Stability of Planar Switched Systems: the Nondiagonalizable Case

Moussa Balde (1) , Ugo Boscain (2)

1 Département de Mathématiques et Informatique, Faculté des Sciences et Techniques
2 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies

Consider the planar linear switched system $\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where $A$ and $B$ are two $2\times2$ real matrices, $x\in~\R^2$, and $u(.):[0,\infty[\to\{0,1\}$ is a measurable function. In this paper we consider the problem of finding a (coordinate-invariant) necessary and sufficient condition on $A$ and $B$ under which the system is asymptotically stable for arbitrary switching functions $u(.)$. This problem was solved in previous works under the assumption that both $A$ and $B$ are diagonalizable. In this paper we conclude this study, by providing a necessary and sufficient condition for asymptotic stability in the case in which $A$ and/or $B$ are not diagonalizable. To this purpose we build suitable normal forms for $A$ and $B$ containing coordinate invariant parameters. A necessary and sufficient condition is then found without looking for a common Lyapunov function but using "worst-trajectory'' type arguments.

Mots clés: switched systems; planar; arbitrary switchings; worst-trajectory; worst-trajectory.

Domaines: Optimisation et contrôle [math.OC]

Moussa Balde, Ugo Boscain. Stability of Planar Switched Systems: the Nondiagonalizable Case. 2006. ⟨hal-00105866⟩