Optimization of the Dirichlet problem for gradient differential inclusions

The paper is devoted to optimization of the gradient differential inclusions (DFIs) on a rectangular area. The discretization method is the main method for solving the proposed boundary value problem. For the transition from discrete to continuous, a specially proven equivalence theorem is provided. To optimize the posed continuous gradient DFIs, a passage to the limit is required in the discrete-approximate problem. Necessary and sufficient conditions of optimality for such problems are derived in the Euler-Lagrange form. The results obtained in terms of the divergence operation of the Euler-Lagrange adjoint inclusion are extended to the multidimensional case. Such results are based on locally adjoint mappings, being related coderivative concept of Mordukhovich.