$A(f)(s) = \frac{{\mathrm{d}}}{{\mathrm{d}}s} f (s)$. We choose $\alpha$ to be the constant function equal to $1$ at every point of the interval $[0,1]$. (We can take for instance $B=(I-\Delta)^{-\frac{1}{2}}$.) Moreover we choose $\beta(s)=\chi_{[0,\frac{1}{2}]}(s) - \chi_{[0,\frac{1}{2}]}(s)$ ($\chi_{\Omega} $ is the characteristic function of a set $\Omega$). Consider an initial datum $(t,x)$ such that $\left\langle \alpha,
x\right\rangle =0$. In view of Proposition \[propqui\] an optimal strategy $u^{op}$ is $$u^{op}(s)=-\left\langle \beta,\bar\alpha(s) \right\rangle =0$$ The related optimal trajectory is $$x^{op}(s)= e^{(s-t)A}y.$$

\[rm:controesempio\] We observe that, using such strategy, $\left\langle
\alpha,x^{op}(t)\right\rangle = 0$ for all $s\geq t$. So the trajectory remains for a whole interval in a set in which the value function is not differentiable. Anyway, applying Theorem \[thdeterministicverification\], the optimality is proved. Moreover $x$ can be chosen out of the domain of $A$ and so the assumptions of the verification theorem given in [@LiYong] (page 263, Theorem 5.5) are not verified in this case.

Sub- and superoptimality principles and construction of $\epsilon$-optimal controls {#subsuper}
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Let $B$ be a bounded linear positive self-adjoint operator on $\mathcal{H}$ such that $A^*B$ bounded on $\mathcal{H}$ and let $c_0\leq 0$ be a constant such that $$\label{bcond} {\left\langle}(A^* B + c_0 B)x,x {\right\rangle}\leq 0 \;\;\;\;\;\; for
\; all 