# References required for "minimal switches to control a switched linear system"

A switched linear system of ODEs is defined as a system of ODEs of the form $$\dot{x}(t) = A_i(t)x(t)$$ , with $$A_i$$ for $$i=1,2,...,N$$ being a fixed number of matrices. More precisely, given a $$\sigma : [0,t] \to \{1,2,...,N\}$$ and an initial point $$\mathbf{x} \in \Bbb R^n$$,we could talk about (assuming existence and uniqueness of solution) the solution to $$\dot{x}(t) = A_{\sigma(t)}x(t)$$ with $$x(0) = \mathbf x$$, and look at its trajectory.

In the research of switched linear systems, often problems of controllability(See here) are considered. Other topics of research include minimizing , among all $$\sigma$$ suitable (e.g. steering to a certain point/region), a quadratic functional of $$\sigma$$.

However, I am not aware of any research looking at the minimum number of switches to reach a point/region.

More precisely, consider a fixed $$\mathbf x \in \mathbb R^n$$, a destination region $$S \subset \mathbb R^n$$ with $$\mathbf x \notin S$$, a fixed $$T>0$$ (think of $$T$$ as being very large) and let $$\Sigma = \{\sigma : [0,T] \to \{1,2,...,N\}, \text{ The solution of \dot{x}(t) = A_{\sigma(t)}x(t) reaches S before time T}\}$$

Assume that $$\Sigma$$ is non-empty. Now, for each $$\sigma \in \Sigma$$, the number of times it switches is its number of discontinuities, so define $$D(\sigma) = n$$ if $$\sigma$$ has $$n$$ discontinuities, and $$D(\sigma) = \infty$$ if $$\sigma$$ has infinitely many discontinuities.

Is there literature available on studying $$\min_{\sigma \in \Sigma}D(\sigma)$$?

In words : Assuming there is a control steering $$\mathbf x$$ to $$S$$, there is a control with at most $$\_\_\_$$ switches also steering $$\mathbf x$$ to $$S$$.

I would like to know if there are papers/textbooks which discuss this exact question, in the context of linear switched systems. I am aware of Pontryagin's maximum principle, but do not know how it would provide structural results for this question, so if anybody could suggest an approach via PMP I would be happy as well.

• I think you should rather look into dynamic programming. PMP is not likely to bring you any further regarding the switching structure. Commented Nov 13, 2020 at 19:12
• Somewhat relevant: math.stackexchange.com/questions/1187337/… Commented Nov 13, 2020 at 23:42
• @Dmitry Thank you for your reply. I shall attempt dynamic programming. But the point is that I need a structural theorem of the kind I mentioned. While DP can help me solve a specific problem, will I be able to use it to write theorems? I have seen it being done (e.g. with monotone operators etc. coming into play) but over there the structure is not of a switched linear system but rather of an ordinary linear system which is easier to analyse. But thank you for your comment. Commented Nov 14, 2020 at 3:29
• @Calculon Thank you for the question. It does have a similarity to the question I ask because I too expect to consider products of matrices of the form $e^{A_it_i}$ for $t_i$ being positive real numbers, but what I search for is whether the ranges of these intersect $S$ for a certain product, and what is the minimal number of matrices involved in that product. I will still look into the kernel stabilizing property, it should be giving me some structure on the problem. Thank you. Commented Nov 14, 2020 at 3:35