Stability of planar switched systems: The nondiagonalizable case

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.