# Controllability Gramian asymptotics for small times

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.

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

• I guess you mean Seidman, T.I. Math. Control Signal Systems (1988) 1: 89. doi.org/10.1007/BF02551238 ? Jun 4, 2019 at 6:53
• Can you expand your comment so that those without access also understand it? Jun 4, 2019 at 7:51
• Indeed, the paper you are referring to contains the estimate of $W(T)^{-1}$ for small $T$. Thanks a lot! Jun 4, 2019 at 11:11

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.

• Thanks, this is very instructive and concise. The asymptotic for $W(t)_{ij}$ also includes $t^{i+j-1}$ right? Jun 4, 2019 at 11:08
• Yes, It Is a typo, sorry. I will fix it. Jun 4, 2019 at 11:18