Bogolyubov's theorem under constraints generated by a controlled second-order evolution system

We prove an analogue of Bogolyubov's theorem with constraints in the form of a controlled second-order evolution system. The main assertion of this theorem deals with relations between the values of an integral functional that is non-convex with respect to control on the solutions of a controlled system with non-convex constraints on the control and the values of the functional convexified with respect to control on the solutions of a controlled system with convexified constraints. This theorem also establishes relations between the solutions of non-convex and convexified controlled systems. We apply the theorem to the problem of minimizing a non-convex integral functional on the solutions of a non-convex controlled system. We consider in detail an example of a non-linear hyperbolic system.