## A converse Lyapunov theorem for a class of dynamical systems which undergo switching.(English)Zbl 0960.93046

The paper considers mainly linear polysystems arising from dynamical systems undergoing switches $$\dot x= A_\gamma x$$, where $$\gamma\in \Gamma$$ is the range of the swtiching path $$s:\mathbb{R}_+\to\Gamma$$, $$s(\cdot)$$ being a piecewise constant function. For such systems converse Lyapunov theorems are obtained in the case of exponential stability. The interesting feature of the paper is that sometimes the Lyapunov function may fail to be quadratic and an upper bound on its degree has not yet been found.

### MSC:

 93D20 Asymptotic stability in control theory 34D20 Stability of solutions to ordinary differential equations 93D30 Lyapunov and storage functions
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