and so the claim in proved.

It is easy to see that both $W$ and the value function $V$ for the problem are continuous on $[0,T]\times \mathcal{H}$ and moreover $\psi=W$ and $\psi=V$ satisfy $$|\psi(t,x)-\psi(t,y)|\le C\|x-y\|_{-1} \quad\hbox{for
all}\,\,t\in[0,T], x,y\in \mathcal{H}$$ for some $C\ge 0$. In particular $W$ and $V$ have at most linear growth as $\|x\|\to\infty$. By Theorem \[thexistence\], the value function $V$ is a a viscosity solution of the HJB equation (\[example1HJB\]) in $(0,T]\times\mathcal{H}$. Moreover, since $\alpha=By$ for some $y\in \mathcal{H}$, comparison holds for equation (\[example1HJB\]) which yields $W=V$ on $[0,T]\times\mathcal{H}$. (Comparison theorem can be easily obtained by a modification of techniques of [@CL5] but we cannot refer to any result there since both $V$ and $W$ are unbounded. However the result follows directly from Theorem 3.1 together with Remark 3.3 of [@Kel]. The reader can also consult the proof of Theorem 4.4 of [@KeSw]. We point out that our assumptions are different from the assumptions of the uniqueness Theorem 4.6 of [@LiYong], page 250).

Therefore we have an explicit formula for the value function $V$ given by $V(t,x)=W(t,x)$. We see that $V$ is differentiable at points $(t,x)$ if $\left\langle \alpha,x\right\rangle \ne  0$ and $$DV(t,x)=\left\lbrace
\begin{array}{ll}
\bar \alpha(t) & if 