Converse control-Lyapunov function theorems for continuous-time switched linear systems

This paper studies switching stabilization problems for continuous-time switched linear systems (SLSs). We consider the most general switching stabilizability defined as the existence of a measurable switching signal under which the resulting time-varying system is asymptotically stable. In addition, we consider feedback stabilizability in Filippov sense defined as the existence of a switching law under which the closed-loop Filippov solution is asymptotically stable. We develop sufficient and necessary conditions for switching stabilizability as well as feedback stabilizability in Filippov sense. It is proved that for continuous-time SLSs, both switching stabilizability and feedback stabilizability in Filippov sense are equivalent to the existence of a piecewise quadratic control-Lyapunov function (CLF). Furthermore, such a CLF can be obtained by taking the pointwise minimum of a finite number of quadratic functions.