Stability of planar nonlinear switched systems

Abstract

Let X and Y be two smooth vector fields on ℝ2, globally asymptotically stable at the origin, and consider the time-dependent nonlinear system q(t) = u(t)X(q(t)) + (1 - u(t))Y(q(t)), where u : [0, ∞) → {0, 1} is an arbitrary measurable function. Analyzing 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 do not change under small perturbations of the vector fields.