-global stability of a Cohen–Grossberg neural network system with nonnegative equilibria☆
Section snippets
Introduction and model description
Cohen–Grossberg neural networks, proposed by Cohen and Grossberg (1983), have been extensively studied both in theory and applications. They have been successfully applied to signal processing, pattern recognition and associative memory, for example.
The success of these applications relies on understanding the underlying dynamical behavior of the models. A thorough analysis of the dynamics is a necessary step towards a practical design of neural networks. Such a model can be formalized as
Preliminaries
In this paper we use the following notations. denotes the matrix transpose of . denotes the symmetric part of matrix . denotes that is symmetric and positive definite. Similarly we denote , , and . denotes the first orthant. We denote the smallest and largest element of a set by and , respectively. denotes the spectrum set of matrix . denotes some norm of vector and matrix. In particular,
Existence and uniqueness of the nonnegative equilibrium
In this section, we discuss the existence and uniqueness of the nonnegative equilibrium in the NCP sense. Theorem 2 Existence and Uniqueness of the Nonnegative Equilibrium Suppose , , and . Let and . If there exists a positive definite diagonal matrix such thatholds, then for each , there exists a unique nonnegative equilibrium of the system(3)in the NCP sense. Proof Let where and are
-global asymptotic stability of the nonnegative equilibrium
In this section, we discuss the -global asymptotic stability of the nonnegative equilibrium defined in the previous section. Let be the nonnegative equilibrium of the system (3) in the NCP sense and . Thus, the system (2) can be rewritten as or in matrix form where for ,
Comparison and numerical example
In Chen and Rong (2003), the authors investigated global stability of delayed Cohen–Grossberg neural networks where amplifier functions are assumed strictly positive. In this way there was not of much difference between dealing with Cohen–Grossberg neural networks and cellular or Hopfield neural networks which do not contain any amplifier functions. The same linear matrix inequality as (16) was presented to guarantee the existence, uniqueness, and global stability of the equilibrium. In this
Conclusions
In this paper, we investigate the dynamics of the positive solutions of the Cohen–Grossberg neural networks with a time delay. Based on the theory of NCP and the LMI technique, a condition is obtained guaranteeing the existence, uniqueness, and global stability of the nonnegative equilibrium in the NCP sense. If the equilibrium is positive, then the stability is globally exponential. Numerical examples verify the viability of the theoretical results.
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Cited by (52)
Input-to-state stability of positive delayed neural networks via impulsive control
2023, Neural NetworksOn global exponential stability of positive neural networks with time-varying delay
2017, Neural NetworksCitation Excerpt :In this paper, we consider a class of positive NNs with bounded time-varying delay. Different from existing results, for example, in Liu and Huang (2008) and Lu and Chen (2007), we derive conditions based on some novel comparison techniques to ensure simultaneously that the system is positive and, for each nonnegative input vector, there exists a unique nonnegative equilibrium point which is globally exponentially stable. The derived conditions are formulated in terms of linear programming which can be solved by various computational tools.
Passivity analysis of Markov jump BAM neural networks with mode-dependent mixed time-delays via piecewise-constant transition rates
2016, Journal of the Franklin InstituteCitation Excerpt :According to the iterations of forward and backward information flows between the two layers, a two-way associative search for stored bipolar vector pairs was constructed. Furthermore, the Cohen–Grossberg-type BAM neural network model, as an important one in BAM neural network models and an extension of the traditional single layer neural networks model, was originally introduced by Cohen and Grossberg [6] in 1983 and investigated by considerable literatures [7–18]. On one hand, time-delays caused by many practical factors do inevitably exist in network models.
Positive invariant sets and global exponential attractive sets of BAM neural networks with time-varying and infinite distributed delays
2014, NeurocomputingCitation Excerpt :And some authors studied the almost periodic solutions by dichotomy and the fixed point principle [15,16]. It is worth mentioning that Lyapunov stability [5–12,32–36,39,40,43] refers to the stability of the equilibrium points which require the existence of equilibrium points, while Lagrange stability [17–24,37,38] refers to the stability of the total system, rather than the stability of the equilibrium points. Moreover, the global stability in Lyapunov sense on unique equilibrium point can be viewed as a special case of stability in Lagrange sense by regarding an equilibrium point as an attractive set. [17].
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This work is supported by the National Science Foundation of China 60074005, 60374018.