On the Hamilton-Jacobi-Bellman equations in Banach spaces

This paper is concerned with a certain class of distributed parameter control problems. The value function of these problems is shown to be the unique viscosity solution of the corresponding Hamiltonian-Jacobi-Bellman equation. The main assumption is the existence of an increasing sequence of compact invariant subsets of the state space. In particular, this assumption is satisfied by a class of controlled delay equations. This research was partly supported by the Institute for Mathematics and Its Applications with funds provided by the National Science Foundation and the Office of Naval Research. The author is indebted to Professor P. L. Lions for stimulating discussions and helpful suggestions.