On Global Uniform Asymptotic Stability of Nonlinear Time-varying Systems in Cascade

In this short paper we deal with the stability analysis problem of non-autonomous non-linear systems, in cascade. In particular we give suucient conditions to guarantee that: (i) a globally uniformly stable (GUS) nonlinear time-varying (NLTV) system remains GUS when it is perturbed by the output of a globally uniformly asymptotically stable (GUAS) NLTV system, under the assumption that the perturbing signal is absolutely integrable; (ii) if in addition the perturbed system is GUAS, it remains GUAS under the cascaded interconnection; (iii) two GUAS systems yield a GUAS cascaded system, under some growth restrictions over the Lyapunov function. Our proofs rely on the second method of Lyapunov, roughly speaking on a \ ? stability analysis". 1 Notation. In this paper the solution of a diierential equation _ x = f(t; x) where f : IR 0 IR n ! IR n , with initial conditions (t 0 ; x 0) 2 IR 0 IR n , is denoted x(t; t 0 ; x 0) or simply x(t) 1. We say that the system _ x = f(t; x), is globally uniformly stable (resp. GUAS) if the trivial solution x(t; t 0 ; x 0) = 0 is globally uniformly stable (resp. GUAS). A continuous function : IR 0 ! IR 0 is said to be of class K, 2 K, if (x) is strictly increasing and (0) = 0; 2 K 1 if in addition (x) ! 1 as x ! 1. A continuous function (t; x) : IR 0 IR 0 ! IR 0 is of class KL if (t;) 2 K for each xed t 0 and (t; x) ! 0 as t ! 1 for each x 0. We also deene n 4 = 1; : : :; n]. In this paper kkk is the usual Euclidean norm of vectors. The space of integrable and square integrable vector functions of dimension n, is denoted L n 1 0; 1) and L n 2 0; 1) respectively; though for simplicity, we may also write L n 1 , L n 2. Correspondingly, the space of bounded functions is denoted L n 1. _ V (#) (t; x) is the time derivative of Lyapunov function V (t; x) along the trajectories represented by the diierential equation (#).