Synchronization of reaction–diffusion Hopfield neural networks with s-delays through sliding mode control

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.

Synchronization of delayed stochastic reaction–diffusion Hopfield neural networks via sliding mode control

“…Other parameters are given as KP = (0.1, 0. With these conditions discussed above, the behavior of ( 17) and ( 18) is exponentially synchronized under (9) in the mean-square sense by Theorem 1. The results can be validated through the simulation results; see Figs.…”

“…Let system (5) satisfy (H1)-(H4). Suppose that the switching surface is given by (8), and the SMC law is set to be (9). Then the solution of (5) is exponentially stable in the mean-square sense on the switching surface described by (8).…”

“…By definition of e t C in Section 2, (1) and ( 2) are exponentially synchronized in the mean-square sense under (9).…”

“…In this case, it is reasonable to study the HNNs with distributed delays 0 −τi f i (u i (t + θ))k i (θ) dθ, k i are kernel functions [19,27,28]. Both of them can be included in the s-delays 0 −τi f i (u i (t + θ)) dκ i (θ) with dκ i satisfying Lebesgue-Stieltjes measure [9]. So the HNNs with s-delay is much more general than other type of delayed HNNs, whether in mathematics or physics.…”