Nonautonomous attractors of switching systems

Switching systems formed by switching between a finite number of autonomous ordinary differential equations are formulated as abstract nonautonomous dynamical systems driven by the shift operator on the space of switching controls. A metric is proposed for such switching controls and the space of switching controls with a given positive dwell time is shown to be compact. This allows known results on pullback attractors for nonautonomous dynamical systems to be applied to switching systems including conditions which ensure that a pullback attractor exists and is also a forward attractor. In particular, this provides a conceptual framework for investigating the asymptotic behaviour of switching systems beyond the usual case in which the asymptotic stability of the zero solution under switching is considered.