---
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.

    **Key words**: optimal control of PDE, verification theorem, dynamic programming, $\epsilon$-optimal controls, Hamilton-Jacobi-Bellman equations.

    **MSC 2000**: 35R15, 49L20, 49L25, 49K20.
author:
- 'G. Fabbri[^1] F. Gozzi[^2] and A. Świȩch[^3]'
title: 'Verification theorem and construction of $\epsilon$-optimal controls for control of abstract evolution equations'
---

Introduction
============

In this paper we investigate several aspects of the dynamic programming approach to optimal control of abstract evolution equations. The optimal control problem we have in mind has the following form. The state equation is $$\label{deterministicstateequation} \left\lbrace
\begin{array}{l}
\dot{x}(t) = Ax(t) + b(t,x(t),u(t)),\\
x(0)=x,
\end{array}
\right.$$ $A$ is a linear, densely defined maximal dissipative operator in a real separable Hilbert space $\mathcal{H}$, and we want to minimize a cost functional $$\label{deterministiccostfunctional} J(x;u(\cdot))= \int_0^T
L(t,x(t),u(t)) {\mathrm{d}}t + h(x(T))$$ over all controls $$u(\cdot)\in\mathcal{U}[0,T]= \{ u\colon [0,T] \to U : \; u \;
\hbox{is measurable} \},$$ where $U$ 