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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 $\theta$,..,$\theta^n$.

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    what do you mean by *input*? In control theory both the term "input" and the letter $u$ are protected and refer to the control applied to the system. The general LTV model for this is $\dot{x} = A(t)x(t) + B(t)u(t)$ and $y(t) = C(t)x(t) + D(t)u(t)$, so this would imply your $A$ is just the zero matrix and thus $x(t) = \int u(\tau) d\tau$. There is no nontrivial transition matrix because the state is uncoupled from its derivative. – 2017-01-14
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    That is right, there is an adyacent system like the one you wrote. But I am interested in a specific condition of the input, the one I wrote above. To define this condition you only take the A and C matrices of your whole system. – 2017-01-20

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