Optimal control of Goursat–Darboux-type polyhedral differential inclusions

UDC 517.96, 517.97 We study the problem of optimal control of the Goursat–Darboux-type differential inclusions (DFIs) given by polyhedral set-valued mappings and, for this purpose, pose an auxiliary problem with discrete Goursat–Darboux inclusion. By using the Farkas theorem, we compute locally adjoint mappings (LAMs) and establish necessary and sufficient optimality conditions for the polyhedral discrete Goursat–Darboux inclusions. Further, by the discretization method, for the Goursat–Darboux-type polyhedral DFI, we formulate (by using solely the specific features of its polyhedral property), first for a discrete-approximate problem and then, for a continuous problem, necessary and sufficient optimality conditions in the form of Euler–Lagrange-type polyhedral inclusions.