Maximum Principle of Semi-Linear Distributed Systems

Copvright © IFAC 3rd Symposium Control of Distributed Parameter Systems Toulouse . France. 1982

MAXIMUM PRINCIPLE OF SEMI-LINEAR DISTRIBUTED SYSTEMS Yao Yun-Long Institute of Mathematics , Fudan University, Shanghai, China

Abstract. This paper considers the optimal problem with general cost functional (a ) and the time optimal problem for the systems governed by nonlinear Vo!terra's integral equations (a ) on a Banach space X. Vnder the conditions 1-3 mentioned in the paper, Ehe Maximum Principle satisfied by optimal control u*(.) is given and applications to semi-linear evolution systems are presented. Keywords. Maximum Principle, nonlinear Volterra's integral equation, cost functional, optimal control, variation of control, reachable region and cone. Introduction

C [O,Tl with mes6 <6 ( for below);

Let X, Y and Z be Banach spaces and T a given positive number. Let V, the control region, be a separable subset of Z and the control set be 'Jr= {u(.), (o,T) .... V strongly measurable}. (a ) l

Suppose x(t) = x(t,u(.» is the solution of the nonlinear Volterra's integral equation on X:

see cond.3

Remark 1. An operator Vet,s) is said to be with property 11 L (Y+X) 11 i f Vet,s) £l(Y,x) for (t,s) £ FT a.~. satisfies cond.l like G(t,s) above. Cond.2 There exist strongly continuous

Fr~chet derivative 3b(t,x,u)/3x = b (t,x,u)

£ot(X,Y) and f (t,x,u) £ X* on [O,TfxDxV; x

x(t) = het) + ftG(t,s)b(s,x(s),u(s»ds, o u(.)£"]t, t~O (a ) 2 where h(.)£C(O,T;D), D - an open subset of X; G(t,s)£~(Y,X) for (t,s)£F = {et,s), T~t~s~O} a.e.; and b(s,x,u)£CtCO,Tl xDXV, Y).

Cond.3 For each u(.) £ 9' and x*(.) £ C(O,T;D) there is £> 0 and ~(.) £ Ll(O,T) ( £ and ~(.) may depend on u(.) and x*(.) ) such that

The admissible control set Vad is V d = { u(.)£ 1', x(t,u(.» Suppo~e Vad is not empty.

~(.)

Ub(t,x(t),u(t»U

exist on (O,TJJ

The cost functional is

<

~(t)

IIf(t,x(t) ,u(t»1I <

~(t)

IIbx(t,x(t) ,u(t» 11 <

~(t)

1/ fx(t,x(t) ,u(t» 11 < ~ (t) (a ) 3 where the real functional f(t,x,u)£C([O,TJxDxV). for t £ (O,TJ and x(.) £ S£(x*(.»C C(O,T;D) ( S£(x*(.», the closed ball with centre at Furthermore,suppose that x*(.) and radius £ in C(O,T;D) ). Cond.l G(t,s) is strongly measurable in Remark 2. From the above conditions 1-3, it (t,s) on (O,T)x(O,T) ( we may define G(t,s) is easy to show that =0, if (t,s)£F ); it is strongly measurable T IIb(t,x (t) ,u(t» - b(t,x (t) ,u(t»" in t on (O,T) for each s£(O,T) - N , N is l 2 a ~-null set in (O,T) and strongl? me~surable ~ ~(t)· 11 xl (t) - x (t) n 2 in s on (O,T) for each t; for all y£Y and t IIf(t,x (t) ,u(t» - f(t,x (t) ,u(t» 11 £ [0, T ), we have l 2 T ~ ~(t)· /I xl (t) - x (t) 11 lim II(G(t,s)-G(t,s»yUds 0; 2 - 0 t -+ t for all x (.) and x (.) £ S£(x*(.». l 2 th~re exists a measurable scalar function For example, it often happens that, in the S et,s) defined on (O,T]x[O,TJand S(t,s) = 0 unbounded control problem if t < s such that UG(t,s)lI< S(t,s) for (t,s) £[O,Tjx[O,T] where S(t,s) i~ also measurable G(t,s) = G (t,s)/(t-s)a and ~ (t) = l/sY o in t and s on (O,T]respectively, besides for where G (t,s) £~(Y,X) for (t,s)£F and any £>0, there exists 6>0 such that T strongl? continuous in (t,s) on FT' In this f ~S (t,s)~(s)ds < £ (as) case, IIG(t,s) rr
=

f!f(s,x(s),u(s»ds

f

81

Y. Yun-Long

82

In this paper we consider the following two problems: Problem I. Find optimal control u*(.) E Uad such that min { J(u(.», u(.) E Uad } = J(u*(.». The second problem is the time optimal one. Suppose the target set R(t) is a bounded convex closed body in X and varies continuously in t on (O,T] and h(O) £ R(O). Furthermore, suppose there exist u(.) E 7r and t E[O,T ] such that x(t ,u(.» E R(t ). Let ~ (t) ge the reachable r~gion of (a2)~ that is,

As special cases of Theorem 1 and 2, we have the following corollaries about semi-linear evolution s ystems. Let us first consider the first-order semilinear equation :

Problem 11. Find optimal time t* and optimal control u*(.) E ~ such that x(t*,u*(.» E R(t*) and dis( ~ (t),R(t» > 0 for 0 ~ t < t*.

Ax(t) + b(t,x(t),u(t»

x(O)

XoE X, u(.) E 1.

Here A is the infinite~!mal generator of a linear C - semigroup e on the Banach s pace X. The m~ld solution of (b ) is given by 7 the integral equation on X x(t) = eAtx

~ (t)

= {x (t,u(.», all u(.)ElJtand tE(O,t ) } max where (O,t ) is the maximum existing interval of the sol~~lon x(t,u(.».

x(t)

+ fteA(t-s)b(s,x(s),u(s»ds

o

0

(b

Corollary 1. The necessary condition for optimal control u*(.) of (b]) with cost (a ) 3 is that u*(.) satisfies (b ), where H(t,u) 3 defined by (b ) and \jJ(t) is the solution of 2 the dual puterbation equation on X*: b~(t» \jJ (t)

wet) =-(A* + THE MAIN RESULTS

ft

s

G(t,1)b (1)U(1,S)d1 x (b ) 1

for all tdO, T J and SE (0, T]a.e. ,where b (1) = b (1,X*(1),U*(1». The integral in (b1~ is in ~he strong s ense. Let H(t,u) = -f(t,x*(t) ,u) + \jJ (t) =

-fT

Here

x

Theorem 2. If u*(.) is the optimal control for Problem 11, then there exists gEX*, IIgll= 1 such that max { < pet) ,b(t,x* (t) ,u», uEU }=

:, for t E(O,t*) where pet) = U*(t*,t)g, and U*(t*,t) is the adjoint operator of U(t*,t). Furthermore, the transversal condition ~

0


p (t)

G*(t*,t)g + t*

ft

G*(s,t)b~(s)p(s)ds

for all h E R(t*)-x*

holds.

x(t) + Ax(t) = b(t,x(t),u(t», t > 0 x(O)

Xo E X, x(O) = Xo E X

(b ) 9

where A is a positive definite, self-adjoint operator with domain D(A) C X. The mild solution of (b ) is the strongly continuous solution of (a 9 ) if het) = cos(A~t)xo + 2 _k ~ -~ ~ A 'sin(A t)x and G(t,s) = A sin(A (t-s». o From theorem 1 and 2 we can derive analogous maximum principles for Problem I and 11 of the s ys tem (b ). 9 Corollary 3. If u*(.) is the optimal control of Problem I for (b g ), then u*(.) satisfies (b ) where vet) is the mild solution of se~ond-order perturbation equation

~ (t) + (A + b (t» \jJ (t) = f (t,x*(t),u*(t» x

\jJ (T) = ~ (T) = 0,

x

for 0 < t < T.

Corollary 4. If u*(.) is the optimal control of Problem 11 for (b a ), then u*(.) satisfies (b ) where pet) is a ' non-zero solution of 4 pet) + Ap(t) = b~(t,x*(t),u*(t» THE PROOF OF THEOREM 1

for t E(O,t*) a.e. where b*(s) = b*(s,x*(s),u*(S» E ~(X,Y) is the adj~int ope~ator of b (s,x*(s),u*(s» and the integral is in th~ sense of p*.

b~(t»p(t).

Let X be a Hilbert space. Let us consider second-order semilinear equation:

) holds.

Remark 3. It can be verified that pet) is an solution of the Volterra's equation on X*:

and fx(t)

Furthermore, the transversal condition

for all hER(t*)-x*

x* = x(t*,u*(.»

b~(t,x*(t),u*(t»

pet) = -(A* +

Theorem 1. If u*(.) is the optimal control for Problem I, then u*(.) satisfies the following relation max { H(t,u), u EU }=H(t,u*(t» for tf;(O,T)

=

fx(t, x*(t),u*(t».

(b ) 2

the integral being on in the s trong sense. The sign<\jJ(t),y> denoting the value of the linear functional \jJ (t) at yE Y.

b~(t)

Corollary 2. If u*(.) is the optimal control of Problem 11 for (b ), then u*(.) satisfies 7 (b ) where pet) E x* is a non-zero solution 4 of the linear equation on X*:

< \jJ (t) ,b(t,x*(t) ,u»

f (s,x*(s),u*(s»U(s,t)ds EY*

t

+ fx(t)

\jJ (T) = 0, t E (O,T).

There is "L (Y-+X)" operator U(t,s) which satisfies theeintegral equation U(t,s) = G(t,s) +

a)

Set

n-1

~o(Tk / n,T(k+6 )/n)

c: (O,T)(C ) 1

Haximum Principle of Semi-linear Distributed Systems where 0 £

83

(0,1), n = 1,2, ... ; r = ~ 0 otherwise for t £ Eo(n); 1 U (C ) 2

e~(t,n) = U

u(t) { u*(t)

for t £ Eo(n); otherwise (C ) 3 where u*(.) and u(.) £ 1. It is clear that uo(·,n)£']r. [O,T] to X and wet,s) =

° for

t < s. Denote

Lemma 3. If x*(t) = x(t,u*(.» [0,t1, then for each u(.) £ 0

Lemma 1. If wet) = W(t,.) £ Ll(O,T;X) for each t £(O,T) and w(.) £ C(O,T;Ll(O,T;X», then for each 0 £

(0,1) there is a natural

exist on

7r

there exists

£ (0,1) and for each 0 £(0,0

there

)

0

exist n(o) such that xo (t) = x(t,u (.» o uo(t,n(o»

wet) = w(t,.).

11.m

0+

+0

(uo (t)

) also exist on [O,t] and

max _\\xr(t)-x*(t) (5 t£ (0, tJ

Ax (t) 11

where t

~x(t)=foU(t,s)(b(s,x*(s),u(s»-

number n( o) such that 11

this proves Lemma 2.

0

Suppose that wet,s) is a mapping from [O,TJx

2£.

<

and

f~ w(t,s)eo(s)ds";; 0 2

(C 4 )

(C ) 9 -b(s,x*(s),u*(s»)ds

proof. wet) is uniformly continuous on the

where U(t,s) is given as (b ). l Proof. By condition 3 mentioned in Introduc-

compact interval (O,TJ. Let

tion, there exist £ >0 and

for t £ [O,T] , where e o (s) = eo(s,n( o».

IIw(t') - w(t") ilL (0 T.X)= T 1 " follw(t',s) - w(t",s)lI ds <0 2 /2 whenever

I

t' - t"

I

x

< A . Set

£ Ll (O,T)

~(.)

such that (CS)

tk = Ak/2

ij b(t,x(t) ,u*(t»/I ::., ~(t)

IIbx(t,x(t),u*(t»II ;;

~(t)

k = 0,1, ... < 2T/A ). By Riemann-Lebesgue Lemmafor x(.) £ S£(x*(.» and t £ [O,tJ. There is a sufficiently large a > 0, by Lemma 2, such its extension from ), we hav~ T that for all t £ [O,T) as n + ex> w(tk,s)eo(s,n)ds + f~ e-a(t-s)S(t,s)~(s)ds < 1/3. (C ) for k = 0,1, •.. < 2T/A so that ll We define a new normll./1 of C(O,t;X) by 2 1/ f~ w(tk,s)eo(s)ds U< 0 /2 a for some natural number n(o), eo(s) at = e o (s,n( o» and for k = 0,1, ... < 2T/A. 11 x(.) fI = max{ e- llx(t)1I ,t £[O,t]} a We see that for every t £[O,T] there exists where x(.) £ C(O,t;X). It is obvious that

°

fo

an tk satisfying 11

I

I

t - tk


11 .U a is equivalent to 11 x(.) lie =

f~ w(t,s)eo(s)ds 11

= max {flx(t)" , t£ [0, tJ } , so that for suf-

;; lIf~

ficiently small

w(tk,s)eo (s)dsll +

+ f~ Uwet,s) - w(tk,s) /I ds < 0~/ 2 + 0 2 /2 = 0 2

•

this proves Lemma 1.

0, the closed ball

~>

S~(x*(.),a) = { x( . ) £ C(O,t;X),

IIx(.)-x*(.)1I

C

S (x*(.». £

Let To: SA(x*(.),a) ~ C(O,t;X) be a family Tox(.) = het) +

lim sup fT e-a(t-s) S (t,s)~(s)dSr O. a+ -too t£Co,TJ o ~s)

< A}

=

of operators with a parameter 0 £ (0,1 J

Lemma 2.

for #(t,s) and

a

see condition 1 in

f

t

G(t,s)b(s,x(s),us(s» ds (C ) 12 for x(.) £ SA(x*(.) ,a) and t £ (O,tJ. 0

Because TOx*(.) = x*(.), it can be verified

INTRODUCTION ).

that there is a sufficiently small 0

Proof. It is obvious that (as) implies

by (CIO)'(C ll ) and Remark 2, such that To is a contraction operator from SA(x*(.),a) to

fT S(t,s)~(s)ds < (T/o + 1 )£ = M o

0

> 0,

for all t £ [O,T]. Set Et = {s£(O,T],

itself for 0 £ [0, 0

S (t,s)~(s)

an unique x (.) £ SA(x*(.),a) c S£(x*(.» o such that Toxo = Xo for 0 £ (0, 0 0 ) •

> 2M/ o } . Then

mesE < fT S(t,s)~(s)dsxo/2M< Mxo/2M <0. t='

0

Hence for a > 2M/o£ fT e-a(t-s)S(t,s)~(s)ds = o

0

)

•

Hence there exists

Set 6x(t,0) = x*(t) + 1.. ( xo(t) - x*(t»

where A £ to,l],

Y. Yun-Long

R4

the uniform convergence. The" Le(Y XR....XXR)"

X (t, A) = x*(t) + A(X (t) - x*(t)) where 8 8 A E (0,11 , then ~x(t, 8 ) is the solution of

operators U(t,s) is the solution of the

the integral equation

equation U(t,s)

~ x(t, 8 )

fto

fl

G(t,s)(

0

x

fto

x(s, 8)d A)ds +

u

+

u

w(t,s)ds + (C 13 )

~xo(t) = f~ (f(s,x*(s),u(s))-

G(t,s)( b(s,x*(s),u(s)) -

-f(s,x*(s),u*(s)))ds +

+ft(ft f (T)U(T,S)dT)(b(s,x*(s),u(s))o s x

By Lermna 1, for each 8 E [0,1 ] there exis ts n( 8) such that

-b(s,x*(s),u*(s)))ds.

f~ w(t,s)e 8 (s)ds/ 8 ~ 8 .

From (C

16 x (t)

Similary, by the Contration Mapping Theorem, ~

the Volterra's equation on (O,t]

here b (s) x

=

x t

fo

+ w(t,s)ds (C 1S ) b (s,x*(s),u*(s)). By the varix

ation of constants formula ( Under the conditions 1-3 in Introduction, it can be proved 1S

XXR(with the norm II(~

x0 8 (T)-x~(T)

implies

17 x 08 (T) - x~(T) 8

Thus,

i.e.

(T)

~xo(T) ~

~

~xo(T)

as 8 .... +0.

0. By (C

) and the defini1S tion of H(t,u) in (b ), we have proved the 2 the inequality "( For u(.) E '} ) (C 19 )

)1\

=

x(t)

o

= het) +

max (nxll, Ixol),

19

)

implies (b ). 3

~(t,s)D(s,x(s),u(s))ds

o

u ( .) E 'Jr. Because the control

for every u

E R - the real field)

ft

~l(O,T)

region U is a separable set in Z, (C

o

where x E X and x

(C

16

)

The proof of theorem 2 Suppose that u*(.), the optimal contr ol of

where (x(t) ) E X xR x (t)

x(t)

(h~.))

fi ( . ) G(t,s)

"'(

- ()

the Problem I, exist on (O,t*l , where t* is optimal time.

C(O,T;XxR)

E

Lermna 4. For each t E (O,t*l, the closure of

1) : YxR .... XxR ° °

(G(t,s)

and

())

o s,x s ,u s

=

the set n et)

(b(s,x(s),u(s))) YXR f(s,x(s),u(s)) E .

It is easy to show that conditions 1-3 of

16

x~(T)

08

t; continuous in u and contains in

Consider the new s ystem on the Banach space

(C

J(u(.)). Thus x

It easy to see that H(t,u) is measurable in

The proof of Theorem 1.

16

f(s,x(s),u(s))d s .

0

f! H(s,u(s))ds ~ f! H(s,u*(s))ds.

) may be represented by

(C ) . 9

(C

)

ft =

=

J(u*(.))

(C

G(t,s)b (s) ~x(s)ds +

the solution of (C

=

Hence xo(T)

13 x(t) is the unique continuous solution of

fto

)' we know that

o

) we can prove (CS) within which

~x(t) =

bx (s) = bx (s,x*(s),

is

-b(s,x*(s),u*(s))).

using (C

a(t,T)5 x- (T "~ U( T ,S)dT

u*(s)). By computing, the last component of ~(t)

where wet,s)

Its

for s E [O,T]a.e. where

f~ w(t,s)e 8 (s)ds/ 8

+

G(t,s) +

b (s,X ~ (S, A ),U~(S))'

) in Introduction hold. Using Lemma 3 to )' we obtain

here

~8 (t) =

z(t,u(.)),U(.)E~ }

i s c onvex

set in X, where z(t,u(.)) defined by z(t,u(.))

=

ftG(t,s)b (s)z(s,u(.))d s + o

x

+ f~w(t,s)dS.

(d l )

Proof. From (b ) and (C ) we have 14 l t z(t,u(.)) = f U(t,s)b(s,x*(s),u(s))d s . (d ) 2

= x(t,u8 ('))' x*(t) = x(t,u*(.)),

Ito Vet,s) (b(s,'X*(s) ,u(s))

{

o

x (t) - i{*(t) 8

~x(t)

=

-

- b(s,x*(s),u*(s)))ds and u*(.) is the optimal control of Problem I. The sign " ~ " means

For each u (.) and u (.) E 'J, s et 2 l ( ) f u 1 (·) for t E E (n); u B .,n = l u (.) otherwise 8 2

for 8 E [0,11. For E (n) see (Cl)' Hence, 8 by Lermna 1, there exist n( 8) such that

Maximum Principle of Semi-linear Distributed Systems z(t,u (. ,n(O») - (oz(t,u (.» + o I + (1-0)z(t,u (·» ) 2 = f~ U(t,s)z(s)eo(s,n(B»ds ~ 0 as n~ where z(s) = b(s,x*(s),u (s» - b(s,x*(s), 1 u (s», that is, for each OE [O,lJ and all 2 u (.) and u (.) f ~ 2 I oz(t,u (,» + (1-0)z(t,U 2 (.»E ~(t), 1 i.e. ~(t) is a convex set in X. Corollary 5. The closure of reachable cone Vet) = { X*(t)+A(Z(t,U(.»-z(t,u*(.»), A~O, u(.)

85

bounded functional g E

X*, vg 11 = 1 such that

,:;, for xEV(t*) and yER(t*). By setting y=x(t*,u*(.»

and x=x(t*,u*) +

A(Z(t*,u(.»-z(t*,u*», we obtain

~



substituting z(t*,u(.»=

f

t*

0

U(t*,s)b(s,x*(s),u(s»ds

and

z(t*,u*(.»=f

t* o

It

U(t*,s)b(s,x*(s),u(s»ds

into the above inequality, according to the similar argument of theorem 1, we obtain

EIJr}

max {
is a convex set in X.

Lemma

5. V(t*) A kerR(t*)

0

Proof. We assume that V(t*)

0

(null set) kerR(t*) # 0, A > 0 such

i.e., there exist uo=uO(.)E1and

o

for s E (o,t*)a.e .. Set p(s)=U*(t*,s)g we obtain (b ). By setting x=x(t*,u*(.»

4

and

z=x(t*,u*)+h ( hER(t*)-x* ) we can derive

that xo=x*(t*)+ AO(Z(t*,uo)-z(t*,u*»

E kerR(t*).

By the definition of kerR(t*), there is a positive number r such that x+xo E kerR(t*) for all xEX and

11

x 11
The proof of Remark 3 Under the conditions 1-3 in the Introduction,

convex set and x*(t*) E R(t*),

there exists a solution U(t,s) of (b ) which 1 satisfies the condition 1 and is given by the

k(x+x )+(l-k)x*(t*) E kerR(t*) o for k E[O,l], or

variation of constants formula

fts

U(t,s) = G(t,s) +

kx+x*(t*)+kAo(Z(t*,uo)-z(t*,u*»EkerR(t*). From (C ) (taking u(.)=u (.», we know that o 8 (x(t*,u kA (.»-x*(t*»/k AO o

-(z(t*,uo)-z(t*,u*»

~

0

as k

Hence for sufficiently small k x(t*,ukA

~

O.

~O,

(.»-(x*(t*)+k Ao(Z(t*,u )o o -z(t*,u*») < kr,

that is , x(t*,u

kA

or

(.»

E kerR(t*)

set varying continuously in t. So does RC(t). Hence dis(~(t,uk A (.»,Rc(t»

is a continuous

function in t on [oO,t*l and there is a

(e ) 1

the operator Goo(t,s) satisfies the following equation G00 (t,s)=G(t,s)b x (s)+ftG(t,T)b (T)G 00 (T,s)dT S x (e )

2

and, by denoting G (t,s) = G(t,s)bx(s), it 1 can be expressed as 2 Goo(t,s) = G (t,s) + G (t,s) + ... + 1 1

+

(.», RC(t*» > O. kA But, by the assugption on R(t), R(t) is convex

00

for s£(0,T1 a.e. and each tE(O,T), where

o

dis(x(t*,u

G (t,T)G(T,S)dT

n G (t,s) 1

+ ...

where

fT Gn1 (t,s)= fTfT 0 0 · · · oG1(t,T1)G1(T1,T2)··· G1(Tn_1,s)dT1dT2···dTn_1· (e ) is absolutely convergent according to 3 the norm in the space £(X) for 't/ t and sa. e ..

positive number T such that dis(x(t*-T,U

kA

x (t *-T, u

(.»,Rc(t*_T» o

kA

(.»

>

0, i.e.

Composing the both sides of (e ) with bx(s) 1 we obtain

*

E R ( t -T) .

o

this is a contradiction, since t* is the optimal time.

U(t,s)bx(s)=G(t,s)bx(s)+

f stG

00

(t,T)G(T,S)b (S)dT x

for sE[O,TJ a.e. and each tElO,TJ. It shows The proof of theorem 2. Since V(t*) f1 kerR(t*) = 0 and kerR(t*) # 0, there is a linear

that, like Goo(t,s), U(t,s)bx(s) is also the solution of (e ). From this we can prove the 2

86

Y. Yun-Lo ng

following

JTo

U(t,1)b (1)G(1,S)d1 x

JT G (t,1)G(1,S)d1 : U(t,s) - G(t,s) o

00

for every tE[O,TJ and a.e. sErO,TJ. Thus t G*(1,s)b*(1)U*(t,1)d1. s x But pes) : U*(t*,s)g, this proves (b 6 )· U*(t,s):G*(t,s)+J

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TeoPHfi Orr:rnMaJIbHOro ynpa-

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paMH,

rrapar.eT-

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