On nouniqueness of solutions of Hamilton-Jacobi-Bellman equations

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.