Cluster synchronization for directed heterogeneous dynamical networks via decentralized adaptive intermittent pinning control

Abstract

In this paper, by proposing a novel adaptive intermittent scheme, we consider the intermittent pinning-control problem for cluster synchronization of directed heterogeneous dynamical networks, i.e., directed networks with nonidentical dynamical nodes. Through constructing a piecewise Lyapunov function and utilizing the analysis technique, some sufficient conditions to guarantee global cluster synchronization are derived. It is noted that the adaptive intermittent strategy developed in this paper is decentralized, which only relies on some local information rather than the global information of the whole network. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.

Adaptive exponential lag synchronization for neural networks with mixed delays via intermittent control

“…It should be stressed that here the control period (t l+1 -t l ) and control width δ l are both non-fixed, and hence this control strategy is more general. Obviously, when t l+1 -t l ≡ T and δ l ≡ δ, l ∈ N + , the adaptive intermittent control type becomes the periodic one, which has been studied in [39][40][41].…”

“…In the case that both the control periods and the control widths are fixed, i.e., t l+1 -t l ≡ T and δ l ≡ δ, for all l ∈ N + , where T and δ are two positive constants, then the control type becomes the adaptive periodically intermittent control, which has been investigated in [39][40][41]. Denote θ = δ/T, then the following result can easily be obtained from Theorem 1.…”

“…In [40,41,50], under adaptive intermittent control, the complete synchronization and cluster synchronization of directed dynamical networks were studied by constructing a piecewise Lyapunov function. However, the piecewise Lyapunov function given in [40,41,50] isn't continuous at t = t l+1 , l ∈ N + , and therefore only some criteria ensuring globally asymptotical synchronization were derived in [40,41,50].…”

“…Intermittent control, as a discontinuous control method, has been adopted in engineering fields in view of its convenient implementation in engineering control [33][34][35][36][37][38]. Recently, an intermittent control scheme with fixed both control period and control width, namely periodically intermittent control scheme, has been developed to study the synchronization problem for chaotic systems as well as dynamical networks; see [12,13,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] and references cited therein. However, in practical applications, the requirement that both control period and control width be fixed is evidently unreasonable.…”

“…A numerical example is finally provided to demonstrate the effectiveness of the proposed control methodology. The main contributions of this paper can be outlined as follows: (1) different from the periodically intermittent control adopted in previous works [12,13,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], here the adaptive intermittent control is nonperiodic with non-fixed both control period and control width, which extends the intermittent control strategy's application scope; (2) unlike in the references [39][40][41]50], the candidate Lyapunov function introduced in this paper is piecewise continuous, and then by means of which, some criteria ensuring globally exponential lag synchronization are established; (3) the derived theoretical results indicate that the lag synchronization criteria are related to the control rates rather than the control periods, which is propitious to selecting the control periods in practical applications.…”