Optimal control of hereditary differential inclusions

Deals with optimal control problems for dynamical systems governed by constrained hereditary differential inclusions. We pay main attention to hereditary systems of the so-called neutral type, which contain time-delays in velocity variables along with delays in state coordinates. To the best of our knowledge, such problems for hereditary differential inclusions have not been considered in the literature. We construct well-posed discrete approximations of the continuous-time problem by a sequence of discrete-time problems, which can be treated by means of powerful tools of modern variational analysis and generalized differentiation. In this way we justify stability of discrete approximations and derive necessary optimality conditions for hereditary inclusions with both convex and nonconvex velocity sets in extended Euler-Lagrange and Hamiltonian forms; the latter one just for the convex case. Our results are expressed in terms of nonconvex sets of generalized normals, subgradients, and coderivatives for nonsmooth sets, functions, and set-valued mappings.