of approximations, it is very explicit and the errors in many cases can be made precise, and it does not require any compactness of the operator $B$. It does however require some weak continuity of the Hamiltonian and uniform continuity of the trajectories, uniformly in $u(\cdot)$. Finally we mention that the sub- and superoptimality inequalities of dynamic programming are interesting on their own.

The paper is organized as follows. Definitions and the preliminary material is presented in Section 2. Section 3 is devoted to the verification theorem and an example where it applies in a nonsmooth case. In Section 4 we prove sub- and superoptimality principles of dynamic programming and show how to construct $\epsilon$-optimal controls.

Notation, definitions and background
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Throughout this paper $\mathcal{H}$ is a real separable Hilbert space equipped with the inner product ${\left\langle}\cdot,\cdot{\right\rangle}$ and the norm $\|\cdot\|$. We recall that $A$ is a linear, densely defined operator such that $-A$ is maximal monotone, i.e. $A$ generates a $C_0$ semigroup of contractions $e^{sA}$, i.e. $$\label{ppp1} \| e^{sA} \| \leq 1 \;\;\; \text{for all $s\geq 0$}$$ We make the following assumptions on $b$ and $L$.

\[hpD2onb\] $$b\colon [0,T] \times \mathcal{H} \times U \to \mathcal{H} \;
\text{is continuous}$$ and 