International Journal of Computational Intelligence Systems

Volume 13, Issue 1, 2020, Pages 472 - 478

Synchronization of Delayed Inertial Cohen–Grossberg Neural Networks Under Adaptive Feedback Controller

Qun Huang, Jinde Cao*, Qingshan Liu
School of Mathematics, Southeast University, Nanjing 211189, China
*Corresponding author. Email:
Corresponding Author
Jinde Cao
Received 29 February 2020, Accepted 31 March 2020, Available Online 17 April 2020.
10.2991/ijcis.d.200402.001How to use a DOI?
CGNNs; Adaptive synchronization; Inertial term; Time-varying delay

This paper investigates the issue on adaptive synchronization of delayed inertial Cohen–Grossberg neural networks (ICGNNs). By adopting the method of variable transformation, the addressed model, which includes the so-called inertial term, is transformed into first-order differential equations. On the basis of the well-known invariant principle of functional differential equations, a novel and analytic scheme which ensures the adaptive synchronization between the drive-response system is proposed in component form. It is worth mentioning that we only need to impose one controller to the spilt systems to realize the adaptive synchronization, which is of less conservatism. At the end of this paper, a numerical example is provided to verify the feasibility of the derived theoretical results. The established figures validate that the numerical simulations coincide well with the developed theoretical results.

© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (


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 [17]. 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 [812]. 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 [1315]. 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 [1619] 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 [2123], 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 [2630]. 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.


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


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,i{1,2,,n}.

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

where ui,vi,i{1,2,,n}.

(A3): There exist positive constants γi such that

where ui,vi,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:

Definition 1.

The drive system (1) and the response system (2) is said to achieve synchronization, if the trivial solution of the error system (5) is asymptotically stable, i.e.,


Remark 1.

Since the drive-response systems (1) and (2) are equivalent to (3) and (4), respectively, the synchronization problem between (1) and (2) can be regarded as the synchronization problem between (3) and (4). In addition, according to the variable transformation, one can see that e2i(t) just plays the role of adjoint variable. Thus we only need to consider the dynamics of e1i(t) when investigating the synchronization problem between systems (1) and (2).

Lemma 2.

[25] For any vectors u,vn, and any positive definite matrix Ξn×n, the following matrix inequality always holds:



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.

Theorem 1.

If the hypotheses (A1)(A4) hold, then the drive-response delayed ICGNNs (1) and (2) are synchronous with the adaptive feedback controller as

the feedback strengths ξi and ηi(i=1,2,,n) are adapted in the light of the following update laws, respectively:
where εi and θi(i=1,2,,n) are any positive scalars.


Constructing the following Lyapunov functional:

where λ1i and λ2i are scalars to be confirmed.

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




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 limtV̇(t)=0, which implies the adaptive synchronization can be reached.

Remark 2.

The synchronization issue of delayed ICGNNs are successfully resolved in Theorem 1. By means of variable transformation method, the original inertial system is split into two subsystems. By imposing an adaptive controller to subsystem 2, Theorem 1 has presented a componentwise scheme to assure the adaptive synchronization.

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


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

Corollary 1.

Suppose that the assumptions (A2) and (A4) hold, the drive-response delayed inertial NNs (14) and (15) are synchronous with the adaptive controller as

the feedback strengths ξi and ηi(i=1,2,,n) are adapted in the light of the following update laws, respectively:
where εi and θi(i=1,2,,n) are arbitrary positive scalars.



Remark 3.

In [31], the authors discussed the synchronization problem of delayed inertial NNs based on the feedback controller. Compared with the model in that paper, Corollary 1 does not impose two controllers to the split drive system, which is of less conservatism.

When d2ui(t)dt2=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)=12i=1nei2(t)+11pj=1nα¯ilj|bij|tτ(t)tej2(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.

Corollary 2.

If the hypotheses (A1)(A4) are satisfied, then the drive-response delayed CGNNs (20) and (21) are synchronous with the controller devised as

the parameter δi(i=1,2,,n) is adapted in light of the following update law:
with μi(i=1,2,,n) are any positive scalars.



Remark 4.

In [16] and [21], the authors studied the synchronization of CGNNs with constant amplification gains, which could be deemed as a special case of Corollary 2. From the above discussion, it is obvious that the model considered in this paper is quite general and our results effectually improve several known ones.


Consider the following two-dimensional ICGNNs with time-varying delay:

where i=1,2. The parameters of system (23) are set as β1=β2=1, a11=0.6, a12=0.3, a21=0.4, a22=0.1, b11=0.1, b12=0.2, b21=0.2, b22=0.05. τ(t)=0.5+0.5sint, J1=J2=0.2. In addition, α1(u1)=211+u12, α2(u2)=2+11+u22, hi(ui)=ui, fi(ui)=tanh(ui) for i=1,2.

The corresponding response system is presented as


By adopting the variable transformation, the above two systems can be rewritten as


It is obvious that 1α1(u1)2, 2α2(u2)3, which implies α¯1=2, α¯2=3. For any u,v, one has

for i=1,2. Hence we take M1=M2=1.

Moreover, for any u,v, we also have

for i=1,2. Thus we choose γ1=γ2=3.

Since fi(ui)=tanh(ui), τ(t)=0.5+0.5sint, one can set fi=li=1(i=1,2) and p=0.5.

Then the assumptions (A1)(A4) are all satisfied. According to (12), after a simple calculation, we take δ1=3, δ2=2, λ11=3.2, λ12=2.125, λ21=9.4, λ22=4.95. According to Theorem 1, the drive-response systems (24) and (25) can achieve synchronization. Figure 1 depicts the state trajectories of systems (24) and (25). Figure 2 further depicts the synchronization errors e1i(t)(i=1,2) between the uncontrolled system and the controlled system. The dynamic behaviors of control parameters ξi and ηi(i=1,2) are illustrated in Figure 3. It is obvious that the numerical simulations coincide well with the developed theoretical results.

Figure 1

State trajectories of drive-response systems (24) and (25).

Figure 2

Dynamic behaviors of error signal between drive system (24) and response system (25).

Figure 3

Dynamic behaviors of parameters ξi and ηi(i=1,2) in the adaptive controller (6).


Generally, the synchronization problem of delayed ICGNNs has been addressed in this work. The introduction of inertial term makes this problem more complicated and challenging. By employing the method of variable transformation, our synchronization criterion is presented in component form, which can be easily verified. It is also worth pointing out that our results comprise some conclusion appeared in the previous literature, as well as reduce restriction on the controller. In the end, a convictive example is proposed to demonstrate the feasibility of the adaptive strategy.


The authors declare that there is no conflict of interests regarding the publication of this article.


Cao and Liu conceived and designed the study. Huang wrote the paper. Cao and Liu reviewed and edited the manuscript. All authors read and approved the manuscript.


This work was partially supported by the National Natural Science Foundation of China under Grants 61573096 and 61833005, the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence under Grant BM2017002. (Corresponding author: Jinde Cao.)


International Journal of Computational Intelligence Systems
13 - 1
472 - 478
Publication Date
ISSN (Online)
ISSN (Print)
10.2991/ijcis.d.200402.001How to use a DOI?
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (

Cite this article

AU  - Qun Huang
AU  - Jinde Cao
AU  - Qingshan Liu
PY  - 2020
DA  - 2020/04/17
TI  - Synchronization of Delayed Inertial Cohen–Grossberg Neural Networks Under Adaptive Feedback Controller
JO  - International Journal of Computational Intelligence Systems
SP  - 472
EP  - 478
VL  - 13
IS  - 1
SN  - 1875-6883
UR  -
DO  - 10.2991/ijcis.d.200402.001
ID  - Huang2020
ER  -