Consider the following linear system
$$\dot{x}=\sum\limits\_{i=1}^{m}{{{\alpha }\_{i}}}\left( t \right)\cdot {{A}\_{i}}\cdot x \quad (1)
$$  

where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, $\sum\limits\_{i=1}^{m}{{{\alpha }\_{i}}}\left( t \right)=1$, ${{\alpha }\_{i}}\left( t \right)$are functions and ${{\alpha }\_{i}}\left( t \right)>0$, ${{A}\_{i}}\in {{\mathbb{R}}^{n\times n}}$ are constant matrices, $i=1,\cdots ,m$. 
Once system (1) is stable , then the Euclidean norm of vector $x$ is bounded. Now, the questions are as the following: 1) what’s the upper bound for $\left\| x \right\|$ when system (1) is stable; 2) what’s the stability condition for system (1)? (Lyapunov function may be an appropriate and conventional method to derive the conditions for the stability of system (1).)



What are the upper bound and stability conditions for the following simple linear system?
This is a more general case than the one of a [switched dynamical system](https://en.wikipedia.org/wiki/Joint_spectral_radius#Applications): it is obtained as the case in which $\alpha\_i(t)$ are constant along discrete time steps of length $h$, and at each of these time steps one of the $\alpha\_i(t)$ is one and all the other ones are zero).


As you can read a few lines above on the same Wikipedia page, that special case already is an undecidable problem, and even approximation algorithms are NP-hard.


