densely defined maximal dissipative operator in $\mathcal{H}$, $R$ is a continuous linear operator $R\colon
\mathbb{R}\to \mathcal{H}$, so it is of the form $R\colon u\mapsto u
\beta$ for some $\beta\in\mathcal{H}$. Let $B$ be an operator as in Section \[subsuper\] satisfying (\[bcond\]). We will be using the notation of Section \[subsuper\].

We will assume that $A^*$ has an eigenvalue $\lambda$ with an eigenvector $\alpha$ belonging to the range of $B$.

We consider the functional to be minimized $$\label{example1costfunctional} J(x,u(\cdot))= \int_t^T
-\left|\left\langle \alpha,x(s)\right\rangle \right| +
\frac{1}{2}u(s)^2 {\mathrm{d}}s.$$ We define $$\bar\alpha(t){\stackrel{def}{=}}\int_t^T e^{(s-t)A^*} \alpha {\mathrm{d}}s$$ and we take $M{\stackrel{def}{=}}\sup_{t\in[0,T]} |{\left\langle}\bar\alpha(t), \beta
{\right\rangle}|$. We consider as control set $U$ the compact subset of $\mathbb{R}$ given by $U=[-M-1, M+1]$. So we specify the general problem characterized by (\[deterministicstateequation\]) and (\[deterministiccostfunctional\]) taking $b(t,x,u)=Ru$, $L(t,x,u)=
-\left|\left\langle \alpha,x(s)\right\rangle \right| + 1/2 u(t)^2$, $h=0$, $U=[-M-1, M+1]$.

The HJB equation (\[deterministicHJB\]) becomes $$\label{example1HJB} \left\lbrace
\begin{array}{l}
v_t + {\left\langle}Dv, Ax {\right\rangle}-\left|\left\langle \alpha,x\right\rangle
\right|+\inf_{u\in U} \left ( {\left\langle}u,R^*Dv{\right\rangle}_{\mathbb{R}} + \frac{1}{2} u^{2} \right)=0\\
v(T,x)=0
\end{array}
\right.$$ Note that the operator $R^*\colon \mathcal{H} \to \mathbb{R}$ can be explicitly expressed using $\beta$ which was used to define the operator $R$: $R^*x=\left\langle\beta,x\right\rangle$.

Now we observe that for $\left\langle \alpha,x\right\rangle<0$ (respectively $>0$) the HJB equation is the same as 