Stability of planar switched systems: The nondiagonalizable case

Abstract

Consider the planar linear switched system ẋ(t) = u(t)Ax(t) + (1 - u(t))Bx(t), where A and B axe 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.