The main purpose of this paper is to discuss the minimization of energy spent in order that a controlled diffusion process reaches a given target, a d-dimensional bounded domain. The exterior Dirichlet problem for the Hamilton-Jacobi-Bellman equation is studied for a class of criteria which includes the case of energy. Extensions to diffusion with jumps, examples and some other reachability problems are considered.
J.-L. Menaldi and M. Robin, On Some Reachability Problems for Diffusion Processes, in Optimal Control and Partial Differential Equations (A Volume in Honor of A. Bensoussan), Eds. J.-L. Menaldi, E. Rofman and A. Sulem, IOS Press, Amsterdam, 2001, 394-403.