Stability Theory by Liapunov’s Direct Method

Rouche, N.; Habets, Patrick; Laloy, M.

doi:10.1007/978-1-4684-9362-7

Cited by 895 publications

(397 citation statements)

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“…The classical Lyapunov direct method is a powerful tool for stability analysis of both continuous-time and discrete-time processes (Refs. [22][23][24]. Roughly speaking, this approach reduces the analysis of stability properties of a process to the analysis of local improvement of this process with respect to some scalar criterion V(-), usually called the Lyapunov function.…”

Section: Generalized Lyapunov Direct Methodsmentioning

confidence: 99%

Error Stability Properties of Generalized Gradient-Type Algorithms

Solodov

Zavries

1998

Journal of Optimization Theory and Applications

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Abstract. We present a unified framework for convergence analysis of generalized subgradient-type algorithms in the presence of perturbations. A principal novel feature of our analysis is that perturbations need not tend to zero in the limit. It is established that the iterates of the algorithms are attracted, in a certain sense, to an e-stationary set of the problem, where e depends on the magnitude of perturbations. Characterization of the attraction sets is given in the general (nonsmooth and nonconvex) case. The results are further strengthened for convex, weakly sharp, and strongly convex problems. Our analysis extends and unifies previously known results on convergence and stability properties of gradient and subgradient methods, including their incremental, parallel, and heavy ball modifications.

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Section: Generalized Lyapunov Direct Methodsmentioning

confidence: 99%

Error Stability Properties of Generalized Gradient-Type Algorithms

Solodov

Zavries

1998

Journal of Optimization Theory and Applications

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“…If this is the case, then a related result also due to Malkin (see [4]) shows that an asymptotic property persists, in the sense that, as time increases, the neighborhood B ǫ may be taken progressively smaller. Liapunov's method can also be used in the more general setting of time dependent systems which are not perturbations of autonomous systems (see [4]). Given a smooth non-autonomous systemẋ = f (x, t), x ∈ lR n , t ∈ lR, and an equilibrium point x 0 such that f (x 0 , t) = 0, if there exists a smooth function…”

Section: Reminder Of the Basic Theoremsmentioning

confidence: 78%

“…This theorem guarantees the stability of x 0 for any sufficiently small time dependent perturbation, but it does not imply that the solution x(t; x(t 0 ), t 0 ) tends to x 0 when t → ∞, since it does not even require that g(x, t) should vanish as t → ∞. If this is the case, then a related result also due to Malkin (see [4]) shows that an asymptotic property persists, in the sense that, as time increases, the neighborhood B ǫ may be taken progressively smaller. Liapunov's method can also be used in the more general setting of time dependent systems which are not perturbations of autonomous systems (see [4]).…”

Section: Reminder Of the Basic Theoremsmentioning

confidence: 99%

Stability analysis of cosmological models through Lyapunov's method

Charters¹,

Nunes²,

Mimoso³

2001

Class. Quantum Grav.

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We investigate the general asymptotic behaviour of Friedman-RobertsonWalker (FRW) models with an inflaton field, scalar-tensor FRW cosmological models and diagonal Bianchi-IX models by means of Liapunov's method. This method provides information not only about the asymptotic stability of a given equilibrium point but also about its basin of attraction. This cannot be obtained by the usual methods found in the literature, such as linear stability analysis or first order perturbation techniques. Moreover, Liapunov's method is also applicable to non-autonomous systems. We use this advantadge to investigate the mechanism of reheating for the inflaton field in FRW models.

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“…Rouche, Habets and Laloy (1977) for a treatment of set stability. The above condition is not the most general, but sufficient for our purpose.…”

Section: State Feedback Set Stabilizationmentioning

confidence: 99%

“…Recalling that asymptotic stability is equivalent to stability and convergence (Rouche et al, 1977), this result is obtained by noting that the semidefiniteness of the derivative of the Lyapunov function implies (set) stability, while convergence follows from LaSalle's invariance principle (LaSalle, 1960).…”

Section: State Feedback Set Stabilizationmentioning

confidence: 99%