On nouniqueness of solutions of Hamilton-Jacobi-Bellman equations

Abstract

An example of a nonunique solution of the Cauchy problem of Hamilton-Jacobi-Bellman (HJB) equation with surprisingly regular Hamiltonian is introduced. The proposed Hamiltonian H(t,x,p) fulfills the local Lipschitz continuity with respect to the triple of variables (t,x,p), in particular, with respect to the state variable x. Moreover, the mentioned Hamiltonian is convex with respect to p and possesses linear growth in p, so it satisfies the classical assumptions. Given HJB equation with the Hamiltonian satisfying the above conditions, two distinct lower semicontinuous solutions with the same final conditions are given. Moreover, one of the solutions is the value function of Bolza Problem. The definition of lower semicontinuous solution was proposed by Frankowska (1993) and Barron-Jensen (1990). The example that is proposed in the current paper allows to understand better the role of Lipschitz-type condition in the uniqueness of the Cauchy problem solution of HJB equation