On Nonuniqueness of Solutions of Hamilton–Jacobi–Bellman Equations
Arkadiusz Misztela · Applied Mathematics & Optimization · 2016
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 Frankowska (SIAM J. Control Optim. 31:257–272, 1993 ) and Barron and Jensen (Commun. Partial Differ. Equ. 15(12):1713–1742, 1990 ). Using the example an analysis and comparison of assumptions in some uniqueness results in HJB equations is provided.