Global Dissipativity on Uncertain Discrete-Time Neural Networks with Time-Varying Delays

Abstract

The problems on global dissipativity and global exponential dissipativity are investigated for uncertain discrete-time neural networks with time-varying delays and general activation functions. By constructing appropriate Lyapunov-Krasovskii functionals and employing linear matrix inequality technique, several new delay-dependent criteria for checking the global dissipativity and global exponential dissipativity of the addressed neural networks are established in linear matrix inequality (LMI), which can be checked numerically using the effective LMI toolbox in MATLAB. Illustrated examples are given to show the effectiveness of the proposed criteria. It is noteworthy that because neither model transformation nor free-weighting matrices are employed to deal with cross terms in the derivation of the dissipativity criteria, the obtained results are less conservative and more computationally efficient.

1. Introduction

In the past few decades, delayed neural networks have found successful applications in many areas such as signal processing, pattern recognition, associative memories, parallel computation, and optimization solvers [1]. In such applications, the qualitative analysis of the dynamical behaviors is a necessary step for the practical design of neural networks [2]. Many important results on the dynamical behaviors have been reported for delayed neural networks; see [1–16] and the references therein for some recent publications.

It should be pointed out that all of the abovementioned literatures on the dynamical behaviors of delayed neural networks are concerned with continuous-time case. However, when implementing the continuous-time delayed neural network for computer simulation, it becomes essential to formulate a discrete-time system that is an analogue of the continuous-time delayed neural network. To some extent, the discrete-time analogue inherits the dynamical characteristics of the continuous-time delayed neural network under mild or no restriction on the discretization step-size, and also remains some functional similarity [17]. Unfortunately, as pointed out in [18], the discretization cannot preserve the dynamics of the continuous-time counterpart even for a small sampling period, and therefore there is a crucial need to study the dynamics of discrete-time neural networks. Recently, the dynamics analysis problem for discrete-time delayed neural networks and discrete-time systems with time-varying state delay has been extensively studied; see [17–21] and references therein.

It is well known that the stability problem is central to the analysis of a dynamic system where various types of stability of an equilibrium point have captured the attention of researchers. Nevertheless, from a practical point of view, it is not always the case that every neural network has its orbits approach a single equilibrium point. It is possible that there is no equilibrium point in some situations. Therefore, the concept on dissipativity has been introduced [22]. As pointed out in [23], the dissipativity is also an important concept in dynamical neural networks. The concept of dissipativity in dynamical systems is a more general concept and it has found applications in the areas such as stability theory, chaos and synchronization theory, system norm estimation, and robust control [23]. Some sufficient conditions checking the dissipativity for delayed neural networks and nonlinear delay systems have been derived, for example, see [23–33] and references therein. In [23, 24], authors analyzed the dissipativity of neural network with constant delays, and derived some sufficient conditions for the global dissipativity of neural network with constant delays. In [25, 26], authors considered the global dissipativity and global robust dissipativity for neural network with both time-varying delays and unbounded distributed delays; several sufficient conditions for checking the global dissipativity and global robust dissipativity were obtained. In [27, 28], by using linear matrix inequality technique, authors investigated the global dissipativity of neural network with both discrete time-varying delays and distributed time-varying delays. In [29], authors developed dissipativity notions for nonnegative dynamical systems with respect to linear and nonlinear storage functions and linear supply rates, and obtained a key result on linearization of nonnegative dissipative dynamical systems. In [30], the uniform dissipativity of a class of nonautonomous neural networks with time-varying delays was investigated by employing 𝑀-matrix and the techniques of inequality. In [31–33], the dissipativity of a class of nonlinear delay systems was considered; some sufficient conditions for checking the dissipativity were given. However, all of the abovementioned literatures on the dissipativity for delayed neural networks and nonlinear delay systems are concerned with continuous-time case. To the best of our knowledge, few authors have considered the problem on the dissipativity of uncertain discrete-time neural networks with time-varying delays. Therefore, the study on the dissipativity of uncertain discrete-time neural networks is not only important but also necessary.

Motivated by the above discussions, the objective of this paper is to study the problem on global dissipativity and global exponential dissipativity for uncertain discrete-time neural networks. By employing appropriate Lyapunov-Krasovskii functionals and LMI technique, we obtain several new sufficient conditions for checking the global dissipativity and global exponential dissipativity of the addressed neural networks.

Notations 1. The notations are quite standard. Throughout this paper, 𝐼 represents the unitary matrix with appropriate dimensions; ℕ stands for the set of nonnegative integers; ℝ^𝑛 and ℝ^𝑛×𝑚 denote, respectively, the 𝑛-dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices. The superscript “𝑇” denotes matrix transposition and the asterisk “*” denotes the elements below the main diagonal of a symmetric block matrix. |𝐴| denotes the absolute-value matrix given by |𝐴|=(|𝑎_𝑖𝑗|)_𝑛×𝑛; the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that 𝑋 and 𝑌 are symmetric matrices, and that 𝑋−𝑌 is positive semidefinite (resp., positive definite). ‖⋅‖ is the Euclidean norm in ℝ^𝑛. For a positive constant 𝑎, [𝑎] denotes the integer part of 𝑎. For integers 𝑎, 𝑏 with 𝑎<𝑏, ℕ[𝑎,𝑏] denotes the discrete interval given by ℕ[𝑎,𝑏]={𝑎,𝑎+1,…,𝑏−1,𝑏}. 𝐶(ℕ[−𝜏,0],ℝ^𝑛) denotes the set of all functions 𝜙: ℕ[−𝜏,0]→ℝ^𝑛. Matrices, if not explicitly specified, are assumed to have compatible dimensions.

2. Model Description and Preliminaries

In this paper, we consider the following discrete-time neural network model

𝑥(𝑘+1)=𝐶𝑥(𝑘)+𝐴𝑔(𝑥(𝑘))+𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+𝑢(2.1) for 𝑘∈ℕ, where 𝑥(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘))^𝑇∈ℝ^𝑛, 𝑥_𝑖(𝑘) is the state of the 𝑖th neuron at time 𝑘; 𝑔(𝑥(𝑘))=(𝑔₁(𝑥₁(𝑘)),𝑔₂(𝑥₂(𝑘)),…,𝑔_𝑛(𝑥_𝑛(𝑘)))^𝑇∈ℝ^𝑛, 𝑔_𝑗(𝑥_𝑗(𝑘)) denotes the activation function of the 𝑗th neuron at time 𝑘; 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇 is the input vector; the positive integer 𝜏(𝑘) corresponds to the transmission delay and satisfies 𝜏≤𝜏(𝑘)≤𝜏 (𝜏≥0 and 𝜏≥0 are known integers); 𝐶=diag{𝑐₁,𝑐₂,…,𝑐_𝑛}, where 𝑐_𝑖 (0≤𝑐_𝑖<1) describes the rate with which the 𝑖th neuron will reset its potential to the resting state in isolation when disconnected from the networks and external inputs; 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is the connection weight matrix; 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛 is the delayed connection weight matrix.

The initial condition associated with model (2.1) is given by

𝑥(𝑠)=𝜑(𝑠),𝑠∈ℕ−𝜏,0.(2.2)

Throughout this paper, we make the following assumption [6].

(H) For any 𝑗∈{1,2,…,𝑛}, 𝑔_𝑗(0)=0 and there exist constants 𝐺^−𝑗 and 𝐺^+𝑗 such that 𝐺^−𝑗≤𝑔^𝑗𝛼₁−𝑔_𝑗𝛼₂𝛼₁−𝛼₂≤𝐺^+𝑗,∀𝛼₁,𝛼₂∈ℝ,𝛼₁≠𝛼₂.(2.3)

Similar to [23], we also give the following definitions for discrete-time neural networks (2.1).

Definition 2.1. Discrete-time neural networks (2.1) are said to be globally dissipative if there exists a compact set 𝑆⊆ℝ^𝑛, such that for all 𝑥₀∈ℝ^𝑛, ∃ positive integer 𝐾(𝑥₀)>0, when 𝑘≥𝑘₀+𝐾(𝑥₀), 𝑥(𝑘,𝑘₀,𝑥₀)⊆𝑆, where 𝑥(𝑘,𝑘₀,𝑥₀) denotes the solution of (2.1) from initial state 𝑥₀ and initial time 𝑘₀. In this case, 𝑆 is called a globally attractive set. A set 𝑆 is called a positive invariant if ∀𝑥₀∈𝑆 implies 𝑥(𝑘,𝑘₀,𝑥₀)⊆𝑆 for 𝑘≥𝑘₀.

Definition 2.2. Let 𝑆 be a globally attractive set of discrete-time neural networks (2.1). Discrete-time neural networks (2.1) are said to be globally exponentially dissipative if there exists a compact set 𝑆^∗⊃𝑆 in ℝ^𝑛 such that ∀𝑥₀∈ℝ^𝑛⧵𝑆^∗, there exist constants 𝑀(𝑥₀)>0 and 0<𝛽<1 such that inf_{𝑥∈ℝ𝑛⧵𝑆∗}‖‖𝑥𝑘,𝑘₀,𝑥₀−𝑥̃‖‖∶𝑥̃∈𝑆^∗≤𝑀𝑥₀𝛽^𝑘−𝑘0.(2.4) Set 𝑆^∗ is called globally exponentially attractive set, where 𝑥∈ℝ^𝑛⧵𝑆^∗ means 𝑥∈ℝ^𝑛 but 𝑥∉𝑆^∗.

To prove our results, the following lemmas are necessary.

Lemma 2.3 (see [34]). Given constant matrices 𝑃, 𝑄, and 𝑅, where 𝑃^𝑇=𝑃, 𝑄^𝑇=𝑄, then 𝑅𝑃−𝑅^𝑇𝑄<0(2.5) is equivalent to the following conditions: 𝑄>0,𝑃+𝑅𝑄⁻¹𝑅^𝑇<0.(2.6)

Lemma 2.4 (see [35, 36]). Given matrices 𝑃, 𝑄, and 𝑅 with 𝑃^𝑇=𝑃, then 𝑃+𝑄𝐹(𝑘)𝑅+𝑅^𝑇𝐹^𝑇(𝑘)𝑄^𝑇<0(2.7) holds for all 𝐹(𝑘) satisfying 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼 if and only if there exists a scalar 𝜀>0 such that 𝑃+𝜀⁻¹𝑄𝑄^𝑇+𝜀𝑅𝑅^𝑇<0.(2.8)

3. Main Results

In this section, we shall establish our main criteria based on the LMI approach. For presentation convenience, in the following, we denote

𝐺₁=diag𝐺⁻¹,𝐺⁻²𝐺,₄…,𝐺^−𝑛,𝐺₂=diag𝐺⁺¹,𝐺⁺²,…,𝐺^+𝑛,𝐺₃=diag𝐺⁻¹𝐺⁺¹,𝐺⁻²𝐺⁺²,…,𝐺^−𝑛𝐺^+𝑛,=diag𝐺⁻¹+𝐺⁺¹2,𝐺⁻²+𝐺⁺²2,…,𝐺^−𝑛+𝐺^+𝑛2,𝛿=𝜏+𝜏2.(3.1)

Theorem 3.1. Suppose that (H) holds. If there exist nine symmetric positive definite matrices 𝑃>0, 𝑊>0, 𝑅>0, 𝑄_𝑖>0(𝑖=1,2,3,4,5,6), and four positive diagonal matrices 𝐷_𝑖>0(𝑖=1,2), 𝐻_𝑖>0(𝑖=1,2), such that the following LMI holds: Π=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣Π¹¹∗ΠΠ₁₂₂₂Π₁₃Π₁₄Π₁₅00∗∗ΠΠ₂₃₃₃Π₂₄000∗∗∗ΠΠ₃₄₄₄0Π₃₆0∗∗∗∗Π0₅₅000Π₅₇∗∗∗∗∗∗∗∗∗∗Π∗₆₆Π0₇₇⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.2) where Π₁₁=𝐶^𝑇𝑃𝐶−𝑃+𝑄₁+(1+𝜏−𝜏)(𝑄₂−2𝐺₁𝐷₁+2𝐺₂𝐷₂)−𝐺₃𝐻₁+(𝐶−𝐼)^𝑇(𝑄₄+𝑄₆)(𝐶−𝐼)−(1/𝛿²)𝑄₄+𝑄₅+𝑊, Π₁₂=𝐶^𝑇𝑃𝐴+(1+𝜏−𝜏)(𝐷₁−𝐷₂)+𝐺₄𝐻₁+(𝐶−𝐼)^𝑇(𝑄₄+𝑄₆)𝐴, Π₁₃=𝐶^𝑇𝑃𝐵+(𝐶−𝐼)^𝑇(𝑄₄+𝑄₆)𝐵, Π₁₄=𝐶^𝑇𝑃+(𝐶−𝐼)^𝑇(𝑄₄+𝑄₆), Π₁₅=(1/𝛿²)𝑄₄, Π₂₂=(1+𝜏−𝜏)𝑄₃−𝐻₁+𝐴^𝑇(𝑃+𝑄₄+𝑄₆)𝐴, Π₂₃=𝐴^𝑇(𝑃+𝑄₄+𝑄₆)𝐵, Π₂₄=𝐴^𝑇(𝑃+𝑄₄+𝑄₆), Π₃₃=−𝑄₃−𝐻₂+𝐵^𝑇(𝑃+𝑄₄+𝑄₆)𝐵, Π₃₄=𝐵^𝑇(𝑃+𝑄₄+𝑄₆), Π₃₆=−𝐷₁+𝐷₂+𝐺₄𝐻₂, Π₄₄=𝑃+𝑄₄+𝑄₆−𝑅, Π₅₅=−𝑄₁−(1/𝛿²)𝑄₄−(1/(𝜏−𝛿)²)𝑄₆, Π₅₇=(1/(𝜏−𝛿)²)𝑄₆, Π₆₆=−𝑄₂+2𝐺₁𝐷₁−2𝐺₂𝐷₂−𝐺₃𝐻₂, and Π₇₇=−𝑄₅−(1/(𝜏−𝛿)²)𝑄₆, then discrete-time neural network (2.1) is globally dissipative, and 𝑆=𝑥∶‖𝑥‖≤𝑢^𝑇𝑅𝑢𝜆_min(𝑊)^1/2(3.3) is a positive invariant and globally attractive set.

Proof. For positive diagonal matrices 𝐷₁>0 and 𝐷₂>0, we know from assumption (H) that 𝑔(𝑥(𝑘))−𝐺₁𝑥(𝑘)^𝑇𝐷₁𝑥(𝑘)≥0,𝐺₂𝑥(𝑘)−𝑔(𝑥(𝑘))^𝑇𝐷₂𝑥(𝑘)≥0.(3.4)
Defining 𝜂(𝑘)=𝑥(𝑘+1)−𝑥(𝑘), we consider the following Lyapunov-Krasovskii functional candidate for model (2.1) as 𝑉(𝑘,𝑥(𝑘))=⁸_𝑖=1𝑉_𝑖(𝑘,𝑥(𝑘)),(3.5) where 𝑉𝑉₁(𝑘,𝑥(𝑘))=𝑥^𝑇(𝑘)𝑃𝑥(𝑘),₂(𝑘,𝑥(𝑘))=^𝑘−1_𝑖=_𝑘−𝛿𝑉𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖),₃(𝑘,𝑥(𝑘))=^𝑘−1_𝑖=𝑘_{−𝜏(𝑘)}𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)+^𝑘−𝜏_{𝑙=𝑘−𝜏+1𝑘−1𝑖=𝑙}𝑉𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖),₄(𝑘,𝑥(𝑘))=^𝑘−1_𝑖=𝑘_{−𝜏(𝑘)}𝑔^𝑇(𝑥(𝑖))𝑄₃𝑔(𝑥(𝑖))+^𝑘−𝜏_{𝑙=𝑘−𝜏+1𝑘−1𝑖=𝑙}𝑉𝑔^𝑇(𝑥(𝑖))𝑄₃𝑔(𝑥(𝑖)),₅(𝑘,𝑥(𝑘))=2^𝑘−1_𝑖=𝑘_{−𝜏(𝑘)}𝑔(𝑥(𝑖))−𝐺₁𝑥(𝑖)^𝑇𝐷₁𝑥(𝑖)+2^𝑘−𝜏_{𝑙=𝑘−𝜏+1𝑘−1𝑖=𝑙}𝑔(𝑥(𝑖))−𝐺₁𝑉𝑥(𝑖)^𝑇𝐷₁𝑥(𝑖),₆(𝑘,𝑥(𝑘))=2^𝑘−1_𝑖=𝑘_{−𝜏(𝑘)}𝐺₂𝑥(𝑖)−𝑔(𝑥(𝑖))^𝑇𝐷₂𝑥(𝑖)+2^𝑘−𝜏_{𝑙=𝑘−𝜏+1𝑘−1𝑖=𝑙}𝐺₂𝑉𝑥(𝑖)−𝑔(𝑥(𝑖))^𝑇𝐷₂𝑥(𝑖),₇(𝑘,𝑥(𝑘))=1𝛿⁻¹_{𝑙=−𝛿𝑘−1𝑖=𝑘+𝑙}𝑉𝜂^𝑇(𝑖)𝑄₄𝜂(𝑖),₈(𝑘,𝑥(𝑘))=⎧⎪⎪⎨⎪⎪⎩^𝑘−1_𝑖=_𝑘−𝜏𝑥^𝑇(𝑖)𝑄₅𝑥(𝑖)+1𝜏−𝛿^{𝑘−𝜏−1}_{𝑙=𝑘−𝛿𝑘−1𝑖=𝑙}𝜂^𝑇(𝑖)𝑄₆𝜂(𝑖),𝜏≤𝜏(𝑘)≤𝛿,_{𝑘−1𝑖=}_𝑘−𝜏𝑥^𝑇(𝑖)𝑄₅𝑥(𝑖)+1𝜏−𝛿^{𝑘−𝛿−1}_{𝑙=𝑘−𝜏𝑘−1𝑖=𝑙}𝜂^𝑇(𝑖)𝑄₆𝜂(𝑖),𝛿<𝜏(𝑘)≤𝜏.(3.6) Calculating the difference of 𝑉_𝑖(𝑘) (𝑖=1,2,3) along the positive half trajectory of (2.1), we obtain
Δ𝑉₁(𝑘,𝑥(𝑘))=+𝑥^𝑇2𝑥(𝑘_𝑇)𝐶^𝑇𝑃𝐶−𝑃𝑥(𝑘)+2𝑥^𝑇(𝑘)𝐶^𝑇𝑃𝐴𝑔(𝑥(𝑘))+𝑔_𝑇(𝑘)𝐶^𝑇𝑃𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+2𝑥^𝑇(𝑘)𝐶^𝑇𝑃𝑢+2𝑔_𝑇(𝑥(𝑘))𝐴^𝑇𝑃𝐴𝑔(𝑥(𝑘))+2𝑔^𝑇(𝑥(𝑘))𝐴^𝑇𝑃𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+2𝑔_𝑇(𝑥(𝑘))𝐴^𝑇𝑃𝑢+𝑔^𝑇(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑃𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))Δ𝑉₂(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑃𝑢+𝑢^𝑇𝑃𝑢,(3.7)(𝑘,𝑥(𝑘))=𝑥^𝑇(𝑘)𝑄₁𝑥(𝑘)−𝑥^𝑇(𝑘−𝛿)𝑄₁𝑥(𝑘−𝛿),(3.8)Δ𝑉₃(𝑘,𝑥(𝑘))=^𝑘_𝑖=𝑘+1_{−𝜏(𝑘+1)}𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)−^𝑘−1_𝑖=𝑘_{−𝜏(𝑘)}+𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)_{𝑘+1−𝜏𝑙=𝑘}_{−𝑘𝜏+2}_𝑖=𝑙𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)−^𝑘−𝜏_{𝑙=𝑘−𝜏+1𝑘−1𝑖=𝑙}=𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)_{𝑘−𝜏𝑖=𝑘+1}_{−𝜏(𝑘+1)}𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)+^𝑘−1_𝑖=𝑘_−𝜏+1𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)+𝑥^𝑇(𝑘)𝑄₂𝑥(𝑘)−^𝑘−1_𝑖=𝑘−_𝜏(𝑘)+1−𝑥_𝑇𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)(𝑘−𝜏(𝑘))𝑄₂𝑥(𝑘−𝜏(𝑘))+𝜏−𝜏𝑥^𝑇(𝑘)𝑄₂𝑥(𝑘)−^𝑘−𝜏_𝑙=𝑘_−𝜏+1≤1+𝑥^𝑇(𝑙)𝑄₂𝑥(𝑙)𝜏−𝜏𝑥^𝑇(𝑘)𝑄₂𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏(𝑘))𝑄₂𝑥(𝑘−𝜏(𝑘)).(3.9)ΔΔ𝑉𝑉₄₅(𝑘,𝑥(𝑘))≤1+𝜏−𝜏𝑔^𝑇(𝑥(𝑘))𝑄₃𝑔(𝑥(𝑘))−𝑔^𝑇(𝑥(𝑘−𝜏(𝑘)))𝑄₃𝑔(𝑥(𝑘−𝜏(𝑘))),(𝑘,𝑥(𝑘))≤−221𝑔+(𝑥𝜏(−𝑘𝜏−𝜏𝑔(𝑘()𝑥))(𝑘−))𝐺−₁𝐺₁𝑥(𝑘)^𝑇𝐷₁𝑥(𝑘)Δ𝑉₆𝑥(𝑘−𝜏(𝑘))^𝑇𝐷₁𝑥(𝑘−𝜏(𝑘)),(𝑘,𝑥(𝑘))≤−221𝐺+₂𝜏−𝜏𝐺₂𝑥(𝑘)−𝑔(𝑥(𝑘))^𝑇𝐷₂𝑥(𝑘)Δ𝑉₇𝑥(𝑘−𝜏(𝑘))−𝑔(𝑥(𝑘−𝜏(𝑘)))^𝑇𝐷₂𝑥(𝑘−𝜏(𝑘)),(𝑘,𝑥(𝑘))=𝜂^𝑇(𝑘)𝑄₄𝜂(𝑘)−1𝛿^𝑘−1_{𝑖=𝑘−𝛿}≤𝜂_𝑇𝜂^𝑇(𝑖)𝑄₄𝜂(𝑖)(𝑘)𝑄₄𝜂(𝑘)−1𝛿^2𝑘−1_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑄_{4𝑘−1𝑖=}_𝑘−𝛿=𝑥_𝑇𝜂(𝑖)+2𝑥(𝑘_𝑇)(𝐶−𝐼)^𝑇𝑄₄(𝐶−𝐼)𝑥(𝑘)+2𝑥^𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₄𝐴𝑔(𝑥(𝑘))+𝑔_𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₄𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+2𝑥^𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₄𝑢+2𝑔_𝑇(𝑥(𝑘))𝐴^𝑇𝑄₄𝐴𝑔(𝑥(𝑘))+2𝑔^𝑇(𝑥(𝑘))𝐴^𝑇𝑄₄𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+2𝑔_𝑇(𝑥(𝑘))𝐴^𝑇𝑄₄𝑢+𝑔^𝑇(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑄₄𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑄₄𝑢+𝑢^𝑇𝑄₄+1𝛿²𝑥(𝑥𝑘(−𝑘)𝛿)^𝑇−𝑄⁴𝑄₄𝑄₄−𝑄₄𝑥(𝑥𝑘(−𝑘)𝛿).(3.10) Similarly, one has 𝜏≤𝜏(𝑘)≤𝛿 When Δ𝑉₈(𝑘,𝑥(𝑘))=+𝑥^𝑇𝛿(−𝑘)𝜏𝑄₅𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏)𝑄₅𝑥(𝑘−𝜏)𝜏−𝛿𝜂^𝑇(𝑘)𝑄₆𝜂(𝑘)−1𝜏−𝛿^{𝑘−𝜏−1}_{𝑖=𝑘−𝛿}≤𝑥_𝑇𝜂^𝑇(𝑖)𝑄₆𝜂(𝑖)+𝜂_𝑇(𝑘)𝑄₅𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏)𝑄₅𝑥(𝑘−𝜏)(𝑘)𝑄₆𝜂(𝑘)−1𝜏−𝛿(𝛿−𝜏)^{𝑘−𝜏−1}_𝑖=_𝑘−𝛿𝜂^𝑇(𝑖)𝑄_{6𝑘−𝜏−1𝑖=}_𝑘−𝛿≤−𝑥_𝑇𝜂(𝑖)+2𝑥_𝑇(𝑘−𝜏)𝑄₅𝑥(𝑘−𝜏)+𝑥^𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₆(𝐶−𝐼)+𝑄₅𝑥(𝑘)+2𝑥_𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₆𝐴𝑔(𝑥(𝑘))+2𝑥^𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₆𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+2𝑔_𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₆𝑢+𝑔^𝑇(𝑥(𝑘))𝐴^𝑇𝑄₆𝐴𝑔(𝑥(𝑘))+𝑔_𝑇(𝑥(𝑘))𝐴^𝑇𝑄₆𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+2𝑔^𝑇(𝑥(𝑘))𝐴^𝑇𝑄₆𝑢+2𝑔_𝑇(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑄₆𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+1(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑄₆𝑢+𝑢^𝑇𝑄₆𝑢𝜏−𝛿²𝑥𝑥((𝑘𝑘−−𝜏𝛿))^𝑇−𝑄⁶𝑄₆𝑄₆−𝑄₆𝑥𝑥((𝑘𝑘−−𝜏𝛿)).(3.11), 𝛿<𝜏(𝑘)≤𝜏 When Δ𝑉₈(𝑘,𝑥(𝑘))=+𝑥^𝑇𝜂_𝑇(𝑘)𝑄₅𝑥(𝑘)−𝑥^𝑇𝑘−𝜏𝑄₅𝑥𝑘−𝜏(𝑘)𝑄₆𝜂(𝑘)−1𝜏−𝛿^{𝑘−𝛿−1}_{𝑖=𝑘−𝜏}≤𝑥_𝑇𝜂^𝑇(𝑖)𝑄₆𝜂(𝑖)+𝜂_𝑇(𝑘)𝑄₅𝑥(𝑘)−𝑥^𝑇𝑘−𝜏𝑄₅𝑥𝑘−𝜏(𝑘)𝑄₆𝜂(𝑘)−1𝜏−𝛿^{2𝑘−𝛿−1}_𝑖=_𝑘−𝜏𝜂^𝑇(𝑖)𝑄_{6𝑘−𝛿−1𝑖=}_𝑘−𝜏=−𝑥_𝑇𝜂(𝑖)+2𝑥_𝑇𝑘−𝜏𝑄₅𝑥𝑘−𝜏+𝑥^𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₆(𝐶−𝐼)+𝑄₅𝑥(𝑘)+2𝑥_𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₆𝐴𝑔(𝑥(𝑘))+2𝑥^𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₆𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+2𝑔_𝑇(𝑘)(𝐶−𝐼)^𝑇𝑄₆𝑢+𝑔^𝑇(𝑥(𝑘))𝐴^𝑇𝑄₆𝐴𝑔(𝑥(𝑘))+𝑔_𝑇(𝑥(𝑘))𝐴^𝑇𝑄₆𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+2𝑔^𝑇(𝑥(𝑘))𝐴^𝑇𝑄₆𝑢+2𝑔_𝑇(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑄₆𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+1(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑄₆𝑢+𝑢^𝑇𝑄₆𝑢𝜏−𝛿²𝑥𝑘−𝑥(𝑘−𝜏𝛿)^𝑇−𝑄⁶𝑄₆𝑄₆−𝑄₆𝑥𝑘−𝑥(𝑘−𝜏𝛿).(3.12), 𝐻₁>0
For positive diagonal matrices 𝐻₂>0 and 0≤𝑔(𝑥𝑥(𝑘(𝑘)))^𝑇−𝐺³𝐻₁𝐺₄𝐻₁0≤𝑔𝑥(𝑥(𝑘(𝑘−−𝜏𝜏(𝑘(𝑘)))))_𝑇𝐺₄𝐻₁−𝐻₁𝑔𝑥(𝑥(𝑘(𝑘))),(3.13)−𝐺³𝐻₂𝐺₄𝐻₂𝐺₄𝐻₂−𝐻₂𝑔𝑥(𝑥(𝑘(𝑘−−𝜏𝜏(𝑘(𝑘))))).(3.14), we can get from assumption (H) that [6] 𝛼(𝑘)=(𝑥^𝑇(𝑘),𝑔^𝑇(𝑥(𝑘)),𝑔^𝑇(𝑥(𝑘−𝜏(𝑘))),𝑢^𝑇,𝑥^𝑇(𝑘−𝛿),𝑥^𝑇(𝑘−𝜏(𝑘)),𝜉^𝑇(𝑘))^𝑇
Denoting 𝜉(𝑘)=𝑥𝑥(𝑘𝑘−−𝜏),𝜏≤𝜏(𝑘)≤𝛿,𝜏,𝛿<𝜏(𝑘)≤𝜏,(3.15), where Δ𝑉(𝑘,𝑥(𝑘))≤𝑥^𝑇(𝑘)𝐶^𝑇𝑃𝐶−𝑃+𝑄₁+1+𝜏−𝜏𝑄₂−2𝐺₁𝐷₁+2𝐺₂𝐷₂−𝐺₃𝐻₁+(𝐶−𝐼)^𝑇𝑄₄+𝑄₆(𝐶−𝐼)−1𝛿²𝑄₄+𝑄₅+2𝑥_𝑇𝑥(𝑘)+2𝑥_𝑇(𝑘)𝐶^𝑇𝑃𝐴+1+𝜏−𝜏𝐷₁−𝐷₂+𝐺₄𝐻₁+(𝐶−𝐼)^𝑇𝑄₄+𝑄₆𝐴𝑔(𝑥(𝑘))+2𝑥_𝑇(𝑘)𝐶^𝑇𝑃𝐵+(𝐶−𝐼)^𝑇𝑄₄+𝑄₆𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))(𝑘)𝐶^𝑇𝑃+(𝐶−𝐼)^𝑇𝑄₄+𝑄₆𝑢+21𝛿²+𝑔_𝑇𝑥^𝑇(𝑘)𝑄₄𝑥(𝑘−𝛿)+2𝑔_𝑇(𝑥(𝑘))𝐴^𝑇𝑃𝐴+1+𝜏−𝜏𝑄₃−𝐻₁+𝐴^𝑇𝑄₄+𝑄₆𝐴𝑔(𝑥(𝑘))+2𝑔_𝑇(𝑥(𝑘))𝐴^𝑇𝑃𝐵+𝐴^𝑇𝑄₄+𝑄₆𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+𝑔_𝑇(𝑥(𝑘))𝐴^𝑇𝑃+𝐴^𝑇𝑄₄+𝑄₆𝑢+2𝑔_𝑇(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑃𝐵−𝑄₃−𝐻₂+𝐵^𝑇𝑄₄+𝑄₆𝐵𝑔(𝑥(𝑘−𝜏(𝑘)))+2𝑔_𝑇(𝑥(𝑘−𝜏(𝑘)))𝐵^𝑇𝑃+𝐵^𝑇𝑄₄+𝑄₆𝑢+𝑢_𝑇(𝑥(𝑘−𝜏(𝑘)))−𝐷₁+𝐷₂+𝐺₄𝐻₂𝑥(𝑘−𝜏(𝑘))𝑃+𝑄₄+𝑄₆𝑢+𝑥^𝑇(𝑘−𝛿)−𝑄₁−1𝛿²𝑄₄−1𝜏−𝛿²𝑄₆+21𝑥(𝑘−𝛿)𝜏−𝛿²+𝑥_𝑇𝑥^𝑇(𝑘−𝛿)𝑄₆𝜉(𝑘)+𝑥_𝑇(𝑘−𝜏(𝑘))−𝑄₂+2𝐺₁𝐷₁−2𝐺₂𝐷₂−𝐺₃𝐻₂𝑥(𝑘−𝜏(𝑘))(𝑘−𝜏)−𝑄₅−1𝜏−𝛿²𝑄₆=−𝑥_𝑇𝑥(𝑘−𝜏)(𝑘)𝑊𝑥(𝑘)+𝑢^𝑇𝑅𝑢+𝛼^𝑇(𝑘)Π𝛼(𝑘).(3.16) it follows from (3.7) to (3.14) that Δ𝑉(𝑘,𝑥(𝑘))≤−𝑥^𝑇(𝑘)𝑊𝑥(𝑘)+𝑢^𝑇𝑅𝑢≤−𝜆_min(𝑊)‖𝑥‖²+𝑢^𝑇𝑅𝑢<0,(3.17) From condition (3.2) and inequality (3.16), we get 𝑥∈ℝ^𝑛⧵𝑆 when 𝑥∉𝑆, that is, 𝑆. Therefore, discrete-time neural network (2.1) is a globally dissipative system, and the set 𝑆=𝑥∶‖𝑥‖≤𝑢^𝑇𝑅𝑢𝜆_min(𝑊)^1/2(3.18) is a positive invariant and globally attractive set as LMI (3.2) holds. The proof is completed.

Next, we are now in a position to discuss the global exponential dissipativity of discrete-time neural network (2.1) as follows.

Theorem 3.2. Under the conditions of Theorem 3.1, neural network (2.1) is globally exponentially dissipative, and 𝑥∈ℝ^𝑛⧵𝑆 is a positive invariant and globally attractive set.

Proof. When 𝑥∉𝑆, that is, Δ𝑉(𝑘,𝑥(𝑘))≤−𝜆_min(−Π)‖𝑥(𝑘)‖².(3.19), we know from (3.16) that 𝑉(𝑘) From the definition of 𝑉(𝑘)≤𝜆_max(𝑃)‖𝑥(𝑘)‖²+𝜌^𝑘−1_𝑖=_𝑘−𝜏‖𝑥(𝑖)‖²,(3.20) in (3.5), it is easy to verify that 𝜌=𝜆_max𝑄₁+1+𝜏+−2𝜆𝜏_max𝜆_max𝑄₂+𝜆_max𝑄₃𝜆_max(𝐿)+2𝜆_max𝐿𝐷₁+4||𝐺₁𝐷₁||+2𝜆_max𝐿𝐷₂+2𝜆_max||𝐺₂𝐷₂||𝜏𝛿𝜆^max𝑄₄+𝜆_max𝑄₅+4𝜏−𝜏𝜆_max𝑄₆.(3.21) where 𝜇>1 For any scalar 𝜇^𝑗+1𝑉(𝑗+1)−𝜇^𝑗𝑉(𝑗)=≤𝜇𝜇^𝑗+1Δ𝑉(𝑗)+𝜇^𝑗(𝜇−1)𝑉(𝑗)_𝑗(𝜇−1)𝜆_max(𝑃)−𝜇^𝑗+1𝜆_min(−Π)‖𝑥(𝑗)‖²+𝜌𝜇^𝑗(𝜇−1)^𝑗−1_𝑖=_𝑗−𝜏‖𝑥(𝑖)‖².(3.22), it follows from (3.19) and (3.20) that 0 Summing up both sides of (3.22) from 𝑘−1 to 𝑗 with respect to 𝜇^𝑘𝑉(𝑘)−𝑉(0)≤(𝜇−1)𝜆_max(𝑃)−𝜇𝜆_min(−Π)^𝑘−1_𝑗=0𝜇^𝑗‖𝑥(𝑗)‖²+𝜌(𝜇−1)^𝑘−1_{𝑗=0𝑗−1𝑖=𝑗−𝜏}𝜇^𝑗‖𝑥(𝑖)‖².(3.23), we have ^𝑘−1_{𝑗=0𝑗−1𝑖=𝑗−𝜏}𝜇^𝑗‖𝑥(𝑖)‖²≤⎛⎜⎜⎝⁻¹_{𝑖=−𝜏𝑖+}^𝜏_𝑗=0+^{𝑘−1−𝜏}_𝑖=0𝑖+^𝜏_𝑗=𝑖+1+^𝑘−1_{𝑖=𝑘−𝜏𝑘−1𝑗=𝑖+1}⎞⎟⎟⎠𝜇_𝑗‖𝑥(𝑖)‖²≤𝜏𝜇^𝜏sup_{𝑠∈𝑁[−𝜏,0]}‖𝑥(𝑠)‖²+𝜏𝜇^𝜏𝑘−1_𝑖=0𝜇^𝑖‖𝑥(𝑖)‖².(3.24) It is easy to compute that 𝑉(0)≤𝜆_max(𝑃)+𝜌𝜏sup_{𝑠∈𝑁[−𝜏,0]}‖𝑥(𝑠)‖².(3.25) From (3.20), we obtain 𝜇^𝑘𝑉(𝑘)≤𝐿₁(𝜇)sup_{𝑠∈𝑁[−𝜏,0]}‖𝑥(𝑠)‖²+𝐿₂(𝜇)^𝑘_𝑗=0𝜇^𝑖‖𝑥(𝑖)‖²,(3.26) It follows from (3.23)–(3.25) that 𝐿₂𝐿₁(𝜇)=𝜆_max(𝑃)+𝜌𝜏+𝜌(𝜇−1)𝜏𝜇^𝜏,(𝜇)=(𝜇−1)𝜆_max(𝑃)−𝜇𝜆_min(−Π)+𝜌(𝜇−1)𝜏𝜇^𝜏.(3.27) where 𝐿₂(1)<0 Since 𝐿₂(𝜇), by the continuity of functions 𝛾>1, we can choose a scalar 𝐿₂(𝛾)≤0 such that 𝐿₁(𝛾)>0. Obviously, 𝛾^𝑘𝑉(𝑘)≤𝐿₁(𝛾)sup_{𝑠∈𝑁[−𝜏,0]}‖𝑥(𝑠)‖².(3.28). From (3.26), we get 𝑉(𝑘)
From the definition of 𝑉(𝑘)≥𝜆_min(𝑃)‖𝑥(𝑘)‖².(3.29) in (3.5), we have 𝑀=√𝐿₁(𝛾)/𝜆_min(𝑃) Let 𝛽=√1/𝛾, 𝑀>0, then 0<𝛽<1, ‖𝑥(𝑘)‖≤𝑀𝛽^𝑘sup_{𝑠∈𝑁[−𝜏,0]}‖𝑥(𝑠)‖(3.30). It follows from (3.28) and (3.29) that 𝑘∈ℕ for all 𝑆, which means that discrete-time neural network (2.1) is globally exponentially dissipative, and the set 𝐺^−𝑗 is a positive invariant and globally attractive set as LMI (3.2) holds. The proof is completed.

Remark 3.3. In the study on dissipativity of neural networks, the assumption (H) of this paper is as same as that in [28], the constants 𝐺^+𝑗 and 𝑗=1,2,…,𝑛 ([𝜏,𝜏]) in assumption (H) of this paper are allowed to be positive, negative, or zero. Hence, assumption (H), first proposed by Liu et al. in [6], is weaker than the assumption in [23–27, 30].

Remark 3.4. The idea of constructing Lyapunov-Krasovskii functional (3.5) is that we divide the delay interval [𝜏,𝛿] into two subintervals [𝛿,𝜏] and 𝜏(𝑘), thus the proposed Lyapunov-Krasovskii functional is different when the time-delay 𝜏(𝑘) belongs to different subinterval. The main advantage of such Lyapunov-Krasovskii functional is that it makes full use of the information on the considered time-delay 𝑥(𝑘+1)=(𝐶+Δ𝐶(𝑘))𝑥(𝑘)+(𝐴+Δ𝐴(𝑘))𝑔(𝑥(𝑘))+(𝐵+Δ𝐵(𝑘))𝑔(𝑥(𝑘−𝜏(𝑘)))+𝑢,(3.31).

Now, let us consider the case when the parameter uncertainties appear in the discrete-time neural networks with time-varying delays. In this case, model (2.1) can be further generalized to the following one:

𝐶 where 𝐴, 𝐵, Δ𝐶(𝑘) are known real constant matrices, and the time-varying matrices Δ𝐴(𝑘), Δ𝐵(𝑘) and Δ𝐶(𝑘)Δ𝐴(𝑘)Δ𝐵(𝑘)=𝑀𝐹(𝑘)𝑁¹𝑁₂𝑁₃,(3.32) represent the time-varying parameter uncertainties that are assumed to satisfy the following admissible condition:

𝑀 where 𝑁_𝑖(𝑖=1,2,3) and 𝐹(𝑘) are known real constant matrices, and 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼.(3.33) is the unknown time-varying matrix-valued function subject to the following condition:

𝑃>0

For model (3.31), we have the following result readily.

Theorem 3.5. Suppose that (H) holds. If there exist nine symmetric positive definite matrices 𝑊>0, 𝑅>0, 𝑄_𝑖>0(𝑖=1,2,3,4,5,6), 𝐷_𝑖>0(𝑖=1,2), and four positive diagonal matrices 𝐻_𝑖>0(𝑖=1,2), Γ=⎡⎢⎢⎢⎢⎢⎢⎣Γ¹Γ₂Γ₃Γ₄0𝜀Γ₅∗∗−∗𝑃−0𝑄₄0𝑃𝑀0∗∗∗−0𝑄₆𝑄₄𝑀0∗∗∗∗∗∗∗∗𝑄−₆∗𝜀𝑀𝐼−00𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎦<0,(3.34), such that the following LMI holds: Γ₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣Ω¹¹∗ΩΩ₁₂₂₂00Ω₁₅00∗∗Ω0₃₃0000∗∗∗Ω0₄₄0Ω₃₆0∗∗∗∗Ω0₅₅000Ω₅₇∗∗∗∗∗∗∗∗∗∗Ω∗₆₆Ω0₇₇Γ₂⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,Γ₃=𝑃𝐶𝑃𝐴𝑃𝐵𝑃000^𝑇,Γ₄=𝑄₄(𝐶−𝐼)𝑄₄𝐴𝑄₄𝐵𝑄₄000^𝑇,Γ=𝑄₆(𝐶−𝐼)𝑄₆𝐴𝑄₆𝐵𝑄₆000₅,=𝑁₁𝑁₂𝑁₃0000,(3.35) where Ω₁₁=−𝑃+𝑄₁+(1+𝜏−𝜏)(𝑄₂−2𝐺₁𝐷₁+2𝐺₂𝐷₂)−𝐺₃𝐻₁−(1/𝛿²)𝑄₄+𝑄₅+𝑊 and Ω₁₂=(1+𝜏−𝜏)(𝐷₁−𝐷₂)+𝐺₄𝐻₁, Ω₁₅=(1/𝛿²)𝑄₄, Ω₂₂=(1+𝜏−𝜏)𝑄₃−𝐻₁, Ω₃₃=−𝑄₃−𝐻₂, Ω₃₆=−𝐷₁+𝐷₂+𝐺₄𝐻₂, Ω₄₄=−𝑅, Ω₅₅=−𝑄₁−(1/𝛿²)𝑄₄−(1/(𝜏−𝛿)²)𝑄₆, Ω₅₇=(1/(𝜏−𝛿)²)𝑄₆, Ω₆₆=−𝑄₂+2𝐺₁𝐷₁−2𝐺₂𝐷₂−𝐺₃𝐻₂, Ω₇₇=−𝑄₅−(1/(𝜏−𝛿)²)𝑄₆,, and 𝑆=𝑥∶‖𝑥‖≤𝑢^𝑇𝑅𝑢𝜆_min(𝑊)^1/2(3.36) then uncertain discrete-time neural network (3.31) is globally dissipative and globally exponentially dissipative, and ⎡⎢⎢⎢⎣Γ¹Γ₂Γ₃Γ₄∗∗−∗𝑃−0𝑄₄0∗∗∗−0𝑄₆⎤⎥⎥⎥⎦+𝜀₋₁⎡⎢⎢⎢⎣𝑄𝑃𝑀₄0𝑄₆𝑀𝑀⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝑄𝑃𝑀⁴0𝑄₆𝑀𝑀⎤⎥⎥⎥⎦^𝑇+𝜀⎡⎢⎢⎢⎣Γ⁵000⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣Γ000⁵⎤⎥⎥⎥⎦^𝑇<0.(3.37) is a positive invariant and globally attractive set.

Proof. By Lemma 2.3, we know that LMI (3.34) is equivalent to the following inequality: ⎡⎢⎢⎢⎣Γ¹Γ₂Γ₃Γ₄∗∗−∗𝑃−0𝑄₄0∗∗∗−0𝑄₆⎤⎥⎥⎥⎦+⎡⎢⎢⎢⎣𝑄𝑃𝑀₄0𝑄₆𝑀𝑀⎤⎥⎥⎥⎦𝐹(𝑘)⎡⎢⎢⎢⎣Γ⁵000⎤⎥⎥⎥⎦^𝑇+⎡⎢⎢⎢⎣Γ⁵000⎤⎥⎥⎥⎦𝐹_𝑇(𝑘)⎡⎢⎢⎢⎣𝑄𝑃𝑀₄0𝑄₆𝑀𝑀⎤⎥⎥⎥⎦^𝑇<0,(3.38) From Lemma 2.4, we know that (3.37) is equivalent to the following inequality: ⎡⎢⎢⎢⎢⎣Γ¹Γ₂+Γ₅𝐹^𝑇(𝑘)𝑀^𝑇𝑃Γ₃+Γ₅𝐹^𝑇(𝑘)𝑀^𝑇𝑄₄Γ₄+Γ₅𝐹^𝑇(𝑘)𝑀^𝑇𝑄₆∗∗−∗𝑃−0𝑄₄0∗∗∗−0𝑄₆⎤⎥⎥⎥⎥⎦<0.(3.39) that is Φ=Γ₂+Γ₅𝐹^𝑇(𝑘)𝑀^𝑇𝑃 Let Ψ=Γ₃+Γ₅𝐹^𝑇(𝑘)𝑀^𝑇𝑄₄, Υ=Γ₄+Γ₅𝐹^𝑇(𝑘)𝑀^𝑇𝑄₆, Γ₁+Φ𝑃⁻¹Φ^𝑇+Ψ𝑄⁴⁻¹Ψ^𝑇+Υ𝑄⁶⁻¹Υ^𝑇<0.(3.40). As an application of Lemma 2.3, we know that (3.39) is equivalent to the following inequality: [Δ𝐶(𝑘)Δ𝐴(𝑘)Δ𝐵(𝑘)]=𝑀𝐹(𝑘)[𝑁₁𝑁₂𝑁₃] By simple computation and noting that Ψ=Φ𝑄=₄𝑃(𝐶+Δ𝐶)𝑃(𝐴+Δ𝐴)𝑃(𝐵+Δ𝐵)𝑃000^𝑇,Υ=𝑄₆(𝐶+Δ𝐶−𝐼)𝑄₄(𝐴+Δ𝐴)𝑄₄(𝐵+Δ𝐵)𝑄₄000^𝑇,(𝐶+Δ𝐶−𝐼)𝑄₆(𝐴+Δ𝐴)𝑄₆(𝐵+Δ𝐵)𝑄₆000^𝑇.(3.41), we have 𝐶+Δ𝐶 Therefore, inequality (3.40) is just the same as inequality (3.2) when we use 𝐴+Δ𝐴, 𝐵+Δ𝐵, 𝐶 to replace 𝐴, 𝐵, 𝑆=𝑥∶‖𝑥‖≤𝑢^𝑇𝑅𝑢𝜆_min(𝑊)^1/2(3.42) of inequality (3.2), respectively. From Theorems 3.1 and 3.2, we know that uncertain discrete-time neural network (3.31) is globally dissipative and globally exponentially dissipative, and 𝑔𝐶=0.0040.001,𝐴=−00.0.01500..0082,𝐵=−00.0.02500..0017,𝑢=−00.1.24₁1,(𝑥)=tanh(0.7𝑥)−0.1sin𝑥,𝑔₂(𝑥)=tanh(0.4𝑥)+0.2cos𝑥,𝜏(𝑘)=7−2sin𝑘𝜋2.(4.1) is a positive invariant and globally attractive set. The proof is then completed.

4. Examples

Example 4.1. Consider a discrete-time neural network (2.1) with 𝐺⁻¹=−0.1
It is easy to check that assumption (H) is satisfied, and 𝐺⁺¹=0.8, 𝐺⁻²=−0.2, 𝐺⁺²=0.6, 𝜏=5, 𝜏=9, 𝐺₁=−00.1−00.2,𝐺₂=00.800.6,𝐺₃=−00.08−00.12,𝐺₄=0.03500.2,𝛿=7.(4.2). Thus, 𝑄𝑄₁=−70.1.0619037−70.3.0512067,𝑄₂=−10.5.0009559−10.6.0303509,𝑄₃=20..2043535602..07313568₄,𝑃==−−5207.03...038001976752537−−51𝐷07.08...079006476362517,,𝑄𝑊₅==−−5700..01..00291142109708−−5700..13..00471111101408,,𝑅𝑄₆==70−2.2.002.44.304560578654701−.10.40.489.40715066₁644,,𝐻=1.208171.70411,𝐷₂=2.101802.30157₁,=19.7094623.70062,𝐻₂=9.1011010.50938.(4.3) By the Matlab LMI Control Toolbox, we can find a solution to the LMI in (3.2) as follows: 𝑆={𝑥∶‖𝑥‖≤0.9553} Therefore, by Theorem 3.1, we know that model (2.1) with above given parameters is globally dissipative and globally exponentially dissipative. It is easy to compute that the positive invariant and global attractive set are 𝑔𝐶=0.0120.068,𝐴=156.3.784.24,𝐵=−199..29−−1299..77,𝑢=−103..94₁,(𝑥)=tanh(0.7𝑥)−0.1sin𝑥,𝑔₂(𝑥)=tanh(0.4𝑥)+0.2cos𝑥,𝜏(𝑘)=7−2sin𝑘𝜋2.(4.4).

The following example is given to illustrate that when the sufficient conditions ensuring the global dissipativity are not satisfied, the complex dynamics will appear.

Example 4.2. Consider a discrete-time neural network (2.1) with 𝐺_𝑖(𝑖=1,2,3,4) It is easy to check that the linear matrix inequality (3.2) with 𝛿 and 𝑥₁(𝑠)=0.5 of Example 4.1 has not a feasible solution. Figure 1 depicts the states of the considered neural network (2.1) with initial conditions 𝑥₂(𝑠)=0.45, 𝑠∈ℕ[−9,0], 𝑥₁(𝑘). One can see from Figure 1 that the chaos behaviors have appeared for neural network (2.1) with above given parameters.

Figure 1 

State responses of 𝑥2(𝑘) and 𝜃.

5. Conclusions

In this paper, the global dissipativity and global exponential dissipativity have been investigated for uncertain discrete-time neural networks with time-varying delays and general activation functions. By constructing appropriate Lyapunov-Krasovskii functionals and employing linear matrix inequality technique, several new delay-dependent criteria for checking the global dissipativity and global exponential dissipativity of the addressed neural networks have been derived in terms of LMI, which can be checked numerically using the effective LMI toolbox in MATLAB. Illustrated examples are also given to show the effectiveness of the proposed criteria. It is noteworthy that because neither model transformation nor free weighting matrices are employed to deal with cross terms in the derivation of the dissipativity criteria, the obtained results are less conservative and more computationally efficient.

Acknowledgments

The authors would like to thank the reviewers and the editor for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China under Grants 60974132, 60874088, and 10772152.