2006/10/12 by Moussa Balde, Balde, Moussa, Ugo Boscain +1
Mathematics · #37N35 #93D20 #FOS: Mathematics #Optimization and Control (math.OC) #math.OC #msc:37N35 #msc:93D20
paper · pdf · doi:10.48550/arxiv.math/0610401
arxiv created 2006/10/12 · arxiv updated 2009/12/01
Consider the planar linear switched system x(t)=u(t)Ax(t)+(1-u(t))Bx(t), where A and B are two 2×2 real matrices, x ∈ \R2, and u(.):[0,∞[→\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.