Synchronization of Neural Networks with Mixed Time Delays under Information Constraints

Abstract

This paper investigates the synchronization problem of neural networks with mixed time delays under information constrains. The designed synchronization scheme is built on the framework of hybrid systems. Besides including nonuniform sampling, some other characteristics, such as quantization, transmission-induced delays, and data packet dropouts, are also considered. The sufficient condition that depended on network characteristics is obtained to guarantee the remote asymptotical synchronization of neural networks with mixed time delays. A numerical example is given to illustrate the validity of the proposed method.

1. Introduction

Recently, neural networks have been widely studied by many scholars due to its potential applications in pattern recognition, image processing, signal processing, biology engineering, and information science [1–3]. Moreover, there exists a broad class of neural networks with mixed time delays. Following the development in this field, many master-slave synchronization schemes for neural networks with mixed time delays have been proposed, such as [4–14].

Generally, some useful approaches can be utilized for the synchronization problem of neural networks, which include passivity analysis [5, 12], impulsive control [15, 16], adaptive control [17, 18], and stochastic method [13]. In recent years, the sampled-data control scheme is utilized for the synchronization of neural networks with mixed time delays, such as [10, 11, 14]. However, these schemes did not consider quantization, transmission-induced delays, and data packet dropouts. Essentially, in the range of deterministic systems, the general framework considering information constrains can be divided into three cases: (1) continuous-time models; (2) discrete-time models; (3) hybrid models. In this paper, the networked controller design for asymptotical synchronization of neural networks with mixed time delays is discussed in the framework of hybrid systems. Besides including nonuniform sampling, we also consider other network characteristics, such as quantization, transmission-induced delays, and data packet dropouts. From the view of helpful technologies, the input delay approach and the free-weighting matrix technology are applied to obtain the less conservative condition.

Notation. Throughout this paper, superscripts 𝑇 and −1 mean the transpose and the inverse of a matrix, respectively, 𝑁 denotes natural number, 𝑍 denotes integer number, 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space, 𝑅^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices, the identity matrices and zero matrices are denoted by 𝐼 and 0, respectively, the notation ∗ always denotes the symmetric block in one symmetric matrix, the standard notation >(<) is used to denote the positive (negative)-definite ordering of matrices, and inequality 𝑋>𝑌 shows that the matrix 𝑋−𝑌 is positive definite.

2. Preliminaries

Consider the following general master-slave neural network with mixed time delays [11]: 𝑥̇(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝑑(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑔(𝑥(𝑠(𝑡),𝑦̇(𝑡)(𝑡)𝑦(𝑡)𝑦((𝑡))𝑑𝑠+𝑉=−𝐶𝑦+𝐴𝑔+𝐵𝑔𝑡−𝑑))+𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑔𝑦(𝑠)𝑑𝑠+𝑉(𝑡)+𝑢(𝑡), (1) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇∈𝑅^𝑛, 𝑦(𝑡)=[𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)]^𝑇∈𝑅^𝑛, and 𝑥_𝑖(𝑡) and 𝑦_𝑖(𝑡) denote the states of the 𝑖th neuron of master and slave neural networks, respectively; 𝑔(𝑥(𝑡))=[𝑔₁(𝑥₁(𝑡)),𝑔₂(𝑥₂(𝑡)),…,𝑔_𝑛(𝑥_𝑛(𝑡))]^𝑇∈𝑅^𝑛, 𝑔(𝑦(𝑡))=[𝑔₁(𝑦₁(𝑡)),𝑔₂(𝑦₂(𝑡)),…,𝑔_𝑛(𝑦_𝑛(𝑡))]^𝑇∈𝑅^𝑛, and 𝑔(𝑥(𝑡)) and 𝑔(𝑦(𝑡)) denote the neuron activation functions of master and slave neural networks, respectively; 𝐶=diag{𝑐₁,𝑐₂,…,𝑐_𝑛}∈𝑅^𝑛 denotes a diagonal matrix with positive entries; 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛, and 𝐷=(𝑑_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛 denote the connection weight matrices; 𝑢(𝑡)∈𝑅^𝑛 denotes the control input; 𝑉(𝑡)=[𝑉₁(𝑡),𝑉₂(𝑡),…,𝑉_𝑛(𝑡)]^𝑇∈𝑅^𝑛 denotes the external input vector; 𝑑(𝑡) and 𝜏(𝑡) denote the discrete delay and the distributed delay, respectively, and satisfy 𝑑̇(𝑡)(𝑡)≤𝜇<1,0≤𝜏≤𝜏, (2) where 𝜇 and 𝜏 are constants.

Assumption 1 (see [7]). For 𝑖∈{1,2,…,𝑛}, the neuron activation functions satisfy 𝐹⁻_𝑖≤𝑔_𝑖𝑠₁−𝑔_𝑖𝑠₂𝑠₁−𝑠₂≤𝐹⁺_𝑖, (3) where 𝐹⁻_𝑖 and 𝐹⁺_𝑖 are some constants, 𝑠₁,𝑠₂∈𝑅, and 𝑠₁≠𝑠₂.

Assumption 2 (see [7]). The neuron activation functions are bounded.

Let the error be 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡). Then, the synchronization error system can be represented as 𝑒̇(𝑡)=−𝐶𝑒(𝑡)+𝐴𝑓(𝑡)+𝐵𝑓(𝑡−𝑑(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑓(𝑠)𝑑𝑠+𝑢(𝑡), (4) where 𝑓(𝑡)∶=𝑔(𝑦(𝑡))−𝑔(𝑥(𝑡)). The networked synchronization controller is designed as 𝑢(𝑡)=𝑞𝐾𝑦𝑡_𝑖𝑘−𝑥𝑡_𝑖𝑘,𝑡_𝑖𝑘+𝜏_𝑖𝑘≤𝑡<𝑡_𝑖𝑘+1+𝜏_𝑖𝑘+1, (5) where 𝑞(⋅)=[𝑞₁(⋅),𝑞₂(⋅),…,𝑞_𝑛(⋅)]^𝑇∈𝑅^𝑛 denotes the quantizer, 𝐾∈𝑅^𝑛×𝑛 denotes the controller gain matrix, 𝑦(𝑡_𝑖𝑘) and 𝑥(𝑡_𝑖𝑘) are available measurements of 𝑦(𝑡) and 𝑥(𝑡) at sampling instant 𝑡_𝑖𝑘, 𝑘=0,1,…,∞, 𝑖_𝑘∈𝑁 denotes the serial number of the available data packet such that {𝑖₀,𝑖₁,𝑖₂,…}⊆{0,1,2,3,…}, 𝑡_𝑖𝑘 denotes the sampling instant of the available data packet, and 𝜏_𝑖𝑘 denotes the network-induced delay calculated from the instant 𝑡_𝑖𝑘.

The above controller (5) is related with the quantitative function so the following definition is introduced.

Definition 3 (see [19]). A quantizer is called logarithmic if the set of quantized levels is characterized by Ω=±𝑢_𝑙,𝑢_𝑙=𝜌^𝑙𝑢₀,𝑙∈𝑍{0}0<𝜌<1,𝑢₀>0. (6)

The quantizer is assumed to be symmetric; that is, 𝑞(−𝜐)=−𝑞(𝜐), as described in [19]. Selected as the logarithmic quantizer, 𝑞(𝜐) is given as 𝑞_𝑖(𝜐)=𝑢_𝑙^(𝑖)𝑢,if^𝑙(𝑖)𝑢1+𝜎<𝜐≤^𝑙(𝑖)1−𝜎,𝜐>0,0,if𝜐=0,−𝑞_𝑖(−𝜐),if𝜐<0, (7) where 𝑖∈{1,2,…,𝑛} and 𝜎=(1−𝜌)/(1+𝜌).

Generally, 𝑡_𝑘 denotes the overall sampling instant and all the data packets are assigned as serial numbers. But, the data packets may be discontinuous due to dropouts such that the sampling is nonuniform. Similar to many existing results, the sensor is clock-driven; the controller and actuator are event-driven. The clocks among all the devices are synchronized.

Assumption 4 (see [20]). There exist three constants ℎ>0, 𝜏_min≥0, and 𝜏_max≥0 such that 𝑡_𝑘+1−𝑡_𝑘𝜏≤ℎ,_min≤𝜏_𝑖𝑘≤𝜏_max,𝑘=0,1,…,∞, (8) where ℎ denotes the upper bound of the interval between two consecutive sampling instants and 𝜏_min and 𝜏_max denote the minimum and maximum of network-induced delays, respectively. It is assumed that 𝛿_max denotes the admitted maximum of successive data packet dropouts in network transmissions. Considering condition (8) and successive data packet dropouts, the following result can be obtained: 𝑡_𝑖𝑘+1+𝜏_𝑖𝑘+1−𝑡_𝑖𝑘≤𝑖_𝑘+1−𝑖_𝑘ℎ+𝜏_max≤1+𝛿_maxℎ+𝜏_max,𝑘=0,1,…,∞. (9)

Set 𝜙₀∶=(1+𝛿_max)ℎ+𝜏_max and then the initial condition of 𝑒(𝑡) on (𝑡₀−𝜙₀,𝑡₀] is supplemented as 𝑒(𝑡)=𝜙(𝑡), 𝑡∈(𝑡₀−𝜙₀,𝑡₀], where 𝜙(𝑡) is a continuous function on (𝑡₀−𝜙₀,𝑡₀]. By the sector bound method as [19], the synchronization error system (4) can be expressed as 𝑒̇(𝑡)=−𝐶𝑒(𝑡)+𝐴𝑓(𝑡)+𝐵𝑓(𝑡−𝑑(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑓(𝑠)𝑑𝑠+(𝐼+Σ)𝐾𝑒𝑡−𝜏_𝑁(𝑡),𝑒(𝑡)=𝜙(𝑡),𝑡∈𝑡₀−𝜙₀,𝑡₀, (10) where 𝜏_𝑁(𝑡)=𝑡−𝑡_𝑖𝑘, Σ=diag{Σ₁,Σ₂,…,Σ_𝑛} and Σ_𝑖∈[−𝜎,𝜎]. According to (8) and (9), 𝜏_min≤𝜏_𝑁(𝑡)<(1+𝛿_max)ℎ+𝜏_max and 𝜏_𝑁(𝑡)<𝑡_𝑖𝑘+1+𝜏_𝑖𝑘+1−𝑡_𝑖𝑘. The control objective is to design the controller gain matrix 𝐾 such that the synchronization error system (10) is asymptotically stable; that is, 𝑒(𝑡)→0 as 𝑡→∞.

3. Main Result

In this section, the stability of the error system (10) will be analyzed by constructing a corresponding Lyapunov functional. Before beginning the proof procedure, two useful lemmas are introduced.

Lemma 5. Let 𝑋 be any 𝑚×𝑛 matrix, for any constant 𝜖>0 and any positive-definite symmetric matrix 𝑇, such that 2𝜁^𝑇𝑋𝜑≤𝜖𝜁^𝑇𝑋𝑇⁻¹𝑋^𝑇1𝜁+𝜖𝜑^𝑇𝑇𝜑 (11) for all 𝜁∈𝑅^𝑚, 𝜑∈𝑅^𝑛, and 𝑇∈𝑅^𝑛×𝑛.

Lemma 6 (see [21]). For any constant symmetric matrix 𝐺∈𝑅^𝑛×𝑛, 𝐺=𝐺^𝑇>0, scalar 𝜐>0, vector function 𝜓∶[𝑡−𝜐,𝑡]→𝑅^𝑛, such that the integrations in the following are well defined, then 𝜐^𝑡_𝑡−𝜐𝜓^𝑇𝛽𝛽≥𝐺𝜓𝑑𝛽_{𝑡𝑡−𝜐}𝜓𝛽𝑑𝛽^𝑇𝐺^𝑡_𝑡−𝜐𝜓𝛽𝑑𝛽. (12)

Theorem 7. Given scalars 𝜏, 𝜇, and 𝜙₀>0 composed of ℎ, 𝛿_max, 𝜏_max, and diagonal matrices Σ=diag{Σ₁,Σ₂,…,Σ_𝑛}, 𝐹⁻=diag{𝐹⁻₁,𝐹⁻₂,…,𝐹⁻_𝑛}, and 𝐹⁺=diag{𝐹⁺₁,𝐹⁺₂,…,𝐹⁺_𝑛}, the synchronization error system (10) is global asymptotically stable in network environments, if there exist matrices 𝑃₁=𝑃^𝑇₁>0, 𝑃₂=𝑃^𝑇₂>0, 𝑃₃=𝑃^𝑇₃>0, 𝑃₄=𝑃^𝑇₄>0, Λ=diag{𝜆₁,𝜆₂,…,𝜆_𝑛}≥0, any matrices 𝑁=(𝑁^𝑇₁,𝑁^𝑇₂,𝑁^𝑇₃)^𝑇, 𝑀=(𝑀^𝑇₁,𝑀^𝑇₂,𝑀^𝑇₃)^𝑇 with appropriate dimensions, and the controller gain matrix 𝐾 such that the following condition holds: ΘΥ₁Υ₂∗−𝜏𝑃₃0∗∗−𝜙₀𝑃₄<0, (13) where ΦΘ=¹¹Φ₁₂Φ₁₃Φ₁₄Φ₁₅∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅∗∗Φ₃₃00∗∗∗Φ₄₄0∗∗∗∗Φ₅₅,Υ₁𝑀00=𝜏𝐷,Υ₂=𝜙₀𝑁00,Φ₁₁=𝑁₁+𝑁^𝑇₁+𝑀₁𝐶+𝐶^𝑇𝑀^𝑇₁−2𝐹⁻Λ𝐹⁺,Φ₁₂=𝑃₁+𝑁^𝑇₂+𝑀₁+𝐶^𝑇𝑀^𝑇₂,Φ₁₃=−𝑁₁+𝑁^𝑇₃−𝑀₁(𝐼+Σ)𝐾+𝐶^𝑇𝑀^𝑇₃,Φ₁₄=−𝑀₁𝐴+𝐹⁺Λ+𝐹⁻ΦΛ,₁₅=−𝑀₁Φ𝐵,₂₂=𝜙₀𝑃₄+𝑀₂+𝑀^𝑇₂,Φ₂₃=−𝑁₂−𝑀₂(𝐼+Σ)𝐾+𝑀^𝑇₃,Φ₂₄=−𝑀₂Φ𝐴,₂₅=−𝑀₂Φ𝐵,₃₃=−𝑁₃−𝑁^𝑇₃−𝑀₃(𝐼+Σ)𝐾−𝐾^𝑇(𝐼+Σ)^𝑇𝑀^𝑇₃,Φ₄₄=𝑃₂+𝜏𝑃₃Φ−2Λ,₅₅=−1−𝜇𝑃₂. (7)

Proof. Construct the following Lyapunov functional as 𝑉(𝑒(𝑡))=𝑒^𝑇(𝑡)𝑃₁𝑒(𝑡)+^𝑡_{(𝑡)𝑡−𝑑}𝑓^𝑇(𝑠)𝑃₂𝑓(𝑠)+𝑑𝑠_{0−𝜏𝑡𝑡+𝜃}𝑓^𝑇(𝑠)𝑃₃𝑓(𝑠)+𝑑𝑠𝑑𝜃_{0−𝜙0𝑡𝑡+𝜃}𝑒^𝑇̇(𝑠)𝑃₄𝑒̇(𝑠)𝑑𝑠𝑑𝜃, (15) where 𝑃₁=𝑃^𝑇₁>0, 𝑃₂=𝑃^𝑇₂>0, 𝑃₃=𝑃^𝑇₃>0, 𝑃₄=𝑃^𝑇₄>0, and 𝜙₀=(1+𝛿_max)ℎ+𝜏_max. Moreover, the following equations hold for any appropriate dimensional matrices 𝑁₁, 𝑁₂, 𝑁₃, 𝑀₁, 𝑀₂, and𝑀₃: 𝑒^𝑇(𝑡)𝑁₁+𝑒^𝑇̇(𝑡)𝑁₂+𝑒^𝑇𝑡−𝜏_𝑁(𝑡)𝑁₃×𝑒(𝑡)−𝑒𝑡−𝜏_𝑁(𝑡)−^𝑡_{𝑡−𝜏𝑁(𝑡)}𝑒̇(𝑠)𝑒𝑑𝑠=0,_𝑇(𝑡)𝑀₁+𝑒^𝑇̇(𝑡)𝑀₂+𝑒^𝑇𝑡−𝜏_𝑁(𝑡)𝑀₃×𝑒̇(𝑡)+𝐶𝑒(𝑡)−𝐴𝑓(𝑡)−𝐵𝑓(𝑡−𝑑(𝑡))−𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑓(𝑠)𝑑𝑠−(𝐼+Σ)𝐾𝑒𝑡−𝜏_𝑁(𝑡)=0. (16) On the other hand, for any Λ=diag{𝜆₁,𝜆₂,…,𝜆_𝑛}≥0, it follows from (3) such that 0≤−2^𝑛_𝑖=1𝜆_𝑖𝑓_𝑖(𝑡)−𝐹⁻_𝑖𝑒_𝑖(𝑡)𝑓_𝑖(𝑡)−𝐹⁺_𝑖𝑒_𝑖(𝑡)=−2𝑒^𝑇(𝑡)𝐹⁻Λ𝐹⁺𝑒(𝑡)−2𝑓^𝑇(𝑡)Λ𝑓(𝑡)+2𝑒^𝑇(𝑡)𝐹⁺Λ𝑓(𝑡)+2𝑒^𝑇(𝑡)𝐹⁻Λ𝑓(𝑡). (17) Combining (2), (16), and (17), the corresponding time derivative of  𝑉(𝑒(𝑡)) is given by 𝑉̇(𝑒(𝑡))≤2𝑒^𝑇̇(𝑡)𝑃₁𝑒(𝑡)+𝑓^𝑇(𝑡)𝑃₂𝑓(𝑡)−1−𝜇𝑓^𝑇(𝑡−𝑑(𝑡))𝑃₂𝑓(𝑡−𝑑(𝑡))+𝜏𝑓^𝑇(𝑡)𝑃₃𝑓(𝑡)−^𝑡_𝑡−𝜏𝑓^𝑇(𝑠)𝑃₃𝑓(𝑠)𝑑𝑠+𝜙₀𝑒^𝑇̇(𝑡)𝑃₄𝑒̇(𝑡)−^𝑡_𝑡−𝜙0𝑒^𝑇̇(𝑠)𝑃₄𝑒̇(𝑠)𝑒𝑑𝑠+2^𝑇(𝑡)𝑁₁+𝑒^𝑇̇(𝑡)𝑁₂+𝑒^𝑇𝑡−𝜏_𝑁(𝑡)𝑁₃×𝑒(𝑡)−𝑒𝑡−𝜏_𝑁(𝑡)−^𝑡_{𝑡−𝜏𝑁(𝑡)}𝑒̇(𝑠)𝑒𝑑𝑠+2^𝑇(𝑡)𝑀₁+𝑒^𝑇̇(𝑡)𝑀₂+𝑒^𝑇𝑡−𝜏_𝑁(𝑡)𝑀₃×𝑒̇(𝑡)+𝐶𝑒(𝑡)−𝐴𝑓(𝑡)−𝐵𝑓(𝑡−𝑑(𝑡))−𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑓(𝑠)𝑑𝑠−(𝐼+Σ)𝐾𝑒𝑡−𝜏_𝑁(𝑡)−2𝑒^𝑇(𝑡)𝐹⁻Λ𝐹⁺𝑒(𝑡)−2𝑓^𝑇(𝑡)Λ𝑓(𝑡)+2𝑒^𝑇(𝑡)𝐹⁺Λ𝑓(𝑡)+2𝑒^𝑇(𝑡)𝐹⁻Λ𝑓(𝑡). (18) Using Lemmas 5 and 6 and the inequality 𝑒^𝑇̇^𝑡_{𝑡−𝜏𝑁(𝑡)}(𝑠)𝑃₄𝑒̇(𝑠)𝑑𝑠≤^𝑡_𝑡−𝜙0𝑒^𝑇̇(𝑠)𝑃₄𝑒̇(𝑠)𝑑𝑠, (19) the following result can be obtained: 𝑉̇(𝑒(𝑡))≤𝜍^𝑇(𝑡)𝑀00Θ+𝜏𝐷𝑃₃⁻¹𝐷^𝑇𝑀00+𝜙₀𝑁00𝑃₄⁻¹𝑁00𝜍(𝑡), (20) where 𝜍(𝑡)=[𝑒^𝑇(𝑡)𝑒^𝑇̇(𝑡)𝑒^𝑇(𝑡−𝜏_𝑁(𝑡))𝑓^𝑇(𝑡)𝑓^𝑇(𝑡−𝑑(𝑡))]^𝑇, matrices Θ, 𝑀, and 𝑁 are defined in (13). It is explicit that if 𝑀00Θ+𝜏𝐷𝑃₃⁻¹𝐷^𝑇𝑀00+𝜙₀𝑁00𝑃₄⁻¹𝑁00<0, (21) then 𝑉̇(𝑒(𝑡))<0 for any nonzero 𝜍(𝑡). Utilizing Schur complement [22], the condition in Theorem 7 can be obtained, and the proof is completed.

Because the uncertain matrix Σ is involved at the condition (13) in Theorem 7, the following theorem is given to obtain a solvable result.

Theorem 8. Given scalars 𝜏, 𝜇, and 𝜙₀>0 composed of ℎ, 𝛿_max, 𝜏_max, and constant diagonal matrices Σ=diag{𝜎²₁,𝜎²₂,…,𝜎²_𝑛}, 𝐹⁻=diag{𝐹⁻₁,𝐹⁻₂,…,𝐹⁻_𝑛}, and 𝐹⁺=diag{𝐹⁺₁,𝐹⁺₂,…,𝐹⁺_𝑛}, the synchronization error system (10) is global asymptotically stable in network environments, if there exist matrices 𝑃₁=𝑃^𝑇₁>0, 𝑃₂=𝑃^𝑇₂>0, 𝑃₃=𝑃^𝑇₃>0, 𝑃₄=𝑃^𝑇₄>0, Λ=diag{𝜆₁,𝜆₂,…,𝜆_𝑛}≥0, any matrices 𝑁=(𝑁^𝑇₁,𝑁^𝑇₂,𝑁^𝑇₃)^𝑇, 𝑀=(𝑀^𝑇₁,𝑀^𝑇₂,𝑀^𝑇₃)^𝑇 with appropriate dimensions, and the controller gain matrix 𝐾 such that the following condition holds: ΘΥ₁Υ₂𝑀₁𝑀₂𝑀₃𝐾∗−𝜏𝑃₃00000∗∗−𝜙₀0000∗∗∗−Σ⁻¹000∗∗∗∗−Σ⁻¹00∗∗∗∗∗−Σ⁻¹0∗∗∗∗∗∗−𝐼<0, (22) where ΦΘ=¹¹Φ₁₂Φ₁₃Φ₁₄Φ₁₅∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅∗∗Φ₃₃00∗∗∗Φ₄₄0∗∗∗∗Φ₅₅,Υ₁𝑀00Υ=𝜏𝐷,₂=𝜙₀𝑁00,𝑀₁=𝑀₁0000,𝑀₂=0𝑀₂000,𝑀₃=00𝑀₃00,00𝐾=√3𝐾^𝑇00,Φ₁₁=𝑁₁+𝑁^𝑇₁+𝑀₁𝐶+𝐶^𝑇𝑀^𝑇₁−2𝐹⁻Λ𝐹⁺,Φ₁₂=𝑃₁+𝑁^𝑇₂+𝑀₁+𝐶^𝑇𝑀^𝑇₂,Φ₁₃=−𝑁₁+𝑁^𝑇₃−𝑀₁𝐾+𝐶^𝑇𝑀^𝑇₃,Φ₁₄=−𝑀₁𝐴+𝐹⁺Λ+𝐹⁻ΦΛ,₁₅=−𝑀₁Φ𝐵,₂₂=𝜙₀𝑃₄+𝑀₂+𝑀^𝑇₂,Φ₂₃=−𝑁₂−𝑀₂𝐾+𝑀^𝑇₃,Φ₂₄=−𝑀₂Φ𝐴,₂₅=−𝑀₂Φ𝐵,₃₃=−𝑁₃−𝑁^𝑇₃−𝑀₃𝐾−𝐾^𝑇𝑀^𝑇₃,Φ₄₄=𝑃₂+𝜏𝑃₃Φ−2Λ,₅₅=−1−𝜇𝑃₂. (23)

Proof. Utilizing Schur formula and matrix inequality 𝑋^𝑇𝑌+𝑌^𝑇𝑋≤𝑋^𝑇𝑋+𝑌^𝑇𝑌, condition (22) in Theorem 8 is obtained. The proof is completed.

Remark 9. In the above process, the free-weighting matrix technology is applied to complete the proof. Moreover, similar to [23], we can select the new Lyapunov-Krasovskii functional to reduce the conservatism, which utilizes the bounds of the network-induced delay. This will be done in the future works.

4. A Numerical Example

In this section, a numerical example is given to demonstrate the effectiveness of the proposed synchronization scheme.

Example 1. Consider the following master-slave neural network with mixed time delays as in [11, 14]: 𝑥̇(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝑑(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑔(𝑥(𝑠(𝑡),𝑦̇(𝑡)(𝑡)𝑦(𝑡𝑦((𝑡))𝑑𝑠+𝑉=−𝐶𝑦+𝐴𝑔)+𝐵𝑔𝑡−𝑑))+𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑔𝑦(𝑠)𝑑𝑠+𝑉(𝑡)+𝑢(𝑡), (24) where ,,𝐶=1001,𝐴=1.8−0.15−5.23.5𝐵=−1.7−0.12−0.26−2.5,𝐷=0.60.15−2−0.12 (25)𝑑(𝑡)=𝑒^𝑡/(𝑒^𝑡+1), 𝜏(𝑡)=0.5sin²(𝑡), 𝑔₁(𝑠)=𝑔₂(𝑠)=tanh(𝑠), and 𝑉(𝑡)=0. Thus, 𝜇=0.25 and 𝜏=0.5. It is assumed that 𝐹⁻₁=𝐹⁻₂=0, 𝐹⁺₁=𝐹⁺₂=1. According to Assumption 1, the following result is satisfied: 𝐹⁻=0000,𝐹⁺=1001. (26)

The chaotic behaviors of the master and slave systems are given in Figures 1 and 2, with the initial condition chosen as 𝑥(0)=[0.2,0.3]^𝑇 and 𝑦(0)=[−0.1,0.6]^𝑇, respectively. Setting 𝜎=0.5, 𝛿_max=5, 𝜏_max=0.01 and appropriate matrices 𝑀_𝑖(𝑖=1,2,3), the condition (22) in Theorem 8 is feasible for ℎ≤0.001, with the controller gain matrix 𝐾=−17.9664−0.24031.6664−17.5357. (27)

Figure 1 

Chaotic behavior of the master system.

Figure 2 

Chaotic behavior of the slave system with 𝑢(𝑡)=0.

Similar to [14], the initial conditions of the master and slave systems are chosen as 𝑥(0)=[0.2,0.3]^𝑇 and 𝑦(0)=[−0.1,0.6]^𝑇, respectively. The response curves of the error system are shown in Figure 3 for the upper bound of sampling interval ℎ=0.001. It shows that the synchronization error converges to zero asymptotically.

Figure 3 

State response curves of the error system.

5. Conclusion

In the present works, the networked synchronization scheme for master-slave neural networks with mixed time delays has been proposed. The error system can be stabilized under information constraints. The obtained result depends on network characteristics. In future works, more performance requirements for synchronization of master-slave neural networks with mixed time delays will be considered in a uniform network topological structure.

Acknowledgments

This work is supported by the Natural Science Foundation of China under Grant 61203076, the Natural Science Foundation of Tianjin City under Grant 13JCQNJC03500, the Natural Science Foundation of Hebei Province under Grant F2012202100, and the Excellent Young Technological Innovation Foundation Project in Hebei University of Technology under Grant 2011005.