Stability of planar nonlinear switched systems

We study the global asymptotic stability of the time dependent nonlinear system x(t)=u(t)F(x(t))+(1-u(t))G(x(t)), where x ε R², F(x) and G(x) are two C^∞ vector fields, globally asymptotically stable at the origin and u(.) : [0, ∞[→ [0,1] is a completely random measurable function. We give a sufficient and a necessary condition for global asymptotic stability. This result extend some of our previous results obtained in the linear case.