Hamilton-Jacobi-Bellman Equations on Multi-domains

A system of Hamilton–Jacobi (HJ) equations on a partition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}$\end{document} is considered, and a uniqueness and existence result of viscosity solution is analyzed. While the notion of viscosity solution is by now well known, the question of uniqueness of solution, when the Hamiltonian is discontinuous, remains an important issue. A uniqueness result has been derived for a class of problems, where the behavior of the solution, in the region of discontinuity of the Hamiltonian, is assumed to be irrelevant and can be ignored (see (Camilli, Siconolfi in Adv. Differ. Equ. 8(6):733–768, 2003)). Here, we provide a new uniqueness result for a more general class of Hamilton–Jacobi equations.