(based on a previous result of [@CaFr2], see also [@CaFr1] for similar results) in which it is required that the trajectory of the system remains in the domain of $A$ a.e. for the admissible control $u(\cdot)$ in question. This is not required here and in fact this is not satisfied in the example of the next section.

It is shown in [@LiYong] (under assumptions similar to Hypotheses \[hpD2onb\] and \[hpD3onLandh\]) that the couple $x(\cdot), u(\cdot))$ is optimal if and only if $$\begin{gathered}
u(s) \in \bigg\lbrace u\in U \, : \, \lim_{\delta \to 0}
\frac{V((s+\delta), x(s)+ \delta(Ax(s) + b(s,x(s),u)) ) - V(s,x(s))
}{\delta} \\
= -L(s,x(s),u) \bigg\rbrace\end{gathered}$$ for almost every $s\in[t,T]$, where $V$ is the value function.

An example
----------

We present an example of a control problem for which the value function is a nonsmooth viscosity solution of the corresponding HJB equation, however we can apply our verification theorem. The problem can model a number of phenomena, for example in age-structured population models (see [@Iannelli95; @Anita01; @Iannelli06]), in population economics [@FeichtingerPrskwetzVeliov04], optimal technology adoption in a vintage capital context [@BarucciGozzi98; @BarucciGozzi01].

Consider the state equation $$\label{example1stateequation} \left\lbrace
\begin{array}{l}
\dot{x}(s) = Ax(s) + Ru(s)\\
x(t)=x
\end{array}
\right.$$ where$A$ is a linear, 