On the Bellman equation for control problems with exit times and unbounded cost functionals

The article is devoted to the study of Hamilton-Jacobi-Bellman equations (HJBs) for a large class of unbounded optimal control problems for fully nonlinear systems. Our hypotheses will be such that these nonlinear systems have a unique solution trajectory, defined on [0,/spl infin/), for each input. We characterize the value functions of the problems presented as the unique viscosity solutions of the associated HJBs among continuous functions with suitable boundary and growth conditions. As a consequence, we show that the FP (Fuller Problem) value function is the unique radially unbounded viscosity solution of the corresponding HJB among functions which are zero at the origin. Value function characterizations of this kind have been studied and applied by many authors for a large number of stochastic and deterministic optimal control problems and for differential games, including problems for which the value function is discontinuous (M. Bardi and I. Capuzzo-Dolcetta, 1997; W.H. Fleming and H.M. Soner, 1993).