Distributed optimal control and viscosity solutions of infinite dimensional Bellman equations

The Bellman equation for two infinite dimensional optimal control problems is studied here in the context of the Crandall and Lions [3] theory of viscosity solutions. We apply the method introduced in Barron and Jensen [1] to derive the Pontryagin maximum principle using the Bellman equation and the fact that the value function is a viscosity supersolution. The specific problems considered are governed by (1) nonlinear differential-difference equations as dynamics or (2) a nonlinear, divergence form, parabolic p.d.e, as dynamics.