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On exponential stability of nonlinear time-varying differential equations

On exponential stability of nonlinear time-varying differential equations,10.1016/S0005-1098(99)00012-6,Automatica,Dirk Aeyels,Joan Peuteman

On exponential stability of nonlinear time-varying differential equations   (Citations: 31)
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Within the Liapunov framework, a sufficient condition for exponential 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. Verification of the conditions of the main theorem may be harder than in the classical case. It is shown that the proposed conditions are useful for the investigation of the exponential 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 exponential stability of the averaged system implies exponential stability of the original fast time-varying system. A comparison of our work with results taken from the literature is included.
Journal: Automatica , vol. 35, no. 6, pp. 1091-1100, 1999
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    • ...(see, e.g., [1, 2, 7, 10, 14, 16]) do not seem well-adapted to persistently excited systems, at least for...

    Yacine Chitouret al. On the Stabilization of Persistently Excited Linear Systems

    • ...More refined tools as multiple and non-monotone Lyapunov functions (see, e.g., [1, 2, 7, 10, 13, 15]) do not seem well-adapted to permanently excited systems, at least for...

    Yacine Chitouret al. On the stabilization of permanently excited linear systems

    • ...Using local charts around p we see now that the conditions of Theorem 3 in [2] hold for pair (F;F ) and hence the time-varying systemF (x;t ) is asymptotically stable for sufficiently large . A change of timescales yields asymptotic stability of Cs for the time-varying system (5) for sufficiently small "...
    • ...One can give quantitative estimates for a sufficiently small " based on the averaging theory [2]...

    Christian Lagemanet al. Synchronization with partial state feedback on SO(n)

    • ...In [39], this condition is relaxed and it is shown that if the Lyapunov function decreases when evaluated at a discrete sequence of time instants, the system is asymptotically stable...
    • ...The following Theorem extends the results of [39] from the deterministic to the stochastic case and it is used in what follows to establish our main claim...

    Maurizio Porfiriet al. Global stochastic synchronization of chaotic oscillators

    • ...Using recent results on partial averaging tecniques [33], [34], well-established global synchronization criteria based on Lyapunov-stability theory [25], [29], and the concept of matrix measure [35], we establish sufficient conditions for pulse synchronization...
    • ...The proof of our claim combines results from global synchronization of coupled oscillators, based on Lyapunovstability theory [25], [29], with stability results from partial averaging techniques [33], [34]...
    • ...In particular, we build on the use of a quadratic Lyapunov function for chaotic systems coupled via a time-invariant coupling [25], [29] and on the stability results established in [33], [34] that are valid for more general Lyapunov functions...
    • ...Nevertheless, global exponential stability of (9a) can be enforced by using the weaker stability conditions presented in [33]...
    • ...Indeed Theorem 1 of [33], applied to the case at hand, states that if there exists ” > 0 such that for any k 2Z + , V (e((k+1)T))¡V (e(kT)) • ¡”ke(kT)k 2 , where e((k+1)T) is the solution of (9a) with initial condition e(kT) at t = kT , then (9a) is globally exponentially stable...
    • ...In what follows, we show that there exists a finite switching period T ⁄ such that conditions of Theorem 1 of [33] apply for any T < T ⁄ . To this aim, we define for every k 2Z +...
    • ...implies that (9a) is globally exponentially stable according to Theorem 1 of [33]...

    Francesca Fiorilliet al. Fast-switching pulse synchronization of chaotic oscillators

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