Semicontinuous solutions for Hamilton-Jacobi equations and theL ?control problem

Abstract

We prove uniqueness for extended real-valued lower semicontinuous viscosity solutions of the Bellman equation forL ∞-control problems. This result is then used to prove uniqueness for lsc solutions of Hamilton-Jacobi equations of the form −u t +H(t, x, u, −Du)=0, whereH(t, x, r, p) is convex inp. The remaining assumptions onH in the variablesr andp extend the currently known results.