Reduction of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations

Abstract

This article is devoted to the study of lower semicontinuous solutions of Hamilton-Jacobi equations with convex Hamiltonians in the gradient variable. Such Hamiltonians do arise in the optimal control theory. We present a necessary and sufficient condition for the reduction of the Hamiltonian satisfying optimality conditions to the case when the Hamiltonian is positively homogeneous and also satisfies optimality conditions. On one hand, it allows us to reduce uniqueness of solutions problem to Barron-Jensen and Frankowska theorems. On the other hand, it shows us the limits of applicability of this reduction. For Hamiltonians which are not liable to the above reduction we prove the new existence and uniqueness theorems.