Stability of Planar Nonlinear Switched Systems

Abstract

We consider the time-dependent nonlinear system $\dot{q}(t)=u(t)X(q(t))+(1-u(t))Y(q(t))$, where $q\in {\mathbb{R}}^2$, $X$ and $Y$ are two %$C^{\infty }$ smooth vector fields, globally asymptotically stable at the origin and $u:[0,\infty )\to \{0,1\}$ 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.