distributed parameter systems began to appear.
The earliest paper seems to be due to Kwakernaak [5],
and was quickly followed by the work of Falb [6], Tzafestas and Nightingale [7-10], Balakrishnan and Lions [11], Thau [12], and Seinfeld [13], all of whom apparently developed their results independently and nearly simultaneously.
Since these early papers, work has
progressed so that state estimators are now available for numerous classes of linear and non-linear distributed parameter systems. Unfortunately, the reported practical applications of distributed parameter estimation algorithms has badly lagged the theory.
This is
perhaps due to the relative complexity of the estimation equations when compared to the classical Kalman-Bucy filter.
Another factor is the
lack of meaningful contact between potential industrial users and research engineers familiar with the capabilities of these techniques. In this section we shall summarize the results available for distributed parameter estimation and describe recent computational and experimental example applications.
In this way it is hoped to provide
the potential user of these techniques with a guide to the theory as well as inspiration for further applications. This survey section, which has been updated from an earlier lecture [14] by the author, shall begin with a discussion of the theoretical results available for both linear and nonlinear systems and then review the example applications which have appeared to date. A.
Theoretical Results
There have been a large number of theoretical approaches taken in the derivation of distributed parameter state estimators.
The most
direct approach is to somehow approximately lump the distributed system and then apply the classical lumped parameter theoretical results [3,4]. One can certainly proceed in this way and obtain useful results (e.g., [15-20]), although in certain cases the distributed parameter formulation does seem to have advantages (e.g., when there is boundary noise).
Because there are so many ways of Pumping a distributed system
(e.g., finite difference, modal representation, splines, etc.) it is hard to compare exact distributed parameter results with lumped approximations, and thus the possible pitfalls of lumping have not been well
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
defined.
233
In the discussion of theoretical results which follows, we
shall deal mainly with results derived for the full distributed parameter system model; however, several cases of "lumping approaches" to filtering will be included in the applications section.
Even excluding lumping, there have been a wide range of approaches taken in arriving at theoretical results ranging from "formal" such as minimizing a given error functional to more "rigorous" approaches explicitly treating the evolution of the probability distribution of the state variables.
While the more rigorous
approaches yield more precise statistical information about the problem, they are limited in the classes of problems which may be treated.
The
formal appt«)ches yield less detailed statistical information, but have t.he,,ability to treat a wide range of both linear and non-linear problems.
Therefore, both methods of analysis have value and in fact are usually found to yield identical filter equations in situations where both techniques apply.
Because this survey section is emphasizing applications, it is not our intention to attempt an extensive review of the theoretical approaches for developing distributed parameter state estimators, but rather shall concentrate on the results which may be of direct interest to the practitioner.
For a fuller discussion of the theoretical ideas
see the review by Tzafestas [21]. 1.
Observability and Measurement Location Perhaps the first consideration in the application of a state
estimator is the question of observability and the optimal location of measurement devices.
Observability, which has been defined for lumped
parameter systems as the ability to determine the initial state based on the system model and measurements available, is a more difficult question for distributed parameter systems because of the strong influence of the location of the sensing devices.
Thus for a fixed
number of sensors, it is possible to choose locations for which (i) the system is not observable, or (ii) the system is.observable, or (iii) for which the system is both observable and the measurement locations yield the maximum amount of information in the presence of noise.
For these
reasons, the choice of type, number, and location of tensors is crucial
W. H. RAY
234
in the design of a distributed parameter state estimator.
Wang [22) seems to be the first to discuss observability for distributed parameter systems and proposed a definition based on the recovery of the. initial state through analogy to lumped parameter systems.
of
Goodson and Klein [23] define observability as the existence
a unique system state based on the model and the observation device
and have developed general criteria for observability for a number of linear partial differential equation systems.
For first order quasi-
linear systems, their criteria includes the requirement that each of the characteristic lines in the system intersect a sensing device.
Yu and
Seinfeld [24J have quantified this requirement and, by applying the known lumped parameter observability theorems along the characteristic lines of the system, have developed criteria specifying which combinations of states must be measured.
Their results take the
familiar form of determining the rank of an observability matrix. Recently Thowsen and Perkins [25] have extended these results to a slightly more general form of the equations, and Sendaula [26] has considered the related problem Of time delay systems. Goodson and Klein [23] also consider linear diffusion and wave equations and are able to develop sufficient conditions for observability based on the choice of sensor placement.
Cannon and Klein [27]
have numerically investigated the optimal placement of the measurement location for a simple linear example problem of heat conduction with specified surface temperatures.
Yu and Seinfeld [28] have developed
sufficient conditions for observability for linear systems amenable to a modal representation.
Their approach is to represent the system in
terms of N eigenfunctions and then apply the known observability conditions [3,4] for lumped parameter systems to the resulting ordinary differential coefficient equations.
The rank of the observability
matrix depends on the values of the eigenfunctions at the measurement locations.
These results were applied to both one and two dimensional
heat conduction and diffusion examples.
In addition, numerical studies
were performed to show the influence of the measurement location on the quality of the estimates.
In spite of these valuable contributions, at present there exists no'general theory of observability for either linear or
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
235
nonlinear distributed parameter systems; there exists only a few results for the particular linear cases studies.
Nevertheless, several recent
papers [29,30] would seem to lead toward such a theory.
Similarly no
general results are available for determining optimal sensor locations; all that one can do is numerically investigate each problem individually for a range of model parameters, noise statistics, etc.
Several workers
have developed systematic algorithms for carrying this out [31-33]. However, further theoretical results would be extremely useful here. 2.
State Estimators Available
State estimators, for smoothing, filtering, and prediction, have been developed for a wide range of distributed, parameter systems.
In
this section we shall discuss explicitly the filtering results; however, the related prediction and smoothing algorithms can be found in a straightforward manner once the filtering equations are known.
Just as
in the case of lumped parameter systems, linear distributed parameter systems allow the possibility of much more rigorous and precise results than-do nonlinear systems.
Not only do the stochastic system modelling
equations have precise meaning, but the estimate covariances arise naturally in the filter calculations and provide valuable statistical information about the quality of the estimates.
On the other hand
nonlinear state estimators, developed by more formal techniques, are much more general, greatly expand the range of practical problems which can be handled, and have shown to reduce to known, rigorously derived, linear filters when the nonlinear equations are made linear.
Thus the
practitioner is advised to take advantage of the rigorous linear filter, with its precise estimate statistics, where possible, and to make use of the more general nonlinear filtering results, when necessary. Because of the fact that the independently derived linear filters appear as special cases of the more general nonlinear results, in what follows we shall group the state estimation results under their general nonlinear form.
A summary of available theoretical results is presented
in Table 1.
The first class of distributed parameter systems to have filters developed were systems with time delays.
Kwakernaak's classic paper [5]
led the way and showed the structure of the distributed parameter
236
W. H. RAY
filtering problem.
Although a number of linear filtering results have
appeared since, [17,34-42], for the purpose of our discussion here these can be treated as special cases of the nonlinear filtering results of Yu et al. [43].
In this general formulation, the filtering problem for
time delays takes the form z = f(x(t), z(r1,t),
--z(r,,t)) +
(1) JI (z(r,t)dr + g(t)
zt(r,t) = -M(r,t),(r,t) + (2)
g(z(r,t),r,t) + t(r,t) N N .y
y(t) = h(x(t), z(r1*,t), --z( Y*,t),t) +
3) j
(z(r,t),r,t)dr + 1(t)
x(0) = xp
(4)
z(r,0)= z,(r)
(5)
z(O,t) = b(x(t))
(6)
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
237
Table I. Summary of Theoretical Results Topic
References
Systems with time
[5,17,34-43,89]
delays
Systems described by
Linear Systems [7-9,11,12,27,
second order partial
28,45-641
differential equations
Nonlinear Systems [10,13,15, 44,58,65-67]
Linear systems described
[6,9,47,48,54,55,57,58]
by integral or integrodiffer4ntial equations Systems described by
[68]
partial differential equations with moving boundaries
Separation principles
[8,35,38,52,57,69,70]
and stochastic feedback
control where Eqs. (1,2) represent the behavior of a coupled lumped and distributed parameter system defined for donain
r E [0,1]
conditions (4-6). defined by (3).
.
A(t)
and
z(r,t)
t > 0
and on the spatial
are state vectors with boundary
Observations consist of the output vector y(t) The quantities
and n(t)
represent
zero mean random processes with arbitrary statistical properties. Within this framework an extremely wide range of time delay problems
may be treated. 'A second class of distributed parameter systems for which a large number of results are available are second order partial differential equations systems. spatial variable is [44]
A very general form of this problem for one
W. H. RAY
238
zt(r,t) =
(r,t) (7)
t>0,0<_r51
where
b,o(t,z,zr) + xo(t) = 0
r=0
(8)
bl(t,z,zr) + yl(t) = 0
r = 1
(9)
,Zr(r,t) = h(r,t,z(r,t)) +,U(r,t)
(10)
z(r,t)
is a vector state variable modeled by (7-9) and X(r,t)
is a vector of measurements defined by (10).
Y(r,t), Wi(t), rl(t)
are
zero mean volume and boundary noise with arbitrary statistical properties and
ry(r,t) is a zero mean measurement error.
A large number of
powerful linear results have appeared for special cases of this class of problems (7-9,11,12,27,28,45-64], and a number of less rigorous but fairly general non-linear estimators have been developed as well [10,13, 15,44,58,65,66].
However, for purposes of classification all of the
resulting filter equations can be considered special cases of the
filter developed by Hwang et al. [44] for the general nonlinear formulation (7-10).
Recently, Ajinkya et at. [67] have extended these
nonlinear results to include systems described by coupled ordinary and second order partial differential equations.
A number of linear results have also been derived for systems described by integral or integro-differential equations (6,9,47,48,54, 55,57,58], but no explicit examples seem to have been discussed. Very recently a general nonlinear state estimator for systems
described by second order partial differential equations and having moving boundaries was developed [68].
As will be discussed later, this
filter allows the simultaneous estimation of the system state as well as the boundary position for a large number of problems of practical
interest. It should also be noted that some work has appeared on separation principles and the feedback control of some types of linear stochastic systems [8,35,38,52,57,69,70].
In addition, there seems to be a
rapidly growing literature on distributed parameter observers [42,71-74].
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
239
From the papers summarized here, it is clear that there are state estimators available for many classes of distributed parameter systems. In the next section we shall discuss a number of practical engineering problems for which these theoretical results are applicable. B.
AN OVERVIEW OF APPLICATIONS Before beginning a detailed discussion of particular applications
of distributed parameter state estimators,_it is useful to consider some practical points in the implementation of such filters.
Firstly,
let us recall that the great majority of filtering algorithms discussed in the previous section assume measurements continuous in time and often even in space.
Certainly this is not the case in practice where
measurements are usually taken at specific spatial locations and quite often at fixed discrete intervals of time.
It is fortunate that this
apparent inconsistency between the available theory and industrial practice is not a serious limitation - due to some simple tricks.
As
has been shown by Meditch [48] one may convert a continuous spatial
measurement device estimator to a discrete spatial measurement result by making use of a Dirac delta function of the form ri*i = 1,2,--
are the actual measurement locations.
8(r-ri*)
where
This immediately
produces the correct estimation equations for both linear [48] and nonlinear [44] systems with discrete spatial observations. trick involving
6(t-tk)
A similar
may be used for data which must be taken at
discrete time intervals, tk [67].
However, in this case the approach
is difficult to justify rigorously and thus must be used with care.
Nevertheless, it has the advantage that it immediately converts continuous time estimation equations to discrete time measurement equations, and can be shown to produce the known rigorously derived discrete time filters (e.g.; [37]) from their continuous time analogs (e.g.; [5]).
A second practical consideration which should be discussed is the development of efficient computational algorithms for the on-line implementation of distributed parameter filters.
Distributed parameter
filtering algorithms require the solution of the distributed parameter modelling equations in real time.
Furthermore the covariance equations
W. H. RAY
240
turn out to be nonlinear partial differential equations in independent variables where the model.
n
2n + 1
is the number of spatial variables in
Clearly the soltulon of such equations can quickly become
unmanageable if some care is not taken in the choice of the computational algorithm.
For linear systems, the problem is simplified because
the covariance equations are independent of the data and system state and may be solved off-line.
In addition, there are powerful analytical
techniques which may be used for solving the state equations.
For first
order hyperbolic differential equations, one may use the method of characteristics to reduce the computations to solving ordinary differential equations along the characteristic lines [75,76].
For second
order partial differential equations such as occur in heat, diffusion,
or wave processes the solution can be represented in terms of the eigenfunctions of the system.
This modal approach has been proposed
by a number of workers [7,16,28,51,52,58,77,78] and reduces both the state and covariance equations to a set of ordinary differential equations.
For nonlinear state estimation problems the computational
problems are not so easily solved; however, this author feels that the use of weighted residual methods (such as 6alerkin or Collocation procedures), which are the nonlinear analogs of eigenfunction expansions, should lead to filter calculations of manageable proportions. In some applications it is the development of a reliable system
model which limits the use of state estimation.
For heat conduction
problems this is seldom a difficulty, but for more complex processes such as chemical reactors, model development is usually the crucial overriding problem.
Fundamentally, state estimation is a compromise
between complete, errorless state measurement and perfect predictive modelling.
Therefore, the less data available, the better the model
must be and vice-versa.
For these reasons, it is essential in practice
that a detailed modelling study be carried out before state estimation begins.
-
The reported applications of distributed parameter state estimation can be classified as follows: 1.
Computer simulations with computer generated data based on a model of more or less physical significance.
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
2.
241
Off-line filtering of actual process data in which one "pretends" to filter in real time.
3.
Actual real time state variable filtering and control of an operating process.
The great majority of the reported case studies are of type 1 [9,10,13,15,17-19,20,34,35,37,42-44,55,57,60,66,67,74,75,77,79-84], very few are of type 2 [85,86] and to the author's knowledge, there is only one paper [78] of type 3.
With a few exceptions, most of the
type 1 applications are example problems designed to demonstrate a new theoretical result rather than being feasibility studies of a genuine practical problem.
The single reported experimental implementation [78]
provides a great deal of insight into practical details of implementation as well as representing a real test of distributed parameter filtering and thus will be discussed in great detail later. In an effort to provide application ideas to the practitioner, the existing literature of suggested, simulated, and experimental applications of distributed parameter state estimation will now be discussed within the framework of several practical state estimation
problems.
A summary of these problems with relevant references is
given in Table 2. 1.
Heat Conduction Processes
The first class of problems, the estimation of the temperature profile in a heat conductor has been used extensively as a demonstration problem for distributed filters [9,10,13,44,55,60,66,67,77,78,81]; however, this problem does arise in a number of industrial processes such as found in metallurgical operations [67], glass making [87], and nuclear reactor control [84,88].
As a particular example, let us
consider the soaking pit operation in steelmaking.
Ingots are heated
in a furnace (soaking pit) preparatory to rolling into slabs. Figure 1 for a sketch of the physical picture.
See
Flame
Tt
Wa il
T, (t) Ingot
Ts(I,t)
243
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
Table 2.
Summary of Example Applications
State Estimation Problem
References
Estimation of the temperature
[9,10,13,44,55,60,66,67,77,78,81]
profile in a heat conductor Estimation of temperature,
[15,18-20,43,44,80,82,83,85,89]
concentrations, and catalyst activities in chemical reactors Estimation of pollutant
[75,76,86,92]
concentrations in air and water pollution monitoring problems Estimation of states and
[68]
boundary position in systems having moving boundaries
The problem is that the initial temperature distribution of the ingots when they go into the furnace is unknown and, at best, only noisy ingot surface termperature measurements are available.
In spite
of this, it is necessary to control the furnace heating rate and to
remove the ingots from the furnace precisely when the temperature
profile is satisfactpry for rolling. The modelling equations for a simple cylindrical geometry with no axial temperature variations are given by
5t{--T+ aT
s
a
2 a Ts
aTs
rar 1
t0rs 1 0
(11)
which together with the bounda ry conditions aTs
ar
r
0
0
(12)
aTs =
ar
f
a (T 4-T 4) + b (T 4-T 4) s s s w s
r = 1
(13)
W. H. RAY
244
describe the temperature dynamics of the metal ingot. and
Tw
The variables Tf
represent the flame temperature and the furnace wall temperature
respectively.
For this problem the ingot surface temperature, T.O.O.
can be measured intermittently (e.g.; by optical pyrometry), but with a large amount of measurement error.
Thus the measurement device can be
written y(tk) = Ts(1,tk) + T)(tk)
k = 1,2,.....
(14)
The nonlinear filter equations developed in [67] were applied to this problem *using simulated data having Gaussian random errors with standard of = 100°F
deviations
in the furnace flame temperature, aw = 25°F
the furnace wall temperature, and
temperature measurements.
as = 50°F
in
in the ingot surface
The results are shown in Figure 2. The solid
lines are the actual process while the dashed lines are the filter As can be seen, the filter performed very well even in the
estimates.
.face of large process and measurement noise.
Similar problems of state estimation arise in slab reheating furnaces and other metallurgical operations.
From the success of this
simulation study and the experimental study [78] to be discussed later, the potential for future applications of these ideas to metals heating problems seems good. 2.
Chemical Reactors
Chemical reactors freqently appear in tubular or other spatially distributed configurations, and often have unmeasurable state variables such as temperatures, concentrations, and catalyst activities.
The
general aspects of this problem have been discussed in great detail in [89], and various case studies may be found in [15,18-20,43-44,80,82, 83,85].
As particular example, let us consider the problem of
estimating the catalyst activity profile in a packed tubular chemical reactor having catalyst deactivation [82,83,85,89] (cf. Fig. 3).
The
problem arises because one can usually measure only the temperature and possibly the concentration profile in a packed tubular reactor, but
would also like to know the state of the catalyst for reasons of feedback control, optimal planning of shutdowns, optimizing catalyst yields, etc.
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
Figure 2.
Performance of the nonlinear distributed parameter filter as = 50°F .
of = 100°F, aW = 25°F,
245
0
Figure 3.
C(z,t)
A packed bed tubular reactor with decaying catalyst
z
.T (z It)
(z it
Tubular Reactor
Catalyst Pellets
L
L
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
247
This problem was treated by simulation in [20,82,83], and an experimental case study, with off-line filter claculations, is reported in [85].
In this first experimental study of chemical reactor estimation,
it was found that the filter, based on gas concentration measurements, improved the estimates of catalyst activity early in the catalyst lifetime and were consistent with the single direct activity measurement available at shutdown. 3.
Air and Water Pollution The monitoring of pollutant concentrations in urban air sheds
or rivers and estuaries is a problem of current importance. stations are costly and thus limited in number.
Monitoring
Governmental agencies
charged with maintaining air and water quality would therefore like to locate these stations to yield the maximum amount of information.
In
addition the state estimators must produce sufficiently reliable estimates in real time to allow effective enforcement of effluent quality laws and timely control action during episodes of substandard air and water quality.
In spite of the natural applicability of distributed parameter state estimation to these problems, the actual implementation has been very slow - principally due to the lack of adequate models.
For air
pollution, the modelling problem is particularly difficult because of the great number and unpredictability of model parameters and the complexity of the stochastic turbulence equations (cf. [90,91]). Nevertheless, a few papers have appeared applying state estimation to both air [86,92] and water pollution [75,76] problems.
Hopefully with
improved models, one will see the successful application of these ideas to a test region followed by the adoption of the use of state estimation on a routine basis by monitoring agencies.t 4.
Problems with Moving Boundaries A class of systems for which there are many practical problems of
current interest are those systems having moving boundaries.
The theory
has just been developed and some potential applications proposed [68]. These
range from the estimation of temperature profiles and boundary
position in melting and solidification problems found in process metallurgy to the estimation of the extent of crude oil spills.
248
W. H. RAY
A problem receiving current attention by the author is the estimation of the solid steel-mushy zone interface in the continuous casting of steel.
The process, sketched in Figure 4,involves pouring molten
steel at the top of a water cooled mold and continuously drawing out a thin-walled steel strand at the bottom.
If the solid steel crust is
too thin when leaving the mold, either due to some process upset or because the withdrawal rates are too high, the molten steel core will "break-out" and the castin machine must be shut down.
By employing a
distributed parameter filter to estimate the steel thickness in real time, one could operate at high average withdrawal rates while detecting potential break-outs before they occur and thus take appropriate control action.
The performance of such a filter is being tested both numeri-
cally and experimentally at present.
There are a number of other areas of potential applications which shall not be discussed here.
Some examples of these would be the
estimation of the state of deformation of elastic structures [74] and the deciphering of signals in sonar detection [93]. III.
A Real Time Case Study: The first reported application of on-line distributed parameter
state estimation and stochastic feedback control [78] involved estimating and subsequently controlling the temperature profile in a heated slab.
The experimental apparatus is sketched in Figure 5.
slab is one meter long, 2 an thick, and 25 cm wide.
The
Heating of the
slab is provided by 20 heating lamps oriented transversely both above and below the slab.
Cooling is provided by water flowing through 20
holes through the center of the slab.
There are 21 thermocouples
located at the center of the slab at 21 positions along the length. The thermocouple signals are multiplexed and sent to the computer while heating control signals are sent from the computer.
Further details of
the equipment may be found in [78,94]. Because of the design of the apparatus, there are only significant temperature variations in the axial z direction. model the system by
Thus one may
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
XI
J
Mold
Molten Core Liquidus
Line
Solidus Line Mush
Solid
Figure 4.
A schematic diagram of a continuous casting machine
249
W. H. RAY
250
Water tubes
to osc iU oscope
Linear
A/D
Linear
AD
nterpdator
interpolator
Figure 5.
The heated slab used for state estimation and control studies
xiz iI w (z 1
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
251
2 a
x(z,t
at (z,t) = a
- x(z,t) + yu(z',t)
(15)
0_ 0
z = 0, z= 1, z=0 where
x = T-Tw,
a =
k
(16)
P = aw/pCp
,
,
y = C/PCp
(17)
PCpz
The measuring device consisted of thermocouple signals taken at one or more points along the axis of the slab. yi(t) = x(zi*,t) + ni(t)
where
qi(t)
i
= 1,2,...m
(18)
represents the measurement error.
Because the system is linear, the filter equations take the standard form aRat t
= a
2 az zt
- z(z,t) + yu(z,t)
az (19)
m
m
E
z
P(z,zi*,t)Qii(t)[.Yi(t)-z(zj*,t)J
i=l j=l
with boundary conditions (16) on the estimates.
The covariance of the
estimates may be computed from
aP z,z' t
= -2PP.(z,z',t) + a
[a2P(zzst) az2
+ a2P z z' t 1 az'
J
(20) m
m
- E
E
i=l j=1
P(z,zi*,t)Qij(t)P(zj*,z',t) + R+(z,z',t)
252
W. H. RAY
aP za,Z ,t
+ a R0-1(t)6(z') = 0
aP za,z,t
+ a Ro-1(t)6(z) = 0
aP zaZ ,t
a
R1-1(t)6(z'-1)
= 0
z = 0
(21)
z' = 0
(22)
z=1
(23)
aP z',z,t
(24)
az
The fact that the equations are linear makes the implementation through eigenfunction decomposition straightforward.
By substitution
of N
R(z,t) =
ai(t).i(z)
.
(25)
i=0
P(z,z',t) =
N
N
E
2
(26)
B1.(t;:i(z),j (z')
i=0 J=0 N
u(z,t) =
E
(27)
ui*(t)'Pi(z)
i=0
into equations (19-24) yields a set of Fourier coefficients
da(t) -a-t - _
a + 8(try-va3
dB(t)
dt -
(N+1)2 + N+1
ODE's in the
ai(t,, Bi (t)
= t.B(t) + B(t)G
u*(t)
--
(28)
B(t)eTQ(t)aB(t)
+ D(t) + DC(t) + D,(t)
(29)
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
The matrix
A
253
is a diagonal matrix of the first N eigenvalues
corresponding. to the first N eigenfunctions:
wo = 1,
gi(z) = cos irrz
The matrix
eT
(30)
=
i
is defined by
a
(31)
= [Z(zl*), X(z2*) ... 9,>(zm*)J
while (32)
4,(t) = 2a2'E(0)R -1(t)'ET(0)
2a2,,8(1)R1-l(t),'ET(1)
D1(t) =
and
(33)
is the matrix of coefficient arising from
D(t)
R+(z,z',t) =
N
N
E
E
di j (*ci(z)cpj(z')
(34)
i=0 j=0
Due to the symmetry of the covariance equations only ODE's for the along with
N+l
N+l 2+ N+l
need be solved (and these can be done off-line)
Bij(t)
ODE's for the estimates, al(t).
The computational
effort may be summarized in tabular form as Table 3
Computational effort for filter implementation Number of
Estimate Eq'ns
Covariance
Total Eons to
Eigenfunctions
(on-line)
Eq'ns (off-line)
be solved
needed, N+l
(N+1)ODE's
N+1 2+ W '10E's
ODE's 5
2
2
3
3
3
6
9
4
4
10
14
5
5
15
20
6
21
27
6
W. H. RAY
254
In the present experimental study, it was found that
N+l = 3
was adequate to represent all the temperature profiles studied; thus a total of 9 ODE's had to be solved.
For the sake of convenience all
9 ODE's were solved on-line in real time and yet the computational effort required was found to be less than 0.5% of real time. The experimental results for this case study were very impressive (cf [78] for more complete details).
Figure 6 shows typical open loop
results when only a single temperature measurement at
z = 0
was
taken and a substantial random error added to the measurement before passing it to the filter.
In this case a random error was taken from
a Gaussian distribution haVing a standard deviation of 8°C.
As can be
seen, the initial condition given the filter was some 30°C in error, and yet the filter converges very quickly to within 1-2°C of the exact
profile. Typical closed loop results are shown in Figure 7.
In this case
the filter, having only one temperature measurement (at z=0) with 8°C std. deviation added measurement error, is coupled with a proportional + integral modal feedback controller making use of the certaintyequivalence principle.
In all cases the stochastic feedback controller
worked well with the estimates and exact profiles converging quickly and both reaching the desired set point.
The results in Figure 7 are
for an untuned filter and controller; thus one would expect that with tuning the rate of convergence of the stochastic controller could be improved still further.
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
Figure 6.
Open loop distributed parameter filter performance for the heated slab with one temperature sensor (z=0) and 8°C standard deviation added measurement errors.
255
100
Desired profile
'C
Filter Measured profile 80
60
T 40
20 t =0 sec.
01 0
Figure 7.
.2
.4
z
6
8
1.0
Closed loop distributed parameter filter and stochastic feedback controller performance for the heated slab having only one temperature sensor (z=O) and 8°C standard deviation added
measurement errors.
APPLICATIONS OF STATE ESTIMATION AND CONTROL THEORY
IY.
257
Concluding Remarks
From the discussion above it should be clear that there are a wide range of rigorous theoretical results available for optimal state estimation in linear distributed parameter systems.
At the same time,
a substantial number of approximate optimal filtering results leading to useful algorithms systems have appeared.
Nevertheless, there are
several areas in which further theoretical work would be of great help in applications.
Some examples which spring to mind would be conditions
for observability which extend to a wider range of systems and which are easy to apply. sensors.
This would greatly aid in the choice of location for
A second area of need is the development of an interaction/
decomposition theory for stochastic feedback control which extends to a wider range of linear and nonlinear systems and to a wider range of error distributions.
If such a theory were able to contain sensitivity
information with regard to sensor and controller placement, this would be extremely helpful in choosing the optimal sensor location and controller application points.
In the applications/implementation sphere, further research is, perhaps, even more urgently needed.
Our experimental study described
in the last section (which seems to be the only experimental work yet to appear) has done much to show that distributed parameter state estimation and feedback control algorithms can be implemented on minicomputers if one is careful in choosing the numerical algorithms. Thus one pressing need is for work developing and demonstrating efficient algorithms for real time implementation. proceeding In this direction.
Our own research is
We currently have under development
algorithms suitable for linear multi-dimensional systems as well as others for nonlinear systems in single or multiple space dimensions. All of these techniques are being applied to experimental model processes through the use of our PDP 11;40 minicomputer system and the results shall be reported elsewhere in due course.
As noted above, nearly all of the applications of distributed parameter state estimation and feedback control ,eported remain potential applications because of the lact of cor,v;ncing
that the wmplex equations arising in real life problems can be
ins
W. H. RAY
258
feasibly solved on-line with minicomputers.
It is our hope that the
encouraging experience from these first experimental results will inspire the potential industrial and governmental user to begin applying these techniques to the solution of relevant real world problems.
This
in turn should provide feedback to the theoretician in the form of new challenging problems and thus bring enhanced vitality to the entire field.
REFERENCES 1. 2.
3. 4. 5. 6. 7.
8.
9.
10.
11.
12.
13. 14.
Rajbman, N. S., "The Application of Identification Methods in the USSR," Automatics, 12, 73 (1976). Goodson, R. E. and M. Polis, "Identification of Parameters in Distributed Systems," Chapter in Identification, Estimation, and Control of Districuted Parameter Systems, edited by W. H. Ray and D. G. Lainiotis, Marcel Dekker (1976). Kalman, R. E., "A New Approach to Linear Filtering and Prediction Problems," Trans. ASME, J. Basic Eng., 82, 35 (1960). Kalman, R. E. and R. S. Bucy, "New Results in Linear Filtering and Prediction Theory," Trans. ASME, J. Basic. Eng., a, 95 (1961) Kwakernaak, H., "Optimal Filtering in Linear Systems with Time Delays," IEEE Trans. Auto. Cont. AC-12, 169 (1967). Falb, P. H., "Infinite Dimensional Filtering: The Kalman-Bucy Filter in Hilbert Space," Information and Control, 11, 102 (1967). Tzafestas, S. G. and J. M. Nightingale, 'Optimal Filtering, Smoothing, and Prediction in Linear Distributed Parameter Systems," Proc. IEE, 11, 1207.(1968). Tzafestas, S. G. and J. M. Nightingale, "Optimal Control of a Class of Linear Stochastic Distributed Parameter Systems," Proc. IEE, 115, 1213 (1968). Tzafestas, S. G. and J. M. Nightingale, "Concerning Optimal Filtering Theory of Linear Distributed Parameter Systems," Proc. IEE, 112, 1737 (1968). G. and J. M. Nightingale, "Maximum Likelihood Approach to the Optimal Filtering of Distributed Parameter Systems," Proc. IEE, 116, 1085 (1969). Balakrishnan, A. V. and J. L. Lions, "State Estimation for Infinite Dimensional Systems," J. Computer and System Sci., 1, 391 (1967). Thau, F. E., "On Optimum Filtering for a Class of Linear Distributed Parameter Systems," Proceedings 1968 Joint Automatic Control Conference, p. 610; also appeared in Trans. ASME, J. Basic Eng., 21, 173 (1969) Seinfeld, J. H., "Non-Linear Estimation for Partial Differential Equations," Chemical Engineering Science, 24, 75 (1969). Ray, W. H., Parameter State Estimation Algorithms and Applications - A Survey," Proceedings 6th IFAC Congress, paper 8.1 (1975).
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16.
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Seinfeld, J. H., G. R. Gavalas and M. Hwang, "Non-Linear Filtering in Distributed Parameter Systems," Trans. ASME, J. Dyn. Sys. Meas. Cont., G93, 157 (1971). Kuhr, D., "Optimale Filter f{r Lineare Systems Mit Verteilten Parametern," Regelungstechnik and ProzeR-Datenverarbeitung, 18, 506 (1972).
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"NUMERICAL SOLUTION OF THE TRANSONIC EQUATION BY THE FINITE ELEMENT METHOD VIA OPTIMAL CONTROL"
M. 0. Bristeau, R. Glowinski, 0. Pironneau
ABSTRACT It is shown that the transonic equation for compressible potential
flow is equivalent to an optimal control problem of a linear distributed parameter system.
This problem can be discretized by the finite element
method and solved by a conjugate gradient algorithm.
Thus a new class
of methods for solving the transonic equation is obtained.
It is
particularly well adapted to problems with complicated two or three dimensional geometries and shocks. 1.
2.
PLAN 2.
Introduction
3.
Statement of the problem
4.
Gelder's algorithm for subsonic flow
5.
Formulation via optimal control
6.
Discretization and numerical solution
7.
Numerical results
8.
Conclusion
INTRODUCTION
The transonic equation is a non linear partial differential equation which has an elliptic behavior in the subsonic regions of the flow and a hyperbolic behavior in the supersonic regions.
At the
interface the normal component of the speed of the flow can be discontinuous (shocks).
Some finite difference methods have been successfully
developed even for flows around simple 3-D objects (Jameson (1974), Garabedian-Korn (1971)).
However the method of finite differences is
265
M. 0. BRISTEAU etaL
266
not well suited to complicated geometries. An alternative approach using finite elements was studied by Gelder (1971), Norries and de Vries (1973), Periaux (1975) but their methods explode at supersonic speeds. Following Gelder's approach we shall replace the transonic equation by the minimization of a functional in an abstract space, a problem which can be solved by the methods of the theory of calculus of variations and optimal control theory. 3.
STATEMENT OF THE PROBLEM Stationary adiabatic monophasic compressible flows, in which the
effects of viscosity are neglected, are well described by the set of equations apu3 apu2 + a + a x2 x3
apul
0
(3.1)
0
a
xl
(1
2)
= 0
1
(3.2)
C*
u = vO (ui =
(3.3)
p
where and
y
(1971).
a axi
i - 1,2,3)
,
is the density, u is the speed of the fluid and where
po,C*
are constants (y=1.4 for di-atomic gas, see Landau-Lifschitz
We shall denote
1
k =
- a = 1/(Y-1). Therefore, if 1
2
C* the region occupied by the fluid, one must solve the nonlinear partial
differential equation: 0
(3.4)
with the boundary conditions (3.5)
(3.6)
OIr1 = Ol
an 'r2
g2
in
2
is
267
NUMERICAL SOLUTION OF THE TRANSONIC EQUATION
where
and
r1
r2
are parts of the boundary
rI U r2 = a2
assume that
rl n r2 = 0
and
.
ac
of
2
.
We shall
In addition, if there
are shocks (i.e. lines or surfaces where the tangential speed of the flow is continuous but the speed normal to these lines or surfaces is discontinuous) then, across the shock: (3.7)
(pu)+ = (pu)
(Rankine-Hugoniot condition)
(3.8)
un
(entropy condition)
un
where it is understood that the particles of the fluid move from - to +. Note the (3.4) multiplied by
w E
and integrated by parts,
C1(62)
leads to (3.9)
fS2(1-klvol2)avo vwdx = Ir (1-kIvfl2)Lg2wdr2 2
Vw ( C1(52) s.t. wlr1 = 0 ; 1r1 = 0
If the notion of derivative is extended and the space
replaced by
H1(s2) = {wEL2(s2)
jvw E(L2(52))3}
weak formulation of (3.4)-(3.6). 4.
GELDER'S ALGORITHM FOR SUBSONIC FLOW g21r2 = 0
functional
(1-kIv412)a+ldx
Eo(4) = - tsa
we shall say that
(4.2)
is
Note that it contains (3.7).
For notational convenience we suppose
(4.1)
C1(52)
then (3.9) is called a
Hol(2)
4s
=
is a stationary point of
Eo
on
lml rl = 0}
if
6Eo = Eo(o+6o,) - E0(4) = o(o) V6,k E Ho1(Q)
.
Consider the
268
M. 0. BR ISTEAU et al.
Since, from (4.1) (4.3)
6E0 = f2 2k(a+1)(1-kIvhI2)avov64,dx + o(6o)
any stationary point of fo
Hol(y)
on
satisfies
f2(1-kIv4,I2)avovwdx = 0 Vw ( Hol(2) Thus all stationary points of
Eo
on
such that
Hol(52)
0Irl
(holrl
=
and which satisfy (3.8) are solutions of our problem. Let us look at 2
2
2ka V v6
2k(a+l)
1 dx
(1-kjvj
d22
)
with our notation the mach number is such that
M2 = 2ka(1-kFv(pI2)-1
therefore, if
Iv(p I2
is the angle between
o
and
vo
v6,
2 dE
_ - 2k(a+l) f2 P(1-M2 cos2
e)Iv6(k I2 dx
This shows that if in some part of the fluid
M > 1
,
E is not convex
and the solution of (3.4)-(3.8) is only a saddle point of E. other hand, if
M < 1
in
then
2
(3.4)-(3.8) is a minimum of
E
.
E
This fact was utilized by Gelder
(1971) and Periaux (1975) for constructing The functional the
H1(s2)-norm
mn+l E H1(st):
E ;
On the
is convex and the solution of
a solution of (3.4)-(3.8).
is minimized by a gradient method with respect to i . e .
(, k
-
is constructed by solving for
f2 Pn VOn+l vwds2 = 0 VW E Ho1(2)
,
(On+l -'hl) I rl = 0
NUMERICAL SOLUTION OF THE TRANSONIC EQUATION
269
This method works very well (less than 15 iterations in most cases) and it is desirable to construct a method as near to it as possible, for supersonic flows. 5.
FORMULATION VIA OPTIMAL CONTROL Along the line of §5 we shall look for functionals which have the
solution of (3.4)-(3.8) for minimum.
Several functionals were studied
in Glowinski-Pironneau (1975) and Glowinski-Periaux-Pironneau (1976). In this presentation we shall study the following functional:
(5.1)
where
E() = fS2 P(IvEI2)Iv4-E)I2 dx, P(IvEI2) = (14177 J2)a
0 = 0(E)
(5.2)
is the solution in
f2 P(IVEJ2) vovwdx = 0
H1(52)
of
Vw E Hol(S2)
, 0Irl = 41
Suppose that (3.4)-(3.8) has a solution.
Proposition 1.
Given e > 0, small,the problem
min (E() It E 9}
(5.3) where
E _
{r,
(
H1(52)) tIrl = 01
has at least one solution and if solution of (3.4)-(3.8).
JvE(x)I <_ k-1/2(1-e)
a.e x c Q}
Vx E 2 , it is a
At(x) < + m
Furthermore any minimizing sequence {En}n>o
of (5.3) has a subsequence which satisfies (3.5)-(3.7) and
(1-kJvtnJ2)avx
lim fn
vwdx = 0
Vw E HQ1(52)
Proof: the first part of the theorem is obvious. Let
that
{En}
be a minimizing sequence of
E
then
En E c
implies
Ilv nll2 < k-1(i-e)2 f2 dx , therefore a subsequence (denoted
also) converging towards a
Ilv(on-tn)II
0.
i
E s
can, be extracted.
Therefore
fQ Pn v(4n-En) Vwdx = f2 Pn Vtn vwdx -+ 0
Furthermore
{En}
M. 0.8RISTEAU et at.
270
for every subsequence such that
pn
converges in the
L-(2) weak star
topology. Remark.
Note that if
solution of (5.3).
is a weak limit of
t-
{gn)
,
may not be a
This, however, does not seem to create problems in
practice. Proposition 2 If
t1r1 = 01
6Ir1 = 0 , then
(5.6) E(?+6ts)-E(
) =
(M12
2fQ p(jvtf2)(l+
2p'p
1IV'012
= +2k,(1-klvml2)-IIvm12)
Proof From (5.1) and (5.2) (5.7)
E(+6r)-E(ti) = 2fS[2a'vg-m1V(d- 0 12-pv(,6-t;)v6t;+pV(O-g) v6,b]dx
+ o(6) + 0(61;) where ka(1-kIVFl2)a-1
p' = -
From (5.3)
(5.8)
fQ pv6ovwdx = - f2 2p'vt;-v6EvO-vwdx+o(6t;)
and since
p(1v(t;+6)12)
Nw E Ho1 (9)
is bounded from below by a positive number,
if a > 2 . there exists K such that IIv60I) 5 in (5.8), (5.7) becomes Therefore, by letting w = d-t 6E = - 2f [ pV(O- ) V6t + P,
I V4, 12-1 v2;12) vt;' vbt;] dx
IQ
and from (5.2) the term
pvov6t;
disappears.
NUMERICAL SOLUTION OF THE TRANSONIC EQUATION
271
Corollary 1 If
is a stationary point of
i,
E
,
it satisfies:
2
(5.9)
in
(1-Iv Zl2 Iv01-2) vZ] = 0
v-[p(1 +
2
2
(5.10)
(1 +
= 0
(1-Iv I2 Iv- -2) a
P
;
lr1 = l
Ir2
Remark: It should be noted that in most cases (5.3) has no other stationary point than the solutions of (3.4)-(3.7).
Indeed let
(xC,yC,zt) be a curvilinear system of coordinate such that
,o,o)
vi:=(a C
Then, from (5.9), (5.10)
(5.17)
ax
[p(l +
e
(1- IvZI2 Ivm1-2)
or
r2(1- IvZ12 IVml-2)
] = 0,
an
Ir2 = 0
Ir2 = -2, ZIrl = 01
This system looks like the one dimensional transonic equation for a, compressible fluid with density
P (1 + 2
(1- IcEI2 IvOl-2))
Therefore, if the t-stream lines meet two boundaries and the shocks and
1+ then
- Z
.
Z (l- IvE12 Iv4s l -2) > 0
at < + m at
M. 0. BRISTEAU et al.
272
DISCRETIZATION AND NUMERICAL SOLUTIONS
6.
Let
be a set of triangles ortetrahedra of 2 where
Th
T.
is
which approximate
rl
Suppose that
the length of the greatest side.
U T c Q ,
h
n T2 =
or a vertex
VT1,T2 E Th
TETh
0
Let
and
2h = U T
parts of ash
rlh' r2h
h
and
r2
Let
Hh
. be an approximation of
is completely determined by the values
Note that any element of Hh that it takes at the nodes of
i
N = n+p+m
nodes
Pi
Th
with
E ]n,n+p[ , and if we define
wi = 1
(6.2)
Then any function
T VT E Th}
linear on
Hh = {wh ( C°(2h)I wh
(6.1)
has
Hl(2):
.
Therefore if we assume that if
Pi E rlh
wi c Hh
i > n+p, Pi
by
at node 1 and zero at all other nodes
w E Hh
(6.3)
0 = Eaiwi
Algori thm
1
is written as
N
Let
Ioh =
E
i;1w.
, then (5.2) becomes
i=1 f9(1-k{VEhI2)aVOhVwidx = 0 (6.4)
Oh =
n+p i E m wi + i=1
N E
n+p+l
i
01wi
i=1,...,n+p
E r2h
Th if
NUMERICAL SOLUTION OF THE TRANSONIC EQUATION
273
and (5.6) becomes
(6.5)
6Eh =
2 (6.6)
o(8 )
E i=1
6Eh = f2[P-P'(Iv0h1 2-Ivdhi2)]v
dx
Consider the following algorithm Step 0
Choose
Step 1
Compute
0hj
Step 2
Compute
{6Ehj,
Step 3
Compute
6C
set
-r,, t,o
j=0
by solving (6.4) with
Ch = Chj
by (6.6)
n+p = h
(6.7)
E 6C 'w. i=1
by solving
p6hvwidx = 6Ehj, i=1,...,n+p h
Step 4 (6.8)
Compute an approximation
min XE[0,1 ]
jr
2h
P()d v(Ch (x)
of the solution of
S,j
-
h
x))I2dx
where N
gh(x) = 1z
Step 5
Set
(4j-x6gh)wi
4hj+1 = 4h(aj), j=j+l
and go to step 1.
Proposition 3 Let
{ghj}j-,.0
be a sequence generated by algorithm 1 such that
vFj(x)I s k-1/2 Vx, vi
.
Every accumulation point of {Chj}j:-.,o
stationary point of the functional
(6.9)
Eh(Ch) = f2hIV(kh-gh)I2dx
is a
M. O. BRiSTEAU et al.
274
where
is the solution of (6.4), in
Oh -Oh(h)
Sh = {Ch ( Hhi lvth(x)l
k-1/2 tlx E 2h)
`:
Proof Algorithm I (6.9) in
`h
is the method of steepest descent applied to minimize
, with the norm
f2 o hP h dx
(6.10)
h
Therefore
{Eh(Chj)}j
decreases until
6Ehj
reaches zero.
Remark 6.1: (6.4) should be solved by a method of relaxation but (6.7) can be factorized once and for all by the method of Choleski. Remark 6.2: Problem (6.8) is usually solved by a Golden section search or a Newton method.
Remark 6.3: Step 5 can be modified so as to obtain a conjugate gradient method.
Remark 6.4: The restriction: Juh (x)j
k-1/2
in theorem 5.1 is
j
not too close to
k-1/2, otherwise one must treat
this restriction as a constraint in the algorithm.
Also, even though
theorem (5.1) ensures the computation of stationary points only, it is a common experience that global minima can be obtained by this procedure if there is a finite number of local minima.
Remark 6.5: The entropy condition numerically.
Let
M(x)
Ath < + -
can be taken into account
be a real valued function then
Ath < M(x)
becomes, from (6.7)
(6.11)
-E 7j 8Ehj 5 M(xi)
i
= 1,...,n+p
Therefore, to satisfy (6.11) at iteration 6Ehj = 0 equality.
in (6.7) for all
i
j+l
,
it suffices to take
such that (6.11) at iteration
This procedure amounts to control
w = at
j
instead of
is an t
NUMERICAL SOLUTION OF THE TRANSONIC EQUATION
7.
275
NUMERICAL RESULTS The method was tested on a nozzle discretized as shown on figure 1,
The Polak-Ribiere method of
(300 triangular elements, 180 nodes).
conjugate gradient was used with an initial control: At = 0
A mono-dimensional optimization subroutine based
(incompressible flow).
on a dichotomic search was given to us by Lemarechal.
Several boundary
conditions were tested 1°) Subsonic mach number
at the entrance, zero potential on
Mm = 0.63
exit, the method had already converged in 10 iterations (to be compared with the Gelder-Periaux method) giving a criterion (Eho =
EhlO = 2 10-13
10-4)
2°) Entrance and exit potential specified. For a decrease of potential of
41 - 02 = 0.7
the method had
converged in 20 iterations without including the entropy condition, giving a criterion of
Eh20 = 5.10-7 , the results are shown on
figure 2. 30) Supersonic mach number
M. = 1.25
The method computes a solution that has a shock at the first section of discretization. Another boundary condition must be added. One iteration of the method takes 3 seconds on an I811370/158 on this example.
A three dimensional nozzle is being tested: the result will be shown at the conference.
20 to 40 iterations are usually sufficient
for the algorithm to converge. the tabulated data. tested.
The results are in good agreement with
Simple and multi-bodies airfoils are also being
For them it is necessary to include the entropy condition;
80 iterations are usually more than sufficient for the convergence. 8.
CONCLUSIONS
Thus this method seems very promising.
It compares very well with
the finite differences method available and it has the advantage of allowing complicated two and three dimensional geometries.
This work
illustrates the fact that optimal control theory is a powerful tool with unexpected applications sometimes.
276
M. 0. BRISTEAU et al.
NUMERICAL SOLUTION OF THE TRANSONIC EQUATION
277
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a
ao
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M. 0. BRISTEAU et al.
278
ACKNOWLEDGMENT
We wish to thank M. Periaux, Perrier and Poirier for allowing us to use their data files and computer,and for their valuable comments. REFERENCES 1. 2. 3.
4.
5.
6. 7.
8.
Garabedian, P. R., Korn, D. G. - Analysis of transonic airfoils. Com. Pure Appl. Math., Vol. 24, pp. 841-851 (1971). Gelder, D. - Solution of the compressible flow equation. Int. J. on Num. Meth. in Eng., Vol. 3, pp. 35-43 (1971). Glowinski, R., Periaux, J., Pironneau, 0. - Transonic flow computation by the finite element method via optimal control. CongrPs ICCAD Porto Fino, June 1976. transsonique Glowinski, R. and Pironneau, 0. - Calcul par des mEthodes finis et de contr8le optimal. Proc. Conf. IRIA, December 1975. Jameson, A. - Iterative solution of transonic flows. Conf. Pure and Applied Math. (1974). Norries, D. H. and G. de Vries - The Finite Element Method. Academic Press, New York (1973) Periaux, J. - Three dimensional analysis of compressible potential flows with the finite element method. Int. J. for Num. Methods in Eng., Vol. 9 (1975). Polak, E. - Computational methods in optimization. Academic
Press (1971).
