On Switching Stabilizability for Continuous-Time Switched Linear Systems

This technical note studies switching stabilization problems for continuous-time switched linear systems. We consider four types of switching stabilizability defined under different assumptions on the switching control input. The most general switching stabilizability is defined as the existence of a measurable switching signal under which the resulting time-varying system is asymptotically stable. Discrete switching stabilizability is defined similarly but requires the switching signal to be piecewise constant on intervals of uniform length. In addition, we define feedback stabilizability in Filippov sense (respectively, sample-and-hold sense) as the existence of a feedback law under which closed-loop Filippov solution (respectively, sample-and-hold solution) is asymptotically stable. It is proved that the four switching stabilizability notions are equivalent and their sufficient and necessary condition is the existence of a piecewise quadratic control-Lyapunov function that can be expressed as the pointwise minimum of a finite number of quadratic functions.