Dynamic Programming for Nonlinear Systems Driven by Ordinary and Impulsive Controls

Abstract

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 d. 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 Barren, 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

Graph completions for impulsive feedback controls

“…As a first step, we prove that the assumption (2.8) implies 17) whereγ and B = Gγ x − I and are evaluated at (t, x, θ) ∈ Σ × [0, 1]. Observe that, by choosing ε > 0 sufficiently small, we can assume that the matrix B is invertible on Σ ε .…”

“…The references [3,7,13,15,16,17] can only provide a partial sample. It is worth mentioning that graph completions have also been used in [11] and in [12] to define "nonconservative products" in connection with hyperbolic systems of PDEs.…”

Hamilton-Jacobi Conditions for an Impulsive Control Problem

“…In (Bressan, Rampazzo, 1994), this commutativity assumption is lifted by noting that a certain quotient control system, obtained by an appropriate nonlinear local change of coordinates in the state space, is an impulsive one satisfying the above mentioned commutative hypothesis. In (Mota, Rampazzo, 1996), a dynamic programming approach for nonlinear systems driven by ordinary and impulsive controls is considered whose solution is given by a value function that depends on the time variable and on the variation of the control measure. A maximum principle is also proved and the relations between the adjoint variable and the value function are established.…”

State-Constrained Optimal Control Problems of Impulsive Differential Equations

“…It is easy to prove that the value function satisfies a classic Dynamic Programming Principle (see [24] for a general DPP). For each τ ∈]0, T [, and every h ∈ [0, T − τ ], we have…”