BOUNDARY CONTROL FOR COOPERATIVE ELLIPTIC SYSTEMS GOVERNED BY SCHRODINGER OPERATOR Текст научной статьи по специальности «Математика»

DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N4, 2006 Electronic Journal, reg. N P23275 at 07.03.97

http://www. neva. ru/journal e-mail: jodiff@mail.ru

Ordinary differential equations

Boundary Control For Cooperative Elliptic 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 elliptic systems governed by Schrodinger operator defined on Rn, then we discuss the optimal control of boundary type for these systems.

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

1 Introduction

We consider the following cooperative elliptic system :

' (-A + q)yi (-A + q)y2

yi

y2

where :

ayi + by2 + f1 in Rn cyi + dy2 + ¡2 in Rn g1 as |x| ^ oo g2 as |x| ^ o,

(i)

a, b, c and 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 to to at infinity. (3)

In [22], Gali et al. proved the existence of optimal control for system like (S) with q(x) = 0 and with positive weight function. Also they found the set of inequalities which described the distributed control for systems (S) with q(x) = 0 and defined on bounded domain [21]. The case of semilinear cooperative system with q(x) = 0 is discussed in [17].

In [16] Fleckinger, obtained the necessary and sufficient conditions for having the maximum principle and the existence of positive solutions for cooperative system (1) which are:

a<A(q^ d < A(q) (A(q) - a)(A(q) - d) > bc,

where A(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 [30], we study the existence of boundary control for system (1). Our model in the problem is Schrodinger operator.

2 Operator equation.

To prove the existence of the state y = {yi,y2} of system (1), we state briefly some results introduced in [15] concerning the eigenvalue problem

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

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

|y||q = / AI2 + q|y|2dx

'Rn

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 A(q) is simple and is associated with an eigen-function which does not change sign in Rn. It is characterized by:

A(q) / |y|2dx </ |Ay|2 + q|y|2dx Vy G 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

n : (Vq(Rn))2 x (Vq(Rn))2 — R

by

n((yi,y2), (01, 02)) = 1 I [AyiA0i + qyi0i]dx + - i [Ay2A02 + qy202]dx

b JRn c JRn

yi02dx--y202dx - ^ yi0idx - y20idx.

'Rn c J Rn b J Rn JRn

(7)

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

2

Theorem 2.1 For /i,/2 G L2(Rn), there exists a unique solution y = {yi?y2} £ (Vq(Rn))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(Rn) the equivalent norm

I|y|l2,m = / [|Ay|2 + (m + q)|y|2]dx

JR"

and we write (7) as:

1 f a + mi f n((yi,y2), (0i,02))^/ [AyiA^i + (q + m)yi0i]dx--— yi0idx

b JR" b JR"

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

JR" c JR"

d + m

--y202dx - yi02dx.

c Jr" JR"

Then

1 f a + mi f

n((yi,y2), (yi,y2)) = T/ [|Ayi|2 + (q + m)|yi|2]dx--— |yi|2dx

b J R" b «/Rn

f ymdx + - I [|Ay2|2 + (q + m)|y2|2]dx

JR" c JR"

d + m f , 2 r

|y21 dx - yiy2dx.

c Jr" JR"

By Cauchy Schwartz inequality, we have

1 / a + mi f

n((yi,y2), (yi,y2)) [|Ayil2 + (q + m)|yi|2]dx--— |yi|2dx

b J R" b J R"

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

c J R" c JR"

1 2

22

- 2( / |yi|2dx |y2|2dx

'R"

from (6), we deduce

1/ a + m V, ll2 1/ d + m n((yi,y2), (yi,y2)) >T 1 - w , , _ I ||yi || 2,m + " 1 -

A(q)+ m/l|yillq'm c\ A(q) + m ' llq'm

2

A + m

q,m || y 2 || q,m•

2

If (5) holds, then

n ((yi, y2) , (yi,y2)) > C (||yi112,m + ||y2 112,m) (8)

which prove the coerciveness of the bilinear form n. Then for f1,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 = {u1,u2} G (L2(r)2, the state y(u) = {y1(u),y2(u)} of the system is given by the solution of:

' (—A + q)yi(u) = ayi(u) + by2(u) + / in Rn

(-A + q)y2(u) = cyi(u) + dy2(u) + /2 in Rn

\ 11 (9)

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

(Nu,u) > n||u||2L2(Rn))2 . (11)

The control problem then is to find

u = {ui,u2} G Uad such that J(u) < J(v)

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

Under the given consideration, we may apply the Theorem 2.4 of Lions [30] to obtain the following result:

Theorem 3.1 Assume that (8) and (11) hold. If the cost function is given by (10), then there exists an optimal control u = {ui,u2}; Moreover it is charac-

terized by the following equations and inequalities:

(—A + q)pi(u) - ap2 (u) - cp2(u) = yi(u) - zid in Rn (—A + q)p2(u) - bpi(u) - dp2(u) = y2(u) - Z2d in Rn p1(u) = 0 p2(u) = 0 on r

/ ^(vi - ui) + ddVr(v2 - U2)dr + (Nu, v - uW))2 > 0 Vv G Uad, together with (9), 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(Mn)2 + (Nu, v - u)L2(r)2 > 0

i.e.,

(yi(u)-Zid,yi(v)-yi (u))L2(Mn)2 + (y2(u)-Z2d,y2(v)-y2(u))L2(Mn)2 + (Nu,v-u)L2(r)2 > 0

(12)

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

A(0 = {0i, 02}) ^ A0 = {(-A + q)0i - a0i - b02, (-A + q)02 - c0i - d02}

for 0 G (V'(Rn))2.

Then

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

= (Pb (-A + q)yi) - a(Pi,yi) - b(Pi,y2) + (-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) — can be written as

(—A + q )pi — opi — cp2 = yi(u) — zid (—A + q)p2 — bpi — dp2 = y2 (u) — Z2d pi(u) = P2(u) = 0.

So (12) is equivalent to

(( —A + q)pi — opi — cp2,yi(v) — yi(u)) + (( —A + q)p2 — bpi — dp2,y2(v) — y2(u))

+ (Nu, v — u)(L2(r))2 > 0

dpi (u)

(pi^ (—A + q)(yi(v) - yi(u))L2(Rn) - ( ^ ,yi(v) - yi(u))L2(r) + (pi(u) d

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

d (u)

(P2(u), ( —A + q)(y2(v) — y2(u))L2(Rn) — (,y2(v) — y2(u))L2(r) + (P2(u), d

(y2(v )—y2(u)))L2(r)—C(p2(u),yi(v)—yi(u))L2(Rn)—d(p2(u),y2(v )— ^2(u))L2(Rn)

dvA

+ (Nu, v — u)(L2(r))2 > 0.

From (9), we obtain

(pi(u),a(yi(v) — yi(u)) + %2(v) — y2(u)) + fi — fi — a(yi(v) — yi(u)))L2(Rn) +

( dz/A ,vi — ui)^2(r) + (0,d^(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(Mn) + p ( u)

( , v2 — u2)£2(r) + (0, ¿^(y2(v) — y2(u))L2(r) — d(p2(u), ^2(v) — ^2 (u))L2(Rn) +

(Nu, v — u)(L2(r))2 > 0.

Then we have

,dpi(u) dp2(u) . .

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

i.e.,

J (d|M(vi—ui) + ddVu)(v2—u2))dr+(Nu,v —u)(L2(r))2 > 0 Vu G Uad,v G Uad. Which completes the proof of the theorem.

Remark 3.2 To study the optimal control for the scalar case

(-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, 0) = I (VyV0 + qy0)dx - a / y0dx

Jr" JR"

As in theorem (1), we can prove n is coercive if a < A(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

(-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 JR" Jr

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

(-A + q)p(u) - ap(u) = yi(u) - zd in Rn p(u) = 0 in r,

/rd|^(v - u)dr + (Nu,v - u)L2(r) > 0, V v G Uad together with (14), where p(u) is the adjoint state.

Acknowledgement. The author is grateful for the reviewers of the Electronic Journal of Differential Equations for their fruitful comments and invaluable suggestions.

REFERENCES

• [1] Adams, R. A.," Sobolev Spaces," Academic Press, New York, (1975).

• [2] Bahaa, G. M.," Quadratic Pareto optimal control of parabolic equation with state-control constraints and an infinite number of variables." IMA Journal of Mathematical Control and Information, 20, 2, (2003), 167-178.

• [3] Bergounioux, M., " Optimal control of problems governed by abstract elliptic variational inequalities with state constraints." SIAM J. Control Optimization, 36, 1, (1998), 273-289.

• [4] Bergounioux, M., and Tiba, D., " Optimal control for the obstacle problem with state constraints, ESAIM : Proceedings, 4, (1998),7-19.

• [5] Bonnans, F., Casas, E., "A boundary Pontryagin's principle for the optimal control of state-constrained elliptic equations," Internat. Ser. Numer. Math., 107, (1992), 241-249.

• [6] Casas, E.," Control of an elliptic problem with pointwise state constraints." SIAM J. Control and Optimization, 24, 6, (1986), 1309-1318.

• [7] Casas, E.," Boundary Control of semilinear elliptic equations with pointwise state constraints." SIAM J. Control and Optimization, 31, 4, (1993), 993-1006.

• [8] Casas, E.," Boundary control problems for quasi-linear elliptic equations: A Pontryagin's principle." Applied Mathematics and Optimization, 33, (1996), 265-291.

• [9] El-Saify, H. A., " Boundary control problem with an infinite number of variables." IJMMS , 28, 1, (2001), 57-62.

• [10] El-Saify H. A., and Bahaa, G. M., "Optimal control for n x n systems of hyperbolic types." Revista de Mathematica Aplicadas, 22, 1&2, (2001), 41-58.

• [11] El-Saify, H. A. and Bahaa, G. M.," Optimal control for n x n hyperbolic systems involving operators of infinite order." Mathematica Slovaca, 52, 4, (2002), 409-424.

• [12] El-Saify, H. A. and Bahaa, G. M.,"Optimal control for n x n coupled systems governed by Petrowsky type equations with control-constrained and infinite number of variables" Mathematica Slovaca, 53, (2003), 291-311.

• [13] El-Saify, H. A., and Bahaa, G. M.," Boundary control for n x n systems of hyperbolic types involving infinite order operators." Accepted for oral in the second International Conference of Mathematics, Islamic University , Gaza, Palastin, Editor. M. S. Al-Atrash, 26-28 Augusts. (2002).

• [14] El-Saify, H. A., Serag, H. M, and Bahaa, G. M.," On optimal control for nx n elliptic system involving operators with an infinite number of variables." Advances in Modelling & Analysis, 37, 4, (2000), 47-61.

• [15] Fleckinger, J.," Estimates of the number of eigenvalues for an operator of Schrodinger type." Proceedings of the Royal Society of Edinburg, 89A, (1981), 355-361.

• [16] Fleckinger, J.," Method of sub-super solutions for some elliptic systems defined on Rn." Preprint UMR MIP, Universite Toulouse, 3, (1994).

• [17] Fleckinger, J., and Serag, H.," Semilinear cooperative elliptic systems on Rn."Rendiconti Di Mathematica Seri. VII, 15 , Roma (1995), 89-108.

• [18] Gali, I. M., and El-Saify, H. A., " Optimal control of a system governed by hyperbolic operator with an infinite number of variables." JMAA, 85, 1, (1982), 24-30.

• [19] Gali, I. M, and El-Saify, H.A.," Distributed control of a system governed by Dirichlet and Neumann problems for a self-adjoint elliptic operator with an infinite number of variables." Journal of Optimization Theory and Applications 39, 2, (1983), 293-298.

• [20] Gali, I. M. and El-Saify, H. A., " Control of system governed by infinite order equation of hyperbolic type." Proceeding of the international conference on "Functional-Differential systems and related topics". III, Poland, (1983), 99-103.

• [21] Gali, I. M. and Serag, H., " Distributed control of cooperative elliptic systems." Accepted for presentation at the UAB-Georgia Tech International Conference on Differential Equations and Mathematical Physics, Birmingham, Alabama, USA, March 13-19, (1994)

• [22] Gali, I. M. and Serag, H.," Optimal control of cooperative elliptic systems on Rn." Journal of Egyptian Mathematics Society, 13, (1995), 33-39.

• [23] Gali, I. M., El-Saify, H. A. and El-Zahaby, S. A., "Distributed Control of a system governed by Dirichlet and Neumann Problems for Elliptic equations of infinite order." Proceeding of the international conference on "Functional -Differential systems and related topics III, Poland," (1983), 83-87.

• [24] Kotarski, W.," Some problems of optimal and Pareto optimal control for distributed parameter systems." Reports of Silesian University, No.1668, Katowice, Poland, (1997), 1-93.

• [25] Kotarski, W.," Optimal control of a system governed by Petrowsky type equation with an infinite number of variables." Acta Univ. Palack. Olomuuc. Fac. Rerum Natur. Math. 35, (1996), 73-82.

• [26] Kotarski, W.," Optimal control of a system governed by a parabolic equation with an infinite number of variables." Journal of Optimization Theory and Applications, 60, (1989), 33-41.

• [27] Kotarski, W., El-Saify, H. A. and Bahaa, G. M.,"Optimal control of parabolic equation with an infinite number of variables for nonstandard functional and time delay." IMA Journal of Mathematical Control and Information, 19, 4, (2002), 461-476.

• [28] Kotarski, W., El-Saify, H. A. and Bahaa, G. M.," Optimal control problem for a hyperbolic system with mixed control-state constraints involving operator of infinite order." International Journal of Pure and Applied Mathematics, 1, 3, (2002), 241-254.

• [29] Kowalewski, A. and Kotarski, W.," On application of Milutin-Dubovicki's theorem to an optimal control problem for systems described by partial differential equations of hyperbolic type with time delay." Systems Sci. 7, 1, (1981), 55-74.

• [30] Lions, J. L., " Optimal Control of Systems Governed by Partial Differential Equations", Springer-Verlag, Band170, (1971).

• [31]Troltzsch, F." Optimality Conditions for Parabolic Control Problems and Applications." Teubner-Texte zur Mathematik, Band 62, Leipzig, (1984).

• [32] Walczak, S." One some control problems." Acta Universitatis Lodziensis. Folia Mathematica, 1, (1984), 187-196.

• [33] Werner, J. "Optimization Theory and Applications." Viewag, Advanced Lectures In Mathematics, Braunschweig, Wiesbaden, (1984).