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.