Boundary control for cooperative parabolic systems governed by Schrodinger operator Текст научной статьи по специальности «Математика»

DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N 1, 2006 Electronic Journal, reg. N P23275 at 07.03.97

http://www.neeva.ru/journMl e-mail: diff@osipenko.stu.neva.ru

Optimal control

Boundary Control For Cooperative Parabolic Systems Governed By Schrödinger Operator

G. M. BAHAA

Mathematics Department, Faculty of Science, Beni-Suef University, Beni- Suef, Egypt e-mail: Bahaa_gm @hotmail.com

35K20, 49J20, 49K20, 93C20

Abstract

In this paper, we study the existence of solutions for a cooperative parabolic systems governed by Schrodinger operator defined on Rra, then we discuss the optimal control of boundary type for this systems.

Keywords: Cooperative parabolic systems in Rn, Schrodinger operator, Existence of solution, Boundary control, Optimality conditions.

1 INTRODUCTION.

The linear quadratic optimal control problem described by a distributed parameter system has a variety of mechanical and technical sources and applications. The fundamental class of optimal controls and its mathematical approaches can be found in Lions (1971). The necessary and sufficient condition of optimality for systems (n x n systems) governed by different types of partial differential operators defined on spaces of functions of infinitely many variables and spaces of infinite order are discussed in El-Saify & Bahaa (2001,2002a,b,2003,), Ko-tarski (1989,1997),Kotarski & Bahaa( 2005), and Kotarski & El-saify & Bahaa( 2002a,b). Interest in the study of this class of operator is stimulated by problems in quantum field theory. Various optimization problems associated with the optimal control of distributed parameter cooperative systems have been studied by Gali & Serag (1994,1995), Fleckinger (1981,1994) and Fleckinger & Serag (1995).

We consider the following cooperative parabolic systems :

f dyi dt dV2

dt

+ (- -A + q)yl = ayi + bv2 + fi in Rn

+ (- -A + q)V2 = cyi + dv2 + f2 in Rn

Vi = gi as \x\ ^ oo

V2 = g2 as \x\ —> o,

Vi(x, 0) = Vi,o(x) in Rn

V2(x, 0) = V2,o(x) in Rn

(1)

where :

(2)

a, b, c&d are given numbers such that b,c> 0 in this case, we say that the system(1)is cooperative

q(x) is a positive function and tending toroat infinity. (3)

In [14], Gali et al. proved the existence of optimal control for system like (1) with q(x) = 0 and with positive weight function. Also they found the set of inequalities which described the distributed control for systems (1) with q(x) = 0 and defined on bounded domain [13]. The case of semilinear cooperative system with q(x) = 0 is discussed in [12].

In [11] Fleckinger, obtained the necessary and sufficient conditions for having the maximum principle and the existence of positive solutions for cooperative system (1) which are:

ia<X(q), d<X(q) \(X(q) - a)(X(q) - d) > bc,

where X(q) is defined later.

Here, we shall use the same conditions (4) to prove the existence of the state of our system (1); then using the theory of Lions [19], we study the existence of boundary control for system (1). Our model in this problem is the Schrodinger operator.

2 Operator equation.

To prove the existence of the state y = {y\,y2} of system (1), we state briefly some results introduced in [10] concerning the eigenvalue problem:

(-A + q = X(q in Rn 0(x) — 0 as \x\ —> oo, ^ > 0.

The associated variational space is Vq(Rn), the completion of D(Rn), with respect to the norm :

\\y\\q =(/ \ Ay \ 2 + q\y\2dx Since the imbedding of Vq(Rn) into L2(Rn)

is compact. Then the operator (—A + q) considered as an operator in L2(Rn) is positive self-adjoint with compact inverse. Hence its spectrum consists of an infinite sequence of positive eigenvalue tending to infinity; moreover the smallest one which is called the principle eigenvalue denoted by X(q) is simple and is associated with an eigen-function which does not change sign in Rn. It is characterized by:

X(q) i \y\2dx <[ \Ay\2 + q\y\2dx Vy e Vq(Rn). (6)

JRn JRn

Now, to study our system (1) we have the embedding

Vq(Rn) x Vq(Rn) — L2(Rn) x L2(Rn)

is continuous and compact then, we define a bilinear form

a : (Vq(1n))2 x (Vq(1n))2 ^ 1

by

n((yi,y2), (0i, 02)) = 1 I [AyiA^i + qy\^i]dx + - i [Ay2A^2 + qy2fa]dx

b J1" c Jln

- yifadx--y202dx -a yi^idx - y20idx.

J i" c J i" b J i" J i"

(7)

It is easy to check that n is a continuous bilinear form; and then by lax Milgram Lemma, we have the following theorem:

Theorem 2.1 For fi,f2 <E L2(d1n), there exists a unique solution y = {yi,y2} ^ (Vq(1n))2 of system (1) if conditions (4) are satisfied.

Proof We choose m large enough such that a + m > 0 and d + m > 0 and define on Vq (1n) the equivalent norm

I|y|l2m = / [lAy|2 + (m + q)|y|2]dx

J1"

and we write (6) as:

1 f a + mi f

п((Уl,У2), (0i,02)) = t/ [AyiA0i + (q + m)yi0i]dx--j— yi0idx

b J i" b J i"

- I y20idx + 1 I [Ay2A02 + (q + m)y202]dx J1" c J1"

d + m

y202dx - yi02dx.

c JRn J R

Then

n

1 / a +1- ^^ / ^((Vi,V2), (Vi,V2)) = T [\Ayi\2 + (q + m)\vi\2]dx--— \yi\2dx

b J Rn b J Rn

-/ ViV2dx + -/ [\Av2\2 + (q + m)\v2\2]dx

JRn c JRn

d + m f , |2 r

\V2\ dx - ViV2dx.

c J Rn J Rn

By Cauchy Schwarz inequality, we have

1 f a + wi f

n((yi,y2), (VUV2)) >T [| Ayi I 2 + (q + m) | yi | 2]dx--— \y1 | 2dx

b J Rn b J Rn

1 f d + ^^ f + -/ [|Ay2|2 + (q + m)|y2|2]dx--y^dx

c JRn c JR"

- 2( / |yi|2dx) ( i y^dx y JRn J \ JRn

from (5), we deduce

// x / xx^ 1 / 1 a + w Vi ii2 lA d + m . 2

n((yi,y2), (yi,y2)) > ^ I- w^Km + ^ I- x^+w )||V2||q-

2 ,, ......

VI|| || y 2 ||

X + m If (4) holds, then

n((yi,y2), (Vi,V2)) > c(||yi||2 + Wv2||2)

q,m ' qm

2 (TU)n

which prove the coerciveness of the bilinear form n. Then for fi,f2 G L2(Rn), system (1) has a unique solution by Lax Milgram lemma.

3 Formulation of the control problem

The space L2(r) x L2(r) is the space of controls. For a control u = {u^u2} G (L2(r))2, the state y(u) = {yl(u),y2(u)} of the system is given by the solution of

^yi(u)+(-A + q)yi(u) = ayi(u) + by^(u) + fi in

dt dV2(u) dt

+ (—A + q)y2(u) = cyi (u) + dy2(u) + /2 in R

(8)

yi = ui as \x\ ^ 00 y2 = u2 as \x\ ^ 0,

The observation equation is given by z(u) = {zi(u), z2(u)} = y(u) = {yi(u),y2(u)}. For given zd = {zdi,zd2} in (L2(Rn))2; the cost function is given

by:

J (v )=/ (yi (v) - Zdi)2 + (y2 (v) - Zd2)2dx + (Nv,v )(L2(r))2 (9)

JRn

where N G L((L2(r))2, (L2(r))2) is hermitian positive definite operator:

(Nu,u) > ^||u||2i2(1„))2. (10)

The control problem then is to find

u = {ui,u2} G Uad such that

J(u) < J(v) (11)

where Uad is a closed convex subset of(L2(r))2.

Under the given consideration, we may apply the theorem of Lions [19] to obtain the following result:

Theorem 3.1 Assume that (7) and (10) hold. If the cost function is given by (9), then there exists an optimal control u = {ui,u2}; Moreover it is characterized by the following equations and inequalities:

+ (-A + q)pi(u) - ap2(u) - cp2(u) = yi(u) - zu in 1n -^dr + (-A + q)p2(u) - bpi(u) - dp2(u) = y2(u) - Z2d in 1n pi(u) = 0 p2(u) = 0 on r

J ^dVr^i - ui) + dpVr(v2 - u2)dr + (Nu, v - u)(L2(r))2 > 0 Vv G Uad together with (8), where p(u) = {pi(u),p2(u)} is the adjoint state.

Proof

The control u is characterized by

J(u)(v - u) > 0 Vu G Uad

which is equivalent to

(y(u) - Zd, y(v) - y(u))(L2(R"))2 + (Nu, v - u)(L2(r))2 > 0

i.e.,

' 5

(yi(u)-Zid,yi(v)-yi (u))L2(R") + (y2 (u)-Z2d, y2 (v)-y2 (u))l2(1")++(Nu, v-u)(L2(r))2 > 0

(12)

Since (A*P,Y) = (P,AY), where

A(0 = {$i,fa}) ^ A0 = {(-A + q)0i - a0i - b02, (-A + q)02 - c0i -

for 0 e (K(Rn))2-

Then

(P, AY) = (Pi, (-A + q)yi - ayi - by2) + (P2, (-A + q)y2 - cyi - dy2)

= (pi, (-A + q)yi) - a(pi,yi) - b(pi,y2) + (P2,(—A + q)y2) - c(P2,yi)

- d(P2,y2)

= ((-A + q)pi,yi) - a(pi,yi) - c(P2,yi) + ((-A + q)P2,y2) - d(P2,y2)

- b(Pi,y2)

= ((-A + q)pi - api - cp2, yi) + ((-A + q)p2 - bpi - dp2,y2) = (A*P, Y)

where

A*(P = {pi,p2>) ^ {(-A + q)pi - api - cp2, (-A + q)p2 - bpi - dp2}

where A* is the adjoint for A, P is the adjoint state. Then A*P = Y(u) - Zd can be written as

dpi(u) /a \ / n

+ (-A + q)pi - api - cp2 = yi(u) - zu

dt dp2(u)

+ (—A + q)p2 - bpi — dp2 = y2(u) - Z2d

dt

pi(u) = p2(u) = 0.

So (12) is equivalent to

, dpi(u) , . , . , ,, , dp2(u) / A \ 7 7

(--dt--K-A+q )pi-api-cp2,yi(v )-yi(u))+(--—--K-A+q)p2-bpi-dp2,

y2 (v) - y2(u)) + (Nu, v - u)(L2(r))2 > 0

(pi(u), ddt(yi(v)-yi(u))+(-A+q)(yi(v)-yi(u))L2(K-)-(^,yi(v) yi(u))L2(r)+ d

(pi(u),—A (yi(v ) - yi(u))L2(r) - a(pi(u),yi(v ) - yi(u)) - b(pi(u),y2 (v) - y2(u)) +

(p2(u),dt (y2(v )-y2(u)) + (-A+q)(y2(v)-y2(u))L2(M")-( ^ ,y2(v)-y2(u))L2(Y) d

+ (p2(u),dVA(y2(v) - y2(u))L2(r) - c(p2(u),yi(v) - yi(u))L2(R") - d(p2(u), y2(v) - y2(u))l2(R") + (Nu, v - u)(L2(r))2 > 0

From (8), we obtain (pi(u),a(yi(v) - yi(u)) + %2(v) - V2(u)) + /1 - /1 - a(yi(v) - yi(u)))L2{Rn) dd^p ()

+ ( ^ , v1 - u1)L2(r) + (0, ^ (yi(v) - yi(u))L2(r) - C(P2(u), yi (v) - yi (u))L2 (Rn) (P2(u),c(yi (v) - yi(u)) + d(y2(v) - y2(u)) + f2 - f2 - c(yi - yi (u)))L2(Rn)

+ ( QyA , V2 - u2)L2(r) + (0, dVA (y2 (v) - y2(u))L2(r) - d(p (u) , y2 (v) - y2(u)^(Rn)

+ (Nu, v - u)(L2(r))2 > 0.

Then we have

/5p1(u) . .3p2(u) , /7.r .

( ^ , vi - ui)L2(r) + ( dVA , v2 - u2)L2(r) + (Nu, v - u)(L2(r))2 > 0.

i.e.,

/(ddVr(vi-ui) + dfVAul(v2-u2))dr+(Nu,v-u)(L2(r))2 > 0 Vu G Uad,v G

Which completes the proof of the theorem.

Remark 3.2 To study the optimal control for the scalar case

dy + (-A + q)y = ay + f in Rn y (x) = g in r,

we define a bilinear form n : Vq (Rn) x Vq (Rn) ^ R by

n(y,$)= (VyV^ + qy^)dx - a y^dx

Rn Rn

As in theorem (1), we can prove n is coercive if a < X(q) and then there exists a unique solution of (13) for f G L2(Rn). Therefore, the state of the system is given by the solution of:

dy

d*; + (-A + q)y(u) = ay(u) + f + u in Rn y(u) = u in r,

where u is given in the space L2(r) of controls. For given zd in L2(Rn), the cost function is given by

J(v) = |y(v) - zd\2dx + (Nv)vdr JRn Jr

where N is a given hermitian positive definite operator. Then we have the following characterization of optimal control for this system :

together with (14), where p(u) is the adjoint state.

Remark 3.3 Also it is evident that by modifying:

- the boundary conditions,

- the nature of the control (distributed, boundary),

- the nature of the observation,

- the initial differential system,

an infinity of variations on the above problem are possible to study.

Acknowledgment The author is grateful for the referees of Differential Equation and Control Processes Electronic Journal for their fruitful comments and invaluable suggestions.

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[2] Bahaa, G. M.(2005)" Time-Optimal control problem for parabolic equations with control constraints and infinite number of variables," IMA Journal of Mathematical Control and Information, 22, 364-375.

[3] Bahaa, G. M.(2005) " Time-Optimal Control Problem For Infinite Order Parabolic Equations With Control Constraints," Differential Equations and Control Processes, 4, 64-81, Electronic Journal, http://www.neva.ru/journal.

[4] Bahaa, G. M. (2006)" Optimal control for cooperative parabolic systems governed by Schrodinger operator with control constraints." IMA Journal of Mathematical Control and Information, 23, 4, 1-12.

dp(u)

-^ + (-A + q)p(u) - ap(u) = yi(u) - zd in Rn

p(u) = 0 in r,

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