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 R. 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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So many optimal control problems governed by partial differential equations have been studied as in [1][2][3].
Boundary control problems for non-cooperative n n  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: the heterogeneous boundary Neumann conditions: and the conjugation conditions: 1 2 0, 0 , , , where we have the following notations: Ω is a domain that consists of two open, non-intersecting and strictly Lipschitz domains Ω 1 and Ω 2 from an n-dimensional real linear space R n i.e.Ω 1 , 2 R   are bounded, continuous, and strictly Lipschitz domains such that , and is a boundary of a domain  , i = 1, 2.
In addition, and  is an ort of an outer normal to .

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Finally, , The model of system (1) is given by: 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.

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 The bilinear form (2) is continuous, since: and are true [3].Then we have: , is The bilinear form on H  , that is, there exists   R, such Now, we have the following lemma: Lemma 1: (2) is coercive that: hence which proves the coerciveness c form (2). Then we have the following theorem: where   L is defined by:    The linear form (5) is continuo ( us, since: since the inequalities and are true 3 , then: Now, let us multiply both sides of first equation of (1.1) by , and the second e n by quatio  then integration over , we have: . By applying Green s formula:  by sum the two equations, then comparing the summatio n: 4) and ( 5) we obtai is the space of controls.For a con- 1) is githe following systems:

Formulation of the Contro
The sp , y u ven by the solution of and the conjugation conditions: Since there exists a generalized solution  , and ven by: (7) The observation equation is gi , namely: For a given    , minimizes the energy functional: H  , and it is the unique solution in 2 to the weakly stated problem of finding an el

   
The co is to find The cost function ( 9) can be written as (see [1]):

 
, where U ad is a subset of In this case, th Now, we 2 thus, we have: .
L  and it is then haracterized by the following equations and inequalities: c together with (6), where by (13), and ( 14): , Now, since: A p y p A y  , then: by using Green's formula, we obtain: Since the adjoint system takes the form [3]: and by using (19), system (15) is pr   (20.3) oved .Green ' s formula the following equations are to the both sides of Equation ( 21), and to the both sides of Equation ( 22), then by ( 15) we obtain: and Now, we transform (18) by using (15) as follows: by ( 23) and (24), we have: from (2), and using Green's

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 n R , 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, ontrol (distributed, boundary),  the nature of the observation,  the initial differential system, many of variations on the above problem are ossible to study with the help of Lions formalism.[2] J. L. Lions, "Some Methods in the Mathematical Analysis of Systems and their Control," Science Press, Beijing, 1981.

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Then by Lax Milgram lemma, the following theorem is proved.Moreover, it gives the necessary and sufficient imalit there exists a unique optima conditions of opt y.Theorem 2: Assume that (3) holds, l control If the constraints are absent, i.e. when M. Bahaa, "Optimal Control for Cooperative Parabolic Systems Governed by Schrodinger Operator with Control Constraints," IMA Journal of Mathematical Control and Information, Vol. 24, No. 1, 2007, pp.1-12.doi:10.1093/imamci/dnl001[5] H. A. El-Saify, H. M. Serag and M. A. Shehata, "Time Optimal Control Problem for Cooperative Hyperbolic Systems Involving the Laplace Operator," Journal of 15 Dynamical and Control Systems, Vol.
[6] I. M. Gali and H. M. Serag, "Optimal Control of C erative Elliptic Systems Defined on R n ," Journal o