Bogolyubov's theorem for a controlled system related to a variational inequality

We consider the problem of minimizing an integral functional on the solutions of a controlled systemdescribed by a non-linear differential equation in a separable Banach space and a variational inequality.The variational inequality determines a hysteresis operator whose input is a trajectory of the controlled system andwhose output occurs in the right-hand side of the differential equation, in the constraint on the control, and in the functionalto be minimized. The constraint on the control is a multivalued map with closed non-convex values and the integrandis a non-convex function of the control. Along with the original problem, we consider the problem of minimizing theintegral functional with integrand convexified with respect to the control, on the solutions of the controlled systemwith convexified constraints on the control (the relaxed problem).By a solution of the controlled system we mean a triple: the output of the hysteresis operator, the trajectory,and the control. We establish a relation between the minimization problem and the relaxed problem. This relationis an analogue of Bogolyubov's classical theorem in the calculus of variations. We also study the relation betweenthe solutions of the original controlled system and those of the system with convexified constraints on the control.This relation is usually referred to as relaxation. For a finite-dimensional space we prove the existence of an optimalsolution in the relaxed optimization problem.

Bogolyubov's theorem, non-convex integrand, non-convex constraints, relaxation, minimizing sequence