Stability criteria for the linear system [Xdot](t) + A(t)X(t—τ) = 0 and an application to a non-linear system

Gopalsamy, K.

doi:10.1080/00207729008910503

Cited by 15 publications

(11 citation statements)

References 19 publications

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“…However, we cannot use the above inequality to conclude that the closed-loop system (2.13) is asymptotically stable by using the Lyapunov stability theorem because the Lyapunov functional V (t, x t ) defined in (2.26) is not strictly positive definite and belongs to a class of degenerated Lyapunov functionals [10]. In what follows, we will show the asymptotic stability by using Barbȃlat's lemma (Lemma A.3 in the appendix).…”

Section: Lemma 22 (Seementioning

confidence: 99%

Global and Semi-Global Stabilization of Linear Systems With Multiple Delays and Saturations in the Input

Zhou¹,

Lin²,

Duan³

2010

SIAM J. Control Optim.

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This paper studies stabilization problems for linear systems with multiple delays in the input. Two types of delays are considered. The first type of delays is constant delays, which can be arbitrarily large, while the second type is time-varying with an arbitrarily large bound. With the first type of delays, under the condition that the open loop system is absolutely controllable with all its eigenvalues on the imaginary axis, (globally) stabilizing state and output feedback laws are constructed based on the solution to a family of parametric Riccati equations, which can be obtained explicitly through the solution of a parametric linear matrix equation. With the second type of delays, under the condition that the open-loop system is absolutely controllable with all its eigenvalues on the imaginary axis being zero, (global) state and output feedback laws are explicitly constructed based on the solution to a similar family of parametric Riccati equations. When the input is also subject to magnitude saturation, it is shown that semiglobal stabilization, instead of global stabilization, can still be achieved. Numerical examples illustrate the effectiveness of the proposed approach.

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Section: Lemma 22 (Seementioning

confidence: 99%

Global and Semi-Global Stabilization of Linear Systems With Multiple Delays and Saturations in the Input

Zhou¹,

Lin²,

Duan³

2010

SIAM J. Control Optim.

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“…Motivated by the above dissatisfaction, and encouraged by the authors' recent work [27,29], where 3/2-type criteria were obtained for the delayed competitive system of Lotka-Volterra type without instantaneous negative feedback, we will establish some criteria of 3/2 type for the global attractivity of the positive equilibrium N * . Note that, owing to the lack of instantaneous negative feedback, the global attractivity of systems 'without instantaneous negative feedback' (or 'of pure-delay type') becomes much more difficult and has been studied by Gopalsamy [3], Gopalsamy and He [5], He [6][7][8], Kuang [12,13], Kuang and Smith [14,15], Smith [19], So et al . [23] and Tang and Zou [28].…”

Section: Theorem 13 Assume That (A1) Holds and Thatmentioning

confidence: 99%

Global Attractivity in a Predator–prey System With Pure Delays

Tang¹,

Zou²

2008

Proceedings of the Edinburgh Mathematical Society

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“…Recently, the stability problem of Lotka-Volterra systems with various types of delays has been studied extensively (see, for example, [13,21]). In both cases, negative feedback terms (delay dependent or independent) are required and standard Lyapunov function(al)s method and/or Razumikhin-type arguments can be used to study global stability (see [12,13,[15][16][17][20][21][22][23][24][26][27][28]). One type contains delay-independent terras (see, for example, [21,[26][27][28]30]), which dominate other intraspecific and interspecific interaction effects (with and/or without delays); the other type contains no such delay-independent dominated terms, and we refer to this type of system as pure-delay type (see, for example, [12,13,[15][16][17]20,[22][23][24]).…”

Section: Jvi(i) = Ni(t) \Bi -An / K N (T -S)ni(s)ds -A I2 K U (T -S)nmentioning

confidence: 99%

“…To overcome these difficulties, Gopalsamy [ [13]), Gopalsamy was able to derive the asymptotic stability of (1.4). Considering the difference from traditional Lyapunov functional, this type of functional defined by (1.5) was called a 'degenerate' Lyapunov functional in [12], and it has recently been developed further in [20]. Considering the difference from traditional Lyapunov functional, this type of functional defined by (1.5) was called a 'degenerate' Lyapunov functional in [12], and it has recently been developed further in [20].…”

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confidence: 99%

Degenerate Lyapunov functionals of a well-known prey–predator model with discrete delays

1999

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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It is commonly believed that, as far as stabilities are concerned, 'small delays are negligible in some modelling processes'. However, to have an affirmative answer for this common belief is still an open problem for many nonlinear equations. In this paper, the classical Lotka-Volterra prey-predator equation with discrete delaysis considered, and, by using degenerate Lyapunov functionals method, an affirmative answer to this open problem on both local and global stabilities of the prey-predator delay equations is given. It is shown that degenerate Lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of constructing such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained. However, to answer this in the affirmative is still a difficult problem in general. It is well known that, for the classical autonomous Lotka-Volterra prey-predator system,with bi, a {j > 0 (i,j = 1, 2), if it has a positive equilibrium N* = (AT*, AT^), then it must be globally asymptotically stable. In order to account for time-lag effects in this model, Cushing [4,5] considered the following prey-predator integrodifferential 'Present address:

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Stability criteria for the linear system [Xdot](t) + A(t)X(t—τ) = 0 and an application to a non-linear system

Cited by 15 publications

References 19 publications

Global and Semi-Global Stabilization of Linear Systems With Multiple Delays and Saturations in the Input

Global and Semi-Global Stabilization of Linear Systems With Multiple Delays and Saturations in the Input

Global Attractivity in a Predator–prey System With Pure Delays

Degenerate Lyapunov functionals of a well-known prey–predator model with discrete delays

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