Variational stability of some optimal control problems describing hysteresis effects

The problem of minimization of an integral 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 is considered. The integrand is nonconvex with respect to the control variable. The coefficients of the equations as well as the constraint depend on the phase variables. One of the equations contains the subdifferential of the indicator function of a given set. This equation describes an input-output relation of hysteresis type. We approximate our problem by a set of auxiliary regular problems depending on a parameter. Along with the original and approximating problems, we consider the problems of minimizing the integral functionals with the integrands convexified with respect to the control. In this case, the minimization is over the solution sets of the same ordinary differential systems for which the control constraints have also been convexified. Under sufficiently general assumptions, we prove that the convexified original and approximating problems have optimal solutions, which are the limits of minimizing sequences for the corresponding initial problems. Moreover, we show that the minimum value of the functionals of the convexified problems is a continuous function of the parameter. Usually, this property is referred to as the variational stability of minimization problems.