Optimal Control of Initial-Boundary Value Problems for Hyperbolic-Type Polyhedral Discrete and Differential Inclusions

This paper is devoted to the optimization of the discrete and hyperbolic differential inclusions for polyhedral control problems with initial-boundary conditions. We formulate necessary and sufficient optimality conditions in the form of an Euler-Lagrange inclusion and transversality condition for the stated discrete problem. Then, to obtain optimality conditions for a discrete approximation problem, we approximate the discrete problem using the difference approximations of partial derivatives and grid functions on a uniform grid. The idea of that discretization method is to combine the discrete problem with the differential problem. Thus, the derivation of sufficient optimality conditions for the continuous problem is implemented by passing formally to the limit as the discrete steps tend to zero in the discrete-approximation problem. Finally, we conduct a numerical example to illustrate the efficiency of our results.