Approximation and optimization of Darboux type differential inclusions with set-valued boundary conditions

The present paper is devoted to an optimal control problem given by hyperbolic discrete (P _D) and differential inclusions (P _C) of generalized Darboux type and ordinary discrete inclusions. The results are extended to non-convex problems. An approach concerning necessary and sufficient conditions for optimality is proposed. In order to formulate sufficient conditions of optimality for problem (P _C) the approximation method is used. Formulation of these conditions is based on locally adjoint mappings. Moreover for construction of adjoint partial differential inclusions the equivalence theorems of locally adjoint mappings are proved. One example with homogeneous boundary conditions is considered.