2004/03/12 by Paolo Mason, Mason, Paolo, Ugo Boscain +3
Mathematics · #37N35 #93D20 #FOS: Mathematics #Optimization and Control (math.OC) #math.OC #msc:37N35 #msc:93D20
paper · pdf · doi:10.48550/arxiv.math/0403209
18 pages, 3 figures
arxiv created 2004/08/10 · arxiv updated 2009/12/01
In this paper, we consider linear switched systems x(t)=Au(t) x(t), x∈\Rn, u∈ 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.