$v(t,x) \leq V(t,x) \leq J(t,x,u(\cdot))\;\;
\forall(t,x) \in (0,T]\times\mathcal{H},
\;u(\cdot)\in\mathcal{U}[t,T]$.

\(b) Let $(t,x)\in (0,T)\times H$ and let $(x_{t,x}(\cdot),
u(\cdot))$ be an admissible couple at $(t,x)$. Assume that there exist $q\in L^1(t,T;\mathbb{R})$, $p_1\in L^1(t,T;D(A^*))$ and $p_2\in L^1(t,T;\mathcal{H})$ such that $$\label{condE} (q(s),p_1(s),p_2(s)) \in E^{1,+}v(s,x_{t,x}(s)) \;
\;\; \text{for almost all } s\in (t,T)$$ and that $$\begin{gathered}
\label{condmin} \int_t^T ({\left\langle}p_1(s) + p_2(s), b(s,x_{t,x}(s),u(s))
{\right\rangle}+ q(s) +
{\left\langle}A^* p_1(s),x_{t,x}(s){\right\rangle}){\mathrm{d}}t \\
\leq \int_t^T - L(s,x_{t,x}(s),u(s)) {\mathrm{d}}s.
$$ Then $(x_{t,x}(\cdot), u(\cdot))$ is an optimal couple at $(t,x)$ and $v(t,x)=V(t,x)$. Moreover we have equality in (\[condmin\]).

It is tempting to try to prove, along the lines of Theorem 3.9, p.243 of [@YongZhou], that a condition like (\[condmin\]) can also be necessary if $v$ is a viscosity solution (or maybe simply a supersolution). However this is not an easy task: the main problem is that $E^{1,+}$ and the analogous object $E^{1,-}$ are fundamentally different so a natural generalization of a result like Theorem 3.9, p.243 of [@YongZhou] does not seem possible. Moreover our verification theorem has some drawbacks. Condition (\[condmin\]) implicitly implies that $<p_2(r),Ax_{t,x}(r)>=0$ a.e. if the trajectory is in the domain of $A$. This follows from the fact that we would then have an additional term $<p_2(r),Ax_{t,x}(r)>$ in the 