Dirichlet Problems for some Hamilton–Jacobi Equations with Inequality Constraints

Abstract

We use viability techniques for solving Dirichlet problems with inequality constraints (obstacles) for a class of Hamilton–Jacobi equations. The hypograph of the “solution” is defined as the “capture basin” under an auxiliary control system of a target associated with the initial and boundary conditions, viable in an environment associated with the inequality constraint. From the tangential condition characterizing capture basins, we prove that this solution is the unique “upper semicontinuous” solution to the Hamilton–Jacobi–Bellman partial differential equation in the Barron-Jensen/Frankowska sense. We show how this framework allows us to translate properties of capture basins into corresponding properties of the solutions to this problem. For instance, this approach provides a representation formula of the solution which boils down to the Lax–Hopf formula in the absence of constraints.