Does anyone have an example of a locally regular input $u$? Say for
$A=\begin{bmatrix}0&u\\0&0\end{bmatrix}$
and
$C=\begin{bmatrix}1&0\end{bmatrix}$.
Given a $n\times n$ real matrix function $A(t)$ and a $1\times n$ real matrix $C$, we say that an input function $u$ is locally regular if there exists positive $\alpha$ such that for big enough $t$ and $\theta$ we have:
$\int_{t-\frac{1}{\theta}}^t\phi(\tau,t)^TC^TC\phi(\tau,t)d\tau\geq\alpha\theta\Delta_\theta^{-2},$
where $\phi$ is the transition matrix solving:
$\frac{\partial\phi(\tau,t)}{\partial \tau}=A(u(t))\phi(\tau,t),$
$\phi(t,t)=I$,
and
$\Delta_\theta$ is the diagonal matrix with entrances $\theta$,..,$\theta^n$.