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In this paper, we consider 2 × 2 non-cooperative elliptic system involving Laplace operator defined on bounded, continuous and strictly Lipschitz domain of R n . 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.

In this paper, some hyperbolic systems involving Schrödinger operator defined on R n have been established. The existence and uniqueness for the state of these systems have been proved, Then the necessary and sufficient conditions of optimality for such systems have been obtained by a set of equations and inequalities.

In this paper, we consider cooperative hyperbolic systems involving Schrödinger operator defined on n R. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.

ABSTRACT. In this paper, we obtain the necessary and sufficient conditions for having the maximum principle and existence of positive solutions for some cooperative systems involving Schrödinger operators defined on unbounded domains. Then, we deduce the existence of solutions for semi-linear systems. Finally we discuss the generalized maximum principle (φ q -positivity) for non cooperative systems.