The value function of a slow growth control problem with state constraints

An optimal control problem with slow growth and a state constraint is considered. The formal extension of the conventional theory of dynamic programming to this kind of control problem involves two major drawbacks: on one hand the formal Hamiltonian may happen to be discontinuous; on the other hand, just as in the case of bounded controls, the imposition of a state constraint possibly gives rise to a discontinuous value function. The authors provide conditions that guarantee the continuity of the value function of the control problem considered. These conditions concern the directions of the vectogram at the boundary of the state constraint set and hence they are of the same nature as conditions deduced before by H. M. Soner. However, the latter are no longer sufficient as soon as unbounded controls are allowed. Two illustrative examples are presented. Subsequently the authors prove that the value function is the unique continuous viscosity solution of a boundary value problem involving a suitable Hamilton-Jacobi-Bellman equation.