Bogolyubov-type theorem with constraints induced by a control system with hysteresis effect

We consider the problem of minimization of an integral functional with nonconvex with respect to the control integrand. We minimize our functional over the solution set of a control system described by two ordinary differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values. The coefficients of the equations and the constraint depend on the phase variables. One of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable. The equation containing the subdifferential describes an input-output relation of hysteresis type. Along with the original problem, we also consider the problem of minimizing the integral functional with the convexified with respect to the control integrand over the solution set of the same system with the convexified control constraint. Under sufficiently general assumptions, we prove that this relaxed problem has an optimal solution, which is the limit of a minimizing sequence of the original problem. The convergence takes place simultaneously with respect to the trajectory, the control and the functional and is uniform in appropriate topologies.