A Krasovskii-LaSalle theorem for arbitrarily switched systems

This paper concerns uniform global asymptotic stability (UGAS) for arbitrarily switched systems. A novel detectability condition is first proposed to guarantee a persistently exciting condition. The UGAS property can then be established for a class of nonlinear and time-varying switched systems that under the constraint of zeroing output have common (non-switching) limiting systems. A recently derived stability criterion is shown to be a special case of the proposed result. When applied to time-invariant systems, the well-known Krasovskii-LaSalle theorem can be deduced. Furthermore, we do not require dwell-time assumptions that are usually assumed in those results generalizing LaSalle invariance principle, thus providing more flexibility for the controller design. An interesting example is employed to demonstrate how the proposed result can be applied to guarantee the UGAS property without finding a common Lyapunov function that is normally not an easy job. This highlights the significance of the achieved results.