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January  2008, 7(1): 1-21. doi: 10.3934/cpaa.2008.7.1

Stability of planar switched systems: The nondiagonalizable case

Moussa Balde 1, and  Ugo Boscain 2,

1. 

LMDAN-LGDA Dept. de Mathématiques et Informatique, UCAD, Dakar-Fann, Senegal
2. 

Le2i, CNRS, Université de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France

Received  January 2007 Revised  October 2007 Published  October 2007

Consider the planar linear switched system $\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where $A$ and $B$ are two $2\times 2$ real matrices, $x\in \mathbb R^2$, 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.
Keywords: arbitrary switchings, worst-trajectory., Switched systems, planar.
Mathematics Subject Classification: Primary: 93D20; Secondary: 37N3.
Citation: Moussa Balde, Ugo Boscain. Stability of planar switched systems: The nondiagonalizable case. Communications on Pure and Applied Analysis, 2008, 7 (1) : 1-21. doi: 10.3934/cpaa.2008.7.1
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