Some time optimal control problems in non-reflexive Banach spaces*

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1990)

Y uwen W ang (Harbin)

Some time optimal control problems in non-reflexive Banach spaces*

Abstract. Some time optimal control problems for distributed parameter systems in non-reflexive Banach spaces are studied. It is shown that for these problems both existence and the maximum principle for the time optimal control are true under some conditions.

Consider the distributed parameter system in a Banach space X dx(t)

(I) —-— = Ax(t) + u(t) for all t > 0, x(0) = x 0 e X , u(t)eU1 a.e., at

where A is the infinitesimal generator of a strongly continuous operator semigroup T(t), t ^ 0, D{A) its domain, and U1 is the unit ball of the Banach space Z = X**/D(A*)1.

X * and X** denote the dual and bidual of X; A is the canonical map from X to X**. For convenience, denote by x*(x) = <x*, x ) the duality pairing between x * e X * and x e X . A* denotes the dual operator, and

DiA*)1 = <x**, x*> = 0 for all x* eD(A*)}.

Let

Wad = (n(-)eL*(Z); u ^ e U^^ a.e.}

where

L“ (Z) = {w( • ): [0, oo)->Z, w*-measurable and

ess sup || m ( 0 II z ^ for any t > 0 }.

0

For any м (* )е^а<1, x(-; и) is said to be a weak solution of the system (I) if x(t; u) e X for a.e. t ^ 0 and for any

) ) is absolutely continuous, and

x{t; и)) = <A*h, x(t; u)} + (u(t), К), x( 0 ; и) = x 0.

at

This research was supported by a National Science Foundation Grant.

2 0 6 Y. W a n g

Let x t eX. Define

W0 = { u ( -) G ^ ad; 3t > 0 such that = x{t; u)}, t(u ) = inf[f > 0 ; x x = x(t; u)}, u (‘ ) e aM0,

t* = mï{t(u); u ( - ) g û& 0} .

t* is called the optimal transition time. The time optimal control problem is to look for U q EWо such that t* = t(u0). u0 is called the time optimal control.

When X, the state space, is reflexive, this problem has been thoroughly investigated (see [1 ,2 , 4, 7]). However, for instance, if the system (I) were to be used as a model for a heat conduction process in a bounded domain Q cz R3 the most “natural” choice for X would be the space C0(Q) of continuous functions that vanish on the boundary dû (if the Dirichlet boundary conditions were used). On the other hand, if (I) were used as a model for a diffusion process in Û, it would be natural to take X = I}(Q). We see that neither C0(Q) nor I}(Q) is reflexive. For X — C0(û), Z = LX(Q), A = к A. Fattorini [4]

proved existence and the maximum principle for the time optimal control. In this paper, we are going to extend these results to non-reflexive Banach spaces X.

I. State trajectory and lemmas. If X is not reflexive, it is not locally weakly compact, and T*(t), t ^ 0, need not be a strongly continuous semigroup. To overcome this difficulty, we introduce the following subspace of X*:

X* = {x*eX*; Lim ||r*(t)x* — x* Ц** = 0}.

t-* о

L

1 [1, 2]. Let Xgx = {x**eX**; <x**, x*> = 0,for all x*eX§}.

Then

--- s --- w*

(1) X§ = D(A*) , X* = D(A*) ; (2) X§ is closed subspace of X*;

(3) X$* = (Xg)* = X**/Xg 1 2 = X**/D(A*)1;

(4) T*(t) = T*(t)\x^ is a strongly continuous semigroup in X*, A% = is the infinitesimal generator of T*(t), t ^ 0.

(5) For any real pGg(A) (the resolvent set), let Вц = pR(p, A); there exists p0 > 0 such that

(i) Il B J ^ C for some C > 0 and all g ^ p0; B^x-^x as oo for all x e X ; B^X <= D(A) for all peg(A);

(ii) B* = pR(g, A*); B*X* cz D(A*) for all real peg(A); B * x * ^ x * for all

x * g X * and B%x* -+ x* g X $ as p -+ c c .

L

2. Let X = AX, x = Ax ( x g X). Then

(1) (X , II • ||x**) is isomorphic to a closed subspace, say (X, || • Ц^**), of X%*\

(2) T**(t)x = AT(t)x for all x g X and t ^ 0.

P ro o f. (1) Let J: X**->X$* = Х**/Х*х be the quotient map. We need only show that J |*, the restriction of J to X, is injective, and there exists C > 0 such that

lx** ^ II -x II у ^ С || x I for all x e X .

Let x lt x 2 e X be such that J x x = J x 2. By the definition of the quotient map, X j - X je lo 1) and hence <x*, Xj — x2> = <x 1 — x2, x*> = 0 for all x*eX$.

For any x*eX*, it follows from Lemma 1 that there exists a net {x*} c= D(A*) such that <x*, x> = Lima<x*, x) for all x e X . Therefore

<x*, x 1 — x2) = Lim<x*, Xj—x2) = 0 a

for all x*eX *, and hence xq = x2.

For any x e l , we have

ЦхЦ*** = sup {<x*, x>; x*eX$} ^ sup {<x*, x>; x * e l* } = ||x||x.

0 11**11 <1 11**11^1

On the other hand, there exists an x * eX * such that ||x*||x* = 1 and IlX | | = <x*, x>.

By Lemma 1, for B = iâ R( h , A) with real /лед(A), there exist ц0 > 0 and C > 0 such that ||B*|| = ||BJ| ^ C for /л ^ /л0 and B*X* c= D(A*) for [лед(А), B * x * ^ x * as /л-> oo. Therefore ||C_ 1 B*x*||x* ^ 1, C~1B*x* e X* and hence

l|x|lx = <x*, x ) = Lim<B*x*, x ) = CLim ( C ~ 1B*x*, x ) ^ C||x||x**-

00 H~* 00 0

L emma 3. For any u(-)etf/ad and t ^ 0, there exists a unique element ytEXf* such that

t

< y t ,

h) =

T<?(t-s)h)ds

о for all heX%\ we write

t

yt =

T**{t — s)u(s)ds.

о

P ro o f. Since T 0 *(t), t ^ 0, is a strongly continuous semigroup, there exists M t > 0 such that

ess sup ||T(s)|| ^ M t.

0 ^s=St

u(-)etftad implies that u(-) is w*-measurable and

^ 1 a.e. and therefore t

Ft(h) = J

T q *(t — s)h)ds for he X$

о

208 Y. W ang

defines a bounded linear functional on X*. Thus there exists a unique element y(eX$* such that Ft(h) = <yf, h) for all heX$.

и

R em ark 1. J T**(t — s)u(s)ds can be defined as ytl—yt2 for tx, t2 > 0.

t2

L emma 4. Let u ( - ) e ^ ad. Then

T t T — t

(1) j T^*(x — s)u(s)ds = T**(t) J T**(T — t — s)u(s)ds,

о о

T t

( 2 ) j Г0**(т —s)w(s)ds = { T0**(t — s ) u ( s + t —f)ds,

r — t

0

for all 0 ^ t < t .

P ro o f. (1) For any И е Х$

т - t T — t

(T**(t) j To*(T — t — s)u(s)ds,hy = f § T<f*(z — t — s)u(s)ds,T<f(t)hy

о о

T — t T — t

= f < m ( s ), T0*( t — s)h}ds = <( J T<?*{T — s)u(s)ds, hy,

о

and ( 1 ) follows. ( 2 ) can be proved in exactly the same way.

L emma 5. I f T*(t)X* <= D(A*) for all t > 0, then for all x* eX * we have

£*T*(t)x* = T<f(t)B*x*, hgq (A), for any t ^ 0, where B* is as in Lemma 1.

P ro o f. For any x *e X * , since T(t)BMx = BMT{t)x for every xeD(A) and any t ^ 0 , we have

(B*T*(t)x*, x > = <x*,T(0£„x> = <x*, B J ( t ) x ) = <T0*(t)B*x*, x>

---s

for all t ^ 0 and all xeD(A). But X = D(A) , and the assertion follows.

If the system (I) has a weak solution x(-; u) for u ( - ) e ^ ad then x(-; u) is called a state trajectory.

T heorem 1. For any the state trajectory x(-; и ) of the system (I) can be expressed as

t

x(t; и) = Т(г)х 0 + Л _1| T**(t — s)u(s)ds о

for t ^ 0 if and only if

t

R{t) = {yt; yt = 1 T**(t — s)u(s)ds, u{ -) E^ ad]

0

is contained in X for all t > 0.

P ro o f. We need only show the sufficiency. Since R(t) с X for t > 0, t

x(t; и) = Г (^х 0 + Л - 1 J T**(t — s)u(s)dsçX о

for all t ^ O and u(‘) e ^ ad. For any heD(A*), we have

— </i, x(t; м)> d

= 4[<^> T{t)x0)} + (h, Л - 1 j T£*{t-s)u(s)ds>]

at о

= j t <h> T (t)xo>+jt i ( u(s)’ T*(t — s)hyds t

= <Л*Ь, T(t)x0> + J <u(s), T q *1 (t — s)Л*/i>ds-|- <u(t), о

= <Л*Л, T(r)xG> + <J T<?*{t — s)u(s)ds, A*hy + (u{t), h>

о

= <Л*й, x(t; u)) + <(n(t), h)

and t

Qi, x(0; n)> = Lim</z, Т(0хо> + Л - 1 f <TG **(t —s)u(s)ds)

»-*o о

t

= </i, T(0)xo) + Lim f

T*(t — s)hyds

*-o о

= <h,'X0).

--- w*

But X* = 1)(Л*) . It follows that x(0; u) — x 0.

R e m ark 2. In view of Theorem 1, we conjecture that R(t) с X for all t > 0 .

Next, we give a sufficient condition for R(t) a X for all t > 0.

T

2. I f T**(t)X** с X /or all t > 0 and T0**( • ) « ( • ) is strongly measurable for every u(-)etf/ad, then R(t) X for all t > 0.

P ro o f. First, consider the quotient map J from X** to X** =

= X^/DI A* )1. It is easy to show that

T**(s)Jx** = JT**(s)x** for all

and s ^ 0.

For any u(-)efflad and every s > 0, u(s)eX$*; hence there exists v(s)eX**

such that u(s) = Jv(s) for each s in [0, t). Therefore T0**(t —s)u(s) = T<?*(t — s)Jv(s) = JT**(t — s)v(s)eX for every s in [0, t).

14 — Commentationes Math. 30.1

210

On the other hand, there exists M t > 0 such that

\\T0**(t-s)u(s)\\K^ ||r 0 **(t-s)|| = \\T0* ( t - s ) \ \ K M t

for all se [0, t]. Since T**(-)u(-) is strongly measurable, it is Bochner integrable over [ 0 , t].

By Lemma 2, (X , II • W is a closed subspace of 2 f**; it follows that jo T<?*(t — s)u(s)dseX.

II. Existence of time optimal control. Let Z = X**. For the system (I), we have

T heorem 3. Let X be a Banach space, and suppose that R(t) c= X for all t > 0 and D(A*) is a separable subspace of X*. I f ф 0 , there exists a time optimal control и0е ^ 0 such that t* = t(u0).

P ro o f. Since t* — inf{t(u); w (-)e ^ 0) and t(u) = inf{r > 0 ; — x(t; u)}, we can choose vne (%0 and tn > 0 such that t{vn)lt* as b->oo and Фп) t(vn)+l/n, x x = x(t„; v„) for n = 1 , 2 , ...

Taking t ^ tn for n = 1 , 2 , . . . , we have

( 2 . 1 )

where

*i = T(tn)x0 + A 1 j T0**(t„-s)v„(s)ds 0

t

= T{tn)x0 + A ~ l J T<f*(tn — s)un(s)ds

и » = Vn(s) for 0 s ^ tn,

0 for tn < s ^ i, and hence un( - ) e ^ 0.

Since D(A*) is separable, X* = D(A*) is also separable. By a result of Dieudonné used by Fattorini [4],

L * (0 , t ; x n = (L 1 (0 ,i; XJ))*.

Since is a bounded set in L*(0, т; X**) and u „ e ^ 0, there exists a subsequence of {un}, again denoted by {un}, and a u 0 e ^ 0 such that un^ u 0 as n-> oo.

It follows from (2.1) and un{s) = 0, tn < s ^ т (n = 1 ,2 ,...), that

tn

<*i, Й) = (AT(tn)x0, h} + J <u„(s), T*(tn — s)h)ds о

t*

= (AT{tn)x0, hy + $<un(s), T*(tn — s)hyds о

tn

+ j<u„(s), T*(tn — s)h}ds

( 2 . 2 )

= <AT{tn)x0, h) + f <u„(s), To(t* — s)h)ds

0

+ J <

„(

) , T<?(tn-s)h-T(?(t*-s)hyds

о

+ j <un(s), T^(tn — s)h}ds for all heX$.

t*

Since T ( t ) , t ^ 0 , and T * ( t ) , t ^ 0, are strongly continuous semigroups, T*(t* — • )/ jg L 1 (0, t*; X*), and there exists a positive number M such that

esssup \\T*(t)h\\ ^ M, ||T*(tn — s)h — T 0 *(t* — s)/i||x* ->0 a.e. as n-*co.

0 <г$т 0

Therefore:

(i) ||Л Т (д х 0 -Л Т (г*)х0||А Г* ^ ||T ( g x o-T (t* )x o||A:->0 as

со.

t* t *

(ii) J T*(t — j.<n 0 (s), T*(t — s)h}ds as со.

о о

(iii) By the Lebesgue dominated convergence theorem, we have t*

|j <nn(s),’T 0 *(t„ — s)h— T*{t* — s)h}ds\^0 as n-> со.

о

fn ^

(iv) |J <u„(s), T*(tn — s)h}ds\ ^ M(tn —1*)->0 as n-*co.

t*

It follows from (2.2) that

t*

<h, Xj> = <x1? fi> = <dT(t*)x0, fi> + J <w 0 (s), T£{t — s)h}ds

о

= <ЛГ( 1 *)х0, fc> + <J T<?*{t*-s)u0{s)ds, h}'

о

t*

= (h, Г(£*)х 0 + Л - 1 J T 0 **(t*-s)w 0 (s)ds>

о

--- w*

for all h£D(A*). Since X* = D(A*) , we obtain t*

Xj = Т(г*)х 0 + Л - 1 J T**(t* — s)u0{s)ds = x(t*; w0)

о

and hence t(w0) = inf(t > 0; x(t; u0) = x x} ^ t*. Once again, by the definition of t*, we have t* = t(u0).

Ex a m p l e

1. Let X = C0(Q), D(A)

k A x e C 0{Q), x\dQ

0}, Л = kA, where A is the Laplace operator, Q c R3. It is easy to show that

X * = D(A*) = L1^ ) and X$* = Lœ(Q).

212

The input space will be Z = L°°(f2).

E xample 2. Let X = EM(Q), the Orlicz subspace, where M (w) is an iV-function [ 6 ]. Suppose that M(u) satisfies the condition A3 so that the complementary AMunction N(v) of M(u) satisfies the condition A2. Let

D(A) = {xe Co(Ü); k A x e E M(Q), x\dn = 0} and A = kA, where A and Q are as above. It can be shown that

X$ = D{A*) = L?N)(Q) and Х Г = L%(Q),

where L%N)(Q) and L*M{Q) are the Orli& spaces generated by N(v) with Luxemburg norm and by M( m ) with Orlicz norm respectively.

X$ = L%N)(Q) is separable, the input space will be L*M(Q).

E xample 3. Let X = /2 (0 ), D(A) = ( x e C q (Q); fczlxeL1^ ) , x\0Q = 0}, A — kA. Then

X% = D(A*) = C 0 (0) and X$* = Wl0(Q)

where S[R 0 (Q) is the space of all regular countably additive measures ц on Q such that \/n\{dn) = 0 (|ju| is the total variation of ц).

III. Maximum principle for the time optimal control. For the system (I), we have

T heorem 4. Suppose that R(t) с: X for all t > 0 and there exists a positive number t0 such that T(t0)X = X. Then:

(i) I f u0 e is the time optimal control, then there exists x* e X* such that (3.1) ^x*, Л -1 J T**( t — s)Ug(s)ds} = Max<(x*, A~1f T0**(z — s)u(s)ds).

о ueWad о

(ii) I f the time optimal control u0 satisfies the condition M 0 (t)ejB 1 a.e., there exists x * e X * such that

(3.2) (T*(T ^— t) x * ,A ~ 1u0(t)y=\\T*(i: — t)x*\\x* a.e.

where x is the optimal transition time, Bx the unit ball of X.

P ro o f, (i) Let

£ЭД={у; у — А ~ г j T**(x — s)u(s)ds, u(-)e<%ad}.

о

It is obvious that O(i) is a convex subset of X. Next, we show that тШ (т) Ф 0.

Since T(t0)X = X , by the Banach inverse operator theorem, T - 1 (t0) is

a bounded linear operator from X into itself.

If t0 ^ t , then for any y e X , taking x = T(t0) xy, we obtain from Lemma 2 у = Л ~1 AT(t0)x = Л - 1 T 0 **(t0)x

= Л - 1 J Г0**(т —s ) |- T 0 **(t 0 + s —т)х Jrfs = Л - 1 { T£*(x — s)u(s)ds

о \ т / о

where

w(s) = T 0 **(t 0 + s — х)х/х, 0 ^ 5 ^ т.

Let М = supo<f<t ||'T(t)||. By Lemma 2, we have Ms)Hn * ^ IMs)lljr = M T (t 0 + 5 -T )x/i||^*

= ||T(t 0 + s-T)x/T||x < M\\xi\x/T ^ M llT ^o)-1!! \\y\\x/x

for 0 < s ^ x. Therefore if ||y||x < t /M ||T(£0)-1 ||, then ||м(011*** < 1 and hence y G int &(т).

If f 0 < t , for any y e X , choose x e X such that y = T(t0)x. It follows from Lemma 2 that

y = T(t0)x = Л - 1 r 0 **(t0)x = Л - 1 J T0**(t0-s)(T<?*(s)x/t0)ds о

= Л~1 ) То**(т-й[То**« + г 0 -т)х А 0] ^ = Л - 1 |Т 0 * * (т - 0 « ( а ^

г —го О

where

,,, fO, 0 < < x ~ t 0, U lTo*(Ç + t0 — x)x/t0, x - t 0 ^ Ç ^ x . We have

N£)llx~ < IlTçf*(Ç — (x — t0))x/t0\\x** ^ M\\x\\x/t0 ^ MWTito)-1 II \\y\\x/t0.

Hence у стШ (т) provided ||y||x < t 0 /M ||T (t0)_1||.

Next, we prove that у 0 = x^ — T(x)x0 edQ{x), where

T

x A — T(x)x0 = Л - 1 j T0**( t — s)u0(s)dseü(x).

о

Suppose not; then, since the semigroup T{t), t ^ 0, is strongly continuous, there exists an rj > 0 and a /? > 1 such that /?(xt — T(t)x0)eint(2(x) for x~t] < t < x. Therefore

T

f3(x1 — T(t)xо) = Л - 1 J T< ?*(x — s)u(s)ds

0

for all t in (1 — 17 , 1 ), and some ue°Uad which depends on t.

214

It follows from Lemma 4 and Remark 2 that

z — t r

(3.3) P(xl — T(t)x0) = Л -1 [ J — s)u(s)ds+ J T**( t — s)u(s)ds]

0 т- f

= A 1[T**(t) J — t — s)u(s)ds+ J T<?*(% — s)u(s)ds]

+ J T^*(x — s)u(s)ds

= Л - 1 j T£*(t — s)

+

— t — 9)u(0)de + u(s +

— t) ds

= A 1J T**(t — s)v(s)ds, where

1 1_t

(3.4) u(s) = - J T(f*(s + % — t — 6)u(6)d0 + u(s + T — t), 0 ^ s ^ t.

t о

Taking M = sup 0 $f^ 2 t l|T 0 *(OII, we get (3.5) - J T**( s + t — t — 6)u(0)d6

t о x**

1 z~{

= sup - j <w(0), T*(s + x — t — 0)hyde.

heX*0 t О

\ \ h \ \ ^ l ^ r , M / t

Choose rj small enough such that 0

r\

(/?

1 )t/M, and define

v(s)/P, 0 ^ s ^ t. Then, by (3.3H3.5), we obtain

K ( s)ll** = ^ Ms)\\x*0

and i

x 1 — T(t)x0 = Л - 1 J T0**(t —s)w 1 (s)ds, о

i.e. m 1 (-) g ^ 0 and = x(t; мД which contradicts the fact that t* is the optimal transition time and u0 the optimal control.

Since

is a convex set with nonempty interior and y 0 = x 1

T(r)x0

g dQ(x), it follows from the separation theorem that there exists an

x*( ^0)627* such that <x*,y> ^ <**, Уо) f°r уеО(т), that is, (3.6) (x*, Л - 1 J T0**(x — s)u(s)ds) ^ <(x*, Л - 1 J T0**( t —s)u 0 (s)ds)

о о

for all u(-)etftad.

(ii) By the assumption, u0(t)eX and ||u 0 (t)||x« 1 a.e. Let У * = {u(-)eWad; u{t)eX and \\u{t)\\x- < 1 a.e.}.

Then $ a d ^ W ad and u 0 ( - ) e # ad.

For any u ( - ) e $ ad, it follows from Lemma 2 that

(x*, Л - 1 j T**(x — s)u(s)dsy = <(x*, Л - 1 j AT(x — s)A~l u(s)dsy

о о

T

= (**, j T(t —S ^ _ 1 u(s)ds]>

0

T

= J <T*( t — s ) x *, Л — *w(s)^ds.

о It follows from (3.6) that

T T

(3.7) J<T*( t — s ) x *, Л ~ 1 u(s)yds ^ J (T*(x — s)x*, Л - 1 и 0 ( 5 )>^$

о о

for all u(-)E$ad.

Since ||T(t)|| is bounded on bounded intervals, it is clear that for u ( - ) e $ ad

<x*, T{x— ■)A~1u(’) ) e L°°(0, т; j R) c= 77(0, т; Я).

Using the standard Lebesgue density arguments, we can deduce from (3.7) that

<T*( t — t)x*, v) ^ (T*(x — t)x*, A ~ 1u0(t)} a.e. on [0, i]

for all v e B 1 (the unit ball in X), and consequently

<T*( t — t)x*, Л - 1 м 0 (г)) = sup< T*{x — t)x*, v)

veB\

• = ||T*( t —r)x*||^* a.e. in [ 0 , т].

T heorem 5. Suppose that R(t) а X for all t > 0 and x0, х хеО(Л),

m 0 ( ') g ^ 0 is file time optimal control and x is the optimal transition time.

(A) For any X e q {A), there exists an x * e X * such that

X

<(x*, (Я7 —Л)Л - 1 j T0**( t —s)u 0 (s)ds)

о

= Max <x*, (Я7 —Л)Л - 1 J T**(x — s)u(s)dsy,

ueWad 0

216

where °Uad = {u(*)e^ad; A 1 jo T0**( t — s)u(s)dseD(A)}.

(B) I f T*(t)X* c= D(A*) and R(t) c: D(A) for ail t > 0, then for any Яе^(Л) there exists an x* eX* such that

(uG(t), (U — A*)T*(x — t)x*y = \\Щ — A*)T*(x — t)x*\\x* a.e.

P ro o f. (A) Since x0, x ± e D(A), we have T( x ) x 0 e D(A), and hence

T

A~i j T0**(i — s)u0(s)ds = Xj — T(x)x0 e D(A).

о Therefore 4lad Ф 0.

Let

T

Q( t ) = {yED{A); y = A ~ l J T 0 **(T-s)u(s)ds, о

It is obvious that Q( t ) is a convex subset of D(T).

Introduce a new norm on D(A) as follows:

l)W ) = max {Il у II x, \\Ay\\x}.

Since A is a closed linear operator, (D(A), | • |С(Л)) is a Banach space. Next, we want to show that £ 2 (т) has nonempty interior in (D(A), | • |D04)).

For any yED(A)

I х 1 T

у — - J T(x — s)y ds + -[xy — J T(x — s)yds]

т о т о

1 т I х d

= - f T(x — s)у ds + - f s— T(x — s)yds

x о To ds

I х I х

= - J T(x — s)yds — j T(x — s)sAyds

т о T 0

T Г

= j T(x — s)((y — sAy)/x)ds = A ~ l j Tf*(x — s)Au{s)ds,

о о

where u(s) = (y — sAy)/x, 0 < s ^ x. If \y\D{A) < т/(1 + t ) it follows from Lemma

2 that

M u(s)||~ ^ \\u(s)\\x ^ -[IMIx + rM yll*] ^ < 1 .

0 X X '

Thus у e int Q( t ).

By an argument similar to that given in Theorem 4, we can deduce that y 0 = — T(x)x0 6 8Q(x). Thus again, by virtue of the separation theorem, there exists an F ( ^ 0) e (D(A), |-| d ( a ))* such that

F ( y ) ^ F ( x 1 — T ( x ) x 0) for all yEÜ(x).

(3.8)

For any Хед(А), XI — A is a bounded linear (one-to-one) operator from (D(A), I • |DU)) to X. Therefore R(X, A) exists and is a bounded linear operator.

It follows that vy-^F(R(X, A)v) is a bounded linear functional on X, and hence there exists an x * e X * such that F(R{X, A)v) = <x*, v} for all veX . For any yeD(A), taking v = (XI —A)y, we may deduce that

(3.9) (x*,(XI-A)y> = F(y)

and hence, for any « e t ad, we have

T

(3.10) (x*, (XI — Л)Л - 1 j T0**( t — s)u(s)dsy

о

T t

= Т(Л - 1 J T0**( t — s)u(s)ds) ^ Т(Л - 1 J T^*(x — s)uQ(s)ds)

о 0

X

= (x*, (Я/ —Л)Л - 1 j T**( t — s)u0(s)dsy.

о

(B) Since K(t) <= D(T) for all t > 0, for any u e ^ ad we have

X

Л - 1 j T0**( t — s)u(s)dseD(X)

о

and hence it follows from (3.10) that

T

(3.11) <(x*, (1/ — Л)Л - 1 j T0**(x — s)u(s)ds)

о

X

^ <(x*, ( // —Л)Л - 1 J Г0**(т — s)u0(s)dsy

o '

for all u E ^ ad.

Let Вд = /rF(^, Л) for all real /хер(Л). Since T*(t)X* с £)(Л*) for any t > 0, we may deduce from Lemmas 1 and 5 that

X

(3.12) <x*, (Я /-Л )Л " 1| T 0 **(T-s)u(s)ds>

0

T

= lim (B*x*, (Я/ — Л)Л — 1 J T**(T — s)u(s)dsy

#*-*■00 о

T

= lim ((XI — A$)B*x*, Л - 1 j Г0**(т —s)u(s)ds)

д-*-оо 0

T

= lim Г0**(т —s)u(s)ds, (Я/-^Л$)В*х*)

о

X

= lim J <Т0**(т — s)n(s), (Я/ — A*)B*x*}ds

fl-ко о

218

T

= lim j T*(x — s)(XI — A^)B*x*}ds

/ 1 - 0 0 о

T

= lim § (u(s), (XI— A$)T<?(x — s)B%x*}ds

д - 0 0 0

X

= lim § (u(s), (XI— A$)B%T(x — s)x*yds

/ x - o o о

T

= lim J <w(s), B*(XI — A*)T*(x — s)x*}ds

/ i - o o о

= lim J<u(s), B*(XI-A*)T*(x-s)x*}ds.

/1-00 0

Since (XI — у 4*)Г*( т — s ) x * e D(A*) s-a.e. in [0, t ], it follows from Lemma

1 that

B%(XI — A*)T*(x — s)x*->(XI — A*)T*(x — s)x* s-a.e. in [0, t ] as oo. Therefore

(3.13) <n(s), B *( XI -A * )T *( x- s) x* )^( u (s) , (XI — A*)T*(x — s)x*) s-a.e. in [ 0 , t ] as ц-*оо. _---

S

Since X = D(A) , for any se [0 , t ] we have

\\(XI-A*)T*(x-s)x*\\x*= sup ( ( X I- A * )T * ( x - s )x * , x )

xeD{A)

IMI^i

and hence we may choose v(s)eD(A) with | |

(

) | |

^ 1 such that (3.14) \\{XI-A*)T*( t - s ) x *\\x* < ((XI — A*)T* (x — s)x*, ф )> + 1.

Note that

ll^^(s)||^r ^ \\v(s)\\x < 1 a.e.

We deduce that Avefflad, and therefore it follows from (3.11) and (3.14) that

X X

(3.15) J ||(XI — A*)T*(x — s)x*||x*ds ^ \ l l + <(XI-A*)T*(x-s)x*, v(s)}]ds

о 0

t t

= x + J <x*, (XI — A)T(x — s)v(s)yds = x + <x*, (XI — A) j T(x — s)v(s)ds}

о о

X

= т + <(х*, (XI — A\ A~1 J T£*(x — s)Av(s)dsy

о

T

^ x + <(x*, (XI — A)A~l J T**(x — s)u0(s)dsy < oo.

о

By Lemma 1, there exist ц0 > 0 and C > 0 such that ||B*|| = ||J3J| ^ C for all fi ^ ц0. Thus, we have

(3.16) K N s), B*(XI — A*)T* (x — s)x*}\

^ ^ C\\(XI-A*)T*(x-s)x*\\x*

for ^ ^ s - a . e . m £ 0 , T^j arid for all fx ^ jx q .

By (3.12), (3.13), (3.16) and (3.15), it follows from the Lebesgue dominated convergence theorem that

T T

<(x*, (XI — A)A~i ^ T**(x — s)u(s)dsy — J<«(s), (XI— A*)T*(x — s)x*}ds

о о

for all ue<%ad. Therefore (3.11) implies that

(3.17) f < m ( s ), (XI — A*) T*(x — s)x*}ds ^ } < m 0( s ), (XI — A*)T*(x — s)x*}ds

о 0

for all u e ^ ad. Thus again, using the standard Lebesgue density argument, we may deduce from (3.17) that

<v, (X I-A * )T * (x -t) x* ) ^ <u0(t), (XI-A*)T*(x-t)x*) for all u e l / j (the unit ball of X), and consequently

<w 0 (t), (XI — A*)T*(x — t)x*y = sup<t>, (XI — A*)T*(x — t)x*}

veU

= \\(XI-A*)T*(x-t)x*\\x<

t-a.e. in [ 0 , т].

R e m a r k 3. For Examples 1-3, we may easily deduce from Theorems 3 and 5 the existence and the maximum principle for the time optimal control.

References

[1] N. U. A h m ed and K. L. T e o , Optimal Control o f Distributed Parameter Systems, North-Holland, New York-Oxford 1981.

[2] N. D u n fo r d and J. T. S c h w a r tz , Linear Operators, Part 1, Wiley, New York 1958.

[3] J. D i e s t e l and J. J. U h l, Vector Measures, Math. Surveys 15, 1977, 121-157.

[4] H. O. F a t t o r i n i, Control in finite time o f differential equations in Banach spaces, Comm. Pure Appl. Math. 19 (1966), 17-34.

[5] —, The time optimal control problems in Banach spaces, Appl. Math. Optim. 1 (1974), 163-183.

[6] M. A. K r a s n o s e l ’s k ii and Ya. B. R u t ic k ii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen 1961.

[7] X. Li and Y. Y a o , Time optimal control o f distributed parameter systems, Sci. Sinica 7 (1980), 619-624 (in Chinese).

DEPARTMENT O F MATHEMATICS

HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY

HARBIN, P. R. CHINA