always reduce the case (\[ppp2\]) to the case when $A$ satisfies (\[ppp1\]).

Following the dynamic programming approach we consider a family of problems for every $t\in[0,T], y\in \mathcal{H}$ $$\label{sydeterministicstate} \left\lbrace
\begin{array}{l}
\dot{x}_{t,x}(s)=A {x}_{t,x}(s) + b(s,{x}_{t,x}(s),u(s))\\
x_{t,x}(t)=x
\end{array}
\right.$$ We will write $x(\cdot)$ for $x_{t,x}(\cdot)$ when there is no possibility of confusion. We consider the function $$\label{sydeterministiccost} J(t,x;u(\cdot))= \int_t^T L(s,x(s),u(s))
{\mathrm{d}}t + h(x(T)),$$ where $u(\cdot)$ is in the set of admissible controls $$\mathcal{U}[t,T]= \{ u\colon [t,T] \to U: \; u \hbox{ is measurable}
\}.$$ The associated value function $V\colon [0,T]\times\mathcal{H} \to
\mathbb{R}$ is defined by $$\label{deterministicvaluefunction} V(t,x)= \inf_{u(\cdot) \in
\mathcal{U}[t,T]} J(t,x;u(\cdot)).$$ The Hamilton-Jacobi-Bellman (HJB) equation related to such optimal control problems is $$\label{deterministicHJB} \left\lbrace
\begin{array}{l}
v_t(t,x) +{\left\langle}Dv(t,x), Ax {\right\rangle}+ H(t,x,Dv(t,x))=0\\
v(T,x)=h(x),
\end{array}
\right.$$ where $$\left\lbrace
\begin{array}{l}
H\colon [0,T]\times\mathcal{H}\times\mathcal{H} \to \mathbb{R},\\
H(t,x,p)=\inf_{u\in U} \left ( {\left\langle}p, b(t,x,u) {\right\rangle}+ L(t,x,u)
\right )
\end{array}
\right.$$

The solution of the above HJB equation is understood in the viscosity sense of Crandall and Lions [@CL4; @CL5] which is slightly modified here. We consider two sets of tests functions: $$\begin{array}{ll}
test1=\{ \varphi \in C^1((0,T)\times\mathcal{H}) \; : & \varphi
\text{ is weakly sequentially lower}\\
& \text{semicontinuous and } A^*D\varphi\in C((0,T)\times
\mathcal{H}) \}
\end{array}$$ and $$\begin{array}{ll}
test2= \{ g\in 