OPTIMAL CONTROL OF A SYSTEM GOVERNED BY VARIATIONAL I N EQUALITIES FOR INFINITE ORDER

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Kyungpook Mathematical Journal Volume 31, Number 2, Oecember, 1991

OPTIMAL CONTROL OF A SYSTEM GOVERNED BY VARIATIONAL I N EQUALITIES FOR INFINITE ORDER

EL LIPTIC OPERATOR

1. M. Gali and S. A. EI-Zahaby

In the present paper

,

using the theory of J. L. Lions [12] a previous

res빠

the

existence of optimal control of a system governed by Variational In- equalities in the case of infinite number of variables. The problem of the systems governed by very strongly nonlinear variational inequalities for infinite order elliptic operator is also considered here.

Introd uction

1. M. Gali et al [9] presented a set of inequalities defining an optimal control of a system governed by self-adjoint elliptic operators with an infinite number of variables.

Subsequently

,

J.L. Lions suggested a problem related to this result

,

but in different direction by taking the case of operators of infinite order with finite dimension. Gali has solved this problem

,

the results have been published in [8].

In a previous work of EI-Zahaby and EI-Dessouky [7] presented the optimal control of systems governed by system of strongly nonlinear elliptic operators of infinite order.

In the present paper using the theory of J. L. Lions [12] and J. P. Yvon [14], we discuss the existence of optimal control of system governed by variational inequalities. Our resu1ts are divided into two parts. First

,

we discuss the existence of optimal control of a system governed by variational Key words: Optimal Control, Elliptic Operators of Infinite Order, and with an infinite number of variables

,

Variational Inequalities.

193

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inequalities in the case of infinite number of variables

,

and in the second part

,

we discuss the problem for the system governed by very strongly nonlinear variational inequalities for infinite order elliptic operator.

1. Optimal Control of a System Governed by Variational In- equalities in the Case of (Infinite Number of Variables).

We shall consider some function spaces of infinitely many variables [2].

For this purpose

,

we introduce the infinite product R∞ = R¹X R¹X ...

with elements

x =

(xμ)응 l' Xμ

E R¹

and denote by dg( x) the product of measures (p( X1 )dxd x (p( X2)dx2) x .. defined on the σ-hull of the cylindrical sets in R∞ generated by the finite dimensional Borel sets; g(R∞) = 1

,

where C¹(R1 )

>

p(t)

>

0 is a fixed weight

,

I .

^p(t)dt⁼ ^1.

J R"

We have a Hilbert space of functions of infinitely many variables

L2(R∞)

=

L2(R∞， dg(x)).

In the following

,

we shall use the chain

We(R∞) 드 L

2

(R∞) 드 W-e(R∞)

where We(R∞) are Sobolev spaces

,

which are the completions of the class

Cg:' (R∞ ) of infinitely differe따iable functions of compact support with re- spect to the scalar product

(u

,

v)e = ε (DOIu

, D

^OI

v)L

2

^(R∞).

101 1 s;e

Here

DOI

=

_- δ101 ^I

(8x

d

^0l¹(8^{X 2})01^{2 •} ’ ¹^0'¹⁼ε O'i.

t=1

The differentiation is taken in the sense of generalized function on R∞ . W-e(R∞) are the duals of We(R∞).

Analogous to the above chain

,

^{we have}a chain of the form

따 (R∞) 드 L

2

^{(R∞) 드}

^{W안 (R∞)}

,

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Optimal Control of a System Governed 195

where WJ(R∞) are the subset of all functions Wl(R∞) which vanish on the boundary

r

of R∞，

W~(R∞) =

{u/u

E Wl(R∞)， D^Ot^μ= 0 on

r , I

^a

I

~

R -

1

,

R

>

1}

and W，감 (R∞) are their d uals.

Let A bea second order self-adjoint elliptic operator with an infinite number of variables [1 J takes the form

∞ 1 f)2 , ~←

Ay(x)

= ε ~슨 ;τ ( JPk(Xk)ν (x))

+

q(x) ν (x)

k:ï ,

^/Pk(Xk)

^ox;.

^’

-

^{ε (D싫 )(}00 ^x)

⁺ ^q(x)y(x)

^---^/^、

ⁿ ^”

where

(DkY)(X)

f수=슨 A~ ， (JPk(Xk) ν (x)) ， _、!Pk(Xk)_{8Xk \}_V_{rκ\ ‘ικ/} and

q( x)

is a real valued function from L2(R∞) such that q( x) 즈 v

,

1 즈 v

> O .

We consider the continuous bilinear form:

π(때)

⁼

훌 4∞ (Dkν)(x)(D때)(x)dg(x) ⁺ k∞ ^q(x)y(x) ^4> ^(x)dg(x).

⁽²⁾

The coerciveness on WJ(R∞) of this bilinear form follows directly from lemma 1 of [9J

,

i.e.

π( 4)，4>) 즈

vll 4> 11

²

, 4>

E W~(R∞) (3) Theorem 1.

Assume that

(3)

is satisfied.

Let]{

be a closed convex subset of

WJ(R∞).

Fo

1'

f

is

given in

W-¹(R∞)，

the

1'

e

exists

a unique

soh따on y ε ]{

of the va

^1'

iational inequalities

π (y ，4>

- y)

즈

(1,4>-

ν)

(4)

fo

1' all

4>

E [{.

The proof is analogous to the proof of theorem of Lions-Stampacchia

[13J.

Formulation of Our Control Problem

Let L2(R∞) be the space of control. For a control μ E L2^(R∞)the state of the system ν(μ) E WJ(R∞) is given by the solution of the variational inequalities

π(y ，

4> - y)

~

(1 + ^u , 4> - y)

for all

4>

E ]{

(5)

(4)

where

f

E Wo^-¹^(R∞).

The observation equation is given by z(v)

=

ν (v)

and the cost function is given by

J(V)

= k∞ (y(v)

- Zd)2dg(x)

+ CIIVIlL2(R∞}l C> μ

^E

L2(R∞)

⁽⁶⁾

Under the given considerations

,

we may apply the results on the optimal control of variational inequalities of J. L. Lions [12] or in J. P. Yvon [14]

to obtain the minimizing

J(

v) on Uad

,

where the set of admissible control Uad is a closed convex subset of L2(R∞)

.

Theorem 2. Let μs assume that (3) holds. If the cost function is given by (6) and [{ closed convex subset of WJ(R=)

,

f E W-¹니(매R∞에) 하 ^ψwe hav

(7)η G

>

0 ⁰^1'Uad is bounded

,

then there exists at least one optimal coπ trol μ ， μ E Uad such that

J( μ ):::;J(v) fo1' all υ E Uad . (7)

A

Brief Sketch of the Proof.

By taking a minimizing sequence {싸 ， Yn }， where Yn is selected among the solution of (5) corresponding to 싸

,

it follows from the hypothesis (7) and the coercivity hypothesis (3) that we can extract a subsequence {μni' Yn‘ } such that

Uni • ^μand Yn‘ • y weakly in

w'(n).

Since the injection map of WJ(R∞) into L2(R∞) is compact allows us to pass to the limit in the inequalities and to prove that Y is a solution corresponding to U of (5). To conclude

,

the weak lower semicontim따y of J(v) on L2(R∞ )xWl(R∞) assure us that {μ ， Y} is an optimal control-state palr.

2. Optimal Control of a System Governed by Very Strongly Nonlinear Variational Inequalities for Infinite Order Elliptic Op- erator

In this problem

,

we shall consider the nonlinear elliptic operator of infinite order in the form

Ay+Bν (8)

(5)

where

Aν (x) = ε ε

(

^-1)la^I^S;aa^D2aν^(x);B E L(WoOO(aa

,

2)

,

W-∞ (aa ， 2)) j=o lal=j

(9) is an elliptic operator of infinite order having self-adjoint closures [4

,

5J.

Sj is a decreasing sequence of positive numbers such that ε NoSj

<

∞，

J

where No is the number of indices with

lal ::;

j and

By(x) = ε

(

^_1)^lâÎ^Dâ^ga(x)Pa(D얘 (x) ) lal<M<j

m

’ / --‘

‘ 、

where ga E L¹(0) and Pa : R^N²• R

,

_N2is the number of multi-indices a with

lal ::;

M

<

j

,

M fixed be a continuous function in all μa E R.

Here

,

^W

ü (

^aa

,

²⁾is the set of all functions of W∞ (aa ， 2) which vanish on the boundary

r

of the open bounded subset 0 of Rn and

W∞ {aa ， 2} = {u(x) E Ca(O): ε aallDaull~

<

∞}

lal=O

is an infinite order Sobolev space of periodic function defined on 0

,

W- ∞ {aa

,

2}

is its conjugate and aa

>

0 is a numerical sequence.

To define the system (8) more precisely

,

let us introduce the following hypotheses :

A1 : There exists a constant Co

>

0 independent of j

,

such that for all j E N

,

all x E 0 and all ι ER

,

ε

Aa(x ， ι)ça 즈 Q

ε aalç^,,¹2

A2 : ^Letga 즈

o

^{for all}x E 0

,

ga E L¹(0) and let Pa be a continuous nondecreasing function in μa E RNo with Pa(O) 0 for each a with

lal ::;

M

<

j.

To set our problem we introduce the following continuous bilinear form

a(y

, qy)

= ε ε

((

^-1)l^a¹^{aa S;}^{D2a y(x)}

^, ^qy(X))L

²^(O)

on WOOO(aa

,

2)(0).

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It is well known that the ellipticity condition (A1 ) is sufficient for the coerciveness of a(y

,

<þ) i.e.

,

a(y

,

y) ~ vllyll~ooo(aa ，2) ^l^‘^‘^/、 ¹^lⁱ

n ”

(see [8] and [10]).

From the condition (A2 ) and

(11) ,

we have

π(때= 흘 |웰 칸ao:DO:y(

^{x )D}^O:^{y( x )dx}

+

ε

/ _

^9o:

^(X

)po: (D얘 (x))DO: y(x)dx (12)

lo:lSM<j‘ “

~ const.llyll~암 (aa ,2)

Theorem 3. Under the condition (A1 - A2 ) and (1 잉 ， let J{ be a closed convex subset ofW~{ao:， 2} and f E W- ∞ {ao:， 2} ， there exists a unique solution y E J{ of the variational inequalities

(Ay+By,<þ-y) 즈 (1, <þ - y) for all <þ E J{.

Proof This is a consequence of results on theory monotone operators cf.

Browder

[3

], Lions

[11]

and Lions Stampacchia

[13].

For more details of the proof see [6].

Our Control Problem

Let L2(n) be the space of control. For a control u the state of the system y(u) E Wo^<x>{ao:,2}(n) is given by the solution of the nonlinear variational inequalities of infinite order

(Ay

+

By,

<Þ-

ν) 즈 (1+ μ ， <þ - y) for all <þ E J{

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where

f

E Wo-∞ {ao:， 2}.

The observation equation is given by Z(v)=y(v) and the cost function is given by

J(v)=k llν (v)

^- ^zd^Il^L(o)

+

^CllvIlL(o)

,

^{C> 0}

⁽¹⁴⁾

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where Zd is a given element in L2

(n).

The problem is to find u E Uad :

J(μ)

:::; J(

^V)

,

v E Uad

where the set of admissible controls Uad is a closed convex subset of L2(n).

Under the given consideration

,

we may apply the theorems of J. L.

Lions [12] and J. P. Yvon [14] and.[7

,

8

,

10] to obtain :

Theorem 4. Assμme that the condition (A1 - A2 ) a뼈 (12)

,

K closed convex subset of WOOO{ ac>, 2}, f E W-∞ {ac>， 2}. The cost functioπ being given by

(14)

and 텐 ψ e have either v

> 0

or Uad is bounded

,

then there exists at least one optimal control u E U ad such that

J(u):::; J(v) for all v E Uad.

The proof is analogous to the proof of Theorem 2.

References

(1) Berezanskii, Ju. M., Self-adjointncss of cl/iptic opcrator with an infinitc numbcr of variab/cs, Ukrain, Math. Z., Vol.27 , No.6, 1975, pp.729-742

(2) Berezanskii, Ju. M. and I. M. Gali, Positivc dcβnitc functioη s of inβ nitc/y many variab/cs in a /aycr, Ukrain. Math. Z., Vo1.24, No. 4 (1972), pp. 351-372

(3) Browder, F., 0η thc un랜 cation of thc ca/cu/us of vaπ ationa/ and thc thco 대 of monotonc non/incar opcrations in Banach spaccs, Proc. Nat. Acad. Sci. U.S.A.

56, 419-425, 1966.

(4) Dubinskii, Ju. A., Somc imbcddiπ 9 thcorcms for Sobo/cv spaccs of i1껴 nitc ordcr, Soviet Math. Dokl., Vol. 19 (1978), pp. 1271-1274

(5) , About 0 η c mcthod for so/ving padia/ diffcrcntia/ cquations, Dokl. Akad.

Nauk. SSSR 258 (1981) pp. 780-784

(6) EI-Dessouky, A.T., Vcry strong/y noη /incar variational incqua/itics for infinitc ordcr clliptic opcrator, J. Mathematical Analysis and Applications, to appear.

(7) EI-Zahaby, S.A. and EI-Dessouky, A.T., Optima/ contro/ of systcms govcrncd by systcm of strong/y noπ /incar clliptic opcrators of infinitc ordcr, Proc. of the BAILV Converence, Shanghai, to be held on 1988.

(8) Gali, I. M., Optima/ contro/ of systcm g01Jcrncd by clliptic opcrators of infinitc ordcr, ordinary and padia/ diffcrcntia/ cquations, Proceedings, Dundee, Lecture Notes in Mathematics 1984, pp. 263-272.

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[9] Gali, 1. M. and El-Saify, H.A., Optima/ contro/ of system govemed by a se/f- adjoint e//iptic operator with aη infinite number of variab/es, Proceedings of the International Conference on Functional Differential Systems and Related Topics 11 Poland, Warsaw, May 1981, pp. 126-133.

[10] G외i ， 1. M., El-Saify, H. A. and EI-Zahaby, S. A., Distπbuted contro/ of a sνstem govemed by Diπ ch/et and Neumann prob/em for e//iptic equatioηs of infinite order,

Proceedings of the International Conference on Functional-Differential Systems and Related Topics III, 22-29, May 1983, Blazejewki, Poland.

[11] Lions, J. L., Que/ques methodes de reso/ution des prob/emes a따 /imites nonlin- eaires, Dund, Gauthier- Vil lars, 1969.

[12] Lions, J. L., Optima/ coη tro/ of system govemed by partia/ differentia/ equations,

Springer-Verlag Series, New York, Band 170 (1971)

[13] Lions, J. L. and Stampacchia, G., Variationa/ inequa/ities, Comm. on Pure and Appl. Math. Vol. XX, pp. 439-519 (1967)

[14] Yvon, J .P., Optimal contro/ of systems govemed by variationa/ inequalities, Lec- ture Notes in Computer Science, V Conference on Optim. Techn. Part 1, Springer,

Berlin, Heidelberg, New York, 1973.

MATHEMATICS DEPARTMENT, FACULTY OF SCIENCE, AL-AzHAR UNIVERSITY, NASR CITY, CAIRO, EGYPT.

OPTIMAL CONTROL OF A SYSTEM GOVERNED BY VARIATIONAL I N EQUALITIES FOR INFINITE ORDER