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

This work is devoted to the study of Hamilton-Jacobi-Bellman equations (HJBs) for a large class of unbounded optimal control problems for fully nonlinear systems. The value functions of these problems are characterized as the unique viscosity solutions of the associated HJBs among continuous functions with suitable boundary and growth conditions. As a result, it is shown that the Fuller Problem (FP) value function is the unique radially unbounded viscosity solution of the corresponding HJB among functions which are zero at the origin