Further results on the Bellman equation for optimal control problems with exit times and nonnegative Lagrangians

In a series of papers, we proved theorems characterizing the value function in exit time optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The results applied to problems which satisfy a positivity condition on the integral of the Lagrangian. This positive integral condition assigned a positive cost for remaining outside the target on any interval of positive length. In this note, we prove a new theorem which characterizes the exit time value function as the unique bounded-from-below viscosity solution of the Bellman equation that vanishes on the target. The theorem applies to problems satisfying an asymptotic condition on the trajectories, including cases where the positive integral condition is not satisfied. Our results are based on an extended version of “Barblat's lemma”. We apply the theorem to variants of the Fuller problem and other examples where the Lagrangian is degenerate.