Nonsmooth dynamical systems: From the existence of solutions to optimal and feedback control

Abstract

In this paper, we investigate a nonlinear and nonsmooth dynamics system (NNDS, for short) involving two multi-valued maps which are a convex subdifferential operator and a generalized subdifferential operator in the sense of Clarke, respectively. Under general assumptions, by using a surjectivity theorem for multi-valued mappings combined with the theory of nonsmooth analysis and arguments on pseudomonotone operators, the existence of a solution to (NNDS) is proved. Then, an optimal control problem governed by (NNDS) is introduced, and a solvability result for the optimal control problem is established. Moreover, we study a nonlinear feedback control system driven by (NNDS) and an u.s.c. multi-valued feedback map, and employ the Kakutani-Ky Fan fixed point theorem to obtain an existence theorem of solutions for the feedback control problem. Finally, we deliver a convergence result in the sense of Kuratowski describing the changes in the set of solutions for the feedback control problem as the initial data x0 is perturbed in Hilbert space H.