Optimal control of volterra equations with impulses

We consider an optimal control problem for a system governed by a Volterra integral equation with impulsive terms. The impulses act on both the state and the control; the control consists of switchings at discrete times. The cost functional includes both, an integrated cost rate (continuous part) and switching costs at the discrete impulse times (discrete part). We prove necessary optimality conditions of a form analogous to a discrete maximum principle. For the particular case of a system governed by impulsive ordinary differential equations, we obtain an impulsive maximum principle as a special case of the necessary optimality conditions for impulsive Volterra equations.