Stability of planar switched systems: the linear single, input case

We study the stability of the origin for the dynamical system, x˙(t)=u(t)Ax(t)+(1-u(t))Bx(t), where A and B are two 2×2 real matrices with eigenvalues having strictly negative real part, x∈R² and u(.):[0, ∞[→ [0, 1] is a completely random measurable function. More precisely, we find a (coordinates invariant) necessary and sufficient condition on A and B for the origin to be asymptotically stable for each function u(.). This bidimensional problem assumes particular interest since linear systems of higher dimensions can be reduced to our situation. Two unpublished examples in the (more difficult) case in which both matrices have real eigenvalues are analyzed in details.