Optimal problem of regular cost function for retarded system

OPTIMAL PROBLEM OF REGULAR COST FUNCTION FOR RETARDED SYSTEM

JONG-YEOUL PARK, JIN-MuN JEONG AND YOUNG-CHEL KWUN

ABSTRACT. We study the optimal control problem of system gov- erned by retarded functional differential

, 1°

x (t) = Aox(t)+A1x(t - h)+ ^-ha(s)A2x(t+s)ds+Bou(t)

in Hilbert spaceH. After the fundamental facts of retarded system and the description of condition so called a weak backward unique- ness property are established, the technically important maximal principle and the bang-bang principle are given. its corresponding linear system.

1. Introduction

In this paper we deal with the control problem for retarded func- tional differential equation:

(1.1)

~x(t)^{=Aox(t)+A1X(t - h)+}

I:

+Bou(t),

(1.2) x(O) =gO, x(s) =g1(s), ^sE [-h,0)

in Hilbert space H. We investigate the optimization of control func- tions appearing as the cost function with particular objective.

Received March 18, 1997.

1991 Mathematics Subject Classification: Primary 49B20; Secondary 93C20.

Key words and phrases: optimal control problem, retarded functional differen- tial equation, maximal principle, bang-bang principle.

This work was supported by the Basic Science Research Institute Program, Ministry of Education, 1996, Project No. BSRI-96-1410.

(2.1)

We solve the optimization problem by introducing the structural operatorF and the transposed dual system in the sense of S. Nakagiri

[7].

In section 2, we consider some the regurality and a formular rep- resentation for functional differential equations in Hilbert spaces. We establish a form of the mild solution which is described by the integral equation in terms of fundamental solution using structural operator.

In section 3, we shall give a cost function, which is called the feedback control law for regulator problem and consider results on the existence and uniqueness of optimal control on some admissible set. After consid- ering the relation between the operatorAl and the structural operator F, we will give the condition so called a weak backward uniqueness property. Maximal principle and bang-bang principle for the given technologically important cost function are also derived.

2. Functional differential equation with time delay

Let H be a Hilbert spaces and V be a dense subspace in H. The norm on V(resp. H) will be denoted by 11 . 11 (resp. I· I) and the corresponding scalar products will be denoted by ((', ·))(resp. (', .)).

Assume that the injection of V into H is continuous. The antidual ofV is denoted by V*, and the norm ofV* by 11 . 11*. Identifying H with its antidual we may consider that H is embedded in V*. Hence we have V c H c V* densely and continuously. Ifthe operator Ao is a bounded linear operator from V to V* and generates an analytic semigroup, then it is easily seen that

H = {x EV* :

I

for the timeT >0 where 11 .11* is the norm of the element ofV* . .The realization ofAo^inH which is the restriction ofAo^to

D(Ao)⁼ {u EV : Aou E H}

is also denoted by Ao. Therefore, in terms of the interpolation theory we can see that

(2.2) (V,V*)J._{2 ,}2=H

and hence we can also replace the interpolation space F in the paper G. Blasio, K. Kunisch and A. Sinestrari [2] with the space H. Hence, from now on we derive the same results of [2]. Leta(u, v) be a bounded sesquilinear form defined inV x V satisfying Garding's inequality

Let Aobe the operator associated with a sesquilinear form (Aou, v) = -a(u, v), u, v E V

Then Aogenerates an analytic semigroup in both H and V* and so the equation (1.1) and (2.2) may be considered as an equation in both Hand V*.

Let the operatorsA₁andA₂be a bounded linear operators fromV to V*. The functiona(·) is assume to be a real valued Holder continuous in [-h,0] and the controller operator Bo is a bounded linear operator from some Hilbert space U to H. Under these conditions, from (2.2) and Theorem 3.3 of [2] we can obtain following result.

Let Z denote the product reflexive space H x L²(-h,0;V) with the norm

The adjoint space Z* of Z is identified with the product space H x L2(0, T;V*) via duality pairing

where (.,.) denotes the duality pairing.

PROPOSITION 2.1. Let 9= ^{(9°,91) E} Z andu E L2(0,T; U). Then for each T >0, a solution x of the equation (1.1) and (1.2) belongs to

L2(0,T;V) nW¹,2(0,T;V*) c C([O,T]; H).

where W^l,2(0,T; V*) denotes the SobolevspaceofV*-valued measur- able functions on (0,T) such that itselfandits distributional derivative belongto£2(0,T; V*).

Let x(t; g,u) be the solution of (1.1) and (1.2) with initial value 9= (gO,gl) EZ and controlu E £2(0,T; U). According to S. Nakagiri [7], we define the fundamental solution W(t) for (1.1) and (1.2) by

W(t) 0 = {X(t;(gO,O),O), t~O

g O t < 0

for gO EH. Here, we note that

x(t; (gO,0), 0) = ^G(t)+

Lt

+[Oh a(-T)A2W(r+s)d-T)ds, t ~ ⁰

where G(t) is an analytic semigroup generated by Ao. Since we as- . sume that a(·) is Holder continuous, the fundamental solution ex- ists as seen in [11]. It is also known that W(t) is strongly contin- uous and AoW(t) and dW(t)/dt are strongly continuous except at t = nh, n = 0, 1, 2, ....

For each t > 0, we introduce the structural operator F(·) from H x £2(0,T; V) to H ^X £2(0,T; V*) defined by

Fg = (gO,Flgl),

Flgl = Algl (-h - s)+

l

-h

for 9= (90,f/) E Z. The solution x(t) = x(t;g,u) of (1.1) and (1.2) is represented by

.

1°

it

-h

°

where

Ut(s) = W(t - s - h)Al +^{iSh W(t -} s+r)a(r)A2dr for t ~O.

PROPOSITION 2.2. IfAl : V --+ V* isan isomorphism, then F : Z --+ Z* isan isomorphism.

Proof For f E Z* the element 9E Z satisfyinggO = fO and gl( -h - s) +ISh a(T)Al^{1A2l(T - S)dT= Allfl(s)}

is the unique solution of Fg = f. The integral equation mentioned above is of Volterra type, and so it can be solved by successive approx-

imation method. 0

The following result isobtained from Lemma 5.1 in [8].

LEMMA 2.1. Letf E V(O,T;H), 1:5p:5 ^00. If

it

then f(t) = 0 a.e. 0 :5t :5T.

THEOREM2.1. LetAl be anisomorphism. Then the solutionx(t;g,0) is identically zero on a positivemeasure containingzeroⁱⁿ [-h,T] for T :2:h ifand onlyifgO = 0 andgl== O.

Proof With the change of variable and Fubini's theorem we obtain [Ohllt (S)gl(S)dS

= [OhW(t - s - h)A₁g^l(S)ds

+ [ :([ShW(t - s+T)a(T)A2dT)l(s)ds

= [OhW(t+s){AlX(_h,O)(s)gl(-h - s)

+J:h a(T)A2(T)gl(T - S)dT}ds

= [OhW(t+s)[Flg¹](s)ds.

Thus the mild solutionx(t;g,0) is represented by

Thus, we have that x(O) = W(O)gO = gO = 0 in H. Because that Al is an isomorphism and, we obtain that PI is isomorphism from Proposition2.2. Therefore from Lemma 2.1 x(t;g,0) = 0 if and only if

gO =^{0 andgl} ==O. 0

Let 1= [0,T], T> 0be a finite interval. We introduce the trans- posed system which is exactly same asin S. Nakagiri[8]. Let qo E X*,

qi E £1(1;H). The retarded transposed system in H is defined by·

(2.3)

dyd(t)_t +Aoy(t) +Aiy(t+h)+

1°

+qi(t) =0 a.e. tEl, (2.4)

y(T) =qo, y(s) =0 a.e. ^sE (T, T +h].

Let W*(t) denote the adjoint of W(t). Then as proved inS. Nakagiri [8], the mild solution of(2.3) and (2.4) is defined as follows:

T

y(t) =W*(T - t)(qo) +

1

for t E 1 in the weak sense. The transposed system is used to present a concrete form of the optimality conditions for control optimization problems.

COROLLARY 2.1. The solution y(t) is identicallyzero on a positive measure containingT in [T, T +h] ifand onlyifqo = 0 andqi ⁼ o.

Ifthe equation (2,3) and (2.4) satisfies the result in Corollary 2.1, the equation(2.3) and (2.4) issaid to have a weak backward uniqueness property.

3. Optimality for the regular cost function

In this section, the optimal control problem is to find a control u which minimizes the cost function

T

J(u) = (Gx(T) , X(T))H +

1

Let us assume that there exists no admissible control which satisfies Gx(T;g,u) # 0.

THEOREM 3.1. LetUad be closed convex in L²(0, T;U). Then there exists a unique element u EU^ad such that

(3.1) J(u) = inf J(V).

VEUa.d

Moreover, it is holds the following inequality:

IT

where yet) is a solution of(2.3) and (2.4) for initial condition that yeT) = Gxu(T) and yes) =

°

(3.2)

dyd(t)_t +^A(jy(t)+^Aiy(t+^h)+

1°

+^D(t)xu(t) =

°

(3.3)

yeT) = Gxu(T), yes) =

°

inthe weak sense.

Proof. Let x(t) = x(t;9, 0). Then it holds that J(v) = 1r(v, v)

where

1r(u,v) =(Gxu(T),xv(T))H

+

I

1r(v,v) _~ cllvl1² V E L²(0,T;U).

Therefore in virtue of Theorem 1.1 of Chapter 1 in [6] there exists a unique uE L²(0,T;U) such that (3.1). Ifu is an optimal control (cf.

Theorem 1.3. Chapter 1 in [6]), then

(3.4) l ^{(u)(v - u)} ~ 0 u E Uad,

where .I(u)v means the Frechet derivative of J at u, applied to v. It is easily seen that

x~(t)(v- u)= (v - u,_x~(t))

= xv(t) - xu(t).

Since

l (u)(v - u)= 2(Gxu(T),xv(T) - xu(T)) + 21

T

(D(t)xu(t), xv(t) - xu(t)) + 2(Q(t)u(t), v(t) - u(t))dt, (3.4)-is equivalent to that

I

T . T

1

^o 1

~ o.

Hence

yes) =W*(T - s)Gxu(T)+

iT

Let the admissible set Uad be

Uad = {uE L²(0,T;U): u(t) E W}

o

where W is a bounded subset in U.

COROLLARY 3.1 (MAXIMAL PRINCIPLE). Let W be bounded and Q= o. ^{Ifu be an} optimal solution for J then

max(v,AuIBoq(t)) = (u(t),AUIBoq(t))

vEW

almost everywhere in 0 :::; t leT where q(t) = -yet) and yet) is given byin Theorem 3.1.

Proof. We note that if Uad is bounded then the set of elements u E Uad such that (3.1) is a nonempty, closed and convex set in Uad.

Let t be a Lebesgue point ofu, v E Uad and t < t +€ < ^T. Further, put

( ) {

V, if t < s< t +^€

v€ S =

u(s), otherwise.

Then Substitutingv€ forv in (3.4) and dividing the resulting inequality by ^€, we obtain

11

- (AUIBoq(s),u(s) - v(s))ds ~O.

€ t

Thus by letting^€--7 0,the proof is complete. o

Thus from Theorem 3.1 the result is obtained.

THEOREM 3.2 (BANG-BANG PRINCIPLE). Let Bo^{be one to one}

mapping. Then the optimal control u(t)isa bang-bangcontrol, Le, u(t) satisfies u(t) E 8W for almost allt where 8W denotes the boundary

ofUad.

Proof On account of Corollary 3.1 it is enough to show that Boq(t)

=1= 0 for almost all t. If Boq(t) = 0 on a set e of positive measure containingT, then q(t) =0 for each tEe. By Corollary 2.1, we have

Gxu(T) = 0, which is a contraction. 0

From now on, we consider the case where Uad = L²(0, T;U). Let xu(t) =x(t; g, 0)+J~W(t - s)Bou(s)ds be solution of (1.1) and (1.2).

DefineT E B(H,L2(0,T;H)) and TT E B(L2(0,T;H),H) by

(T4»(t) = I t W(t - s)4>(s)ds,

TT4>= IT W(T - s)4>(s)ds.

Then we can write the cost function as (3.5)

J(u) =(G(x(T; g,0) +TTBou) , (x(T; g,0)+TTBou))H

+(D(x(·;g,O) +TBou),x(·;g,O) +TBou)p(O,T;H)

+(Qu,U)P(O,T;U)"

The adjoint operators T* and TT are given by

(T*4»(t) = iT W*(s - t)4>(s)ds, (T;4»(t) = W*(T - t)4>.

THEOREM 3.3. Let Uad ^{= L2(0, T;U).} Then there exists a unique control u such that (4.1) and

u(t) = _A-¹Boy(t)

foralmostallt, where A = Q+BoT* DTB o+BoT;GT;Bo and where y( t) is a solution of (2.3) and (2.4) for initial condition that y(T) = Gx(T) and y(s) = 0 for sE (T, T +h] substituting qi(t) by Dx(t).

Proof The optimal control for J is unique solution of (3.6)

From (3.5) we have

i (u)v =o.

i (u)v =2(G(x(T; g, 0) +TTBOU) , TTBov»

+2(D(x(·; g,0) +TBou) , TBov)

+2(Qu, v)

=2«Q+_{BoT* DTBo}+BT;GTTBo)u,v)

+2(BoT* Dx(·;g,0)+BoT;Gx(T;g,O),v).

Hence (3.6) is equivalent to that

(Au+BoT* Dx(t; g, 0)+BoT;Gx(T; g, 0), v) = 0

since A-I E B=(O,T;H,U) (see Appendix of [3]). Hence from The definitions ofT and TT it follows that

yet) = T* Dx(t; g,0)+T;Gx(T; g, 0)

= W*(T - t)Gx(T) +iT W*(s - t)Dx(t)ds.

Therefore, the proof is complete.

References

o

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Jong-Yeoul Park

Department of Mathematics Pusan National University Pusan 609-739, Korea Jin-Mun Jeong

Division of Mathematical Sciences Pukyong National University Pusan 608-737, Korea

Young-Chel Kwun

Department of Mathematics Dong-A University

Pusan 604-714, Korea