On nouniqueness of solutions of Hamilton-Jacobi-Bellman equations

TL;DR

This paper presents an example of multiple solutions to a Hamilton-Jacobi-Bellman equation with a regular Hamiltonian, challenging the common assumption of uniqueness in such equations.

Contribution

It provides a concrete example demonstrating nonuniqueness of solutions for HJB equations with regular Hamiltonians, and analyzes the assumptions behind uniqueness results.

Findings

Existence of two distinct solutions with the same final conditions.

One solution corresponds to the value function of a Bolza problem.

Analysis of assumptions in existing uniqueness theorems.

Abstract

An example of a nonunique solution of the Cauchy problem of Hamilton-Jacobi-Bellman (HJB) equation with surprisingly regular Hamiltonian is presented. The Hamiltonian H(t,x,p) is locally Lipschitz continuous with respect to all variables, convex in p and with linear growth with respect to p and x. The HJB equation possesses two distinct lower semicontinuous solutions with the same final conditions; moreover, one of them is the value function of the corresponding Bolza problem. The definition of lower semicontinuous solution was proposed by Barron-Jensen (1990) and Frankowska (1993). Using the example an analysis and comparison of assumptions in some uniqueness results in HJB equations is provided.