Uniqueness results for the value function via direct trajectory-construction methods

We present some new results, together with a number of particularly simple and user-friendly versions of results obtained in recent years by the author and M. Malisoff, on the uniqueness of solutions of the Hamilton-Jacobi-Bellman equation (HJBE) for deterministic finite-dimensional optimal control problems under non-standard hypotheses. Our approach is completely control-theoretic and totally self-contained, using the systematic construction of special trajectories of various kinds, and not involving any PDE methods. We donot assume that the Lagrangian is positive, or that the dynamics is Lipschitz-continuous.