An extension of LaSalle's invariance principle for switched systems

In this paper we address invariance principles for a certain class of switched nonlinear systems. We provide an extension of LaSalle's invariance principle for these systems and state asymptotic stability criteria. We also present some related results on the compactness of the trajectories of these switched systems.     

Some results on the stabilization of switched systems

This paper deals with the stabilization of switched systems with respect to (w.r.t.) compact sets. We show that the switched system is stabilizable w.r.t. a compact set by means of a family of switched signals if and only if a certain control affine system whose admissible controls take values in a polytope is asymptotically controllable to that set. In addition we present a control algorithm that based on a family of open-loop controls which stabilizes the aforementioned control system, a model of the system and the states of the switched system, generates switching signals which stabilize the switched system in a practical sense. We also give results about the convergence and the robustness of the algorithm.     

A converse Lyapunov theorem for nonlinear switched systems

In this paper we present a converse Lyapunov theorem for uniform asymptotic stability of switched nonlinear systems. Its proof is a simple consequence of some results on converse Lyapunov theorems for systems with bounded disturbances obtained by Lin et al. (SIAM J. Control Optim. 34 (1996) 124–160), once an association of the switched system with a nonlinear system with disturbances is established.     

A condition for the stability of switched nonlinear systems

In this paper, we present a sufficient condition for the global asymptotic stability of a switched nonlinear system composed of a finite family of subsystems. We show that the global asymptotic stability of each subsystem and the pairwise commutation of the vector fields that define the subsystems (i.e., the Lie bracket of any pair of them is zero) are sufficient for the global asymptotic stability of the switched system. We also show that these conditions are sufficient for the existence of a common Lyapunov function.     

On converse Lyapunov theorems for ISS and iISS switched nonlinear systems

In this paper we present converse Lyapunov theorems for ISS and integral input to state stable (iISS) switched nonlinear systems. Their proofs are based on existing converse Lyapunov theorems for input–output to state stable and iISS nonlinear systems, and on the association of the switched system with a nonlinear system with inputs and disturbances that take values in a compact set.