**Set-up.** Consider the following linear controlled system 


$$
\dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1)
$$
where $y$ is the state of the system, $y(t) \in R ^n$, $A \in R ^{n\times n}$, $B \in R^{n \times m}$, $u$ is the control taking values in $R^m$, $m>n$. Assume that the Kalman condition is satisfied:
$$
\text{rank}[B, AB, ..., A ^{n-1}B] = n .
$$


It is known that in this case any point in $R^n$ is reachable from any other within an arbitrary small time $T\_s>0$, that is, for any $y\_0, y \_1 \in R^n$ and $T\_s>0$ there exists a signal $u:[0,T\_s] \to R^m$ such that $(1)$ has a solution with $T = T\_s$, $y(0) = y\_0 $, $y(T) = y\_1$. One such control $u$ can be written down explicitly via the controllability Gramian, see e.g. R.W. Brockett, *Finite Dimensional Linear Systems*, Theorem 1, Section 13. Set 
$$
W(T) = \int\limits \_0 ^{T } e ^{t A} B B'e ^{t A'} dt,
$$
$\eta = [W(T)]^{-1} (y\_0 - e^{T A}y\_1 )$, and 
$$
u(t) = -B'e^{t A} \eta.
$$
The matrix $W(T) = W(T\_s)$ is invertible since the Kalman condition is satisfied. If $T\_s$ is small, then the operator norm of $[W(T\_s)]^{-1}$ is large.


**Question.** Is there something known about the asymptotic of $[W(T\_s)]^{-1}$ when $T\_s \to 0$? In particular, I am interested in the bounds on the largest eigenvalue of $[W(T\_s)]^{-1}$ of the form say 
$$\lambda \_1 ([W(T\_s)]^{-1}) \leq f(T\_s),$$
 where $\lambda \_1 (M)$ is the largest eigenvalue of a matrix $M$, and $f(t) \to \infty$ as $t \to 0$. 


**Thoughts.** I do not have a lot of experience in linear systems. I tried a little bit to prove something directly, using e.g. that the matrix exponential is in fact a polynomial, but to no avail. I also tried looking up some references suggested to me and which I found with a search engine, but I didn't find an answer to my question.



Controllability Gramian asymptotics for small times
Did you try with 
"How violent are fast controls?"
by Thomas I. Seidman?
It should include the estimates you are looking for. 


