various verification theorems and results on existence and explicit forms of optimal feedback controls in particular cases are established. However, despite of these results, so far the use of the dynamic programming approach in the resolution of the general optimal control problems in infinite dimensions has been rather limited. Infinite dimensionality of the state space, unboundedness in the equations, lack of regularity of solutions, and often complicated notions of solutions requiring the use of sophisticated test functions are only some of the difficulties.

We will discuss two aspects of the dynamic programming approach for a fairly general control problem: a verification theorem which gives a sufficient condition for optimality, and the problem of construction of $\epsilon$-optimal feedback controls.

The verification theorem we prove in this paper is an infinite dimensional version of such a result for finite dimensional problems obtained in [@Zh]. It is based on the notion of viscosity solution (see Definitions \[defdeterministicsubsol\]-\[defdeterministicsol\]). Regarding previous result in this direction we mention [@CaFr1; @CaFr2] and the material in Chapter 6 §5 of [@LiYong], in particular Theorem 5.5 there which is based on [@CaFr1]. We briefly discuss this result in Remark \[remliyo\].

The construction of $\epsilon$-optimal controls we present here is 