Global asymptotic stability of delayed neural networks with discontinuous neuron activations

doi:10.1016/j.neucom.2013.02.021

Neurocomputing

Volume 118, 22 October 2013, Pages 322-328

https://doi.org/10.1016/j.neucom.2013.02.021 Get rights and content

Abstract

In this paper, we integrate a class of delayed neural networks with discontinuous activations, which are not supposed to be bounded or nondecreasing. Conditions of existence of an equilibrium point are established by means of the Leray–Schauder theorem of set-valued maps. Then, the existence of solutions is proved based on viability theorem. Furthermore, global asymptotical stability of the networks is studied by using Lyapunov–Krasovskii stability theory. The results of global asymptotical stability are in term of linear matrix inequality. The obtained results extend previous works on global stability of delayed neural networks with discontinues activations.

Introduction

In the past decade, neural networks have extensive applications in optimization, classification, solving nonlinear algebraic equations, signal and image processing, pattern recognition, automatic control, associative memories and other areas [1], [2], [3], [4]. Therefore, a great deal of attentions have been paid to the study of neural networks recently, and various issues for neural networks have been investigated [5], [6], [7]. It is well known that implementations of neural networks depend heavily on the dynamical behaviors of the networks. However, time delays often occur in the processing of information storage and transmission which may create bad dynamical behaviors of the networks for example oscillatory and instability. Hence, it is necessary to study the dynamical behaviors of delayed neural networks. And, the stability of delayed neural networks has received considerable interests [8], [9], [10].

All the above literatures [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] and some other articles such in [11], [12], [13], [14] are based on assuming that the activation functions are continuous or even Lipschitz continuous. Nevertheless, it is well known that neural networks with discontinuous activations may be better for designing and applying an artificial neural networks. For example, considering the classical Hopfield neural networks [15], the standard assumption is that the activations are used in high-gain limit where they closely approach a discontinuous hard-comparator function. Therefore, the dynamical behavior of discontinuous neural networks have attracted much attention and many results have been presented [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. To our knowledge, Forti and Nistri in [16] have firstly studied the global stability of a neural network modeled by a differential equation with discontinuous activation functions based on Lyapunov diagonally stable matrix and constructing suitable Lyapunov function, and some stability conditions were proposed. Afterwards, a more general delayed neural network model with discontinuous activation functions is proposed in [17], the author analyzes global exponential stability and global convergence of the system in which the activation functions are unbounded. Since results given in many articles are based on the assumption that the activation functions are nondecreasing, Wang, Huang and Guo [23] discuss the dynamics behaviors of neural networks with discontinuous activations without assuming the boundedness and monotonicity of the neuron activations, where the neural networks are without delay.

Motivated by the above discussions, we will discuss the global asymptotic stability of delayed neural networks with discontinuous activations, where the activation functions are not assumed to be bounded or/and nondecreasing. Some sufficient conditions for the global stability of delay neural networks are established, which extend previous works on delayed neural networks with discontinuous activations.

The structure of this paper is outlined as follows. Due to the consideration of differential equations with discontinuous right-hand sides, in Section 2, we introduce the definitions and some lemmas we need use. We discuss the existence of equilibrium points and solutions in Section 3. In Section 4, the sufficient conditions for global asymptotic stability are established. Some numerical examples are given to show the effectiveness of our proposed results in Section 5. At last, we conclude this paper in Section 6.

Section snippets

Model descriptions and preliminaries

First of all, some mathematical notations used throughout this paper are presented. $R$ denotes the set of real numbers. For $x \in R^{n}$ , x^T denotes its transpose. The vector norm is defined as $∥ x ∥ = \sqrt{x^{T} x}$ . $R^{n \times m}$ are the set of real matrices. For $A \in R^{n \times n}$ , $λ_{\max} (A)$ and $λ_{\min} (A)$ are the maximum and the minimum eigenvalues of A, the spectral norm used will be $∥ A ∥ = {(λ_{\max} (A^{T} A))}^{1 / 2}$ .

In this paper, we consider delayed neural networks model with discontinuous neuron activations described by the following equation: $\dot{x} (t) = -$

Existence of equilibrium and viability

In this part, we will first prove existence of equilibrium for the system (1) by equilibrium theorem. Firstly, we give some necessary definitions concerning equilibrium theorem to set-value maps.

Definition 8

K is a convex subset of $R^{n}$ . The tangent cone T_K(x) to K at $x \in K$ is defined as $T_{K} (x) = \underset{h > 0}{⋃^{¯}} \frac{K - x}{h}$ where $⋃^{¯}$ is the closure of the union set.

Lemma 2

Aubin and Frankowska [29]

Suppose $Ω$ is a convex subset of $R^{n}$ and a nonempty convex se-value map $F : [0, 1] \times Ω ⊸ R^{n}$ is upper semicontinuous. If the set-valued map $x ⊸ F (0, x)$ satisfies the tangential

Global asymptotic stability and convergence

In this section, we will discuss the global asymptotical stability. The corresponding results will be presented in the below.

Note that, in view of Theorem 1, suppose that x^⁎ is an equilibrium point of (1), $γ^{⁎}$ is the corresponding OEP. Let $y (t) = x (t) - x^{⁎}$ , then the neural networks (1) can be transformed as below $\dot{y} (t) = - Cx (t) + Af (y (t)) + Bf (y (t - τ))$ where $f (y) = {(f_{1} (y_{1}), \dots, f_{n} (y_{n}))}^{T}$ , and $f_{i} (y_{i}) = g_{i} (y_{i} + x_{i}^{⁎}) - γ_{i}^{⁎}, γ^{⁎} \in K [g (x^{⁎})]$ . Obviously, $\exists \bar{γ} (t) \in K [f (y (t))]$ , such that $\dot{y} = - Cy (t) + A \bar{γ} (t) + B \bar{γ} (t - τ)$ Then, under the

Example

Example 1

Consider a two-dimensional delayed neural network (1) described by the following: $A = [\begin{matrix} - 1.5 & - 0.05 \\ 0.05 & - 1.5 \end{matrix}], B = [\begin{matrix} 0.8 & - 0.1 \\ 0.1 & 0.8 \end{matrix}], g_{i} (s) = \{\begin{matrix} s + 1 & s > 0 \\ - s - 1 & s \leq 0 \end{matrix}, i = 1, 2, C = diag (1.5, 1.5), τ = 5$ It is clear that the activation function $g (x) = (g_{1} (x), g_{2} (x))$ is discontinuous and unbounded. Now take $L_{i} = 1$ and $P = diag (1, 1)$ , then $P_{i} = 1$ , $\bar{λ} = 1.4$ , $∥ C^{- 1} (A + B) ∥ = 0.2278$ , so we have $\frac{L_{i} P_{i} {\bar{c}}^{2} ∥ C^{- 1} (A + B) ∥^{2}}{\bar{λ} \underset{Ì²}{c}} = 0.2441 < \frac{1}{4}$ Thus, the Assumption 1, Assumption 2, Assumption 3, Assumption 4 hold. Using the Matlab LMI and Control Toolbox, we can know the

Conclusion

This paper studies global asymptotic stability of neural networks with discontinuous activation functions where the activation may be described as unbounded or/and nonmonotonic functions. This activation functions are extensive in practice. The generalized Lyapunov–Krasovskii stability theory has been applied to prove the results based on the Filippov theory. The criteria are described in the form of LMIs. Some numerical examples are proposed to show the effectiveness of the results.

Acknowledgments

The work is supported by the Natural Science Foundation of China under Grants 60974021 and 61125303, the 973 Program of China under Grant 2011CB710606 and the Fund for Distinguished Young Scholars of Hubei Province under Grant 2010CDA081.

Jian Xiao received his BS degree from Changchun University, Changchun, China, and his MS degree from Guangxi University, Nanning, China, in 2007 and 2010, respectively. He is currently a doctoral candidate in the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China, and also in the Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China. His current research interests include stability theory,

References (33)

Y.Q. Yang et al.
A feedback neural network for solving convex constraint optimization problems
Appl. Math. Comput.
(2008)
X.X. Liao et al.
Global exponential stability in Lagrange sense for recurrent neural networks with time delays
Nonlinear Anal. Real World Appl.
(2008)
Z.W. Liu et al.
Novel stability criterions of a new fuzzy cellular neural networks with time-varying delays
Neurocomputing
(2009)
L.L. Wang et al.
Complete stability of cellular neural networks with unbounded time-varying delays
Neural Networks
(2012)
B.Z. Du et al.
Neural networks letterstability analysis of static recurrent neural networks using delay-partitioning and projection
Neural Networks
(2009)
W.B. Zhang et al.
Stability of delayed neural networks with time-varying impulses
Neural Networks
(2012)
A.L. Wu et al.
Dynamic behaviors of memristor-based recurrent neural networks with time-varying delays
Neural Networks
(2012)
M. Forti et al.
Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipchitz activations
Physica D
(2006)
W.L. Lu et al.
Dynamical behaviors of Cohen–Grossberg neural networks with discontinuous activation functions
Neural Networks
(2005)
J.F. Wang et al.
Global asymptotic stability of neural networks with discontinuous activations
Neural Networks
(2009)

Z.W. Cai et al.

On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions

Neural Networks

(2012)

J.F. Wang et al.

Dynamical behavior of delayed Hopfield neural networks with discontinuous activations

Appl. Math. Model.

(2009)

E. Kaslika et al.

Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis

Neural Networks

(2011)

Y. Shen et al.

Almost sure exponential stability of recurrent neural networks with Markovian switching

IEEE Trans. Neural Networks

(2009)

Y.R. Liu et al.

Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays

IEEE Trans. Neural Networks

(2009)

Z.W. Liu et al.

Novel stability analysis for recurrent neural networks with multiple delays via line integral-type L–K functional

IEEE Trans. Neural Networks

(2010)

Cited by (12)

Finite-time synchronization of discontinuous fuzzy neural networks with mixed time-varying delays and impulsive disturbances
2023, Results in Control and Optimization
In this paper, the finite-time synchronization issue is addressed for discontinuous fuzzy neural networks with mixed time-varying delays and impulsive disturbances. The dynamic impulsive gain about the impulsive moment is proposed as an alternative to the previous static impulsive gain. The time-varying impulsive gain can combine the synchronizing and desynchronizing impulses to directly observe their effects on the system, avoiding the classification discussion. In order to study the finite-time synchronization of the drive–response system, we obtain an equivalent system from the original fuzzy neural networks based on the concepts of Filippov solutions and differential inclusion theory. Furthermore, by constructing new Lyapunov functions that combine the 1-norm and the comparative system method, some algebraic sufficient conditions for the synchronization of the drive system and the response system in finite time are established, which are based on two types of novel non-chattering controllers. In addition, the parameters of the system itself can simply demonstrate the sufficient conditions that have been obtained her e, avoiding the time-consuming calculation of matrix inequalities. Moreover, the settling time estimation scheme have been established. Eventually, numerical examples are provided to demonstrate the effectiveness of the control scheme.
Periodicity and global exponential periodic synchronization of delayed neural networks with discontinuous activations and impulsive perturbations
2021, Neurocomputing
Citation Excerpt :
For example, the paper [9] studied the existence and global asymptotic stability of periodic solution for discrete and distributed time-varying delayed neural networks with discontinuous activations by using Filippov’s differential inclusion theory, the Leray–Schauder alternative theorem in multivalued analysis and generalized Lyapunov-like approach. The authors in paper [11] investigated the existence and global asymptotical stability of equilibrium point for a class of delayed neural networks with discontinuous activations by Leray–Schauder theorem of set-valued maps and Lyapunov–Krasovskii stability theory. In [14], the authors studied the dissipativity and synchronization problems of a class of delayed bidirectional associative memory (BAM) neural networks in which neuron activations are modeled by discontinuous bivariate functions.
In this paper, periodic dynamics and periodic synchronization of a class of delayed neural networks with discontinuous neuron activations and impulsive perturbations are investigated. Firstly, based on Filippov’s functional differential inclusion and impulsive differential inclusions theory, the existence of solutions of the considered neural network system is studied. Secondly, according to Krasnoselskii’s fixed point theorem on cones of set-valued analysis, sufficient conditions are provided for the existence of at least one periodic solution and at least two periodic solutions for the neural network systems. Furthermore, we also use Kakutani’s fixed point theorem to give sufficient criteria for the existence of at least one periodic solution of the neural network systems. These results shown that there are still one or more periodic solutions for the neural networks with discontinuous activations under the impulsive effects. Moreover, sufficient criteria are given to the exponential periodic synchronization under the switching state-feedback control strategy with or without delay. Finally, two numerical simulation examples are presented to verify the correctness and validity of our main conclusions.
Global exponential stability of uncertain neural networks with discontinuous Lurie-type activation and mixed delays
2016, Neurocomputing
This paper deals with the problem of the global exponential stability of a class of uncertain neural networks with discontinuous Lurie-type activation and mixed delays. By establishing a new sufficient condition, we first prove the existence of the equilibrium point by using the Leray–Schauder alternative theorem. Then, by employing a new Lyapunov functional, we obtain the global exponential stability of the equilibrium point of the uncertain neural network. In the end, some comparisons and numerical examples are given to show the improvement of the conclusions in this paper.
Stability analysis of fractional-order Hopfield neural networks with discontinuous activation functions
2016, Neurocomputing
Citation Excerpt :
Under the assumption that the activation functions are nondecreasing, the global exponential stability and global convergence of the network system were analyzed in Refs. [13,14]. Further, without assuming the boundedness and monotonicity of the activation functions, the dynamics behaviors of the discontinuous neural networks were discussed by using Lyapunov stability theory in Refs. [15,16]. Note that all the above mentioned works just studied integer-order models of neural networks.
Fractional-order Hopfield neural networks are often used to model the information processing of neuronal interactions. For a class of such networks with discontinuous activation functions, it is needed to investigate the existence and stability conditions of their solutions. Under the framework of Filippov solutions, a growth condition is firstly given to guarantee the existence of their solutions. Then, some sufficient conditions are proposed for the boundedness and stability of the solutions of such discontinuous networks by employing the Lyapunov functionals. Finally, a numerical example is presented to demonstrate the effectiveness of the theoretical results.
Dynamics in fractional-order neural networks
2014, Neurocomputing
Citation Excerpt :
Neural networks have received much attention in recent years [1–14].
This paper investigates a general class of neural networks with a fractional-order derivative. By using the contraction mapping principle, Krasnoselskii fixed point theorem and the inequality technique, some new sufficient conditions are established to ensure the existence and uniqueness of the nontrivial solution. Moreover, uniform stability of the fractional-order neural networks is proposed in fixed time-intervals. Finally, some examples are given to illustrate the effectiveness of theoretical results.
Almost periodic dynamical behaviors for generalized Cohen-Grossberg neural networks with discontinuous activations via differential inclusions
2014, Communications in Nonlinear Science and Numerical Simulation
In this paper, we investigate the almost periodic dynamical behaviors for a class of general Cohen–Grossberg neural networks with discontinuous right-hand sides, time-varying and distributed delays. By means of retarded differential inclusions theory and nonsmooth analysis theory with generalized Lyapunov approach, we obtain the existence, uniqueness and global stability of almost periodic solution to the neural networks system. It is worthy to pointed out that, without assuming the boundedness or monotonicity of the discontinuous neuron activation functions, our results will also be valid. Finally, we give some numerical examples to show the applicability and effectiveness of our main results.

View all citing articles on Scopus

Zhigang Zeng received his BS degree from Hubei Normal University, Huangshi, China, and his MS degree from Hubei University, Wuhan, China, in 1993 and 1996, respectively, and his PhD degree from Huazhong University of Science and Technology, Wuhan, China, in 2003.

He is a professor and PhD advisor in the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China, and also in the Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China. His current research interests include neural networks, switched systems, computational intelligence, stability analysis of dynamic systems, pattern recognition and associative memories.

Wenwen Shen received her BS degree from Wuhan University of Technology, Wuhan, China, in 2009. She is currently a doctoral candidate in the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China, and also in the Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China. His current research interests include stability theory, artificial neural networks and switching control.

View full text

LettersGlobal asymptotic stability of delayed neural networks with discontinuous neuron activations

Abstract

Introduction

Section snippets

Model descriptions and preliminaries

Existence of equilibrium and viability

Aubin and Frankowska [29]

Global asymptotic stability and convergence

Example

Conclusion

Acknowledgments

Appl. Math. Comput.

Nonlinear Anal. Real World Appl.

Neurocomputing

Neural Networks

Neural Networks

Neural Networks

Neural Networks

Physica D

Neural Networks

Neural Networks

Neural Networks

Appl. Math. Model.

Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis

Neural Networks

Almost sure exponential stability of recurrent neural networks with Markovian switching

IEEE Trans. Neural Networks

Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays

IEEE Trans. Neural Networks

Novel stability analysis for recurrent neural networks with multiple delays via line integral-type L–K functional

IEEE Trans. Neural Networks

Letters
Global asymptotic stability of delayed neural networks with discontinuous neuron activations