Lack of BV bounds for impulsive control systems

For a nonlinear impulsive control system, we extend the so-called graph completion approach, originally developed for controls with bounded variation, to controls with possibly infinite variation. Precisely, we introduce a notion of generalized solution associated to a control whose total variation is bounded on [0, t] for every t < T , but possibly unbounded on [0, T ]. We prove existence, consistency with classical solutions and well-posedness of this solution. In particular, we characterize it as a pointwise limit of certain regular solutions. The notion that we consider provides the natural setting for controllability questions and for some non-coercive optimal control problems, where chattering phenomena at the final time are expected. More in general, it is well suited to describe the evolution of control systems subject to a train of impulses where no a-priori bounds on the number and the amplitude of the impulses are imposed.