Global Exponential Stability of Antiperiodic Solutions for Discrete-Time Neural Networks with Mixed Delays and Impulses

Abstract

The problem on global exponential stability of antiperiodic solution is investigated for a class of impulsive discrete-time neural networks with time-varying discrete delays and distributed delays. By constructing an appropriate Lyapunov-Krasovskii functional, and using the contraction mapping principle and the matrix inequality techniques, a new delay-dependent criterion for checking the existence, uniqueness, and global exponential stability of anti-periodic solution is derived in linear matrix inequalities (LMIs). Two simulation examples are given to show the effectiveness of the proposed result.

1. Introduction

Over the past decades, delayed neural networks have found successful applications in many areas such as signal processing, pattern recognition, associative memories, and optimization solvers. In such applications, the qualitative analysis of the dynamical behaviors is a necessary step for the practical design of neural networks [1]. Many important results on the dynamical behaviors have been reported for delayed neural networks, see [1–5] and the references therein for some recent publications. Although neural networks are mostly studied in the continuous-time setting, they are often discretized for experimental or computational purposes. The dynamic characteristics of discrete-time neural networks have been extensively investigated, for example, see [6–10] and the references cited therein.

Impulsive differential equations are mathematical apparatus for simulation of process and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth [11, 12]. Consequently, many neural networks with impulses have been studied extensively, and a great deal of the literature is focused on the existence and stability of an equilibrium point [13–17]. In [18, 19], the authors discussed the existence and global exponential stability of periodic solution of a class of neural networks with impulse.

The study of antiperiodic solutions for nonlinear differential equations is closely related to the study of periodic solutions, and it was initiated by Okochi in 1988 [20]. Arising from problems in applied sciences, it is well-known that the existence and stability of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations as a special periodic solution [21, 22]. As pointed out in [23], antiperiodic solutions arise naturally in the mathematical modeling of various physical processes. For example, antiperiodic trigonometric polynomials are often studied in interpolation problems [24, 25], the signal transmission process of neural networks can often be described as an antiperiodic process [26], and antiperiodic wavelets are discussed in [27]. During the past twenty years antiperiodic problems of nonlinear differential equations have been extensively studied by many authors, for example, see [28–30] and references therein. Recently, the problem of antiperiodic solutions for neural networks with or without time delays has received considerable research interest, see for example [26, 31–38] and references therein. In [31, 32], using some analysis skills and Lyapunov method, the authors studied the existence and exponential stability of antiperiodic solutions for shunting inhibitory cellular neural networks with time-varying discrete delays or distributed delays. In [33], some sufficient conditions have been established for checking the existence and exponential stability of antiperiodic solutions of high-order Hopfield neural networks with time-varying delays. In [34], the recurrent neural networks with time-varying delays and continuously distributed delays have been considered, and some sufficient conditions for the existence and exponential stability of the antiperiodic solutions have been given. In [35, 36], the existence and exponential stability of antiperiodic solutions have been studied for Cohen-Grossberg neural networks with time-varying delays and continuously distributed delays. In [37], the authors obtained some sufficient conditions to ensure existence and exponential stability of the antiperiodic solutions for a class of Hopfield neural networks with periodic impulses. In [38], several sufficient conditions are established for the existence and global exponential stability of antiperiodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scale.

However, to the best of our knowledge, there are few papers published on the existence and exponential stability of antiperiodic solutions for discrete-time neural networks with mixed delays and impulses. Motivated by the above discussion, the objective of this paper is to study the existence and exponential stability of antiperiodic solutions for impulsive discrete-time recurrent neural networks with time-varying discrete delays and distributed delays. Under more general description on the activation functions, we obtain a new criterion for checking the existence, uniqueness, and global exponential stability of antiperiodic solution, which can be checked numerically using the effective LMI toolbox in MATLAB. Two simulation examples are given to show the effectiveness and less conservatism of the proposed criteria.

Notations
The notation used here is quite standard. ℝ^𝑛 and ℝ^𝑚×𝑛 denote, respectively, the 𝑛-dimensional Euclidean space and the set of all 𝑚×𝑛 real matrices. The superscript “T” denotes matrix transposition. 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. If 𝐴 is a matrix, denote by 𝜆_max(𝐴) (resp., 𝜆_min(𝐴)) the largest (resp., smallest) eigenvalue of 𝐴. For integers 𝑎, 𝑏, and 𝑎<𝑏, 𝑁[𝑎,𝑏] denotes the discrete interval given by 𝑁[𝑎,𝑏]={𝑎,𝑎+1,…,𝑏}. 𝒞(𝑁[−𝜏,0],ℝ^𝑛) denotes the set of all functions 𝜙∶𝑁[−𝜏,0]→ℝ^𝑛. In symmetric block matrices, the symbol ∗ is used as an ellipsis for terms induced by symmetry. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.

2. Model Description and Preliminaries

In this paper, we consider the following impulsive discrete-time neural networks with time-varying discrete delays and distributed delays𝑥_𝑖(𝑘+1)=𝑎_𝑖𝑥_𝑖(𝑘)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗(𝑘)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑘−𝜏₁(𝑘+_𝑛𝑗=1)𝑑_𝑖𝑗^{𝑘+𝜏2−1}_𝑚=𝑘𝑔_𝑗𝑥_𝑗𝑚−𝜏₂+𝐼_𝑖(𝑘),𝑘≠𝑘_𝑟𝑥_𝑖𝑘_𝑟+1=𝑥_𝑖𝑘_𝑟+𝑒_𝑖𝑟𝑥_𝑖𝑘_𝑟,𝑟=1,2,… (2.1) or, in an equivalent vector form𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑔(𝑥(𝑘))+𝐶𝑔𝑥𝑘−𝜏₁(𝑘)+𝐷^{𝑘+𝜏2−1}_𝑚=𝑘𝑔𝑥𝑚−𝜏₂+𝐼(𝑘),𝑘≠𝑘_𝑟𝑥𝑘_𝑟+1=𝑥𝑘_𝑟+𝑒_𝑟𝑥𝑘_𝑟,𝑟=1,2,… (2.2) for 𝑘=1,2,…, 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; 𝐼(𝑘)=(𝐼₁(𝑘),𝐼₂(𝑘),…,𝐼_𝑛(𝑘))^𝑇∈ℝ^𝑛, 𝐼_𝑖(𝑘) represents the external input on the 𝑖th neuron at time 𝑘; the positive integer 𝜏₁(𝑘) corresponds to the transmission delay and satisfies 𝜏̌≤𝜏₁(𝑘)≤𝜏 (𝜏̌ and 𝜏 are known integers such that 𝜏>𝜏̌≥0); the positive integer 𝜏₂ describes the distributed time delay; 𝐴=diag(𝑎₁,𝑎₂,…,𝑐_𝑛), 𝑎_𝑖 (0≤𝑎_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; 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛 is the connection weight matrix, 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛 and 𝐷=(𝑑_𝑖𝑗)_𝑛×𝑛 are the delayed connection weight matrices; 𝑘_𝑟 are the impulse instants satisfying 0=𝑘₀<𝑘₁<⋯<𝑘_𝑟<𝑘_𝑟+1<⋯ and 𝑘_𝑟+1−𝑘_𝑟>1; 𝑒_𝑟∶ℝ^𝑛→ℝ^𝑛 (𝑟=1,2,…) denotes a sequence of jump operators.

Remark 2.1. The delay term ∑^{𝑘+𝜏2−1}_𝑚=𝑘𝑔(𝑥(𝑚−𝜏₂)) in the system (2.2) is the so-called distributed delay in the discrete-time setting, which can be regarded as the discretization of the integral form ∫^𝑡_𝑡−𝜏2𝑔(𝑥(𝑠))𝑑𝑠 for the continuous-time system.
The initial condition associated with system (2.2) is given by 𝑥(𝑠)=𝜙(𝑠),𝜙∈𝒞𝑁[−𝜏,0],ℝ^𝑛, (2.3) where 𝜏=max{𝜏,𝜏₂}.
Throughout this paper, we make the following assumptions.(H1)𝑔(𝑢) and 𝑒_𝑟(𝑢) are odd functions, 𝜏₁(𝑘) and 𝐼(𝑘) are 𝜔-periodic function and 𝜔-antiperiodic function, respectively, that is,𝑔(−𝑢)=−𝑔(𝑢),𝑒_𝑟(−𝑢)=−𝑒_𝑟(𝑢),𝑢∈ℝ^𝑛𝜏,𝑟=1,2,…,₁(𝑘)=𝜏₁(𝑘+𝜔),𝐼(𝑘)=−𝐼(𝑘+𝜔),𝑘=1,2,…. (2.4) And there exists a positive integer 𝑝 such that 𝑘_𝑟+𝑝=𝑘_𝑟+𝜔,𝑒_𝑟+𝑝(𝑢)=𝑒_𝑟(𝑢),𝑢∈ℝ^𝑛,𝑟=1,2,…. (2.5)(H2) There exist constants 𝑙_𝑗̌ and 𝑙_𝑗̂ (𝑗=1,2,…,𝑛) such that𝑙_𝑗̌≤𝑔_𝑗𝛼₁−𝑔_𝑗𝛼₂𝛼₁−𝛼₂≤𝑙_𝑗̂,∀𝛼₁,𝛼₂∈ℝ,𝛼₁≠𝛼₂. (2.6)(H3) There exist constants ℎ̌_𝑖 and ℎ_𝑖 (𝑖=1,2,…,𝑛) such thatℎ̌_𝑖≤𝑒_𝑖𝑟𝛼₁−𝑒_𝑖𝑟𝛼₂𝛼₁−𝛼₂≤ℎ_𝑖,∀𝛼₁,𝛼₂∈ℝ,𝛼₁≠𝛼₂,𝑟=1,2,…. (2.7)

Definition 2.2. The antiperiodic solution 𝑥^∗(𝑘) of system (2.2) with (2.3) is said to be globally exponentially stable if there exist two positive constants 𝑀>0 and 0<𝜀<1 such that‖𝑥(𝑘)−𝑥^∗(𝑘)‖≤𝑀𝜀^𝑘sup_{[]𝑘∈𝑁−𝜏,0}‖𝜙(𝑘)−𝜙^∗(𝑘)‖, (2.8) for all 𝑘=1,2,…, where 𝑥(𝑘) is any solution of system (2.2) with (2.3), 𝜙(𝑘) and 𝜙^∗(𝑘) are the initial functions of solutions 𝑥(𝑘) and 𝑥^∗(𝑘), respectively.

Lemma 2.3. Let 𝑀∈ℝ^𝑛×𝑛 be positive definite matrix, 𝛼_𝑖∈ℝ^𝑛(𝑖=1,2,…,𝑚). Then the following inequality holds: ^𝑚_𝑖=1𝛼_𝑖^𝑇𝑀^𝑚_𝑖=1𝛼_𝑖≤𝑚^𝑚_𝑖=1𝛼^𝑇_𝑖𝑀𝛼_𝑖. (2.9)

The proof of Lemma 2.3 can be carried out by following a similar line as in [10], and hence it is omitted.

3. Main Result

The main objective of this section is to obtain sufficient conditions on the existence, uniqueness, and exponential stability of antiperiodic solution for system (2.2). For presentation convenience, in the following, we denotě𝜏𝜏=𝜏+2𝐿₁=diag𝑙₁̌,𝑙₂̌,…,𝑙_𝑛̌,𝐿₂=diag𝑙₁̂,𝑙₂̂,…,𝑙_𝑛̂,𝐿₃=diag𝑙₁̌𝑙₁̂,𝑙₂̌𝑙₂̂,…,𝑙_𝑛̌𝑙_𝑛̂,𝐿₄𝑙̌=diag¹+𝑙₁̂2,𝑙₂̌+𝑙₂̂2𝑙̌,…,^𝑛+𝑙_𝑛̂2,𝐻₁=diagℎ₁,ℎ₂,…,ℎ_𝑛,𝐻₂=diagmaxℎ̌₁,ℎ₁,maxℎ̌₂,ℎ₂,…,maxℎ̌_𝑛,ℎ_𝑛. (3.1)

Theorem 3.1. Under assumptions (H1), (H2), and (H3), there exit exactly one 𝜔-antiperiodic solution of system (2.2) with (2.3) and all other solutions of system (2.2) with (2.3) converge exponentially to it as 𝑘→+∞, if there exist ten 𝑛×𝑛 symmetric positive define matrices 𝒫, ℛ_𝑖(𝑖=1,2,3), 𝒬_𝑖, (𝑖=1,2,3,4,5,6), and four 𝑛×𝑛 positive diagonal matrix 𝒯_𝑖(𝑖=1,2), 𝒮_𝑖(𝑖=1,2) such that the following LMIs Π+Ω_𝑖<0,Ξ+Ω_𝑖<0,𝑖=1,2 (3.2) hold or Π+Ω_𝑖<0,Ξ+Ω_𝑖<0,𝑖=3,4 (3.3) hold, where ΠΠ=¹¹Π₁₂Π₁₃Π₁₄00ℛ₁0∗Π₂₂Π₂₃Π₂₄0000∗∗Π₃₃Π₃₄Π₃₅000∗∗∗Π₄₄0000∗∗∗∗Π₅₅000∗∗∗∗∗−𝒬₁00∗∗∗∗∗∗−𝒬₂0∗∗∗∗∗∗∗−𝒬₃,ΞΞ=¹¹Ξ₁₂0000ℛ₁0∗Ξ₂₂000000∗∗Ξ₃₃0Ξ₃₅000∗∗∗Ξ₄₄0000∗∗∗∗Ξ₅₅000∗∗∗∗∗−𝒬₁00∗∗∗∗∗∗−𝒬₂0∗∗∗∗∗∗∗−𝒬₃,Ω₁=00000000∗0000000∗∗000000∗∗∗00000∗∗∗∗−3ℛ₂2ℛ₂ℛ₂0∗∗∗∗∗−2ℛ₂−ℛ₃0ℛ₃∗∗∗∗∗∗−ℛ₁−ℛ₂0∗∗∗∗∗∗∗−ℛ₃,Ω₂=00000000∗0000000∗∗000000∗∗∗00000∗∗∗∗−3ℛ₂ℛ₂2ℛ₂0∗∗∗∗∗−ℛ₂−ℛ₃0ℛ₃∗∗∗∗∗∗−ℛ₁−2ℛ₂0∗∗∗∗∗∗∗−ℛ₃,Ω₃=00000000∗0000000∗∗000000∗∗∗00000∗∗∗∗−3ℛ₃ℛ₃02ℛ₃∗∗∗∗∗−ℛ₂−ℛ₃ℛ₂0∗∗∗∗∗∗−ℛ₁−ℛ₂0∗∗∗∗∗∗∗−2ℛ₃,Ω₄=00000000∗0000000∗∗000000∗∗∗00000∗∗∗∗−3ℛ₃2ℛ₃0ℛ₃∗∗∗∗∗−ℛ₂−2ℛ₃ℛ₂0∗∗∗∗∗∗−ℛ₁−ℛ₂0∗∗∗∗∗∗∗−ℛ₃, (3.4) with Π₁₁=𝐴𝒫𝐴−𝒫+𝒬₁+𝒬₂+𝒬₃+(1+𝜏−𝜏̌)𝒬₄−2𝐿₁𝒮₁+2𝐿₂𝒮₂+(𝐴−𝐸)ℛ(𝐴−𝐸)−ℛ₁−𝐿₃𝒯₁,Π₁₂=𝐴𝒫𝐵+(1+𝜏−𝜏̌)𝒮₁−𝒮₂+(𝐴−𝐸)ℛ𝐵+𝐿₄𝒯₁,Π₁₃()Π=𝐴𝒫𝐶+𝐴−𝐸ℛ𝐶,₁₄()Π=𝐴𝒫𝐷+𝐴−𝐸ℛ𝐷,₂₂=𝐵^𝑇𝒫𝐵+𝐵^𝑇ℛ𝐵+(1+𝜏−𝜏̌)𝒬₅+𝜏₂𝒬₆−𝒯₁,Π₂₃=𝐵^𝑇𝒫𝐶+𝐵^𝑇Πℛ𝐶,₂₄=𝐵^𝑇𝒫𝐷+𝐵^𝑇Πℛ𝐷,₃₃=𝐶^𝑇𝒫𝐶+𝐶^𝑇ℛ𝐶−𝒬₅−𝒯₂,Π₃₄=𝐶^𝑇𝒫𝐷+𝐶^𝑇Πℛ𝐷,₃₅=𝒮₂−𝒮₁+𝐿₄𝒯₂,Π₄₄=𝐷^𝑇𝒫𝐷+𝐷^𝑇1ℛ𝐷−𝜏₂𝒬₆,Π₅₅=−𝒬₄+2𝐿₁𝒮₁−2𝐿₂𝒮₂−𝐿₃𝒯₂,Ξ₁₁=𝐻₁𝒫+𝒫^𝑇𝐻₁+𝐻₂𝒫𝐻₂+𝒬₁+𝒬₂+𝒬₃+(1+𝜏−𝜏̌)𝒬₄−2𝐿₁𝒮₁+2𝐿₂𝒮₂+𝐻₂ℛ𝐻₂−ℛ₁−𝐿₃𝒯₁,Ξ₁₂=(1+𝜏−𝜏̌)𝒮₁−𝒮₂+𝐿₄𝒯₁,Ξ₂₂=(1+𝜏−𝜏̌)𝒬₅+𝜏₂𝒬₆−𝒯₁,Ξ₃₃=−𝒬₅−𝒯₂,Ξ₃₅=Π₃₅,Ξ₄₄1=−𝜏₂𝒬₆,Ξ₅₅=Π₅₅,ℛ=𝜏̌²ℛ₁+(𝜏−𝜏̌)²ℛ₂+(𝜏−𝜏)²ℛ₃. (3.5)

Proof. Let 𝒳={𝜙∣𝜙∈𝒞(𝑁[−𝜏,0],ℝ^𝑛)}. For 𝜙∈𝒳, define 𝜙=sup_{[]𝑠∈−𝜏,0}𝜙(𝑠), (3.6) then 𝒳 is a Banach space with the topology of uniform convergence.
For 𝜙,𝜓∈𝒳, let 𝑥(𝑘,𝜙) and 𝑥(𝑘,𝜓) be the solutions of system (2.2) with initial values 𝜙 and 𝜓, respectively. Define 𝑥^(𝑘)𝜙(𝑡)=𝑥𝑘+𝑡,𝜙,∀𝑡∈𝑁[−𝜏,0],𝑘=1,2,…, (3.7) and then 𝑥^(𝑘)(𝜙)∈𝒳 for all 𝑘=1,2,…. It follows from system (2.2) that 𝑥𝑥𝑔𝑥𝑥𝑔𝑥𝑘+1,𝜙−𝑥𝑘+1,𝜓=𝐴𝑘,𝜙−𝑥𝑘,𝜓+𝐵𝑘,𝜙−𝑔𝑘,𝜓+𝐶𝑘−𝜏₁(𝑘),𝜙−𝑔𝑥𝑘−𝜏₁(𝑘),𝜓+𝐷^𝜏2_𝑚=1𝑔𝑥𝑘−𝑚,𝜙−𝑔𝑥𝑘−𝑚,𝜓,𝑘≠𝑘_𝑟𝑥𝑘_𝑟+1,𝜙−𝑥𝑘_𝑟+1,𝜓=𝑥𝑘_𝑟,𝜙−𝑥𝑘_𝑟,𝜓+𝑒_𝑟𝑥𝑘_𝑟,𝜙−𝑒_𝑟𝑥𝑘_𝑟,𝜓,𝑟=1,2,…. (3.8) Letting 𝑦(𝑘),𝑓(𝑘)𝑥𝑥,𝑢=𝑥𝑘,𝜙−𝑥𝑘,𝜓=𝑔𝑘,𝜙−𝑔𝑘,𝜓_𝑟𝑦𝑘_𝑟=𝑒_𝑟𝑥𝑘_𝑟,𝜙−𝑒_𝑟𝑥𝑘_𝑟,𝜓, (3.9) system (3.8) can then be simplified as 𝑦(𝑘+1)=𝐴𝑦(𝑘)+𝐵𝑓(𝑘)+𝐶𝑓𝑘−𝜏₁(𝑘)+𝐷^𝜏2_𝑚=1𝑓(𝑘−𝑚),𝑘≠𝑘_𝑟𝑦𝑘_𝑟+1=𝑦𝑘_𝑟+𝑢_𝑟𝑦𝑘_𝑟,𝑟=1,2,…. (3.10)
Defining 𝜂(𝑘)=𝑦(𝑘+1)−𝑦(𝑘), we consider the following Lyapunov-Krasovskii functional candidate for system (3.10) as 𝑉(𝑘)=⁸_𝑖=1𝑉_𝑖(𝑘), (3.11) where 𝑉₁(𝑘)=𝑦^𝑇(𝑘)(𝑘),𝑉𝒫𝑦₂(𝑘)=^𝑘−1_{𝑗=𝑘−𝜏}𝑦^𝑇𝑗𝒬₁𝑦𝑗+^𝑘−1_{̌𝜏𝑗=𝑘−}𝑦^𝑇𝑗𝒬₂𝑦𝑗+^𝑘−1_𝑗=𝑘−𝜏𝑦^𝑇𝑗𝒬₃𝑦𝑗,𝑉₃(𝑘)=^𝑘−1_{𝑗=𝑘−𝜏1(𝑘)}𝑦^𝑇𝑗𝒬₄𝑦𝑗+^{̌𝜏𝑘−}_𝑚=𝑘−𝜏₊₁^{𝑘−1𝑗=𝑚}𝑦^𝑇𝑗𝒬₄𝑦𝑗,𝑉₄(𝑘)=^𝑘−1_{𝑗=𝑘−𝜏1(𝑘)}𝑓^𝑇𝑗𝒬₅𝑓𝑗+^{̌𝜏𝑘−}_𝑚=𝑘−𝜏₊₁^{𝑘−1𝑗=𝑚}𝑓^𝑇𝑗𝒬₅𝑓𝑗,𝑉₅(𝑘)=^𝜏2_𝑚=1^𝑘−1_{𝑗=𝑘−𝑚}𝑓^𝑇𝑗𝒬₆𝑓𝑗,𝑉₆(𝑘)=2^𝑘−1_{𝑗=𝑘−𝜏1(𝑘)}𝑓(𝑗)−𝐿₁𝑦(𝑗)^𝑇𝒮₁𝑦𝑗+2^{̌𝜏𝑘−}_𝑚=𝑘−𝜏₊₁^{𝑘−1𝑗=𝑚}𝑓(𝑗)−𝐿₁𝑦(𝑗)^𝑇𝒮₁𝑦𝑗,𝑉₇(𝑘)=2^𝑘−1_{𝑗=𝑘−𝜏1(𝑘)}𝐿₂𝑦𝑗−𝑓𝑗^𝑇𝒮₂𝑦𝑗+2^{̌𝜏𝑘−}_𝑚=𝑘−𝜏₊₁^{𝑘−1𝑗=𝑚}𝐿₂𝑦𝑗−𝑓𝑗^𝑇𝒮₂𝑦𝑗,𝑉₈(𝑘)=𝜏̌^𝑘−1_{̌𝜏𝑚=𝑘−}^𝑘−1_𝑗=𝑚𝜂^𝑇𝑗ℛ₁𝜂𝑗+(𝜏−𝜏̌)^{̌𝑘−𝜏−1}_{𝑚=𝑘−𝜏}^𝑘−1_𝑗=𝑚𝜂^𝑇𝑗ℛ₂𝜂𝑗+(𝜏−𝜏)^{𝑘−𝜏−1}_𝑚=𝑘−𝜏^{𝑘−1𝑗=𝑚}𝜂^𝑇𝑗ℛ₃𝜂𝑗. (3.12)
We proceed by considering two possible cases of 𝑘≠𝑘_𝑟 and 𝑘=𝑘_𝑟.
Case 1 (𝑘≠𝑘_𝑟 (𝑟=1,2,…)). Calculating the difference of 𝑉_𝑖(𝑘) (𝑖=1,2,…,8) along the system (3.10), by Lemma 2.3 we have Δ𝑉₁(𝑘)=𝑦^𝑇(𝑘+1)𝒫𝑦(𝑘+1)−𝑦^𝑇(𝑘)𝒫𝑦(𝑘)=𝑦^𝑇(𝑘)𝐴^𝑇𝒫𝐴−𝒫𝑦(𝑘)+2𝑦^𝑇(𝑘)𝐴^𝑇𝒫𝐵𝑓(𝑘)+2𝑦^𝑇(𝑘)𝐴^𝑇𝒫𝐶𝑓𝑘−𝜏₁(𝑘)+2𝑦^𝑇(𝑘)𝐴^𝑇𝒫𝐷^𝜏2_𝑚=1𝑓(𝑘−𝑚)+𝑓^𝑇(𝑘)𝐵^𝑇𝒫𝐵𝑓(𝑘)+2𝑓^𝑇(𝑘)𝐵^𝑇𝒫𝐶𝑓𝑘−𝜏₁(𝑘)+2𝑓^𝑇(𝑘)𝐵^𝑇𝒫𝐷^𝜏2_𝑚=1𝑓(𝑘−𝑚)+𝑓^𝑇𝑘−𝜏₁(𝑘)𝐶^𝑇𝒫𝐶𝑓𝑘−𝜏₁(𝑘)+2𝑓^𝑇𝑘−𝜏₁(𝑘)𝐶^𝑇𝒫𝐷^𝜏2_𝑚=1𝑓()+𝑘−𝑚_𝜏2𝑚=1𝑓(𝑘−𝑚)^𝑇𝐷^𝑇𝒫𝐷^𝜏2_𝑚=1𝑓(𝑘−𝑚),Δ𝑉₂(𝑘)=𝑦^𝑇(𝑘)𝒬₁+𝒬₂+𝒬₃𝑦(𝑘)−𝑦^𝑇(𝑘−𝜏)𝒬₁𝑦(𝑘−𝜏)−𝑦^𝑇(𝑘−𝜏̌)𝒬₂𝑦(𝑘−𝜏̌)−𝑦^𝑇(𝑘−𝜏)𝒬₃𝑦(𝑘−𝜏),Δ𝑉₃(𝑘)=^{̌𝜏𝑘−}_{𝑗=𝑘+1−𝜏1()𝑘+1}𝑦^𝑇𝑗𝒬₄𝑦𝑗+^𝑘−1_{̌𝑗=𝑘−𝜏+1}𝑦^𝑇𝑗𝒬₄𝑦𝑗+𝑦^𝑇(𝑘)𝒬₄𝑦(𝑘)−^𝑘−1_{𝑗=𝑘−𝜏1(𝑘)+1}𝑦^𝑇𝑗𝒬₄𝑦𝑗−𝑦^𝑇𝑘−𝜏₁(𝑘)𝒬₄𝑦𝑘−𝜏₁(𝑘)+(𝜏−𝜏̌)𝑦^𝑇(𝑘)𝒬₄𝑦(𝑘)−^{̌𝜏𝑘−}_𝑗=𝑘−𝜏₊₁𝑦^𝑇𝑗𝒬₄𝑦𝑗≤(1+𝜏−𝜏̌)𝑦^𝑇(𝑘)𝒬₄𝑦(𝑘)−𝑦^𝑇𝑘−𝜏₁(𝑘)𝒬₄𝑦𝑘−𝜏₁(𝑘),Δ𝑉₄(𝑘)≤(1+𝜏−𝜏̌)𝑓^𝑇(𝑘)𝒬₅𝑓(𝑘)−𝑓^𝑇𝑘−𝜏₁(𝑘)𝒬₅𝑓𝑘−𝜏₁(𝑘),Δ𝑉₅(𝑘)=^𝜏2_𝑚=1𝑓^𝑇(𝑘)𝒬₆𝑓(𝑘)−𝑓^𝑇(𝑘−𝑚)𝒬₆𝑓(𝑘−𝑚),≤𝜏₂𝑓^𝑇(𝑘)𝒬₆𝑓(𝑘)−1𝜏₂^𝜏2_𝑚=1𝑓(𝑘−𝑚)^𝑇𝒬₆^𝜏2_𝑚=1𝑓(𝑘−𝑚),Δ𝑉₆(𝑘)≤2(1+𝜏−𝜏̌)𝑓(𝑘)−𝐿₁𝑦(𝑘)^𝑇𝒮₁𝑦(𝑘)−2𝑓𝑘−𝜏₁(𝑘)−𝐿₁𝑦𝑘−𝜏₁(𝑘)^𝑇𝒮₁𝑦𝑘−𝜏₁(𝑘),Δ𝑉₇(𝑘)≤2(1+𝜏−𝜏̌)𝐿₂𝑦(𝑘)−𝑓(𝑘)^𝑇𝒮₂𝑦(𝑘)−2𝐿₂𝑦𝑘−𝜏₁(𝑘)−𝑓𝑘−𝜏₁(𝑘)^𝑇𝒮₂𝑦𝑘−𝜏₁(𝑘),Δ𝑉₈(𝑘)=𝜂^𝑇(𝑘)ℛ𝜂(𝑘)−𝜏̌^𝑘−1_{̌𝜏𝑗=𝑘−}𝜂^𝑇𝑗ℛ₁𝜂𝑗−(𝜏−𝜏̌)^{̌𝑘−𝜏−1}_{𝑗=𝑘−𝜏}𝜂^𝑇𝑗ℛ₂𝜂𝑗−(𝜏−𝜏)^{𝑘−𝜏−1}_𝑗=𝑘−𝜏𝜂^𝑇𝑗ℛ₃𝜂𝑗. (3.13)(3.14)(3.15)(3.16)(3.17)(3.18)(3.19)(3.20)
From the definition of 𝜂(𝑘) and (3.10), we have 𝜂^𝑇(𝑘)ℛ𝜂(𝑘)=𝑦^𝑇(𝑘)(𝐴−𝐸)^𝑇ℛ(𝐴−𝐸)𝑦(𝑘)+2𝑦^𝑇(𝑘)(𝐴−𝐸)^𝑇ℛ𝐵𝑓(𝑘)+2𝑦^𝑇(𝑘)(𝐴−𝐸)^𝑇ℛ𝐶𝑓𝑘−𝜏₁(𝑘)+2𝑦^𝑇(𝑘)(𝐴−𝐸)^𝑇ℛ𝐷^𝜏2_𝑚=1𝑓(𝑘−𝑚)+𝑓^𝑇(𝑘)𝐵^𝑇ℛ𝐵𝑓(𝑘)+2𝑓^𝑇(𝑘)𝐵^𝑇ℛ𝐶𝑓𝑘−𝜏₁(𝑘)+2𝑓^𝑇(𝑘)𝐵^𝑇ℛ𝐷^𝜏2_𝑚=1𝑓(𝑘−𝑚)+𝑓^𝑇𝑘−𝜏₁(𝑘)𝐶^𝑇ℛ𝐶𝑓𝑘−𝜏₁(𝑘)+2𝑓^𝑇𝑘−𝜏₁(𝑘)𝐶^𝑇ℛ𝐷^𝜏2_𝑚=1𝑓()+𝑘−𝑚_𝜏2𝑚=1𝑓(𝑘−𝑚)^𝑇𝐷^𝑇ℛ𝐷^𝜏2_𝑚=1𝑓(𝑘−𝑚). (3.21) From Lemma 2.3, it can be shown that the following inequality holds: −𝜏̌^𝑘−1_{̌𝜏𝑗=𝑘−}𝜂^𝑇𝑗ℛ₁𝜂𝑗≤−^𝑘−1_{̌𝜏𝑗=𝑘−}𝜂𝑗^𝑇ℛ₁^𝑘−1_{̌𝜏𝑗=𝑘−}𝜂𝑗=̌𝑦(𝑘)𝑦(𝑘−𝜏)^𝑇−ℛ₁ℛ₁∗−ℛ₁𝑦(𝑘)𝑦(𝜏̌).𝑘− (3.22)
When 𝜏̌≤𝜏₁(𝑘)≤𝜏, let 𝑎(𝑘)=(𝜏₁(𝑘)−𝜏̌)/(𝜏−𝜏̌). Then 0≤𝑎(𝑘)≤1. It is easy to get that −(𝜏−𝜏̌)^{̌𝑘−𝜏−1}_{𝑗=𝑘−𝜏}𝜂^𝑇𝑗ℛ₂𝜂𝑗=−(𝜏−𝜏̌)^{𝑘−𝜏1(𝑘)−1}_{𝑗=𝑘−𝜏}𝜂^𝑇𝑗ℛ₂𝜂𝑗−(𝜏−𝜏̌)^{̌𝑘−𝜏−1}_{𝑗=𝑘−𝜏1(𝑘)}𝜂^𝑇𝑗ℛ₂𝜂𝑗≤−𝜏−𝜏₁(𝑘)^{𝑘−𝜏1(𝑘)−1}_{𝑗=𝑘−𝜏}𝜂^𝑇𝑗ℛ₂𝜂𝑗−𝑎(𝑘)𝜏−𝜏₁(𝑘)^{𝑘−𝜏1(𝑘)−1}_{𝑗=𝑘−𝜏}𝜂^𝑇𝑗ℛ₂𝜂𝑗−(1−𝑎(𝑘))𝜏₁(𝑘)−𝜏̌^{̌𝑘−𝜏−1}_{𝑗=𝑘−𝜏1(𝑘)}𝜂^𝑇𝑗ℛ₂𝜂𝑗−𝜏₁(𝑘)−𝜏̌^{𝑘−𝜏1(𝑘)−1}_{𝑗=𝑘−𝜏}𝜂^𝑇𝑗ℛ₂𝜂𝑗≤−^{𝑘−𝜏1(𝑘)−1}_{𝑗=𝑘−𝜏}𝜂^𝑇𝑗ℛ₂^{𝑘−𝜏1(𝑘)−1}_{𝑗=𝑘−𝜏}𝜂𝑗−𝑎(𝑘)^{𝑘−𝜏1(𝑘)−1}_{𝑗=𝑘−𝜏}𝜂^𝑇𝑗ℛ₂^{𝑘−𝜏1(𝑘)−1}_{𝑗=𝑘−𝜏}𝜂𝑗−(1−𝑎(𝑘))^{̌𝑘−𝜏−1}_{𝑗=𝑘−𝜏1(𝑘)}𝜂^𝑇𝑗ℛ₂^{̌𝑘−𝜏−1}_{𝑗=𝑘−𝜏1(𝑘)}𝜂𝑗−^{̌𝑘−𝜏−1}_{𝑗=𝑘−𝜏1(𝑘)}𝜂^𝑇𝑗ℛ₂^{̌𝑘−𝜏−1}_{𝑗=𝑘−𝜏1(𝑘)}𝜂𝑗=𝑦(𝑘−𝜏₁(𝑘))𝑦(𝑘−̌𝜏)𝑦(𝑘−𝜏)^𝑇−2ℛ₂ℛ₂ℛ₂∗−ℛ₂0∗∗−ℛ₂𝑦𝑘−𝜏₁(𝑘)𝑦(𝑘−𝜏)𝑦(𝜏̌)(𝑘)𝑘−+𝑎𝑦(𝑘−𝜏₁(𝑘))𝑦(𝑘−𝜏)^𝑇−ℛ₂ℛ₂∗−ℛ₂𝑦𝑘−𝜏₁(𝑘)𝑦(𝑘−𝜏)+(1−𝑎(𝑘))𝑦(𝑘−𝜏₁̌(𝑘))𝑦(𝑘−𝜏)^𝑇−ℛ₂ℛ₂∗−ℛ₂𝑦𝑘−𝜏₁(𝑘)𝑦(𝜏̌),−(𝑘−𝜏−𝜏)^{𝑘−𝜏−1}_𝑗=𝑘−𝜏𝜂^𝑇𝑗ℛ₃𝜂𝑗≤𝑦(𝑘−𝜏)𝑦(𝑘−𝜏)^𝑇−ℛ₃ℛ₃∗−ℛ₃𝑦(𝑘−𝜏)𝑦(𝜏).𝑘− (3.23)(3.24)
For positive diagonal matrices 𝒯₁>0 and 𝒯₂>0, we can get from assumption (H2) that 0≤𝑦(𝑘)𝑓(𝑘)^𝑇−𝐿₃𝒯₁𝐿₄𝒯₁∗−𝒯₁𝑦(𝑘)𝑓(𝑘)𝑦,0≤𝑘−𝜏₁(𝑘)𝑓𝑘−𝜏₁(𝑘)^𝑇−𝐿₃𝒯₂𝐿₄𝒯₂∗−𝒯₂𝑦𝑘−𝜏₁(𝑘)𝑓𝑘−𝜏₁(𝑘). (3.25)(3.26) Denoting 𝛼(𝑘)=𝑦^𝑇(𝑘),𝑓^𝑇(𝑘),𝑓^𝑇𝑘−𝜏₁(𝑘),^𝜏2_𝑚=1𝑓^𝑇(),𝑦𝑘−𝑚_𝑇𝑘−𝜏₁(𝑘),𝑦^𝑇(𝑘−𝜏),𝑦^𝑇(𝑘−𝜏̌),𝑦^𝑇(𝑘−𝜏)^𝑇, (3.27) it follows from (3.13)–(3.26) that Δ𝑉(𝑘)=⁸_𝑖=1Δ𝑉_𝑖(𝑘)≤𝑦^𝑇(𝑘)𝐴𝒫𝐴−𝒫+𝒬₁+𝒬₂+𝒬₃+(1+𝜏−𝜏̌)𝒬₄−2𝐿₁𝒮₁+2𝐿₂𝒮₂+(𝐴−𝐸)ℛ(𝐴−𝐸)−ℛ₁−𝐿₃𝒯₁𝑦(𝑘)+2𝑦^𝑇(𝑘)𝐴𝒫𝐵+(1+𝜏−𝜏̌)𝒮₁−𝒮₂+(𝐴−𝐸)ℛ𝐵+𝐿₄𝒯₁𝑓(𝑘)+2𝑦^𝑇(𝑘)[𝐴𝒫𝐶+(𝐴−𝐸)ℛ𝐶]𝑓𝑘−𝜏₁(𝑘)+2𝑦^𝑇(𝑘)[𝐴𝒫𝐷+(𝐴−𝐸)ℛ𝐷]^𝜏2_𝑚=1𝑓(𝑘−𝑚)+𝑓^𝑇(𝑘)𝐵^𝑇𝒫𝐵+𝐵^𝑇ℛ𝐵+(1+𝜏−𝜏̌)𝒬₅+𝜏₂𝒬₆−𝒯₁𝑓(𝑘)+2𝑓^𝑇(𝑘)𝐵^𝑇𝒫𝐶+𝐵^𝑇ℛ𝐶𝑓𝑘−𝜏₁(𝑘)+2𝑓^𝑇(𝑘)𝐵^𝑇𝒫𝐷+𝐵^𝑇ℛ𝐷^𝜏2_𝑚=1𝑓(𝑘−𝑚)+𝑓^𝑇𝑘−𝜏₁(𝑘)𝐶^𝑇𝒫𝐶+𝐶^𝑇ℛ𝐶−𝒬₅−𝒯₂𝑓𝑘−𝜏₁(𝑘)+2𝑓^𝑇𝑘−𝜏₁(𝑘)𝐶^𝑇𝒫𝐷+𝐶^𝑇ℛ𝐷^𝜏2_𝑚=1𝑓()+𝑘−𝑚_𝜏2𝑚=1𝑓(𝑘−𝑚)^𝑇𝐷^𝑇𝒫𝐷+𝐷^𝑇1ℛ𝐷−𝜏₂𝒬₆^𝜏2_𝑚=1𝑓(𝑘−𝑚)+2𝑦^𝑇(𝑘)ℛ₁𝑦^𝑇(𝑘−𝜏̌)+2𝑓^𝑇𝑘−𝜏₁(𝑘)𝒮₂−𝒮₁+𝐿₄𝒯₂𝑦𝑘−𝜏₁(𝑘)+𝑦^𝑇𝑘−𝜏₁(𝑘)−𝒬₄+2𝐿₁𝒮₁−2𝐿₂𝒮₂−𝐿₃𝒯₂−3ℛ₂𝑦𝑘−𝜏₁(𝑘)+2𝑦^𝑇𝑘−𝜏₁(𝑘)ℛ₂+𝑎(𝑘)ℛ₂𝑦(𝑘−𝜏)+2𝑦^𝑇𝑘−𝜏₁(𝑘)ℛ₂+(1−𝑎(𝑘))ℛ₂𝑦(𝑘−𝜏̌)−𝑦^𝑇(𝑘−𝜏)𝒬₁+(1+𝑎(𝑘))ℛ₂+ℛ₃𝑦(𝑘−𝜏)+2𝑦^𝑇(𝑘−𝜏)ℛ₃𝑦(𝑘−𝜏)−𝑦^𝑇(𝑘−𝜏̌)𝒬₂+ℛ₁+ℛ₂+(1−𝑎(𝑘))ℛ₂𝑦(𝑘−𝜏̌)−𝑦^𝑇(𝑘−𝜏)𝒬₃+ℛ₃𝑦(𝑘−𝜏)=𝛼^𝑇(𝑘)𝑎(𝑘)Π+Ω₁+(1−𝑎(𝑘))Π+Ω₂𝛼(𝑘). (3.28)
When 𝜏≤𝜏₁(𝑘)≤𝜏, let 𝑏(𝑘)=(𝜏−𝜏₁(𝑘))/(𝜏−𝜏). Then 0≤𝑏(𝑘)≤1. In the similitude of the proof of inequality (3.23), we have −(𝜏−𝜏)^{𝑘−𝜏−1}_𝑗=𝑘−𝜏𝜂^𝑇𝑗ℛ₃𝜂𝑗≤𝑦𝑘−𝜏₁(𝑘)𝑦(𝜏)𝑦(𝑘−𝑘−𝜏)^𝑇−2ℛ₃ℛ₃ℛ₃∗−ℛ₃0∗∗−ℛ₃𝑦𝑘−𝜏₁(𝑘)𝑦(𝜏)𝑦(𝑘−𝑘−𝜏)(𝑘)𝑦+𝑏𝑘−𝜏₁(𝑘)𝑦(𝑘−𝜏)^𝑇−ℛ₃ℛ₃∗−ℛ₃𝑦𝑘−𝜏₁(𝑘)𝑦(𝜏)+((𝑘𝑦𝑘−1−𝑏))𝑘−𝜏₁(𝑘)𝑦(𝑘−𝜏)^𝑇−ℛ₃ℛ₃∗−ℛ₃𝑦𝑘−𝜏₁(𝑘)𝑦(𝑘−𝜏),−(𝜏−𝜏̌)^{̌𝑘−𝜏−1}_{𝑗=𝑘−𝜏}𝜂^𝑇𝑗ℛ₂𝜂𝑗≤𝑦(𝑘−𝜏̌)𝑦(𝑘−𝜏)^𝑇−ℛ₂ℛ₂∗−ℛ₂𝑦(𝜏̌)𝑦(𝑘−𝑘−𝜏). (3.29) By using similar method in (3.28), it follows from (3.13) to (3.22), and (3.25) to (3.26), and (3.29) that Δ𝑉(𝑘)≤𝛼^𝑇(𝑘)𝑏(𝑘)Π+Ω₃+(1−𝑏(𝑘))Π+Ω₄𝛼(𝑘). (3.30)Case 2 (𝑘=𝑘_𝑟 (𝑟=1,2,…)). Note that the inequalities from (3.13) to (3.26) except (3.13) and (3.21) hold for 𝑘=𝑘_𝑟. Calculating Δ𝑉₁(𝑘_𝑟) and 𝜂^𝑇(𝑘_𝑟)ℛ𝜂(𝑘_𝑟) along the system (3.10), we have Δ𝑉₁𝑘_𝑟=𝑦^𝑇𝑘_𝑟+1𝒫𝑦𝑘_𝑟+1−𝑦^𝑇𝑘_𝑟𝒫𝑦𝑘_𝑟=𝑦𝑘_𝑟+𝑢_𝑟𝑦𝑘_𝑟^𝑇𝒫𝑦𝑘_𝑟+𝑢_𝑟𝑦𝑘_𝑟−𝑦^𝑇𝑘_𝑟𝒫𝑦𝑘_𝑟≤𝑦^𝑇𝑘_𝑟2𝐻₁𝒫+𝐻₂𝒫𝐻₂𝑦𝑘_𝑟𝜂^𝑇𝑘_𝑟ℛ𝜂𝑘_𝑟=𝑢^𝑇_𝑟𝑦𝑘_𝑟ℛ𝑢_𝑟𝑦𝑘_𝑟≤𝑦^𝑇𝑘_𝑟𝐻₂ℛ𝐻₂𝑦𝑘_𝑟. (3.31)
By using similar method in Case 1, we can obtain that Δ𝑉(𝑘)=𝛼^𝑇(𝑘)𝑎(𝑘)Ξ+Ω₁+(1−𝑎(𝑘))Ξ+Ω₂𝛼(𝑘) (3.32) or Δ𝑉(𝑘)=𝛼^𝑇(𝑘)𝑏(𝑘)Ξ+Ω₃+(1−𝑏(𝑘))Ξ+Ω₄𝛼(𝑘). (3.33)
Combining the above discussions in Case 1 and 2, we obtain from (3.2), (3.3), (3.28), (3.30), (3.32), and (3.33) that Δ𝑉(𝑘)≤−𝛾₁‖𝛼(𝑘)‖²≤−𝛾₁‖𝑦(𝑘)‖², (3.34) where 𝛾₁=min_𝑖=1,2,3,4{𝜆_min(−Π−Ω_𝑖),𝜆_min(−Ξ−Ω_𝑖)}>0.
From the definition of 𝑉(𝑘), it is easy to verify that 𝑉(𝑘)≤𝜆_max(𝒫)𝑦(𝑘)²+𝛾₂^𝑘−1_{𝑗=𝑘−𝜏}𝑦𝑗², (3.35) where 𝛾₂=𝜆_max𝒬₁+𝜆_max𝒬₂+𝜆_max𝒬₃+(1+𝜏−𝜏̌)𝜆_max𝒬₄+𝜆_max𝒬₅𝜆_max𝐿^𝑇𝐿+4𝜆_max𝐿𝒮₁+4𝜆_max𝐿𝒮₂+𝜏₂𝜆_max𝒬₆+4𝜏̌²𝜆_maxℛ₁+4(𝜏−𝜏̌)²𝜆_maxℛ₂+4(𝜏−𝜏)²𝜆_maxℛ₃, (3.36) where 𝐿=diag(max{|𝑙₁̌|,|𝑙₁̂|},max{|𝑙₂̌|,|𝑙₂̂|},…,max{|𝑙_𝑛̌|,|𝑙_𝑛̂|}).
For any scalar 𝜌≥1, it follows from (3.34) and (3.35) that 𝜌^𝑖+1𝑉(𝑖+1)−𝜌^𝑖𝑉(𝑖)=𝜌^𝑖+1Δ𝑉(𝑖)+𝜌^𝑖𝑉(𝑖)≤𝜌𝜌−1_𝑖𝜌−1𝜆_max(𝒫)−𝛾₁𝜌^𝑖+1𝑦(𝑖)²+𝛾₂𝜌^𝑖𝜌−1^𝑖−1_{𝑗=𝑖−𝜏}𝑦𝑗². (3.37) Summing up both sides of (3.37) from 0 to 𝑘−1 with respect to 𝑖, we have 𝜌^𝑘𝑉(𝑘)−𝑉(0)≤𝜌−1𝜆_max(𝒫)−𝛾₁𝜌^𝑘−1_𝑖=0𝜌^𝑖𝑦(𝑖)²+𝛾₂𝜌−1^𝑘−1_𝑖=0^𝑖−1_{𝑗=𝑖−𝜏}𝜌^𝑖𝑦𝑗². (3.38) Note that ^𝑘−1_𝑖=0^𝑖−1_{𝑗=𝑖−𝜏}𝜌^𝑖𝑦𝑗²=⁻¹_𝑗=−𝜏^𝑗+𝜏_𝑖=0+^{𝑘−1−𝜏}_𝑗=0^𝑗+𝜏_𝑖=𝑗+1+^𝑘−2_{𝑗=𝑘−𝜏}^𝑘−1_𝑖=𝑗+1𝜌^𝑖𝑦𝑗²≤𝜏𝜌^𝜏−1_𝑗=−𝜏𝑦𝑗²+𝜏𝜌^{𝜏𝑘−1−𝜏}_𝑗=0𝜌^𝑗𝑦𝑗²+𝜏𝜌^𝜏𝑘−2_{𝑗=𝑘−𝜏}𝜌^𝑗𝑦𝑗²≤𝜏²𝜌^𝜏sup_{𝑠∈𝑁[−𝜏,0]}𝑦(𝑠)²+𝜏𝜌^𝜏𝑘−2_𝑗=0𝜌^𝑗𝑦𝑗². (3.39) From (3.35), we have 𝑉(0)≤𝜆_max(𝒫)+𝛾₂𝜏sup_{[]𝑠∈𝑁−𝜏,0}𝑦(𝑠)². (3.40) It follows from (3.38)–(3.40) that 𝜌^𝑘𝑉(𝑘)≤𝐾₁𝜌sup_{𝑠∈𝑁[−𝜏,0]}𝑦(𝑠)²+𝐾₂𝜌^𝑘−1_𝑗=0𝜌^𝑗𝑦𝑗², (3.41) where 𝐾₁𝜌=𝜆_max(𝒫)+𝛾₂𝜏+𝛾₂𝜌−1𝜏²𝜌^𝜏,𝐾₂𝜌=𝜌−1𝜆_max(𝒫)−𝛾₁𝜌+𝛾₂𝜌−1𝜏𝜌^𝜏. (3.42) Since 𝐾₂(1)<0, by the continuity of function 𝐾₂(𝜌), we can choose a scalar 𝜉>1 such that 𝐾₂(𝜉)<0. Obviously, 𝐾₁(𝜉)>0. From (3.41), we obtain 𝜉^𝑘𝑉(𝑘)≤𝐾₁(𝜉)sup_{𝑠∈𝑁[−𝜏,0]}𝑦(𝑠)². (3.43)
From the definition of 𝑉(𝑘), we have 𝑉(𝑘)≥𝜆_min(𝒫)𝑦(𝑘)². (3.44) Let 𝑀=𝐾₁(𝜉)/𝜆_min(𝒫), 𝜀=1/𝜉, then 𝑀>0, 0<𝜀<1. It follows form (3.43) and (3.44) that 𝑦(𝑘)≤𝑀𝜀^𝑘sup_{𝑠∈𝑁[−𝜏,0]}𝑦(𝑠). (3.45) That is 𝑥𝑘,𝜙−𝑥𝑘,𝜓≤𝑀𝜖^𝑘𝜙−𝜓. (3.46) We can choose a positive integer 𝑁 such that 𝑀𝜀^𝑁𝜔≤14. (3.47) Define a Poincaré mapping Γ∶𝒳→𝒳 by Γ𝜙=−𝑥^(𝜔)𝜙. (3.48) Then, we can derive from (3.46) and (3.47) that Γ^(𝑁)𝜙−Γ^(𝑁)𝜓≤14𝜙−𝜓, (3.49) which shows that Γ^(𝑁) is a contraction mapping and therefore there exits a unique fixed point 𝜙^∗∈𝒳 of Γ^(𝑁), which is also the unique fixed point of Γ such that Γ𝜙^∗=𝜙^∗. (3.50) Therefore, −𝑥𝑤+𝑘,𝜙^∗=𝜙^∗(𝑘),∀𝑘∈𝑁[−𝜏,0]. (3.51)
Let 𝑥(𝑘,𝜙^∗) be the solution of system (2.2) through (0,𝜙^∗). From assumption (H1) we know that −𝑥(𝑘+𝜔,𝜙^∗) is also a solution of system (2.2). It follows from (3.51) that −𝑥(𝑘+𝜔,𝜙^∗) is also through (0,𝜙^∗). By the uniqueness of solution we can know 𝑥𝑘,𝜙^∗=−𝑥𝑘+𝜔,𝜙^∗, (3.52) for 𝑘=1,2,…, which indicates that 𝑥(𝑘,𝜙^∗) is exactly one 𝜔-antiperiodic solution of system (2.2). To this end, it is easy to see that all other solutions converge exponentially to it as 𝑘→+∞. The proof is completed.

Remark 3.2. The conditions are dependent on both the lower bound and upper bound of delays. It has been shown that the delay-dependent stability conditions are generally less conservative than the delay-independent ones, especially when the size of the delay is small.

Remark 3.3. In this paper, the model includes both discrete and distributed delays simultaneously, and can be used to describe some well-known neural networks owing to its generality. In [38], only one kind of delay has been considered, which is a special case of neural networks with mixed delays. Furthermore, in [38], the activations were assumed to be bounded functions, while the boundedness condition is removed in this paper.

4. Examples

In this section, some examples and numerical simulations are provided to illustrate our results.

Example 4.1. Consider a discrete-time neural networks (2.2) with two neurons, where ,𝑔𝐴=0.2000.3,𝐵=0.060.01−0.060.1,𝐶=0.1−0.05−0.05−0.02,𝐷=−0.020.02−0.050.07₁(𝑢)=arctan(2𝑢),𝑔₂(𝑢)=−arctan(4𝑢),𝐼₁(𝑘)=−3sin𝑘𝜋8,𝐼₂(𝑘)=4cos𝑘𝜋8,𝜏₁(𝑘)=3+sin𝑘𝜋4,𝜏₂𝑘=2,_𝑟=8𝑟−3,𝑒_1𝑟(𝑢)=𝑒_2𝑟(𝑢)=0.08sin(𝑢)−0.68𝑢,𝑟=1,2,…. (4.1) It can be verified that assumptions (H1), (H2), and (H3) are satisfied with 𝑙₁̌=0, 𝑙₁̂=2, 𝑙₂̌=−4, 𝑙₂̂=0, 𝜏̌=2, 𝜏=4, 𝜔=8, ℎ̌₁=ℎ̌₂=−0.76, ℎ₁=ℎ₂=−0.6. Thus, 𝐿₁=diag(0,−4), 𝐿₂=diag(2,0), 𝐿₃=diag(0,0), 𝐿₄=diag(1,−2), 𝐻₁=diag(−0.6,−0.6), 𝐻₂=diag(0.76,76). By the MATLAB Control Toolbox, we find a solution to the LMI in (3.2) as follows: 𝒫=3.5412−0.8274−0.82748.5025,ℛ₁=0.0254−0.0149−0.01490.0106,ℛ₂=,ℛ0.1931−0.0932−0.09320.1155₃=0.1941−0.0937−0.09370.1158,𝒬₁=0.1396−0.0553−0.05530.0807,𝒬₂=,𝒬0.1762−0.0693−0.06930.0917₃=0.1877−0.0771−0.07710.0986,𝒬₄=0.0230−0.0077−0.00770.0222,𝒬₅=,𝒬0.11970.05540.05540.0274₆=0.1370−0.0709−0.07090.1201,𝒯₁=2.5687000.8572,𝒯₂=,𝒮1.2461000.0252₁=0.0056000.0102,𝒮₂=.0.0756000.0034 (4.2)Therefore, by Theorem 3.1, we know that system (2.2) with above given parameters has exactly one 8-antiperiodic solution and all other solutions of the system converge exponentially to it as 𝑘→+∞, which is further verified by the simulation given in Figure 1.

Figure 1 

State responses of the discrete-time neural networks with initial conditions (𝑥1(𝑠),𝑥2(𝑠))𝑇=(−1,0)𝑇, 𝑠∈𝑁[−4,0].

Example 4.2. Consider a discrete-time neural networks (2.2) with three neurons, where ,,𝑔𝐴=0.150000.20000.3,𝐵=0.050.04−0.040.080.04−0.04−0.04−0.080.04;𝐶=−0.04−0.020.060.09−0.020.020.060.010.01,𝐷=−0.020.050.02−0.050.02−0.010.02−0.05−0.02₁(𝑢)=𝑔₂(𝑢)=𝑔₃(𝑢)=sin(2𝑢)−𝑢,𝐼₁(𝑘)=−2sin𝑘𝜋,𝐼12₂(𝑘)=𝐼₃(𝑘)=cos𝑘𝜋12,𝜏₁(𝑘)=2+cos𝑘𝜋6,𝜏₂𝑘=2,_𝑟=12𝑟−12+3(−1)^𝑟+7,𝑒_1𝑟(𝑢)=𝑒_2𝑟(𝑢)=𝑒_3𝑟(𝑢)=−𝑢,𝑟=1,2,…. (4.3) It can be verified that assumptions (H1), (H2), and (H3) are satisfied with 𝑙₁̌=𝑙₂̌=𝑙₃̌=−3, 𝑙₁̂=𝑙₂̂=𝑙₃̂=1, 𝜏̌=1, 𝜏=3, 𝜔=12, ℎ̌₁=ℎ̌₂=ℎ̌₃=−1, ℎ₁=ℎ₂=ℎ₃=−1. Thus, 𝐿₁=diag(−3,−3,−3), 𝐿₂=diag(1,1,1), 𝐿₃=diag(−3,−3,−3), 𝐿₄=diag(−1,−1,−1), 𝐻₁=diag(−1,−1,−1), 𝐻₂=diag(1,1,1). By the MATLAB Control Toolbox, we find a solution to the LMI in (3.3) as follows: 𝒫=41.8020−4.34734.5690−4.347330.07570.26824.56900.268225.8887,ℛ₁=,ℛ0.4005−0.29580.1172−0.29580.81180.29420.11720.29420.6825₂=0.5178−0.2326−0.1006−0.23260.68430.2146−0.10060.21460.6101,ℛ₃=,𝒬0.4881−0.2381−0.1038−0.23810.70320.2240−0.10380.22400.6280₁=1.0470−0.15150.4772−0.15150.6361−0.16550.4772−0.16551.0423,𝒬₂=,𝒬1.0644−0.16260.4402−0.16260.6310−0.15970.4402−0.15970.9820₃=1.2250−0.14860.6129−0.14860.8285−0.18640.6129−0.18641.2586,𝒬₄=,𝒬1.7504−0.31240.7392−0.31240.3251−0.45400.7392−0.45400.9975₅=0.26370.0305−0.03000.03050.0846−0.1291−0.0300−0.12910.2397,𝒬₆=,𝒯0.5803−0.19630.1481−0.19630.82320.32960.14810.32960.3495₁=3.49230003.46320002.3878,𝒯₂=,𝒮0.95950000.45670000.3671₁=0.65750000.48530000.2337,𝒮₂=.0.37090000.01630000.0507 (4.4)
Therefore, by Theorem 3.1, we know that system (2.2) with the above-given parameters has exactly one 12-antiperiodic solution and all other solutions of the system converge exponentially to it as 𝑘→+∞, which is further verified by the simulation given in Figure 2.

Figure 2 

State responses of the discrete-time neural networks with initial conditions (𝑥1(𝑠),𝑥2(𝑠),𝑥3(𝑠))𝑇=(−1,0,1)𝑇, 𝑠∈𝑁[−3,0].

5. Conclusions

In this paper, the discrete-time neural networks with mixed delays and impulses have been studied. A delay-dependent LMI criterion for the existence and global exponential stability of antiperiodic solutions has been established by constructing an appropriate Lyapunov-Krasovskii functional, and using the contraction mapping principle and the matrix inequality techniques. Moreover, two examples are given to illustrate the effectiveness of the results.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 60974132 and the Natural Science Foundation Project of CQ CSTC2011jjA00012.