On the stability and stabilization of time-varying nonlinear control systems

Many stability and stabilization problems of nonlinear time-varying systems lead to asymptotic behavior of (-K + L)-type systems, which consist of a K-function and an L-function. The stability of these systems is of fundamental importance for a series of stabilization problems of time-varying nonlinear control systems. Even though the asymptotical stability of such systems has been used widely and (in most cases) implicitly, we do not find a rigorous proof, in the literature, and the existing proof for a particular case is questionable. Under quite general conditions, we prove that the solution of these systems tends to 0 as t -> infinity. Some generalizations are also obtained. As an immediate consequence, a general theorem is obtained for the stabilization of time-varying systems. Using the new framework, we examine several stability and stabilization problems. First of all, for cascade systems, two sets of sufficient conditions are obtained for uniformly asymptotical stability and globally asymptotical stability, respectively. Then we consider the stability of ISS and IISS systems. A new concept, namely, strong IISS, is proposed. Several stability properties for autonomous systems are extended to time-varying systems. Finally, we consider stabilization via detection. A rigorous proof is given for a smooth state feedback time-varying system with weak detectability to be stabilizable by means of an observer.