TA01 10:20 zy Proceedings zyxwvutsrqponm of the 36th Conference on Decision zyxwvutsrqpon & Control San Diego, Califomla USA December 1997 New criteria for exponential and uniform asymptotic stability of nonlinear time-variant differential equations Dirk Aeyels Joan Peuteman SYSTeMS Universiteit Gent Technologiepark-Zwijnaarde, 9 9052 GENT (Zwijnaarde) BELGIUM e-mail : Dirk.AeyelsQrug.ac.be e-mail : Joan.PeutemanQrug. ac. be Abstract Within the Liapunov framework, sufficient conditions for exponential and uniform asymptotic stability of or- dinary differential equations are proposed. Unlike with classical Liapunov theory, the time derivative of the Li- apunov function, taken along solutions of the system, may have positive and negative values. Verification of the conditions of the main theorems may be harder than in the classical case. It is shown that the proposed conditions are useful for the investigation of the expo- nential stability of fast time-varying systems. This sets the stability study by means of averaging in a Liapunov context. In particular, it is established that exponen- tial stability of the averaged system implies exponential stability of the original fast time-varying system. zyxwvu 1 Introduction The classical Liapunov approach to uniform asymp- totic stability of the null solution of a dynamical sys- tem i(t) = zyxwvuts f(z,t) requires the existence of a positive definite, decrescent Liapunov function zyxwvutsr V zyxwvu (x, t) whose derivative along the solutions of the system is nega- tive definite. When this derivative is negative semi- definite, stability rather than asymptotic stability fol- lows. When the derivative of the Liapunov function is negative semi-definite and the differential equation is autonomous, the Barbashin-Krasovskii theorem or LaSalle's invariance principle may be helpful in prov- ing asymptotic stability; extension to the periodic case is possible. For nonperiodic systems, Narendra and An- naswamy [7] show that with V(z,t) i 0 uniform asym- totic stability can be proven if there exists a T such that where zyxwvutsr y(.) is a strictly increasing continuous function on R+ which is zero at the origin, and where x(t + T) is the solution of the system at t + T with initial con- Vt : V(z(t + T),t + T) - V(~(t),t) 5 -y(llz(t)ll) < 0 dition z(t) at t. Weaker conditions than the Narendra- Annaswamy conditions still implying asymptotic sta- bility have recently been obtained [l, 2, 31. The present paper and [3] are to a great deal inspired by the result of Narendra-Annaswamy. In the present paper, we ba- sically claim that in the asymptotic stability theorem of Narendra-Annaswamy thre negative semi-definiteness condition on the time-derivative o,f th,e Liapunov func- tion can be dispensed with: the origin of a dynami- cal system is uniformly asymptotically stable under the condition that for a positive definite decrescent V(x,t), 3T > 0 and a sequence of times ti such that V(~(ti+~), - V(z(ti), ti) F -r(llz(ti)ll) where y(.) is a strictly increasing continuous function on R+ which is zero at the origin and tz+l-t; 5 T zyx 'dk E Z and ti -+ 03 as k --+ 03 and ti -+ -03 as k -+ -03. Com- pared to [7], V(z,t) 5 0 is no longer required. Com- pared to [3], the asymptotic stability condition needs to be satisfied only for a sequence tz, not for all t. In the present paper, we also show that the origin of a dynamical system is exponentially stable under the condition that for a positive definite decrescent V(x, t) with XminzTz 5 V(x,t) 5 Xm,azxTx: (Xm?:n,Xm,az E RZ), 3T > 0 and a sequence of times ti such that V(x(t~+l),t~+l) - V(x(ti),ti) 5 -~l/z(tz)11~ with zy v E R: and ti+l - ti 5 T Vk E Z and ti --f 03 as k + 03 and ti -+ -03 as k -+ -03. In Section 2 of this paper, the main theorems give new sufficient conditions which guarantee exponential or uniform asymptotic stability of a nonlinear dynam- ical system. In Section 3, exponential stability of fast time-varying systems is discussed using the main the- orem of Section 2 for exponential stability. In Section 4, it is established that exponential stability of the av- eraged system guarantees exponential stability of the original fast time-varying system. The way of prov- ing this result is different from the approach which can be found in the literature. In Section 5, the results of this paper are compared with the results found in the 0-7803-3970-8197 $10.00 0 1997 IEEE 1712