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Common Polynomial Lyapunov Functions for Linear Switched Systems

TitleCommon Polynomial Lyapunov Functions for Linear Switched Systems
Publication TypeJournal Article
Year of Publication2006
AuthorsMason, P, Boscain, U, Chitour, Y
JournalSIAM J. Control Optim. 45 (2006) 226-245

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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