Dynamic Programming for Nonlinear Systems Driven by Ordinary and Impulsive Controls
Monica Motta, Franco Rampazzo · SIAM Journal on Control and Optimization · 1996
A dynamic programming approach is considered for a class of minimum problems with impulses. The minimization domain consists of trajectories satisfying an ordinary differential equation whose right-hand side depends not only on a measurable control v but also on a second control u and on its time derivative $\dot u$. For this reason, the control u and the differential equation are called impulsive. The value function of the considered minimum problem turns out to depend on the time, the state, the u variable, and the variation allowed to the impulsive control. It is shown that the value function satisfies, in a generalized sense, a dynamic programming equation (DPE), which is obtained from a dynamic programming principle involving space–time trajectories. Moreover the value function is the unique map-solving equation (DPE) satisfying either an inequality condition or a supersolution condition at each point of the boundary. Incidentally this extends a result by Barron, Jensen, and Menaldi [Nonlinear Anal., 21 (1993), pp. 241–268], where the impulsive control is scalar monotone and the corresponding vector field is independent of the state variable. Next, a maximum principle is proved, and the well-known relationship between adjoint variables and value function is suitably extended to impulsive control systems. A fully elaborated example concludes the paper.