Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions

Abstract

In this paper, we consider 2 × 2 non-cooperative elliptic system involving Laplace operator defined on bounded, continuous and strictly Lipschitz domain of Rn. First we prove the existence and uniqueness for the state of the system under conjugation conditions; then we discuss the existence of the optimal control of boundary type with Neumann conditions, and we find the set of equations and inequalities that characterize it.

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Qamlo, A. and Mohammed, B. (2013) Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions. Intelligent Control and Automation, 4, 280-286. doi: 10.4236/ica.2013.43032.

1. Introduction

So many optimal control problems governed by partial differential equations have been studied as in [1-3].

Systems governed by elliptic, parabolic, and hyperbolic operators have been considered, some of which are of distributed type as in [4-12], while some others are of boundary type as in [13-17].

Boundary control problems for non-cooperative elliptic systems involving Laplace operator have been discussed in [17].

Here, using the theory of [3], we study the boundary control problem for 2 × 2 non-cooperative elliptic systems involving Laplace operator but under conjugation conditions.

Let us consider the following elliptic equations:

(1.1)

the heterogeneous boundary Neumann conditions:

(1.2)

and the conjugation conditions:

(1.3)

(1.4)

where we have the following notations:

Ω is a domain that consists of two open, non-intersecting and strictly Lipschitz domains Ω1 and Ω2 from an n-dimensional real linear space Rn i.e. Ω1, are bounded, continuous, and strictly Lipschitz domains such that

Furthermore, is a boundary of a domain, , and is a boundary of a domain, i = 1, 2.

,

r1 = constant, and n is an ort of an outer normal to G.

Finally,

The model of system (1) is given by:

system (1) is called non-cooperative, since the coefficients took the previous form.

We first prove the existence and uniqueness for the state of system (1), then we formulate the control problem. We also prove the existence and uniqueness of the optimal control of boundary type, and we discuss the necessary and sufficient conditions of the optimality.

2. The Existence and Uniqueness for the State of System (1)

Since, then by Cartesian product we have the following chain [1]:

.

On, we define the following bilinear form:

(2)

The bilinear form (2) is continuous, since:

since the inequalities

are true

[3]. Then we have:

Now, we have the following lemma:

Lemma 1:

The bilinear form (2) is coercive on, that is, there exists l Î R, such that:

Proof:

(3)

which proves the coerciveness condition of the bilinear form (2). Then we have the following theorem:

Theorem 1:

For a given, there exists a unique solution for system (1).

Proof:

Since (3) is hold, then by Lax-Milgram lemma, there exists a unique element

such that

, (4)

where is defined by:

. (5)

The linear form (5) is continuous, since:

Now, let us multiply both sides of first equation of (1.1) by, and the second equation by then integration over W, we have:

,

.

By applying Greens formula:

by sum the two equations, then comparing the summation with (2), (4) and (5) we obtain:

then we deduce (1.2), which completes the proof.

3. Formulation of the Control Problem

The space is the space of controls. For a control, the state

of system (1) is given by the solution of the following systems:

(6.1)

(6.2)

and the conjugation conditions:

(6.3)

(6.4)

Since there exists a generalized solution

to the boundary value problem (6)then such solution is reasonable on of, and

. (7)

The observation equation is given by:

where, namely:

i.e.

(8)

For a given, the cost function is given by

(9)

where

The function is specified on the domain , minimizes the energy functional:

(10)

on, and it is the unique solution in

to the weakly stated problem of finding an element that meets the following integral equation:

. (11)

The control problem then is to find such that, where Uad is a closed convex subset of.

The cost function (9) can be written as (see [1]):

(12)

In this case, the bilinear form and the linear form are expressed as:

(13)

(14)

Now, we prove the continuity of and

on as follows [3]:

Let and be solutions from to problem (11) under f = 0 and g = 0. Then from the bilinear form which is given by (2), we can derive the following inequality:

Since, then:

i.e. the function is continuously dependent on u.

Then the continuity of and L(v) on is proved.

The bilinear form is coercive on

sinceThus:

Then by Lax Milgram lemma, the following theorem is proved. Moreover, it gives the necessary and sufficient conditions of optimality.

Theorem 2:

Assume that (3) holds, there exists a unique optimal control that is closed convex subset of and it is then characterized by the following equations and inequalities:

(15.1)

(15.2)

(15.3)

(15.4)

(16)

together with (6), where

Proof:

The optimal control is characterized by (see [1])

(17)

by (13), and (14):

thus:

this inequality can be written as

(18)

Now, since:, then:

by using Green’s formula, we obtain:

then

(19)

Since the adjoint system takes the form [3]:

(20.1)

(20.2)

(20.3)

and by using (19), system (15) is proved .

From Greens formula the following equations are true:

(21)

(22)

to the both sides of Equation (21), and

to the both sides of Equation (22), then by (15) we obtain:

(23)

and

(24)

Now, we transform (18) by using (15) as follows:

(25)

by (23) and (24), we have:

by using (2):

from (2), and using Green’s formula:

from (6), we obtain:

which proves (16).

Remark:

If the constraints are absent, i.e. when

, then the equality:

follows from condition (16).

Hence

. (26)

4. Conclusions

The main result of the paper contains necessary and sufficient conditions of optimality (of Pontryagin’s type) for 2 ´ 2 elliptic systems under Neumann conjugation conditions involving Laplace operator defined on bounded, continuous and strictly Lipschitz domain of, that give characterization of optimal control.

We can consider boundary control problems for 2 ´ 2 and n ´ n elliptic distributed systems with Dirichlet conjugation boundary conditions. Also we can consider boundary control problems for parabolic and hyperbolic distributed systems with Dirichlet and Neumann conjugation boundary conditions. The ideas mentioned above will be developed in forthcoming papers.

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 systemmany of variations on the above problem are possible to study with the help of Lions formalism.

5. Acknowledgements

The authors would like to express his gratitude to Professor H. M. Serag, Mathematics Department, Faculty of Sciences, AL-Azhar University for suggesting the problem and critically reading the manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

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