Exponential stability of a class of complex-valued neural networks with time-varying delays

doi:10.1016/j.neucom.2015.02.024

Neurocomputing

Volume 164, 21 September 2015, Pages 293-299

https://doi.org/10.1016/j.neucom.2015.02.024 Get rights and content

Abstract

This paper studies a class of complex-valued neural networks with time-varying delays. By using the conjugate system of the complex-valued neural networks and Brouwer׳s fixed point theorem, sufficient conditions to guarantee the existence and uniqueness of an equilibrium are obtained. Some criteria on globally exponential stability of the equilibrium of the complex-valued neural networks are also established by using a delay differential inequality. These results are easy to apply to the study of the complex-valued neural networks whether their activation functions are explicitly expressed by separating their real and imaginary parts or not. Two examples with numerical simulations are given to highlight the effectiveness of the obtained results.

Introduction

Recent years, there is a growing interest in studying complex-valued neural networks (CVNNs), which can be seen from a large number of relevant publications (e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and the references therein). In fact, CVNNs have the potential to solve problems that cannot be solved by their real-valued counterparts. For example, in [7], the XOR problem and the detection of symmetry problem cannot be solved with a single real-valued neuron (i.e., a two-layered real-valued neural network), but they can be solved by a single complex-valued neuron (i.e., a two-layered complex-valued neural network) with the orthogonal decision boundaries, which reveals the potential computational power of complex-valued neurons. In addition, CVNNs have been found of great use in extending the scope of applications of artificial neural networks in optoelectronics, filtering, imaging, speech synthesis, computer vision, remote sensing, quantum devices, spatiotemporal analysis of physiological neural devices and systems, and artificial neural information processing [1], [8], [9], [10], [11], [12], [13], [14].

In applications of neural networks, it is essential to ensure that the designed neural networks are stable. In past decades, the stability of real-valued neural networks has been widely studied and there are a large number of related publications (see, e.g., [15], [16], [17], [18], [19], [20], [21], [22], [23], [24] and the references therein). It is worth mentioning that, compared with real-valued neural networks, it is more difficult to study the stability of CVNNs. Because CVNNs have more complicated properties than the real-valued counterparts. Recently, the stability of CVNNs has caught great attention (see [25], [26], [27], [28], [29], [30]). For example, in [25], a class of CVNNs model on time scales was investigated and some sufficient conditions for ψ-global exponential stability were obtained. In [26], the discrete-time delayed neural networks with complex-valued linear threshold neurons were studied, and several criteria on boundedness and global exponential stability of equilibrium were obtained. In [27], [28], discrete CVNNs were studied and some sufficient conditions on the existence of a unique equilibrium pattern and its global exponential stability were established. Particularly, in [29], the authors studied two types of complex-valued recurrent neural networks whose activation functions are separated into their real and imaginary parts or not. In [30], CVNNs with mixed time delays whose activation functions are expressed by separating their real and imaginary parts were studied. In [29], [30], the n-dimensional complex-valued neural networks were transformed into the 2n-dimensional real-valued ones, and some sufficient conditions ensuring the existence and uniqueness of equilibrium and its global exponential stability were achieved. However, it required the existence, continuity, and boundedness of the partial derivatives of the activation functions about the real and imaginary parts of the state variables, which limits the applications of the obtained results.

Motivated by discussion above, in this paper, we aim at dealing with the stability problem for a class of complex-valued neural networks with time-varying delays. Some sufficient conditions of the existence, uniqueness, and global exponential stability of equilibrium are obtained. In these sufficient conditions, the activation functions only need to satisfy the Lipschitz condition. It removes the restrictions on the existence, continuity, and boundedness of the partial derivatives of the activation functions about their real and imaginary parts. Our method is feasible and more efficient than that in [29], [30].

The rest of this paper is organized as follows. In Section 2, model description and preliminaries are given. In Section 3, several criteria are derived for the exponential stability of unique equilibrium of a class of CVNNs with time-varying delays. Then, in Section 4, two examples are given to illustrate the effectiveness of the main results. Finally, in Section 5, some conclusions are drawn.

Section snippets

Model description and preliminaries

To begin with, we would like to introduce some notations. By $R$ we denote the set of real numbers. Let $z = a + ib$ be a complex number, where $i = \sqrt{- 1}$ , $a, b \in R$ , $| z | = \sqrt{a^{2} + b^{2}}$ . $\bar{z}$ denotes the conjugate complex number of z, $\bar{z} = a + i (- b)$ . Let $C^{n}$ be the n ( $n \geq 1$ ) dimensional complex vector space.

In this paper, we consider a model of complex-valued neural networks with time-varying delays, which can be described by $\frac{{dz}_{k} (t)}{dt} = - d_{k} z_{k} (t) + \sum_{j = 1}^{n} (w_{kj} f_{j} (z_{j} (t)) + v_{kj} g_{j} (z_{j} (t - τ_{j} (t)))) + J_{k}, t \geq t_{0},$ where $z_{k} (t) = x_{k} (t) + {iy}_{k} (t)$ , $k, j = 1, \dots, n$ . For

Main results

In this section, we will first establish a criterion to ensure that Eq. (2.1) has a unique equilibrium point. Then we will derive some criteria that guarantee other solutions of (2.1) converge to its equilibrium point with exponential rates.

Theorem 3.1

Suppose that Assumption 1 is satisfied. Then Eq. (2.1) has a unique equilibrium point if $D - | W | L^{f} - | V | L^{g}$ is an M matrix, where $| W | = {(| w_{ji} |)}_{n \times n}$ , $| V | = {(| v_{ji} |)}_{n \times n}$ , $L^{f} = diag (l_{1}^{f}, \dots, l_{n}^{f})$ , $L^{g} = diag (l_{1}^{g}, \dots, l_{n}^{g})$ .

Proof

Firstly, We shall prove the existence of the equilibrium point

Examples

In this section, we will give two numerical examples to demonstrate the above results.

Example 1

Consider a two-neuron complex-valued neural network described by $\frac{{dz}_{k} (t)}{dt} = - d_{k} z_{k} (t) + \sum_{j = 1}^{2} (w_{kj} f_{j} (z_{j} (t)) + v_{kj} g_{j} (z_{j} (t - τ_{j} (t)))) + I_{k}, t \geq t_{0},$ k=1,2, where $z_{k} (t) = x_{k} (t) + {iy}_{k} (y)$ , $I_{1} = 1.5 - 2.5 i$ , $I_{2} = - 1 - 0.5 i$ , and the interconnected matrices are given as $D = (\begin{matrix} 9 & 0 \\ 0 & 8 \end{matrix}), W = (\begin{matrix} 2 + 3 i & 3 - i \\ 4 - 2 i & 1 + 2 i \end{matrix}), V = (\begin{matrix} - 1 + 2 i & 2 + i \\ 3 - 4 i & - 2 + 2 i \end{matrix}) .$ It is assumed that the activation functions of Eq. (4.1) are $f_{j} (z_{j}) = 0.5 | y_{j} | + 0.5 i | x_{j} |, g_{j} (z_{j}) = \frac{1 - e^{- y_{j}}}{1 + e^{- y_{j}}} + i \frac{1}{1 + e^{- x_{j}}}, j = 1, 2 .$ By

Conclusion

In this paper, the problems of existence and uniqueness of equilibrium point and its global exponential stability have been studied for a class of complex-valued neural networks with time-varying delays. Some sufficient conditions are established by applying conjugate system of CVNNs, Brouwer׳s fixed point theorem, contraction mapping principle, and a delay differential inequality. The results obtained are not only easy to apply for determining the dynamic behavior of the equilibrium point of

Acknowledgments

The research for this paper was supported by the State Scholarship Fund of China (No. 201208515054).

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Jie Pan received the M.Sc. degree in mathematics from Sichuan Normal University, Chengdu, China, in 2004, and the Ph.D. degree, in applied mathematics, from University of Electronic Science and Technology of China, Chengdu, China, in 2010. From 2013 to 2014, he was a Visiting Scholar in the Department of Mathematics in University of Waterloo, Canada. He joined the Department of Applied Mathematics, Sichuan Agricultural University, Ya׳an, China, as a Lecturer in 2004, where he became an Associate Professor in 2005 and a Professor in 2010. His research interests include stability theory, impulsive dynamical systems, complex-valued neural networks, and population growth models. He is the author or coauthor of over 20 research papers.

Xinzhi Liu received the B.Sc. degree in mathematics from Shandong Normal University, Jinan, China, in 1982, and the M.Sc. and Ph.D. degrees, all in applied mathematics, from University of Texas, Arlington, in 1987 and 1988, respectively. He spent two years as a Post-Doctoral Fellow at the University of Alberta, Edmonton, Canada, and then joined the Department of Applied Mathematics, University of Waterloo, Waterloo, Canada, as an Assistant Professor in 1990, where he became an Associate Professor in 1994 and a Full Professor in 1997. His research areas include stability and control, hybrid/impulsive dynamical systems, complex networks, and communication security. He is the author or coauthor of over 300 research papers and two research monographs and 20 edited books.

Weichau Xie received his B.Eng. degree in precision engineering from Shanghai Jiao-Tong University, Shanghai, China, in 1984 and his M.A.Sc. and Ph.D. degrees in civil engineering from the University of Waterloo, Waterloo, Ontario, Canada, in 1987 and 1990, respectively. He was a Stress Analyst and Design Engineer at the Atomic Energy of Canada Limited, Mississauga, Ontario, Canada, from September 1990 to December 1991. He joined the Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada, as an Assistant Professor in January 1992, where he became an Associate Professor and a Full Professor in 1997 and 2002, respectively. His principal areas of research include dynamic stability of structures, structural dynamics and random vibration, nonlinear dynamics, and stochastic mechanics, seismic analysis and design of engineering structures, reliability and safety analysis of engineering systems. He is the author of the books Dynamic Stability of Structures (Cambridge University Press, 2006), Differential Equations for Engineers (Cambridge University Press, 2010). He won the Distinguished Teacher Award from the University of Waterloo in 2007.

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Exponential stability of a class of complex-valued neural networks with time-varying delays

Abstract

Introduction

Section snippets

Model description and preliminaries

Main results

Examples

Conclusion

Acknowledgments

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