Abstract
Synchronization of reaction–diffusion Hopfield neural networks with s-delays via sliding mode control (SMC) is investigated in this paper. To begin with, the system is studied in an abstract Hilbert space C([–r; 0];U) rather than usual Euclid space Rn. Then we prove that the state vector of the drive system synchronizes to that of the response system on the switching surface, which relies on equivalent control. Furthermore, we prove that switching surface is the sliding mode area under SMC. Moreover, SMC controller can also force with any initial state to reach the switching surface within finite time, and the approximating time estimate is given explicitly. These criteria are easy to check and have less restrictions, so they can provide solid theoretical guidance for practical design in the future. Three different novel Lyapunov–Krasovskii functionals are used in corresponding proofs. Meanwhile, some inequalities such as Young inequality, Cauchy inequality,Poincaré inequality, Hanalay inequality are applied in these proofs. Finally, an example is given to illustrate the availability of our theoretical result, and the simulation is also carried out based on Runge–Kutta–Chebyshev method through Matlab.
Highlights
Hopfield neural networks (HNNs) are intensively studied since it was first postulated in [10] due to their successful applications in numerous areas such as pattern recognition,parallel computation, and associative memory [22]
After a scrutiny scan of published work on delayed HNNs, we find that most authors either concentrate on system with discrete delays or distributed delays independently
After a scrutiny scan of the latest works of synchronization for delayed or reaction–diffusion HNNs [2, 4, 7, 11, 12, 17, 18, 30, 37,38,39,40], we find that these criteria are expressed in the form of LMI toolbox, which heavily rely on the Schur complement theorem and optimization method
Summary
Hopfield neural networks (HNNs) are intensively studied since it was first postulated in [10] due to their successful applications in numerous areas such as pattern recognition,parallel computation, and associative memory [22]. As an important collective behavior, synchronization of HNNs becomes a hot topic in recent decades due to their potential applications in secure communications, signal processing, distributed computation [7, 11, 17, 18, 28, 30, 31, 35, 36, 42]. It means that solution of drive system converges to the desired trajectory under appropriate control strategy [26]. These criteria are explicitly expressed when compared with LMIs
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