Locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions

The present paper deals with discrete approximation on a uniform grid of the first boundary value problem (PC)(PC) for differential inclusions of hyperbolic type. In the form of Euler–Lagrange inclusions, necessary and sufficient conditions for optimality are derived for the discrete (PD)(PD) and continuous (PC)(PC) problems on the basis of new concepts of locally adjoint mappings. The results obtained are generalized to the multidimensional case with a second order elliptic operator.