Further results on the Bellman equation for exit time optimal control problems with nonnegative Lagrangians: the case of Fuller´s problem

Malisoff (1999) gave a uniqueness characterization for viscosity solutions of Bellman equations for exit time problems whose Lagrangians vanish for some points outside the target. The result of that paper applies to a very general class of problems whose dynamics give positive running costs over any interval where the state is outside the target, including Fuller´s problem, and shows that the value function is the unique proper viscosity solution of the Bellman equation which vanishes at the target. This paper gives a different approach which improves special cases of the result of Malisoff by proving that the value function for a class of problems including Fuller´s problem is the unique viscosity solution of the Bellman equation that vanishes at the target and is bounded below. We use the fact that all trajectories of these problems whose total running costs over (0, ∞) are finite tend to the origin