Existence and Global Asymptotic Stability of Fuzzy Cellular Neural Networks with Time Delay in the Leakage Term and Unbounded Distributed Delays

Abstract

Fuzzy cellular neural network (FCNN) structures are based on the uncertainties in human cognitive processes and in modeling neural systems, and they provide an interface between a human expert and classical cellular neural networks (CNNs). In this paper, an existence and global asymptotic stability analysis of the equilibrium point of FCNNs with time delay in the leakage term and unbounded distributed delays is investigated. Based on the Lyapunov-Krasovskii functional with free-weighting matrix, and using the homeomorphism mapping principle and linear matrix inequalities (LMIs), a new set of stability criteria for FCNNs is obtained with time delay in the leakage term, time-varying delays and unbounded distributed delays. The proposed results can be easily checked via the LMI Control Toolbox in MATLAB. Moreover, it is well known that the stability behavior of FCNNs is very sensitive to the time delay in the leakage term. In the absence of the leakage term, a new stability criterion is also derived by employing a Lyapunov-Krasovskii functional using an LMI approach. Numerical examples are provided to illustrate the effectiveness and reduced conservativeness of the developed techniques.

Novel Delay-Dependent Stability Criteria for Fuzzy Neural Networks Using Delay-Decomposition Approach

“…Remark 2 As pointed out in Remark 1, the Wirtinger-based integral inequality provides more accurate upper bound than Jensen inequality for dealing with the cross-term arisen when manipulating the derivative of Lyapunov-Krasovskii functional; therefore, the proposed method in Theorem 1 with Wirtinger-based inequality will lead to less conservative results than those by using Jensen inequality [2,3,9].…”

“…However, it is verified that the stability criteria in [2,3,9] admit no feasible solutions, that is, for this example, the conditions of Ref. [2,3,9] fail to verify the stability.…”

“…[2,3,9] fail to verify the stability. Therefore, we can say that for this system, the results in this paper are much effective and less conservative than those in [2,3,9].…”

“…where Since the coefficient 1 2 (M 4 − M 5 ) of τ (t) 2 in Φ(t) is nonnegative definite, based on Lemma 7, Φ(t) < 0 is true if and only if the following two inequalities hold simultaneously Φ(τ (t)) + Φ a + Φ 1 + Φ 2 + ne T 11 L 1 Υ L −1 2 Υ L 1 e 11 < 0,…”

“…Moreover, in some practical applications and hardware implementations of neural networks, the inevitable time delay may be unbounded [13]. Under the assumptions that the activation functions are Lipschitz continuous or monotonically non-decreasing, Tan [19] derived two LMI-based stability criteria for a class of fuzzy cellular neural networks with constant discrete and unbounded distributed delays; with assuming the activation functions being Lipschitz continuous, Balasubramaniam et al [2,3] obtained the existence and global asymptotic stability criteria of the equilibrium point for two class of fuzzy neural networks with leakage delay, time-varying discrete and unbounded distributed delays by using the homeomorphism mapping principle and the Jensen integral inequality; with the activation functions being Lipschitz continuous, Li et al [9] obtained the existence, uniqueness and global asymptotic stability criteria of the equilibrium point for a class of fuzzy cellular neural networks with leakage delay, time-varying discrete and unbounded distributed delays under impulsive perturbations by utilizing the contraction mapping theorem, topological degree theory and the Jensen integral inequality.…”