Global exponential stability in Lagrange sense for recurrent neural networks with time delays

Liao, Xiaoxin; Luo, Qi; Zeng, Zhigang; Guo, Yunxia

doi:10.1016/j.nonrwa.2007.03.018

Cited by 93 publications

(52 citation statements)

References 34 publications

(54 reference statements)

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“…Definition 1 (see [57]). The neural network (2) is said to be uniformly stable in Lagrange sense, if, for any > 0, there is a positive constant = ( ) > 0 such that ‖ ( , )‖ < for any…”

Section: Assumption (A)mentioning

confidence: 99%

“…Definition 2 (see [57]). If there exists a radially unbounded and positive definite function (⋅), a nonnegative continuous function (⋅), and two positive constants and such that, for any solution ( ) of neural network (2), ( ( )) > implies ( ( )) − ≤ ( ) − for any ≥ 0 and ∈ , then the neural network (2) is said to be globally exponentially attractive (GEA) with respect to ( ( )), and the compact set Ω = { ( ) ∈ R : ( ( )) ≤ } is said to be a GEA set of (2).…”

Section: Assumption (A)mentioning

confidence: 99%

“…Definition 3 (see [57]). The neural network (2) is globally exponentially stable (GES) in Lagrange sense, if it is both uniformly stable in Lagrange sense and GEA.…”

Section: Assumption (A)mentioning

confidence: 99%

See 2 more Smart Citations

Exponential Lagrange Stability for Markovian Jump Uncertain Neural Networks with Leakage Delay and Mixed Time-Varying Delays via Impulsive Control

Yogambigai

Ali

Zhu

et al. 2018

Mathematical Problems in Engineering

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The problem of exponential Lagrange stability analysis of Markovian jump neural networks with leakage delay and mixed timevarying delays is studied in this paper. By utilizing the Lyapunov functional method, employing free-weighting matrix approach and inequality techniques in matrix form, we establish several novel stability criteria such that, for all admissible parameter uncertainties, the suggested neural network is exponentially stable in Lagrange sense. The derived criteria are expressed in terms of linear matrix inequalities (LMIs). A numerical example is provided to manifest the validity of the proposed results.

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“…Definition 1 (see [57]). The neural network (2) is said to be uniformly stable in Lagrange sense, if, for any > 0, there is a positive constant = ( ) > 0 such that ‖ ( , )‖ < for any…”

Section: Assumption (A)mentioning

confidence: 99%

Section: Assumption (A)mentioning

confidence: 99%

See 1 more Smart Citation

Exponential Lagrange Stability for Markovian Jump Uncertain Neural Networks with Leakage Delay and Mixed Time-Varying Delays via Impulsive Control

Yogambigai

Ali

Zhu

et al. 2018

Mathematical Problems in Engineering

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show abstract

“…It is worth to mention that unlike Lyapunov stability, Lagrange stability refers to the stability of the total system, rather than the stability of the equilibriums, because the Lagrange stability is considered on the basis of the boundedness of solutions, which depend on the existence of global attractive sets (see [12,18,19,[21][22][23][24][25]). We also note that Lagrange stability has attracted phenomenal worldwide attention.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the study of Lagrange stability of neural networks with delays is practically required, and it has been extensively studied. For example, in [18,19], Liao et al apply Lyapunov functions to study Lagrange stability for recurrent neural networks with constant time delays and time-varying delays. In [20], Yang and Cao consider stability in Lagrange sense of a class of feedback neural networks for optimization problems.…”

Section: Introductionmentioning

confidence: 99%

New LMI-based Criteria for Lagrange Stability of Cohen-Grossberg Neural Networks with General Activation Functions and Mixed Delays

Wang¹,

Qi²

2013

IJCIS

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In this paper, the problem on Lagrange stability of Cohen-Grossberg neural networks (CGNNs) with both mixed delays and general activation functions is considered. By virtue of Lyapunov functional and Halanay delay differential inequality, several new criteria in linear matrix inequalities (LMIs) form for the global exponential stability in Lagrange sense of CGNNs are obtained. Meanwhile, the limitation on the activation functions being bounded, monotonous and differentiable is released, which generalizes and improves those existent results. Moreover, detailed estimations of the globally exponentially attractive sets are given out. It is also verified that outside the globally exponentially attractive set, there is no equilibrium state, periodic state, almost periodic state, and chaos attractor of the CGNNs. Finally, two numerical examples are given to demonstrate the theoretical results.

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Global lagrange stability of complex‐valued neural networks of neutral type with time‐varying delays

Cao

Alsaedi

et al. 2016

Complexity

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In this article, the problem of global exponential stability in Lagrange sense of neutral type complex‐valued neural networks (CVNNs) with delays is investigated. Two different classes of activation functions are considered, one can be separated into real part and imaginary part, and the other cannot be separated. Based on Lyapunov theory and analytic techniques, delay‐dependent criteria are provided to ascertain the aforementioned CVNNs to be globally exponentially stable GES in Lagrange sense. Moreover, the proposed sufficient conditions are presented in the form of linear matrix inequalities which could be easily checked by Matlab. Finally, two simulation examples are given out to demonstrate the validity of theory results. © 2016 Wiley Periodicals, Inc. Complexity 21: 438–450, 2016

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Global exponential stability in Lagrange sense for recurrent neural networks with time delays

Cited by 93 publications

References 34 publications

Exponential Lagrange Stability for Markovian Jump Uncertain Neural Networks with Leakage Delay and Mixed Time-Varying Delays via Impulsive Control

Exponential Lagrange Stability for Markovian Jump Uncertain Neural Networks with Leakage Delay and Mixed Time-Varying Delays via Impulsive Control

New LMI-based Criteria for Lagrange Stability of Cohen-Grossberg Neural Networks with General Activation Functions and Mixed Delays

Global lagrange stability of complex‐valued neural networks of neutral type with time‐varying delays

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