$\dot x= A_{\sigma}x+B_{\sigma}u$ where $\sigma:\mathbb{R}\to\{1,2,\dots,p\}\subsetneq\mathbb{N}$ is the switching signal, $A_i$ is $n\times n$, $B_i$ is $n\times m$ matrix, $p$ is the number of different subsystems, $x(t):\mathbb{R}\to\mathbb{R}^n$ is a state variable, $u:\mathbb{R}\to\mathbb{R}^m$ is the input. In other words, the system is modeled as a time varying linear differential equation whose co-efficient matrices are piecewise constant. The time variance follows from the action of the switches present in the system.
In general switches or component faults induce jumps in certain state variables, in order to allow jumps in the solution we can make our solution space in distributional solution framework. So, in general solutions may consists of Dirac impulses and/or their derivatives?
Now my question is how to get a globally defined solution for the switched ODE for a given $\sigma$?