there exist a constant $M>0$ and a local modulus of continuity $\omega(\cdot,\cdot)$ such that $$\begin{array}{ll}
\|b(t,x,u) - b(s,y,u)\| \leq M \|x-y\| + \omega(|t-s|,\|x\|\vee \|y\|)\\
\hskip 5cm \text{for all $t,s\in [0,T], \; u\in U \; x,y\in\mathcal{H}$}\\
\|b(t,0,u)\| \leq M \;\; \text{for all $(t,u)\in [0,T] \times U$}
\end{array}$$

\[hpD3onLandh\] $$L\colon [0,T] \times \mathcal{H} \times U \to \mathbb{R} \;\;\; and
\;\;\; h\colon \mathcal{H} \to \mathbb{R} \;\;\; \text{are
continuous}$$ and there exist $M>0$ and a local modulus of continuity $\omega(\cdot,\cdot)$ such that $$\begin{array}{ll}
|L(t,x,u) - L(s,y,u)|, \; |h(x)-h(y)| \leq \omega(\|x-y\|+|t-s|,\|x\|\vee \|y\|)\\
\hskip 5cm \text{for all $t,s\in [0,T], \; u\in U \; x,y\in\mathcal{H}$}\\
|L(t,0,u)|, |h(0)| \leq M \;\; \text{for all $(t,u)\in [0,T] \times
U$}
\end{array}$$

Notice that if we replace $A$ and $b$ by $\tilde A=A-\omega I$ and $b(t,x,u)$ with $\tilde b(t,x,u)= b(t,x,u) + \omega x$ the above assumptions would cover a more general case $$\label{ppp2} \| e^{sA} \| \leq e^{\omega s} \;\;\; \text{for all
$s\geq 0$}$$ for some $\omega \geq 0$. However such $\tilde b$ does not satisfy the assumptions of Section 4 and may not satisfy the assumptions needed for comparison for equation (\[deterministicHJB\]). Alternatively, by making a change of variables $\tilde
v(t,x)=v(t,e^{\omega t}x)$ in equation (\[deterministicHJB\]) (see [@CL4], page 275) we can 