An exponential H8 synchronization method is addressed for a class of uncertain master and slave neural networks with mixed time-delays, where the mixed delays comprise different neutral, discrete and distributed time-delays. An appropriate discretized Lyapunov-Krasovskii functional and some free weighting matrices are utilized to establish some delay-dependent sufficient conditions for designing a delayed state-feedback control as a synchronization law in terms of linear matrix inequalities under less restrictive conditions. The controller guarantees the exponential H8 synchronization of the two coupled master and slave neural networks regardless of their initial states. Numerical simulations are provided to demonstrate the effectiveness of the established synchronization laws.
TopIntroduction
In the last few years, synchronization in neural networks (NNs), such as cellular NNs, Hopfield NNs and bi-directional associative memory networks, has received a great deal of interest among scientists from various fields (Chen & Dong,1993 ; Sun et al., 2007; Wang et al., 2008; Cheng et al., 2006; Cao et al., 2007). In order to better understand the dynamical behaviours of different kind of complex networks, an important and interesting phenomenon to investigate is the synchrony of all dynamical nodes. In fact, synchronization is a basic motion in nature that has been studied for a long time, ever since the discovery of Christian Huygens in 1665 on the synchronization of two pendulum clocks. The results of chaos synchronization are utilized in biology, chemistry, secret communication and cryptography, nonlinear oscillation synchronization and some other nonlinear fields. The first idea of synchronizing two identical chaotic systems with different initial conditions was introduced by Pecora and Carroll (Pecora & Carroll, 1990), and the method was realized in electronic circuits. The methods for synchronization of the chaotic systems have been widely studied in recent years, and many different methods have been applied theoretically and experimentally to synchronize chaotic systems, such as feedback control (Fradkov & Pogromsky, 1996; Gao et al., 2006; Karimi & Maass, 2009; Wen et al., 2006; Hou et al., 2007; Lu & van Leeuwen, 2006), adaptive control (Liao & Tsai, 2008; Feki, 2003; Wang et al., 2006a; Fradkov & Markov, 1997; Fradkov et al., 2000), backstepping (Park, 2006) and sliding mode control (Yan et al., 2006; García-Valdovinos et al., 2007). Recently, the theory of incremental input-to-state stability to the problem of synchronization in a complex dynamical network of identical nodes, using chaotic nodes as a typical platform was studied in (Cai & Chen, 2006).