1. INTRODUCTION

Neural networks(NNs) can be extensively applied in different fields, such as pattern recognition, parallel computing, image processing and so on. Therefore, they have been diffusely investigated in the past decades [1–7]. In order to realize these applications, it is necessary for people to study the dynamical behaviors of NNs, which greatly motivates the qualitative analysis of different dynamical behaviors [8–12]. In addition to stability, many other dynamical behaviors, including periodic oscillation, bifurcation, dissipation and synchronization, exist in real applications. Among them, the synchronization of NNs has developed into a issue of both theoretical and practical significance since synchronization is one of the most important issues related to their dynamic behaviors [13–15]. Moreover, time delays are inevitably encountered during the hardware implementation, which should be brought into the network model. Therefore, the synchronization problems of NNs with time delays have caused considerable attention and a great deal of works have been published, see [16–19] and references therein.

Particularly, the Cohen–Grossberg neural network(CGNN) model was initially established in 1983 [20]. It comprises a great deal of famous NNs, including Hopfield NNs, bidirectional associative memory NNs and cellular NNs. Consequently, the synchronization problem of CGNNs has also been proverbially studied due to their underlying applications in information processing, distributed computation and secure communication. For example, in [21–23], the synchronization problems of CGNNs with constant amplification gains and different delays were discussed. In [24], some synchronization criteria for delayed CGNNs were proposed by employing a periodic intermittent controller. In [25], Gan studied the adaptive synchronization of CGNNs with both mixed delays and unknown parameters.

Nevertheless, the existing studies mainly focus on the NNs with only first derivative of the states, and the effect caused by an inertial term is not taken into account. The introduction of the inertial term can be deemed as a key factor in generating chaotic behaviors, bifurcation and some other complicated dynamics in a networked system [26–30]. In addition, some synchronization conditions for inertial NN with or without Markovian jumping parameters were proposed in [31] and [32]. Based on the above discussion, it is obvious that the theoretical results for the synchronization of delayed ICGNNs are limited, which motivates the present research.

In the light of the well-known invariant principle, an adaptive feedback controller is provided to achieve the synchronization goal. Compared with previous research, the main contributions of the work are summarized in three perspectives: (i) The introduction of inertial term makes this problem more challenging, which has been successfully resolved based on the variable transformation method. (ii) We only need to impose one controller to the spilt systems, which is of less conservatism. (iii) The model considered in this paper is rather general, thus it comprises many existing results as its special cases.

The remaining part of the paper is organized as follows. The model description, necessary definition, lemma as well as assumptions are given in Section 2. A novel and analytic adaptive controller is presented in Section 3 to realize the synchronization of delayed ICGNNs. Moreover, an illustrative example is presented to validate the feasibility of the developed synchronization strategy in Section 4. At last, the conclusion of this work is made in Section 5.

2. MODEL DESCRIPTION AND PRELIMINARIES

In this paper, the delayed ICGNNs model can be described by the following differential equations:

for i∈{1,2,⋯,n}. In which ui(t) denotes the state of the ith neuron at time t and the second derivative of ui(t) represents the inertial term, aij and bij stand for the connection strength and the time-varying delay connection strength, respectively. fj(⋅) represents the the activation function, βi>0 is a constant, αi(⋅) is an amplification function, hi(⋅) denotes an appropriate behaved function, Ji stands for the external input and τ(t) represents the time-varying delay.

Considering (1) as the drive system, the corresponding response system is devised as

where Ui(t) is an appropriate controller that will be designed in the sequel.

The variable transformation is introduced as

where δi>0 is a constant.

Then, the above two systems are equivalent to

and

Denote the error signals as e1i(t)=xi(t)−ui(t) and e2i(t)=yi(t)−vi(t). Then, the error dynamics between the uncontrolled system (3) and the controlled system (4) are derived as follows:

To proceed further, the following hypotheses are utilized:

(A1): There exist positive constants α¯i and Mi such that

for all ui,vi∈ℝ,i∈{1,2,⋯,n}.

(A2): There exist positive constants li and fi such that the activation functions fi(⋅) satisfy

where ui,vi∈ℝ,i∈{1,2,⋯,n}.

(A3): There exist positive constants γi such that

where ui,vi∈ℝ,i∈{1,2,⋯,n}.

(A4): The time-varying transmission delay τ(t) is supposed to satisfy

Before moving on, we present the following definition and lemma:

3. MAIN RESULTS

In this part, a new criterion is presented to achieve the adaptive synchronization of ICGNNs with time-varying delay. Besides, the corresponding corollaries of inertial NNs and delayed CGNNs are proposed.

Calculating V̇(t) along the trajectory of error system (5) results in

Considering the assumptions (A1)−(A4) and the adaptive feedback controller (6), V̇(t) undergoes the following estimation:

According to Lemma 2, one can derive that

Substituting (10) into the right side of (9), one can further obtain that

Letting

In light of (11) and (12), one has

where e1(t)=(e11(t),e12(t),⋯,e1n(t))T. It is apparent that V̇(t)=0 if and only if e1(t)=0. Based on Definition 1 and the famous invariant principle, no matter what the initial value is, the trajectory of e1(t) would converge asymptotically to the largest invariant set E={e1(t)=0} contained in limt→∞V̇(t)=0, which implies the adaptive synchronization can be reached.

If we take αi(ui(t))=1 and hi(ui(t))=ciui(t), then the model (1) degenerates to

As a consequence, the corresponding response system is

Following the aforementioned variable transformation, the above two systems can be rewritten as

and

Following the same line as in Theorem 1, the following corollary can be readily obtained:

When d 2ui(t)dt 2=0, βi=1, the system (1) is further degenerated to

For the drive system (20), the corresponding response system can be designed as

Analogously, we denote the error signal as ei(t)=xi(t)−ui(t). By constructing a slightly different Lypunov functional V(t)=12 ∑i=1nei 2(t)+11−p∑j=1nα¯ilj|bij|∫t−τ(t)tej 2(s)ds+1μi(δi+λi) 2 and following the similar way as in Theorem 1, one can easily derive the adaptive synchronization criterion for systems (20) and (21), which is described in the following corollary without detailed proof.