Mathematics of Control, Signals and Systems

, Volume 15, Issue 1, pp 42-70

Exponential Stability of Nonlinear Time-Varying Differential Equations and Partial Averaging

  • Joan PeutemanAffiliated withSYSTeMS, Universiteit Gent, Technologiepark-Zwijnaarde 9, 9052 Gent (Zwijnaarde), Belgium.¶Joan Peuteman is presently working at the KHBO, Departement Industrieële Wetenschappen en Technologie, Zeedijk 101, 8400 Oostende, Belgium. Joan.Peuteman@kh.khbo.be.
  • , Dirk AeyelsAffiliated withSYSTeMS, Universiteit Gent, Technologiepark-Zwijnaarde 9, 9052 Gent (Zwijnaarde), Belgium.

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Abstract.

In this paper we formulate, within the Liapunov framework, a sufficient condition for exponential stability of a differential equation. This condition gives rise to a new averaging result referred to as “partial averaging”: exponential stability of a system \(\), with α sufficiently large, is implied by exponential stability of a time-varying system \(\).

Key words. Differential equations, Exponential stability, Liapunov stability, Averaging, Circle criterion.