Stability and Synchronization of Discrete-Time Markovian Jumping Neural Networks With Mixed Mode-Dependent Time Delays

• Published in: IEEE transactions on neural networks Vol. 20; no. 7; pp. 1102 - 1116
• Main Authors: Liu, Yurong, Wang, Zidong, Liang, Jinling, Liu, Xiaohui
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2009; Institute of Electrical and Electronics Engineers
• Subjects: Algorithms; Applied sciences; Artificial Intelligence; Biological neural networks; Computer science; control theory; systems; Computer Simulation - trends; Connectionism. Neural networks; Councils; Delay effects; Discrete-time neural networks (DNNs); Exact sciences and technology; Jumping; Linear matrix inequalities; linear matrix inequality; Linear Models; Lyapunov method; Markov Chains; Markovian jumping parameters; Mathematical Computing; Mathematical models; Mathematics; mixed time delays; Neural networks; Neural Networks (Computer); Software; Software Validation; Stability; Stability analysis; Stochastic Processes; stochastic stability; Sufficient conditions; Synchronism; Synchronization; Time delay; Time Factors; Markov process; Mode coupling; Anonymity; Lyapunov method; Kronecker product; Stochastic stability; Neural network; Synchronization; Delay; Markov chain; Time dependence; mixed time delays; Discrete time; Linear matrix inequality; Markovian jumping parameters; Matrix method; Delay time; Discrete-time neural networks (DNNs); Jumping; Computer security; Lyapunov function
• ISSN: 1045-9227; 1941-0093; 1941-0093
• DOI: 10.1109/TNN.2009.2016210

In this paper, we introduce a new class of discrete-time neural networks (DNNs) with Markovian jumping parameters as well as mode-dependent mixed time delays (both discrete and distributed time delays). Specifically, the parameters of the DNNs are subject to the switching from one to another at different times according to a Markov chain, and the mixed time delays consist of both discrete and distributed delays that are dependent on the Markovian jumping mode. We first deal with the stability analysis problem of the addressed neural networks. A special inequality is developed to account for the mixed time delays in the discrete-time setting, and a novel Lyapunov-Krasovskii functional is put forward to reflect the mode-dependent time delays. Sufficient conditions are established in terms of linear matrix inequalities (LMIs) that guarantee the stochastic stability. We then turn to the synchronization problem among an array of identical coupled Markovian jumping neural networks with mixed mode-dependent time delays. By utilizing the Lyapunov stability theory and the Kronecker product, it is shown that the addressed synchronization problem is solvable if several LMIs are feasible. Hence, different from the commonly used matrix norm theories (such as the M-matrix method), a unified LMI approach is developed to solve the stability analysis and synchronization problems of the class of neural networks under investigation, where the LMIs can be easily solved by using the available Matlab LMI toolbox. Two numerical examples are presented to illustrate the usefulness and effectiveness of the main results obtained.