Application of viscosity solutions of infinite-dimensional Hamilton-Jacobi-Bellman equations to some problems in distributed optimal control

We apply the recently developed Crandall and Lions theory of viscosity solutions for infinite-dimensional Hamilton-Jacobi equations to two problems in distributed control. The first problem is governed by differential-difference equations as dynamics, and the second problem is governed by a nonlinear divergence form parabolic equation. We prove a Pontryagin maximum principle in each case by deriving the Bellman equation and using the fact that the value function is a viscosity supersolution.

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