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| This work deals with a class of optimal control problems with impulsive controls. The optimization problem consists of trajectories of bounded variation satisfying an impulsive differential equation involving a measurable control and a second control with bounded variation. The dynamics of the trajectories depend not only on these two controls but also on the time derivative of the second control, which leads to the impulses in the trajectories. The definition of solutions to the impulsive dynamical system was introduced by Bressan and Rampazzo using the concept of Graph Completion and it has been studied recently by Aronna and Rampazzo in a more general setting. By means of the graph completion and reparametrization techniques, we study an auxiliary control problem and the properties of the value function. The characterization of the value function is provided via the Hamilton-Jacobi-Bellman approach and a verification theorem is given to test the optimality of a given control. Furthermore, we will discuss the problem with state constraints and conclude with some applications on mechanical systems. |
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