Optimal control problems with states of bounded variation and hysteresis

This paper is concerned with an optimal control problem for measure-driven differential equations with rate independent hysteresis. The latter is modeled by an evolution variational inequality equivalent to the action of the play operator generalized to the case of discontinuous inputs of bounded variation. The control problem we consider is, a priori, singular in the sense that it does not have, in general, optimal solutions in the class of absolutely continuous states and Lebesgue measurable controls. Hence, we extend our problem and consider it in the class of the so-called impulsive optimal control problems. We introduce the notion of a solution for the impulsive control problem with hysteresis and establish a relationship between the solution set of the impulsive system and that of the corresponding singular system. A particular attention is paid to approximation of discontinuous solutions with bounded total variation by absolutely continuous solutions. The existence of an optimal solution to the impulsive control problem with hysteresis is proved. We also propose a variant of the so-called space-time reparametrization adapted to optimal impulsive control problems with hysteresis and discuss a method for obtaining optimality conditions for impulsive processes.