Generalized Projective Synchronization between Two Different Neural Networks with Mixed Time Delays

Abstract

The generalized projective synchronization (GPS) between two different neural networks with nonlinear coupling and mixed time delays is considered. Several kinds of nonlinear feedback controllers are designed to achieve GPS between two different such neural networks. Some results for GPS of these neural networks are proved theoretically by using the Lyapunov stability theory and the LaSalle invariance principle. Moreover, by comparison, we determine an optimal nonlinear controller from several ones and provide an adaptive update law for it. Computer simulations are provided to show the effectiveness and feasibility of the proposed methods.

1. Introduction

Over the past two decades, the investigation on the synchronization of complex networks has attracted a great deal of attention due to its potential applications in various fields, such as physics, mathematics, secure communication, engineering, automatic control, biology, and sociology [1–9]. In the literature, there are many widely studied synchronization patterns, which define the correlated in-time behaviors among the nodes in a dynamical network, for example, complete synchronization [10–12], lag synchronization [13–15], anti-synchronization [16–18], phase synchronization [19–21], projective synchronization [22–32], and so on. Projective synchronization reflects a kind of proportionality between the synchronized states, so it is an interesting research topic and has many applications. For instance, if this proportional feature is applied to M-nary digital communication, the communication speed can be accelerated substantially. In view of this merit, many researchers throw themselves into generalized projective synchronization.

Recent years have witnessed many written achievements on projective synchronization between two identical complex dynamical networks [22–30]. We introduce three typical references here. In [28], Chen et al. studied projective synchronization of time-delayed chaotic systems in a driven-response complex network, where the nodes are not partially linear and the scale factors are different from each other. In [29], Feng et al. investigated projective-anticipating and projective-lag synchronization on complex dynamical networks composed of a large number of interconnected components, in which the dynamics of the nodes of the complex networks were time-delayed chaotic systems without the limitation of the partial linearity. In [30], Wang et al. explored the problem of outer synchronization between two complex networks with the same topological structure and time-varying coupling delay with a new mixed outer synchronization behavior. In addition, a novel nonfragile linear state feedback controller is designed to realize the mixed outer synchronization between two networks and proved analytically by using the Lyapunov-Krasovskii stability theory.

However, in real world, studying the phenomena of synchronizing two different complex networks is closer to reality. The so-called different implies that the drive and response networks have different node dynamics, different number of nodes, or different topological structures. Recently, some related works have come out, such as [31, 32]. In [31], Zheng et al. probed into adaptive projective synchronization between two complex networks with different topological structures, although its systems contained time-varying delays. Some results on topology identification were obtained which can be also seen as a highlight of this paper. In [32], the generalized projective synchronization with the above three differences was investigated based on the LaSalle invariance principle. However, the model of complex network only has linear coupling and coupling time delay.

Due to the finite information transmission and processing speeds among the units, the connection delays in realistic modeling of many large networks with communication must be taken into account. Therefore, it is important to study the effect of time delay in synchronization of coupled systems. Usually, time delay involves two parts. One is the delay inside the systems, called internal delay. The other is caused by the exchange of information between systems referred to as coupling delay. Moreover, nonlinear functions can display more nature phenomena. Hence, the internal delay and nonlinear functions are introduced into the considered neural networks in this writing. What is more, the nonlinear functions in the drive and response networks are also different. In particular, three comparable nonlinear controllers are presented to realize the GPS based on the LaSalle invariance principle and some basic inequalities. On the contrary, an optimal nonlinear controller is produced eventually. In the last theorem, we use an adaptive control technique for the optimal nonlinear controller in order to make the feedback control gain small enough.

Notation. Throughout this paper, 𝑅^𝑛 and 𝑅^𝑚×𝑛 denote 𝑛-dimensional Euclidean space and the set of 𝑚×𝑛 real matrices, respectively. 𝜆_min(𝐴) represents the smallest eigenvalue of a symmetric matrix A. ⊗ is the Kronecker product. The superscript 𝑇 of 𝑥^𝑇 or 𝐴^𝑇 denotes the transpose of the vector 𝑥∈𝑅^𝑛 or the matrix 𝐴∈𝑅^𝑚×𝑛. 𝐼_𝑛 is identity matrix with 𝑛 nodes.

2. Model Description and Preliminaries

Consider a general neural network with mixed time delays consisting of 𝑁₁ nodes and nonlinear couplings, which is described as follows: 𝑥̇_𝑖(𝑡)=−𝐶₁𝑥_𝑖(𝑡)+𝐴₁𝑓₁𝑥_𝑖(𝑡)+𝐵₁𝑔₁𝑥_𝑖𝑡−𝜏₁+^𝑁1_𝑗=1+𝐺_𝑖𝑗Γ₁ℎ₁𝑥_𝑗(𝑡)_𝑁1_𝑗=1𝐺^𝜏𝑖𝑗Γ₂ℎ₁𝑥_𝑗𝑡−𝜏₂,𝑖∈ℐ₁,(2.1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈𝑅^𝑛,(𝑖∈ℐ₁={1,2,…,𝑁₁}) are the state variables of the 𝑖th node at time 𝑡; 𝐶₁=diag{𝑐₁₁,𝑐₁₂,…,𝑐_1𝑛} is the decay constant matrix with 𝑐_1𝑚>0(𝑚∈{1,2,…,𝑛}); 𝐴₁=(𝑎^1𝑖𝑗)_𝑛×𝑛 and 𝐵₁=(𝑏^1𝑖𝑗)_𝑛×𝑛 are system matrices; 𝑓₁(𝑥_𝑖(𝑡))=[𝑓¹¹(𝑥_𝑖1(𝑡)),𝑓¹²(𝑥_𝑖2(𝑡)),…,𝑓^1𝑛(𝑥_𝑖𝑛(𝑡))]^𝑇, 𝑔₁(𝑥_𝑖(𝑡))=[𝑔¹¹(𝑥_𝑖1(𝑡)),𝑔¹²(𝑥_𝑖2(𝑡)),…,𝑔^1𝑛(𝑥_𝑖𝑛(𝑡))]^𝑇 and ℎ₁(𝑥_𝑖(𝑡))=[ℎ¹¹(𝑥_𝑖1(𝑡)),ℎ¹²(𝑥_𝑖2(𝑡)),…,ℎ^1𝑛(𝑥_𝑖𝑛(𝑡))]^𝑇 are the continuous functions of the neurons; the positive constants 𝜏₁ and 𝜏₂ are internal delay and coupling delay, respectively; Γ₁ and Γ₂ are the inner coupling matrices at time 𝑡 and 𝑡−𝜏₂, respectively, which describe the way of linking the components in each pair of connected two nodes; 𝐺=(𝐺_𝑖𝑗)_𝑁1×𝑁1 and 𝐺^𝜏=(𝐺^𝜏𝑖𝑗)_𝑁1×𝑁1 are coupling configuration matrices which are not necessarily irreducible and symmetric.

In this paper, the neural network (2.1) is used as the drive network, and the response neural network consisting of 𝑁₂ nodes is expressed by 𝑦̇_𝑖(𝑡)=−𝐶₂𝑦_𝑖(𝑡)+𝐴₂𝑓₂𝑦_𝑖(𝑡)+𝐵₂𝑔₂𝑦_𝑖𝑡−𝜏₁+^𝑁2_𝑗=1+𝐻_𝑖𝑗Γ₁ℎ₂𝑦_𝑗(𝑡)_𝑁2_𝑗=1𝐻^𝜏𝑖𝑗Γ₂ℎ₂𝑦_𝑗𝑡−𝜏₂+𝑢_𝑖(𝑡),𝑖∈ℐ₂,(2.2) where 𝑦_𝑖(𝑡)=(𝑦_𝑖1(𝑡),𝑦_𝑖2(𝑡),…,𝑦_𝑖𝑛(𝑡))^𝑇∈𝑅^𝑛,𝑖∈ℐ₂={1,2,…,𝑁₂} are the state variables of the 𝑖th node at time 𝑡. Without loss of generality, we suppose 𝑁₁≥𝑁₂>0. 𝐶₂=diag{𝑐₂₁,𝑐₂₂,…,𝑐_2𝑛} is the decay constant matrix with 𝑐_2𝑚>0(𝑚∈{1,2,…,𝑛}); 𝐴₂=(𝑎^2𝑖𝑗)_𝑛×𝑛 and 𝐵₂=(𝑏^2𝑖𝑗)_𝑛×𝑛 are system matrices; 𝑓₂(𝑦_𝑖(𝑡))=[𝑓²¹(𝑦_𝑖1(𝑡)),𝑓²²(𝑦_𝑖2(𝑡)),…,𝑓^2𝑛(𝑦_𝑖𝑛(𝑡))]^𝑇, 𝑔₂(𝑦_𝑖(𝑡))=[𝑔²¹(𝑦_𝑖1(𝑡)),𝑔²²(𝑦_𝑖2(𝑡)),…,𝑔^2𝑛(𝑦_𝑖𝑛(𝑡))]^𝑇 and ℎ₂(𝑦_𝑖(𝑡))=[ℎ²¹(𝑦_𝑖1(𝑡)),ℎ²²(𝑦_𝑖2(𝑡)),…,ℎ^2𝑛(𝑦_𝑖𝑛(𝑡))]^𝑇 are the continuous functions of the neurons; 𝜏₁, 𝜏₂, Γ₁, and Γ₂ have the same meaning as those in (2.1). 𝐻=(𝐻_𝑖𝑗)_𝑁2×𝑁2 and 𝐻^𝜏=(𝐻^𝜏𝑖𝑗)_𝑁2×𝑁2 are coupling configuration matrices which are not necessarily irreducible and symmetric either.

Now, two mathematical definitions for the generalized projective synchronization are introduced as follows.

Definition 2.1. If there is a nonzero constant 𝜎 such that lim_𝑡→+∞‖‖𝑦_𝑖(𝑡)−𝜎𝑥_𝑖(𝑡)‖‖=0,𝑖∈ℐ₂,(2.3) the GPS between neural networks (2.1) and (2.2) is said to be achieved. The parameter 𝜎 is called a scaling factor.

Definition 2.2. A continuous function 𝑓(⋅)∶𝑅→𝑅 is said to be the nonnegative-bound function class, denoted as 𝑓∈NBF(𝜉), if there exists a positive scalar 𝜉, such that 0≤𝑓(𝑥)−𝑓(𝑦)𝑥−𝑦≤𝜉(2.4) holds for any 𝑥,𝑦∈𝑅.
The following hypothesis is used throughout the paper.

Assumption 2.3. For activation functions 𝑓^1𝑖,𝑓^2𝑖,𝑔^1𝑖,𝑔^2𝑖,ℎ^1𝑖,ℎ^2𝑖(𝑖∈{1,2,…,𝑛}), there exist positive constants 𝜃_𝑖,𝛽_𝑖,𝜇_𝑖,𝛾_𝑖,𝜋_𝑖,𝛿_𝑖(𝑖∈{1,2,…,𝑛}), such that 𝑓^1𝑖(⋅)∈NBF(𝜃_𝑖),𝑓^2𝑖(⋅)∈NBF(𝛽_𝑖),𝑔^1𝑖(⋅)∈NBF(𝜇_𝑖), 𝑔^2𝑖(⋅)∈NBF(𝛾_𝑖), ℎ^1𝑖(⋅)∈NBF(𝜋_𝑖), ℎ^2𝑖(⋅)∈NBF(𝛿_𝑖).
For better convenience, we denote that 𝜃𝛽==mmaaxx_{1≤𝑖≤𝑛1≤𝑖≤𝑛}𝜃_𝑖;𝜇=max_{1≤𝑖≤𝑛}𝜇_𝑖;𝜋=max_{1≤𝑖≤𝑛}𝜋_𝑖;𝛽_𝑖;𝛾=max_{1≤𝑖≤𝑛}𝛾_𝑖;𝛿=max_{1≤𝑖≤𝑛}𝛿_𝑖.(2.5)

Lemma 2.4 (see [33]). For a matrix 𝐵=(𝑏_𝑖𝑗)∈𝑅^𝑝×𝑞, denote 𝛼(𝐵)=(1/2)max[𝑝,𝑞]max_𝑖,𝑗|𝑏_𝑖𝑗|, then 𝑥^𝑇𝐵𝑦≤𝛼(𝐵)𝑥^𝑇𝑥+𝑦^𝑇𝑦(2.6) holds for all 𝑥∈𝑅^𝑝,𝑦∈𝑅^𝑞.

3. GPS between Two Different Neural Networks with Mixed Time Delays

In this section, we will make a study of GPS between two different neural networks with mixed time delays by means of the LaSalle invariance principle; the, Lyapunov direct method, and nonlinear feedback control technique.

Define the synchronization errors between the drive network (2.1) and the response network (2.2) as 𝑒_𝑖(𝑡)=𝑦_𝑖(𝑡)−𝜎𝑥_𝑖(𝑡),𝑖∈ℐ₂, then we have the following error system: 𝑒_𝑖̇(𝑡)=−−𝜎𝐶𝐵₂𝑒_𝑖1(𝑡)+𝐶₁−𝐶₂𝜎𝑥_𝑖(𝑡)+𝐴₂𝑓₂𝑦_𝑖(𝑡)−𝜎𝐴₁𝑓₁𝑥_𝑖(𝑡)+𝐵₂𝑔₂𝑦_𝑖𝑡−𝜏₁𝑔₁𝑥_𝑖𝑡−𝜏₁+^𝑁2_𝑗=1𝐻_𝑖𝑗Γ₁ℎ₂𝑦_𝑗(𝑡)−𝜎^𝑁1_𝑗=1+𝐺_𝑖𝑗Γ₁ℎ₁𝑥_𝑗(𝑡)_𝑁2_𝑗=1𝐻^𝜏𝑖𝑗Γ₂ℎ₂𝑦_𝑗𝑡−𝜏₂−𝜎^𝑁1_𝑗=1𝐺^𝜏𝑖𝑗Γ₂ℎ₁𝑥_𝑗𝑡−𝜏₂+𝑢_𝑖(𝑡),𝑖∈ℐ₂.(3.1)

Theorem 3.1. Suppose Assumption 2.3 holds; if the nonlinear controllers are chosen as follows: 𝑢_𝑖(𝑡)=𝜎𝐶₂𝑥_𝑖(𝑡)−𝐶₁𝑦_𝑖(𝑡)−𝑘_𝑖𝑒_𝑖(𝑡)−𝐴₂𝑓₂𝜎𝑥_𝑖(𝑡)+𝜎𝐴₁𝑓₁𝑦^𝑖(𝑡)𝜎−𝐵²+𝜎𝐵₁𝑔₂𝜎𝑥_𝑖𝑡−𝜏₁𝑔₁𝑦^𝑖𝑡−𝜏₁𝜎−^𝑁2_𝑗=1𝐻_𝑖𝑗+𝜎_𝑁2Γ₁ℎ₂𝜎𝑥_𝑗(𝑡)+𝐻^𝜏𝑖𝑗Γ₂ℎ₂𝜎𝑥_𝑗𝑡−𝜏₂_𝑗=1𝐺_𝑖𝑗Γ₁ℎ₁𝑦^𝑗(𝑡)𝜎+𝜎^𝑁2_𝑗=1𝐺^𝜏𝑖𝑗Γ₂ℎ₁𝑦^𝑗𝑡−𝜏₂+𝜎_𝑁1𝜎_𝑗=_𝑁2+1𝐺_𝑖𝑗Γ₁ℎ₁𝑥_𝑗(𝑡)+𝐺^𝜏𝑖𝑗Γ₂ℎ₁𝑥_𝑗𝑡−𝜏₂,𝑖∈ℐ₂,(3.2) where 𝑘_𝑖 are the feedback control gains, let 𝑘=min_𝑖∈ℐ2{𝑘_𝑖}, and when 𝑘≥−𝜆_min𝐶₁+𝐶₂+𝛼𝐴₂𝛽²+1+𝛼𝐵₂1+𝛾²+𝛼𝐴₁|𝜎|+𝜃²+𝛼𝐵₁|𝜎|+𝛼𝐺⊗|𝜎Γ|₁+𝛼𝐻⊗Γ₁𝛿²+1+𝛼𝐻^𝜏⊗Γ₂1+𝛿²|𝜎|+𝜋|𝜎²|+𝛼𝐺^𝜏⊗Γ₂|𝜎|+𝛼𝐵¹𝜇²|𝜎|+𝛼𝐺^𝜏⊗Γ₂𝜋²|𝜎|+𝜖¹,(3.3) where 𝜖₁ is a positive constant, then the GPS between the two neural networks (2.1) and (2.2) can be achieved.

Proof. Consider the Lyapunov functional candidate 𝑉(𝑡)=12^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝛼𝐵¹𝜇²|𝜎|+𝛼𝐵²𝛾²^𝑁2_𝑖=1^{𝑡𝑡−𝜏}₁+𝛼𝐺𝑒^𝑇𝑖(𝑠)𝑒_𝑖(𝑠)𝑑𝑠_𝜏⊗Γ₂𝜋²|𝜎|+𝛼𝐻^𝜏⊗Γ₂𝛿²^𝑁2_𝑖=1^{𝑡𝑡−𝜏}₂𝑒^𝑇𝑖(𝑠)𝑒_𝑖(𝑠)𝑑𝑠.(3.4) Calculating 𝑉̇ with respect to 𝑡 along the solution of (3.1), and noticing the nonlinear feedback controllers (3.2), one has 𝑉̇(𝑡)∣₍₃₎=^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝛼𝐵¹𝜇²|𝜎|+𝛼𝐵²𝛾²^𝑁2_𝑖=1𝑒^𝑇𝑖+𝛼𝐺(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₁𝑒_𝑖𝑡−𝜏₁_𝜏⊗Γ₂𝜋²|𝜎|+𝛼𝐻^𝜏⊗Γ₂𝛿²^𝑁2_𝑖=1=𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₂𝑒_𝑖𝑡−𝜏₂_𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡+)𝐵−₂𝐶₁+𝐶₂𝑒_𝑖(𝑡)−𝑘_𝑖𝑒_𝑖(𝑡)+𝐴₂𝑓₂𝑦_𝑖(𝑡)−𝑓₂𝜎𝑥_𝑖(𝑡)+𝑔₂𝑦_𝑖𝑡−𝜏₁−𝑔₂𝜎𝑥_𝑖𝑡−𝜏₁_𝑁2_𝑗=1+𝐻_𝑖𝑗Γ₁ℎ₂𝑦_𝑗(𝑡)−ℎ₂𝜎𝑥_𝑗(𝑡)_𝑁2_𝑗=1+𝜎𝐴𝐻^𝜏𝑖𝑗₁Γ₂ℎ₂𝑦_𝑗𝑡−𝜏₂−ℎ₂𝜎𝑥_𝑗𝑡−𝜏₂𝑓¹𝑦^𝑖(𝑡)𝜎−𝑓¹𝑥_𝑖(𝑡)+𝜎𝐵₁𝑔¹𝑦^𝑖𝑡−𝜏₁+𝜎_𝑁2𝜎−𝑔¹𝑥_𝑖𝑡−𝜏₁_𝑗=1𝐺_𝑖𝑗Γ₁ℎ¹𝑦^𝑗(𝑡)𝜎−ℎ¹𝑥_𝑗+𝜎_𝑁2(𝑡)_𝑗=1𝐺^𝜏𝑖𝑗Γ₂ℎ¹𝑦^𝑗𝑡−𝜏₂+𝛼𝐵𝜎−ℎ¹𝑥_𝑗𝑡−𝜏₂₁𝜇²|𝜎|+𝛼𝐵²𝛾²^𝑁2_𝑖=1𝑒^𝑇𝑖+𝛼𝐺(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₁𝑒_𝑖𝑡−𝜏₁_𝜏⊗Γ₂𝜋²|𝜎|+𝛼𝐻^𝜏⊗Γ₂𝛿²^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₂𝑒_𝑖𝑡−𝜏₂.(3.5) By utilizing Lemma 2.4 and Assumption 2.3, we have the following four inequalities: the first one is ^𝑁2_𝑖=1𝑒^𝑇𝑖≤(𝑡𝛼)𝐴𝐴₂₂𝑓₂𝑦_𝑖(𝑡)−𝑓₂𝜎𝑥_𝑖(𝑡)≤𝛼𝐴₂^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝑓₂𝑦_𝑖(𝑡)−𝑓₂𝜎𝑥_𝑖(𝑡)^𝑇𝑓₂𝑦_𝑖(𝑡)−𝑓₂𝜎𝑥_𝑖(𝑡)𝛽²+1^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡).(3.6) Denote 𝐻₂(𝑒(𝑡))=[(ℎ₂(𝑦₁(𝑡))−ℎ₂(𝜎𝑥₁(𝑡)))^𝑇,(ℎ₂(𝑦₂(𝑡))−ℎ₂(𝜎𝑥₂(𝑡)))^𝑇,…,(ℎ₂(𝑦_𝑁2(𝑡))−ℎ₂(𝜎𝑥_𝑁2(𝑡)))^𝑇]^𝑇 and 𝑒(𝑡)=(𝑒^𝑇1(𝑡),𝑒^𝑇2(𝑡),…,𝑒^𝑇𝑁₂(𝑡))^𝑇, and then we can get the second inequality: ^𝑁2_𝑁𝑖=1²_𝑗=1=𝑒𝑒^𝑇𝑖_𝑇(𝑡)𝐻_𝑖𝑗Γ₁ℎ₂𝑦_𝑗(𝑡)−ℎ₂𝜎𝑥_𝑗(𝑡)≤𝛼(𝐻𝑡)⊗𝐻Γ⊗₁Γ₁𝐻₂(𝑒(𝑡))=𝛼𝐻⊗Γ₁𝑒^𝑇(𝑡)𝑒(𝑡)+𝐻^𝑇2(𝑒(𝑡))𝐻₂(𝑒(𝑡))≤𝛼𝐻⊗Γ₁^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+ℎ₂𝑦_𝑖(𝑡)−ℎ₂𝜎𝑥_𝑖(𝑡)^𝑇ℎ₂𝑦_𝑖(𝑡)−ℎ₂𝜎𝑥_𝑖(𝑡)𝛿²+1^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡),(3.7) and the third one is as follows: ^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝜎𝐴₁𝑓₁𝑦^𝑖(𝑡)𝜎−𝑓¹𝑥_𝑖≤|𝜎|𝛼𝐴₁(𝑡)^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝑓₁𝑦^𝑖(𝑡)𝜎−𝑓¹𝑥_𝑖(𝑡)^𝑇𝑓¹𝑦^𝑖(𝑡)𝜎−𝑓¹𝑥_𝑖≤|𝜎|𝛼𝐴₁(𝑡)^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝜃²𝜎²=|𝜎|+𝜃₂𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)|𝜎|𝛼𝐴¹^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡).(3.8) Let 𝐻₁(𝑒(𝑡))=[(ℎ₁(𝑦₁(𝑡)/𝜎)−ℎ₁(𝑥₁(𝑡)))^𝑇,(ℎ₁(𝑦₂(𝑡)/𝜎)−ℎ₁(𝑥₂(𝑡)))^𝑇,…,(ℎ₁(𝑦_𝑁2(𝑡)/𝜎)−ℎ₁(𝑥_𝑁2(𝑡)))^𝑇]^𝑇; thus, we have the last one: 𝜎^𝑁2_𝑁𝑖=1²_𝑗=1𝑒^𝑇𝑖(𝑡)𝐺_𝑖𝑗Γ₁ℎ¹𝑦^𝑗(𝑡)𝜎−ℎ¹𝑥_𝑗=𝜎𝑒_𝑇(𝑡)≤|𝜎|𝛼(𝑡)𝐺𝐺⊗⊗Γ₁Γ₁𝐻₁(𝑒(𝑡))^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+ℎ₁𝑦^𝑖(𝑡)𝜎−ℎ¹𝑥_𝑖(𝑡)^𝑇ℎ¹𝑦^𝑖(𝑡)𝜎−ℎ¹𝑥_𝑖≤|𝜎|𝛼𝐺⊗Γ₁(𝑡)^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝜋²𝜎²=|𝜎|+𝜋₂𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)|𝜎|𝛼𝐺⊗Γ¹^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡).(3.9)
Similarly, we can obtain the following four inequalities: ^𝑁2_𝑖=1𝑒^𝑇𝑖≤(𝑡𝛼)𝐵𝐵₂₂𝑔₂𝑦_𝑖𝑡−𝜏₁−𝑔₂𝜎𝑥_𝑖𝑡−𝜏₁^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝛾²𝑒^𝑇𝑖𝑡−𝜏₁𝑒_𝑖𝑡−𝜏₁,(3.10)_𝑁2_𝑁𝑖=1²_𝑗=1≤𝑒𝛼^𝑇𝑖(𝐻𝑡)𝐻^𝜏𝑖𝑗_𝜏Γ₂ℎ₂𝑦_𝑗𝑡−𝜏₂−ℎ₂𝜎𝑥_𝑗𝑡−𝜏₂⊗Γ₂^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝛿²𝑒^𝑇𝑖𝑡−𝜏₂𝑒_𝑖𝑡−𝜏₂,(3.11)_𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝜎𝐵₁𝑔₁𝑦^𝑖𝑡−𝜏₁≤|𝜎|𝛼𝐵₁𝜎−𝑔¹𝑥_𝑖𝑡−𝜏₁^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝜇²𝜎²𝜎𝑒^𝑇𝑖𝑡−𝜏₁𝑒_𝑖𝑡−𝜏₁,(3.12)_𝑁2_𝑁𝑖=1²_𝑗=1𝑒^𝑇𝑖(𝑡)𝐺^𝜏𝑖𝑗Γ₂ℎ¹𝑦^𝑗𝑡−𝜏₂≤|𝜎|𝛼𝐺_𝜏𝜎−ℎ¹𝑥_𝑗𝑡−𝜏₂⊗Γ₂^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝜋²𝜎²𝑒^𝑇𝑖𝑡−𝜏₂𝑒_𝑖𝑡−𝜏₂.(3.13)
Substituting (3.6)–(3.13) into (3.5), we can obtain 𝑉̇(𝑡)≤−𝜆_min𝐶₁+𝐶₂+𝛼𝐴₂𝛽²+1+𝛼𝐵₂1+𝛾²+𝛼𝐴₁|𝜎|+𝜃²+𝛼𝐻⊗Γ₁|𝜎|+𝛼𝐵¹|𝜎|+𝛼𝐵𝛿²+1+𝛼𝐻^𝜏⊗Γ₂1+𝛿²+𝛼𝐺⊗Γ₁|𝜎|+𝜋|𝜎²|+𝛼𝐺^𝜏⊗Γ₂|𝜎|−𝑘₁𝜇²|𝜎|+𝛼𝐺^𝜏⊗Γ₂𝜋²|𝜎|𝑒^𝑇(𝑡)𝑒(𝑡),(3.14)
Taking account of condition (3.3), we have 𝑉̇(𝑡)≤−𝜖₁𝑒^𝑇(𝑡)𝑒(𝑡)≤0.
Clearly, 𝑆={𝑒_𝑖(𝑡)=0,𝑘_𝑖=𝑘,𝑖∈ℐ₂} is the largest invariant set contained in {𝑉̇(𝑡)=0}={𝑒_𝑖(𝑡)=0,𝑖∈ℐ₂}. In terms of the LaSalle invariant principle, the trajectory asymptotically converges to the largest invariant set 𝑆 with any initial value of (3.1), namely, lim_𝑡→+∞‖𝑒_𝑖(𝑡)‖=0,𝑖∈ℐ₂. Hence, the GPS between neural networks (2.1) and (2.2) is realized. The proof is completed.

In Theorem 3.1, the generalized projective synchronization of two different neural networks with time delay has been investigated by choosing suitable nonlinear feedback controllers. However, it requires feedback control gains to be large in 𝑢_𝑖 given by (3.2), which is not practical. Hence, it is desirable to improve the scheme for reducing the feedback control gains to be as small as possible. Now, we give the following improvement scheme.

Theorem 3.2. Suppose Assumption 2.3 holds; if the nonlinear controllers are chosen as follows: 𝑢_𝑖(𝑡)=+𝜎𝜎𝐶𝐵₂₁−𝐶₁𝑥_𝑖(𝑡)−𝐴₂𝑓₂𝜎𝑥_𝑖(𝑡)+𝜎𝐴₁𝑓₁𝑥_𝑖(𝑡)−𝐵₂𝑔₂𝜎𝑥_𝑖𝑡−𝜏₁𝑔₁𝑥_𝑖𝑡−𝜏₁−^𝑁2_𝑗=1𝐻_𝑖𝑗+𝜎_𝑁1Γ₁ℎ₂𝜎𝑥_𝑗(𝑡)+𝐻^𝜏𝑖𝑗Γ₂ℎ₂𝜎𝑥_𝑗𝑡−𝜏₂_𝑗=1𝐺_𝑖𝑗Γ₁ℎ₁𝑥_𝑗(𝑡)+𝐺^𝜏𝑖𝑗Γ₂ℎ₁𝑥_𝑗𝑡−𝜏₂−𝑘_𝑖𝑒_𝑖(𝑡),𝑖∈ℐ₂,(3.15) where 𝑘_𝑖 are the feedback gains, denote 𝑘=min_𝑖∈ℐ2{𝑘_𝑖}, and if 𝑘≥+−𝛼𝜆𝐻_min⊗𝐶Γ₂₁+𝛼𝐴₂𝛽²+1+𝛼𝐵₂1+𝛾²𝛿²+1+𝛼𝐻^𝜏⊗Γ₂1+𝛿²+𝜖₂,(3.16) where 𝜖₂ is a positive constant, then the GPS between neural networks (2.1) and (2.2) can be achieved.

Proof. Select the following Lyapunov functional candidate: 𝑉(𝑡)=12^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝛼𝐵₂𝛾^2𝑁2_𝑖=1^{𝑡𝑡−𝜏}₁𝑒^𝑇𝑖(𝑠)𝑒_𝑖(𝑠)𝑑𝑠+𝛼𝐻^𝜏⊗Γ₂𝛿^2𝑁2_𝑖=1^{𝑡𝑡−𝜏}₂𝑒^𝑇𝑖(𝑠)𝑒_𝑖(𝑠)𝑑𝑠.(3.17) Differentiating 𝑉 with respect to time along (3.1), we have 𝑑𝑉𝑑𝑡∣⁽³⁾=^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝛼𝐵₂𝛾^2𝑁2_𝑖=1𝑒^𝑇𝑖+𝛼𝐻_𝜏(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₁𝑒_𝑖𝑡−𝜏₁⊗Γ₂𝛿^2𝑁2_𝑖=1𝑒^𝑇𝑖=(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₂𝑒_𝑖𝑡−𝜏₂_𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡+)𝐵−₂𝐶₂𝑒_𝑖(𝑡)−𝑘_𝑖𝑒_𝑖(𝑡)+𝐴₂𝑓₂𝑦_𝑖(𝑡)−𝑓₂𝜎𝑥_𝑖(𝑡)+𝑔₂𝑦_𝑖𝑡−𝜏₁−𝑔₂𝜎𝑥_𝑖𝑡−𝜏₁+^𝑁2_𝑗=1𝐻_𝑖𝑗Γ₁ℎ₂𝑦_𝑗(𝑡)−ℎ₂𝜎𝑥_𝑗(𝑡)_𝑁2_𝑗=1+𝛼𝐵₂𝐻^𝜏𝑖𝑗Γ₂ℎ₂𝑦_𝑗𝑡−𝜏₂−ℎ₂𝜎𝑥_𝑗𝑡−𝜏₂+𝛼𝐻_𝜏𝛾^2𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₁𝑒_𝑖𝑡−𝜏₁⊗Γ₂𝛿^2𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₂𝑒_𝑖𝑡−𝜏₂.(3.18)
Substituting (3.6), (3.7), (3.10), and (3.11) into (3.18), we can obtain 𝑉̇(𝑡)≤−+𝜆𝛼𝐻_min⊗𝐶₂Γ₁+𝛼𝐴₂𝛽²+1+𝛼𝐵₂1+𝛾²𝛿²+1+𝛼𝐻^𝜏⊗Γ₂1+𝛿²−𝑘𝑒^𝑇(𝑡)𝑒(𝑡).(3.19) By condition (3.16), we have 𝑉̇(𝑡)≤−𝜖₂𝑒^𝑇(𝑡)𝑒(𝑡)≤0. Similarly, in light of the proof of Theorem 3.1, the GPS between neural networks (2.1) and (2.2) can be achieved under nonlinear controllers (3.15) too.

Remark 3.3. Comparing the infimum of the feedback control gain 𝑘 in Theorem 3.2 with that in Theorem 3.1 implies that 𝑘Γ_res2=−𝜆_min(𝐶₁)+𝛼(𝐴₁)(|𝜎|+𝜃²/|𝜎|)+𝛼(𝐵₁)|𝜎|+𝛼(𝐺⊗Γ₁)(|𝜎|+𝜋²/|𝜎|)+𝛼(𝐺^𝜏⊗Γ₂)|𝜎|+𝛼(𝐵₁)𝜇²/|𝜎|+𝛼(𝐺^𝜏⊗)𝜋²/|𝜎| is the extra part compared with the needed feedback gain in Theorem 3.2. Usually, it can be considered that 𝑘_res>0, which will be demonstrated in simulation.
Furthermore, in order to obtain much smaller feedback control gains, we choose more suitable controllers as follows.

Theorem 3.4. Suppose Assumption 2.3 holds. Under the nonlinear controllers 𝑢_𝑖(𝑡)=+𝜎𝐶𝜎₂𝐵𝑥_𝑖1(𝑡)−𝐶₁𝑦_𝑖(𝑡)−𝐴₂𝑓₂𝜎𝑥_𝑖(𝑡)+𝜎𝐴₁𝑓₁𝑥_𝑖(𝑡)−𝐵₂𝑔₂𝜎𝑥_𝑖𝑡−𝜏₁𝑔₁𝑥_𝑖𝑡−𝜏₁−^𝑁2_𝑗=1𝐻_𝑖𝑗+𝜎_𝑁1Γ₁ℎ₂𝜎𝑥_𝑗(𝑡)+𝐻^𝜏𝑖𝑗Γ₂ℎ₂𝜎𝑥_𝑗𝑡−𝜏₂_𝑗=1𝐺_𝑖𝑗Γ₁ℎ₁𝑥_𝑗(𝑡)+𝐺^𝜏𝑖𝑗Γ₂ℎ₁𝑥_𝑗𝑡−𝜏₂−𝑘_𝑖𝑒_𝑖(𝑡),𝑖∈ℐ₂,(3.20) where 𝑘_𝑖 are the feedback control gains, denote 𝑘=min_𝑖∈ℐ2{𝑘_𝑖}, and if 𝑘≥+−𝛼𝜆𝐻_min𝜏𝐶₁+𝐶₂+𝛼𝐴₂𝛽²+1+𝛼𝐵₂1+𝛾²+𝛼𝐻⊗Γ₁𝛿²+1⊗Γ₂1+𝛿²+𝜖₃,(3.21) where 𝜖₃ is some positive constant, then the GPS between the two neural networks (2.1) and (2.2) can be achieved.

Proof. Choose the same Lyapunov functional as (3.17) in the proof of Theorem 3.2, and then we can get 𝑑𝑉𝑑𝑡∣(3)=^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡+)𝐵−₂𝐶₁+𝐶₂𝑒_𝑖(𝑡)−𝑘_𝑖𝑒_𝑖(𝑡)+𝐴₂𝑓₂𝑦_𝑖(𝑡)−𝑓₂𝜎𝑥_𝑖(𝑡)+𝑔₂𝑦_𝑖𝑡−𝜏₁−𝑔₂𝜎𝑥_𝑖𝑡−𝜏₁+^𝑁2_𝑗=1𝐻_𝑖𝑗Γ₁ℎ₂𝑦_𝑗(𝑡)−ℎ₂𝜎𝑥_𝑗(𝑡)_𝑁2_𝑗=1+𝛼𝐵₂𝐻^𝜏𝑖𝑗Γ₂ℎ₂𝑦_𝑗𝑡−𝜏₂−ℎ₂𝜎𝑥_𝑗𝑡−𝜏₂+𝛼𝐻_𝜏𝛾^2𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₁𝑒_𝑖𝑡−𝜏₁⊗Γ₂𝛿^2𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝑒^𝑇𝑖𝑡−𝜏₂𝑒_𝑖𝑡−𝜏₂.(3.22)
Combining (3.6), (3.7), (3.10), (3.11), and (3.22), we could change inequality (3.19) into 𝑉̇(𝑡)≤+−𝛼𝜆𝐻_min⊗𝐶₁Γ₁+𝐶₂+𝛼𝐴₂𝛽²+1+𝛼𝐵₂1+𝛾²𝛿²+1+𝛼𝐻^𝜏⊗Γ₂1+𝛿²−𝑘𝑒^𝑇(𝑡)𝑒(𝑡).(3.23) Taking into account condition (3.21), the GPS can be realized under nonlinear controllers (3.18).

Remark 3.5. It is easy to find that the infimum of 𝑘 in (3.21) has one more term −𝜆_min(𝐶₁) than (3.16). Because 𝐶₁ is diagonally positive definite, −𝜆_min(𝐶₁)<0 demonstrates the required 𝑘 in Theorem 3.4 is smaller than the one in Theorem 3.2.
If the two neural networks (2.1) and (2.2) have identical number of nodes, node dynamics, and topological structure, that is, 𝑁₁=𝑁₂, 𝐶₁=𝐶₂, 𝐴₁=𝐴₂, 𝐵₁=𝐵₂, 𝑓₁=𝑓₂, 𝑔₁=𝑔₂, 𝐺_𝑖𝑗=𝐻_𝑖𝑗, 𝐺^𝜏𝑖𝑗=𝐻^𝜏𝑖𝑗, and Γ₁=Γ₂, the error system (3.1) can be rewritten as follows: 𝑒_𝑖̇(𝑡)=+−𝐶𝑢_1𝑖𝑒_𝑖(𝑡)+𝐴₁𝑓₁𝑦_𝑖(𝑡)−𝜎𝑓₁𝑥_𝑖(𝑡)+𝐵₁𝑔₁𝑦_𝑖𝑡−𝜏₁−𝜎𝑔₁𝑥_𝑖𝑡−𝜏₁(𝑡)+^𝑁1_𝑗=1+𝐺_𝑖𝑗Γ₁ℎ₁𝑦_𝑗(𝑡)−𝜎ℎ₁𝑥_𝑗(𝑡)_𝑁1_𝑗=1𝐺^𝜏𝑖𝑗Γ₂ℎ₂𝑦_𝑗𝑡−𝜏₂−𝜎ℎ₂𝑥_𝑗𝑡−𝜏₂,𝑖∈ℐ₂.(3.24)
Thus, we can obtain the following corollary for synchronizing the error system (3.24).

Corollary 3.6. Suppose the two neural networks (2.1) and (2.2) have identical number of nodes, node dynamics, and topological structure. If the following nonlinear controllers: 𝑢_𝑖(𝑡)=−−𝐴𝑘_1𝑖𝑓₁𝜎𝑥_𝑖(𝑡)−𝜎𝑓₁𝑥_𝑖(𝑡)−𝐵₁𝑔₁𝜎𝑥_𝑖𝑡−𝜏₁−𝜎𝑔₁𝑥_𝑖𝑡−𝜏₁𝑒_𝑖(𝑡)−^𝑁1_𝑗=1−𝐺_𝑖𝑗Γ₁ℎ₁𝑦_𝑗(𝑡)−𝜎ℎ₁𝑥_𝑗(𝑡)_𝑁1_𝑗=1𝐺^𝜏𝑖𝑗Γ₂ℎ₂𝑦_𝑗𝑡−𝜏₂−𝜎ℎ₂𝑥_𝑗𝑡−𝜏₂,𝑖∈ℐ₁,(3.25) are employed, where 𝑘_𝑖 are the feedback control gains, denote 𝑘=min_𝑖∈ℐ2{𝑘_𝑖}, and when 𝑘≥−𝐶₁+𝛼(𝐴₁)(𝛽²+1)+𝛼(𝐵₁)(1+𝛾²), the error system (3.24) can be synchronized.

Proof. We construct the Lyapunov function as follows: 𝑉(𝑡)=12^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝛼𝐵₁𝛾^2𝑁2_𝑖=1^{𝑡𝑡−𝜏}₁𝑒^𝑇𝑖(𝑠)𝑒_𝑖(𝑠)𝑑𝑠.(3.26) We can conclude that 𝑘≥−𝐶₁+𝛼(𝐴₁)(𝛽²+1)+𝛼(𝐵₁)(1+𝛾²) by using the same method as that in the above theorems.
Furthermore, let us assume that system (3.24) is without time-delay terms; then, system (3.24) reduces to 𝑒_𝑖̇(𝑡)=+−𝐶₁𝑒_𝑖(𝑡)+𝐴₁𝑓₁𝑦_𝑖(𝑡)−𝜎𝑓₁𝑥_𝑖(𝑡)_𝑁1_𝑗=1𝐺_𝑖𝑗Γ₁ℎ₁𝑦_𝑗(𝑡)−𝜎ℎ₁𝑥_𝑗(𝑡)+𝑢_𝑖(𝑡),𝑖∈ℐ₁.(3.27)
Then, a simpler corollary can be produced.

Corollary 3.7. The following controllers: 𝑢_𝑖(𝑡)=−𝐴₁𝑓₁𝜎𝑥_𝑖(𝑡)−𝜎𝑓₁𝑥_𝑖(𝑡)−^𝑁1_𝑗=1𝐺_𝑖𝑗Γ₁ℎ₁𝑦_𝑗(𝑡)−𝜎ℎ₁𝑥_𝑗(𝑡)−𝑘_𝑖𝑒_𝑖(𝑡),𝑖∈ℐ₁,(3.28) can be applied to synchronize the system (3.27) when the feedback control gain 𝑘 satisfies the inequality 𝑘≥−𝐶₁+𝛼(𝐴₁)(𝛽²+1).
Similarly, in the drive network (2.1) and the response network (2.2), if 𝐴₁=𝐼_𝑁1, 𝐴₂=𝐼_𝑁2, 𝐵₁=𝐵₂=0, ℎ₁, ℎ₂ are linear functions and 𝐺=𝐻=0, the error system (3.1) can be rewritten as follows: 𝑒_𝑖̇(𝑡)=−𝐶₁+𝐶₂𝑒_𝑖(𝑡)+𝑓₂𝑒_𝑖(𝑡)+^𝑁2_𝑗=1𝐻^𝜏𝑖𝑗𝜎₂𝑒_𝑖𝑡−𝜏₂−𝑘_𝑖𝑒_𝑖(𝑡),𝑖∈ℐ₂.(3.29)

Thus, we can obtain the following corollary for synchronizing the error system (3.24).

Corollary 3.8. By applying the nonlinear controllers 𝑢_𝑖(𝑡)=𝜎𝐶₂𝑥_𝑖(𝑡)−𝐶₁𝑦_𝑖(𝑡)−𝑓₂𝜎𝑥_𝑖(𝑡)+𝜎𝑓₁𝑥_𝑖(𝑡)−𝜎^𝑁2_𝑗=1+𝜎_𝑁1𝐻^𝜏𝑖𝑗Γ₂𝑥_𝑗𝑡−𝜏₂_𝑗=1𝐺^𝜏𝑖𝑗Γ₂𝑥_𝑗𝑡−𝜏₂−𝑘_𝑖𝑒_𝑖(𝑡),𝑖∈ℐ₂,(3.30) where 𝑘_𝑖 are the feedback control gains, denote 𝑘=min_𝑖∈ℐ2{𝑘_𝑖}, and when 𝑘≥−𝜆_min(𝐶₁+𝐶₂)+2𝛼(𝐻^𝜏⊗Γ²)+1, the GPS between the two neural networks (2.1) and (2.2) can be achieved.

Proof. We construct the Lyapunov function as follows: 𝑉(𝑡)=12^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝛼𝐻^𝜏⊗Γ²^{𝑡𝑡−𝜏}₂𝑒^𝑇𝑖(𝑠)𝑒_𝑖(𝑠)𝑑𝑠.(3.31) We can conclude that 𝑘≥−𝜆_min(𝐶₁+𝐶₂)+2𝛼(𝐻^𝜏⊗Γ²)+1 by using the same method as that in the above theorems. Meanwhile, this conclusion also contains the result of [32].
It is easy to see that the theoretical feedback gains given in the above results (Theorems 3.1–3.4) are too conservative, usually much larger than the needed value; clearly, it is desirable to make the feedback gains as small as possible. Here, the adaptive technique is adopted to achieve this goal.

Theorem 3.9. Suppose that Assumption 2.3 holds and the feedback controllers are chosen as (3.20). If the feedback control gains satisfy the update law 𝑘̇_𝑖=𝜚_𝑖𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡),𝑖∈ℐ₂,(3.32) (𝜚_𝑖 are arbitrary positive constants), then the GPS between the neural network (2.1) and (2.2) can be realized.

Proof. We construct the Lyapunov function as follows: 𝑉(𝑡)=12^𝑁2_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝛼𝐵₂𝛾^2𝑁2_𝑖=1^{𝑡𝑡−𝜏}₁𝑒^𝑇𝑖(𝑠)𝑒_𝑖(𝑠)𝑑𝑠+12^𝑁2_𝑖=11𝜚_𝑖𝑘_𝑖−𝜚²+𝛼𝐻^𝜏⊗Γ₂𝛿^2𝑁2_𝑖=1^{𝑡𝑡−𝜏}₂𝑒^𝑇𝑖(𝑠)𝑒_𝑖(𝑠)𝑑𝑠,(3.33) where 𝜚 is a sufficient large positive constant to be determined.
Let 𝑉₁(𝑡)=12^𝑁2_𝑖=11𝜚_𝑖𝑘_𝑖−𝜚².(3.34) Then we have 𝑉₁̇(𝑡)=^𝑁2_𝑖=1𝑘_𝑖−𝜚𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡).(3.35)
Combining (3.18) and (3.35) with (3.33), one can obtain the following inequality: 𝑉̇(𝑡)≤−+𝜆𝛼𝐻_min⊗𝐶₁Γ+₁𝐶₂+𝛼𝐴₂𝛽²+1+𝛼𝐵₂1+𝛾²𝛿²+1+𝛼𝐻^𝜏⊗Γ₂1+𝛿²−𝜚𝑒^𝑇(𝑡)𝑒(𝑡).(3.36) Therefore, according to the above proof of the three theorems, it is easy to verify the conclusion of Theorem 3.9.

4. Numerical Simulations

In this section, several examples are given to verify the conclusions established above. Consider the two neural networks (2.1) and (2.2) with the following parameters 𝐶₁=⎡⎢⎢⎣10𝐵01⎤⎥⎥⎦,𝐶₂=⎡⎢⎢⎣0.09710.1⎤⎥⎥⎦,𝐴₁=⎡⎢⎢⎣−25−30..211⎤⎥⎥⎦,𝐴₂=⎡⎢⎢⎣−25.1.1−30.2₁.1⎤⎥⎥⎦,𝐺==⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⎢⎢⎣−11111−−501.−1.11111685−−01110023.1.4−⎤⎥⎥⎦111104,−110114𝐵−₂110013=⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,⎡⎢⎢⎣−−01..1𝐺55_𝜏−20.3⎤⎥⎥⎦,Γ₁=Γ₂=⎡⎢⎢⎣1001⎤⎥⎥⎦,𝐻=⎡⎢⎢⎢⎢⎢⎢⎣−1113−1113−1113−1113⎤⎥⎥⎥⎥⎥⎥⎦,𝐻_𝜏=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−000001−010111−101001−000001−010001−000001⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣−1001−0111−1001−0001⎤⎥⎥⎥⎥⎥⎥⎦,(4.1) and 𝑔^𝑗𝑖(𝑥)=𝑓^𝑗𝑖(𝑥)=tanh(𝑥(𝑡)) for 𝑗=1,2, 𝑁₁=6 and 𝑁₂=4.

Simulation 1
The four graphs in Figure 1 represent the motion trace of drive and response systems. Comparing Figures 1(b), 1(c), and 1(d) with Figure 1(a), we can find a trivial conclusion. Because the parameters in response system and the ones in drive system are very similar, the shape of the motion orbit of response systems looks like the one of drive system. But the biases between the four traces and the “0” orbit is rather different.

(a)

(b)

(c)

(d)

Figure 1 

(a) Trajectory of the drive system. (b) trajectory of the response system with 𝛼=0.7. (c) trajectory of the response system with 𝛼=−0.7. (d) Trajectory of the response system with 𝛼=−1.

Simulation 2
Setting 𝜃=1, 𝜇=1, 𝜋=1, 𝛽=1, 𝛾=1, 𝛿=1, it is easy to verify that Assumption 2.3 holds. Choose 𝑥_𝑖𝑗(𝑡)=100(12−3𝑖−𝑗) and 𝑥_𝑖𝑗(𝑡)=100(−8+3𝑖+𝑗) for 𝑡≤0 as the initial values and define 𝐸_𝑥=_𝑖>𝑗‖‖𝑥^𝑖(𝑡)−𝑥^𝑗(𝑡)‖‖,𝐸_𝑦=_𝑖>𝑗‖‖𝑦^𝑖(𝑡)−𝑦^𝑗(𝑡)‖‖,𝐸_𝑥𝑦=⁶_4𝑖=1_𝑗=1‖‖𝑥^𝑖(𝑡)+𝑦^𝑗(𝑡)‖‖,(4.2) for measuring the process of projective synchronization.
In Figure 2, the top three plots present the synchronization process of drive system with 𝛼=0.7,−0.7,−1, respectively; the middle three graphs demonstrate the change trend of 𝐸_𝑦 as 𝑡→∞ with the former three projective factors; the bottom three ones represent the synchronization trace of the two neural networks for the three 𝛼s. The nine figures say that, for different 𝛼, the required synchronization times are all in the interval [4, 6]. However, the change amplitudes of the nine curves show obvious differences. By the same way, for the three projective factors, we can provide the three infimums of feedback control gains that are 𝑘≥40.65 with 𝛼=0.7, 𝑘≥40.65 with 𝛼=−0.7 and 𝑘≥39.33 with 𝛼=−1, respectively.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 2 

The change process of 𝐸𝑥, 𝐸𝑦, and 𝐸𝑥𝑦 as time 𝑡→∞ for different projective factors.

Remark 4.1. The reason why we apply three different nonlinear controllers for obtaining Theorems 1, 2, and 3 is that we want to find smaller feedback control gain. By computing, we have 𝑘≥40.65 for Theorem 3.1, 𝑘≥21.43 for Theorem 3.2, and 𝑘≥20.43 for Theorem 3.4, respectively. It shows that our conjecture and simulation meet these results.

Simulation 3
In this simulation, we want to verify the synchronization process of the two neural networks when an adaptive control is applied to the response system. Figure 3 shows the time evolution of 𝐸_𝑥, 𝐸_𝑦, 𝐸_𝑥𝑦 and 𝑘(𝑡), as well as the fourth plot Figure 3(d) tell us that the feedback control gain trends towards 11.15 approximately as 𝑡→∞ which is far lower than 39.33. Actually, this proves that the adaptive control method can diminish the feedback control gain.

(a)

(b)

(c)

(d)

Figure 3 

(a), (b), and (c) show the change process of 𝐸𝑥, 𝐸𝑦, and 𝐸𝑥𝑦 with the adaptive control (3.32); (d) shows the change curve of the adaptive control gain.

5. Conclusion

The GPS between two neural networks with mixed time delays and different parameters was investigated in this paper. By means of the Lyapunov stability theories, the GPS was realized under control of three nonlinear controllers. By comparison, we found that the nonlinear controller in Theorem 3.4 was simpler and easier to ensure that the GPS was achieved. Therefore, it was also applied in practical design. According to the optimal nonlinear controller, an adaptive update technique was designed to guarantee that the feedback control gain was sufficiently small. Eventually, several numerical simulations verified the validity of those results.

Acknowledgments

The authors thank the referees and the editor for their valuable comments on this paper. This work was supported by the National Science Foundation of China under Grant no. 61070087, the Guangdong Education University Industry Cooperation Projects (2009B090300355), the Foundation for Distinguished Young Talents in Higher Education of Guangdong (LYM11115), the Shenzhen Basic Research Project (JC200903120040A, JC201006010743A), and the Shenzhen Polytechnic Youth Innovation Project (2210k3010020).