**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.