Optimal control of discrete and differential inclusions with distributed parameters in the gradient form

This paper is devoted to optimization of so-called first-order differential (P _C ) inclusions in the gradient form on a square domain. As a supplementary problem, discrete-approximation problem (P _A ) is considered. In the Euler-Lagrange form, necessary and sufficient conditions are derived for the problems (P _A ) and partial differential inclusions (P _C ), respectively. The results obtained are based on a new concept of locally adjoint mappings. The duality theorems are proved and duality relation is established.