Stability of switched systems: The 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 ϵ 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.