Common Polynomial Lyapunov Functions for Linear Switched Systems

Details Common Polynomial Lyapunov Functions for Linear Switched Systems Common Polynomial Lyapunov Functions for Linear Switched Systems

2004-03-12

arxiv.org/abs/math/0403209v2

Mathematics

Optimization and Control

18 pages, 3 figures

Scientific paper

In this paper, we consider linear switched systems $\dot x(t)=A_{u(t)} x(t)$, $x\in\R^n$, $u\in U$, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching ({\bf UAS} for short). We first prove that, given a {\bf UAS} system, it is always possible to build a common polynomial Lyapunov function. Then our main result is that the degree of that common polynomial Lyapunov function is not uniformly bounded over all the {\bf UAS} systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given.

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