Copyright 0 IF AC System Structure and Control, Nantes, France, 1998 AVERAGING RESULTS FOR HOMOGENEOUS DIFFERENTIAL EQUATIONS J oan Peuteman, Dirk Aeyels SYSTeMS Universiteit Gent Technologiepark-Zwijnaarde, 9 9052 GENT (Zwijnaarde) BELGIUM e-mail : Joan.Peuteman@rug.ac. be e-mail : Dirk.Aeyels@rug.ac. be Abstract : Within the Liapunov framework, a sufficient condition for uniform asymp- totic stability of ordinary differential equations is proposed. Unlike with classical Liapunov theory, the time derivative of the Liapunov function, taken along solutions of the system, may have positive and negative values. It is shown that the proposed condition is useful for the study of uniform asymptotic stability of homogeneous systems with order T > O. In particular, it is established that asymptotic stability of the averaged homogeneous system implies local uniform asymptotic stability of the original time-varying homogeneous system. This shows that averaging techniques may play a prominent role in the study of homogeneous -not necessarily fast time- varying- systems. Serniglobal stability results may be obtained by 'speeding up' the system by means of a change of time-scale. Copyright © 1998IFAC Resume: Dans le contexte de Liapunov, nous proposons une condition suffisante de stabilite uniformement asymptotique pour une equation differentielle ordinaire. La derivee de la function de Liapunov, dans la direction des solutions du systeme, peut prendre des valeurs positives et negatives. Cette condition va servir a. etudier la stabilite uniformement asymptotique des systemes homogenes avec un degre r > O. En particulier, nous montrons que la stabilite asymptotique du systeme homogene moyennise implique la stabilite uniformement asymptotique dans le voisinage du point d'equilibre. Ce result at montre que les techniques de moyennisation permettent d' etudier les systemes homogenes a. l'aide d'un champ de vecteurs dont la dependence du temps n'est pas necessairement acceleree. Des proprietes de stabilite semi-globale sont obtenues si on accelere l'axe reel du temps du champ de vecteurs. Keywords: Nonlinear systems, time-varying systems, asymptotic stability, averaging control. 1. INTRODUCTION The classical Liapunov approach to uniform asymp- totic stability of the null solution of a dynamical system x(t) = !(x, t) requires the existence of a positive definite, decrescent Liapunov function 65 V(x, t) whose derivative along the solutions of the system is negative definite. When this deriva- tive is negative semi-definite, stability rather than asymptotic stability follows. When the derivative of the Liapunov function is negative semi-definite and the differential equation is autonomous, the