BV solutions of rate independent processes driven by impulsive controls

An extension to discontinuous inputs of bounded variation of the play operator is presented. The motivation for such an extension comes from optimal impulsive control problems with hysteresis. The latter is introduced by means of variational inequalities modeling the action of the play operator on the BV-inputs. Our extension is applied to the graph completions of BV-functions and it allows us to study impulsive control systems characterized by measure-driven differential equations with nonlinear terms of hysteresis type. Some approximation results for impulsive processes are also provided.