Controllability Gramian asymptotics for small times

Question

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.

Answer 1

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

Answer 2

There is a direct answer to your question, by explicit computations. This must be contained as an intermediate result in the paper suggested by Mario Sigalotti, but since I do not know it, I give below a self-consistent proof of what you need.

Let us look at the derivative $\dot{W}(t) = e^{tA} B B^*e^{t A^*} = V(t) V(t)^*$, where the star denotes the transpose and $V(t) = e^{tA} B$. Define then the following filtration of subspaces of $\mathbb{R}^n$:

$$ E_i= \mathrm{span}\{V(0),V^{(1)}(0),\dots,V^{(i-1)}(0)\}, \qquad i \geq 1, $$

that is the space generated by the set of columns of $V$ and their derivative at $t=0$, up to order $i-1$. Clearly $E_i \subset E_{i+1}$. It is an exercise using Kalman condition that there exists some $N\geq 1$ such that $E_{N} = \mathbb{R}^n$ (and we can assume letting $E_0 = \{0\}$ that $E_{N-1} \neq\mathbb{R}^n$, so that $N$ is the minimal number with this property).

Let $k_i = \dim E_i$. Choose coordinates of $\mathbb{R}^n$ adapted to this flag, that is a basis $\{e_1,\dots,e_n\}$ such that $\mathrm{span}\{e_1,\dots,e_{k_i}\}= E_i$. In these coordinates we must have

$$V(t) = \left(\begin{array}{c} \hat{v}_1\\ t\hat{v}_2 \\ \vdots\\ t^{N-1} \hat{v}_N \end{array}\right) + \left(\begin{array}{c} O(t)\\ O(t^2) \\ \vdots\\ O(t^N) \end{array}\right)$$

where $\hat{v}_i$ is a $d_i \times n$ matrix, and we let $d_i=k_i-k_{i-1}$. Notice also that the $\hat{v}_i$ have rank $d_i$. Now using the formula $\dot{W}(t) = V(t)V(t)^*$, and the previous asymptotic for $V(t)$, you get that the $d_i\times d_j$ block $W(t)_{ij}$ of $W(t)$ has the following asymptotic:

$$ W(t)_{ij} = \frac{\hat{v}_i \hat{v}_j^*}{i+j-1}t^{i+j-1}+ O(t^{i+j}). $$

Let $\chi$ be the $n\times n$ block-matrix whose $ij$-th block, for $i,j=1,\dots,N$, is given by $\hat{v}_i \hat{v}_j^*/(i+j-1)$. Notice that $\chi$ is symmetric and $\chi>0$, again as a consequence of the Kalman assumption. From this you easily get a similar block-wise asymptotic for the inverse

$$ W(t)^{-1}_{ij} = \frac{\chi^{-1}_{ij}}{t^{i+j-1}}\left(1 + O(t)\right) $$

This implies $t^{2N-1}W(t)^{-1} = \chi^{-1}+O(t)$, which implies the desired estimate.