On Reflecting Boundary Problem for Optimal Control

Abstract

International audienceThis paper deals with Mayer's problem for controlled systems with reflection on the boundary of a closed subset K. The main result is the characterization of the possibly discontinuous value function in terms of a unique solution in a suitable sense to a partial differential equation of Hamilton–Jacobi–Bellman type

A Maximum Principle for the Controlled Sweeping Process

“…Not mentioning some works scattered in the mechanical engineering literature, some early theoretical results appeared in [20] (a Hamilton-Jacobi characterization of the value function, with C constant, later generalized in [14]) and in [17,18] (existence and discrete approximation of optimal controls, in the related framework of rate independent processes). More recently, the papers [11,12,13] are devoted to the case where the control acts on the moving set, which in turn is required to have a polyhedral structure.…”

Optimality conditions for reflecting boundary control problems

“…We can prove that n(y u (·)) is in N K (y u (·)) ∩ B(0,M ) (see [31]) and moreover we have: Consequently, …”

“…Here K is a nonempty closed subset of R N , U is a compact metric space, f is a bounded function from R N × U into R N and N K (x) is the normal cone to K at x ∈ K. We notice that N K (x) = {0} whenever x ∈ • K; f is only modified on the boundary of K, such that (1) is a problem with reflection at the boundary (see [31]). …”

“…A detailed study of the differential inequality (1) is presented in [31]. For the reader's convenience, we recall some properties of the set of solutions to this controlled system.…”

“…That is equivalent with saying that V is nondecreasing along any admissible trajectory and is constant along optimal trajectories. This properties are used to prove that the value function V is the only solution (in viscosity sense) of the following Hamilton-Jacobi inclusion (see [31]):…”