"Spanning" of solutions of ordinary differential equations Suppose we have a switched ODE
$$\dot{x} = A_{\sigma(t)}x,$$
where $A_{\sigma(t)}$ is a constant matrix given $\sigma(t)\in\mathcal{M}=\{1,2,\cdots,m\}$.
If we fix the initial condition and can arbitrarily switch the signal $\sigma(t)$, what would the reachable set of $x(t)$ look like?
Intuitively, I think it is something like the spanning of the solutions of $\dot{x} = A_1x, \dot{x} = A_2x, \cdots,\dot{x} = A_mx$. I wonder whether there are tools available for analyzing such a space.

It is not true in general that the spanning of the switched system can be obtained from the spanning of the individual systems. Actually even if all individual systems are asymptotically stable the switched system can become unstable from the switching.
Example, taken from Liberzon, D. Switching in systems and control. Vol. 190. Boston: Birkhauser, 2003.
$\dot{x}(t)=A_1x(t)$ and $\dot{x}(t) = A_2x(t)$ with
$$
A_1=\begin{pmatrix}-0.1&-1\\2&-0.1\end{pmatrix}, A_2=\begin{pmatrix}-0.1&-2\\1&-0.1\end{pmatrix}
$$
Both matrices are Hurwitz as they have eigenvalues $-0.1 \pm j\sqrt{2}$.
But now take $\dot{x}(t)=A_{\sigma(t)}x(t)$ and as switching signal use
$$
\sigma(t)=\begin{cases}1& \lfloor t \rfloor \text{ is even}\\2& \lfloor t \rfloor \text{ is odd}\end{cases}
$$
where $\lfloor t \rfloor $ is the floor function. Then take $x_1(0)=1,x_2(0)=2$ for example. After a quick simulation you get:

You can see that although the individual systems are stable the switched system is not and so also the spanning of the switched system is completly different than the spanning of the individual systems.
