A note on stability conditions for planar switched systems

This paper is concerned with the stability problem for the planar linear switched system $\dot{x}(t)=u(t)A_{1}x(t)+(1-u(t))A_{2}x(t)$, where the real matrices $A_1,A_2\in {\mathbb{R}}^{2\times 2}$ are Hurwitz and $u(\cdot) [0,\infty[\to{0,1}$ is a measurable function. We give coordinate-invariant necessary and sufficient conditions on $A_1$ and $A_2$ under which the system is asymptotically stable for arbitrary switching functions $u(\cdot )$. The new conditions unify those given in previous papers and are simpler to be verified since we are reduced to study 4 cases instead of 20. Most of the cases are analyzed in terms of the function $\Gamma(A_1,A_2)={1/2}(\tr(A_1) \tr(A_2)- \tr(A_1A_2))$.