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 system described 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 and whose output occurs in the right-hand side of the differential equation, in the constraint on the control, and in the functional to be minimized. The constraint on the control is a multivalued map with closed non-convex values and the integrand is a non-convex function of the control. Along with the original problem, we consider the problem of minimizing the integral functional with integrand convexified with respect to the control, on the solutions of the controlled system with 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 relation is an analogue of Bogolyubov's classical theorem in the calculus of variations. We also study the relation between the 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 optimal solution in the relaxed optimization problem.