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.
