# What are the upper bound and stability conditions for the following simple linear system?

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).)

This is a more general case than the one of a switched dynamical system: 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).
I realize I came across this question late, but decided to answer in case it's of any help. Thinking the original dynamics as a linear time varying system $\dot{x}(t) = A(t) x(t)$ with $A(t) = \sum_{i=1}^{m}\alpha_{i}(t)A_{i}$, the state can be solved as $x(t) = \Phi\left(t,0\right)x_{0}$, where $x_{0}$ is the initial state, and $\Phi\left(t,\tau\right)$ is the state transition matrix. It is well known that if $A(t)A(\tau) = A(\tau)A(t)$ for all $t, \tau$, then the state transition matrix $\Phi(t,\tau) = \exp\left(\int_{\tau}^{t}A(\sigma)d\sigma\right)$.
If the OP's matrices $A_{i}$ pairwise commute, then indeed $A(t)A(\tau) = A(\tau)A(t)$, and the state transition matrix $\Phi(t,0) = \prod_{i=1}^{m}\exp\left(A_{i}\beta_{i}(t)\right) = \exp\left(\sum_{i=1}^{m}A_{i}\beta_{i}(t)\right)$, where $\beta_{i}(t) = \int_{0}^{t}\alpha_{i}(\sigma)d\sigma > 0$. The stability condition then boils down to the norm of the state transition matrix being bounded: $\parallel \Phi(t,0)\parallel \leq \gamma$ (uniform stability), $\parallel \Phi(t,0)\parallel \leq \gamma \exp\left(-\delta t\right)$, for some $\gamma, \delta > 0$ (uniform asymptotic stability).