Optimal control of hyperbolic type discrete and differential inclusions described by the Laplace

The paper is devoted to the optimization of a first mixed initial-boundary value problem for hyperbolic differential inclusions (DFIs) with Laplace operator. For this, an auxiliary problem with a hyperbolic discrete inclusion is defined and, using locally conjugate mappings, necessary and sufficient optimality conditions for hyperbolic discrete inclusions are proved. Then, using the method of discretization of hyperbolic DFIs and the already obtained optimality conditions for discrete inclusions, the optimality conditions for the discrete approximate problem are formulated in the form of the Euler-Lagrange type inclusion. Thus, using specially proved equivalence theorems, which are the only tool for constructing Euler-Lagrangian inclusions, we establish sufficient optimality conditions for hyperbolic DFIs. Further, the way of extending the obtained results to the multidimensional case is indicated. To demonstrate the above approach, some linear problems and polyhedral optimization with hyperbolic DFIs are investigated.