Stability of Nonlinear Switched Systems on the Plane

We consider the time-dependent nonlinear system q̇(t) = u(t)X(q(t)) + (1 - u(t))Y(q(t)), where q ∈ R2, X and Y are two smooth vector fields, globally asymptotically stable at the origin, and u: [0, ∞) ↦ {0, 1} is an arbitrary measurable function. Analysing the topology of the set where X and Y are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to u(.). Such conditions can be verified without any integration or construction of a Lyapunov function, and they are robust under small perturbations of the vector fields.