Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints

Abstract

This article is devoted to the Hamilton–Jacobi partial differential equation

$$\left\{\begin{array}{lll}\frac{\partial V}{\partial t} = H\left(t, x, - \frac{\partial V}{\partial x}\right) & \hbox{on} & [0, 1]\times {\overline{\Omega}}\\V(1, x) = g(x) & \hbox{on}& {\overline{\Omega}},\end{array}\right.$$

where the Hamiltonian ${{H:[0, 1] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}}}$ is convex and positively homogeneous with respect to the last variable, ${{\Omega \subset \mathbb{R}^n}}$ is open and ${{g : \mathbb{R}^n \to \mathbb{R} \cup \{+ \infty\}}}$ is lower semicontinuous. Such Hamiltonians do arise in the optimal control theory. We apply the method of generalized characteristics to show uniqueness of lower semicontinuous solution of this first order PDE. The novelty of our setting lies in the fact that we do not ask regularity of the boundary of Ω and extend the Soner inward pointing condition in a nontraditional way to get uniqueness in the class of lower semicontinuous functions.