\mathbb{R}\times \mathcal{H} \; : \;
p=p_1+p_2 \; with \; (q,p_1,p_2) \in E^{1,+}v(t,x) \}$$ then $E^{1,+}_1v(t,x) \subseteq D^{1,+}v(t,x)$ and in the finite dimensional case we have $E^{1,+}_1v(t,x) =
D^{1,+}v(t,x)$. Here we have to use $E^{1,+}v(t,x)$ instead of $E^{1,+}_1v(t,x)$ because of the different roles of $g$ and $\varphi$. It is not clear if the sets $E^{1,+}v(t,x)$ and $E^{1,+}_1v(t,x)$ are convex. However if we took finite sums of functions $\eta(t)g_0(\|x\|)$ as $test2$ functions then they would be convex. All the results obtained are unchanged if we use the definition of viscosity solution with this enlarged class of $test2$ functions.

A trajectory-strategy pair $\left(x(\cdot), u(\cdot) \right)$ will be called an [admissible couple]{} for $(t,x)$ if $u\in{\cal
U}[t,T]$ and $x(\cdot)$ is the corresponding solution of the state equation (\[sydeterministicstate\]).

A trajectory-strategy pair $\left(x^*(\cdot),u^*(\cdot) \right)$ will be called an [optimal couple]{} for $(t,x)$ if it is admissible for $(t,x)$ and if we have $$-\infty < J(t,x;u^*(\cdot))\leq J(t,x;u(\cdot))$$ for every admissible control $u(\cdot) \in {\cal U}[t,T]$.

We can now state and prove the verification theorem.

\[thdeterministicverification\] Let Hypotheses \[hpD2onb\], \[hpD3onLandh\] and \[D4deterministiccomparison\] hold. Let $v\in\mathcal{G}$ be a subsolution of the HJB equation (\[deterministicHJB\]) such that $$\label{terminalconditionvertheorem} v(T,x)=h(x) \;\;\; for\; all\;
x\; in \; \mathcal{H}.$$

\(a) We have 