Abstract
For a nonlinear impulsive control system, we extend the so-called graph completion approach and introduce a notion of generalized solution x associated to a control u 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.
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