Optimization of the Dirichlet problem for gradient differential inclusions

Abstract

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.