Verification theorem and construction of $\epsilon$-optimal controls for control of abstract evolution equations

TL;DR

This paper develops a verification theorem and explicit methods for constructing epsilon-optimal controls in the optimal control of abstract evolution equations, including semilinear PDEs, advancing the dynamic programming approach.

Contribution

It introduces a verification theorem and explicit epsilon-optimal control construction for abstract evolution equations, enhancing dynamic programming techniques.

Findings

Established a verification theorem for optimality conditions.

Proved sub- and superoptimality principles in dynamic programming.

Provided explicit construction methods for epsilon-optimal controls.

Abstract

We study several aspects of the dynamic programming approach to optimal control of abstract evolution equations, including a class of semilinear partial differential equations. We introduce and prove a verification theorem which provides a sufficient condition for optimality. Moreover we prove sub- and superoptimality principles of dynamic programming and give an explicit construction of $\epsilon$-optimal controls.