On Less Conservative Stability Criteria for Neural Networks with Time-Varying Delays Utilizing Wirtinger-Based Integral Inequality

Abstract

This paper investigates the problem of stability analysis for neural networks with time-varying delays. By utilizing the Wirtinger-based integral inequality and constructing a suitable augmented Lyapunov-Krasovskii functional, two less conservative delay-dependent criteria to guarantee the asymptotic stability of the concerned networks are derived in terms of linear matrix inequalities (LMIs). Three numerical examples are included to explain the superiority of the proposed methods by comparing maximum delay bounds with the recent results published in other literature.

1. Introduction

Since neural networks are generally recognized as one of the simplified models of neural processing in the human brain and can provide their good performance and strong capability of information processing, they have been successfully applied in many fields such as image and signal processing, pattern recognition, fixed-point computations, optimization, feedback control, medical diagnosis, financial applications, and other scientific areas [1–3]. Due to the finite switching speed of amplifiers and the inherent communication time between the neurons, it is well known that time delays exist and may cause oscillation or deteriorate system performance. Therefore, during the last few decades, many researchers [4–9] put their times and efforts into stability analysis of neural networks with time delays because it is a prerequisite and an important job to check whether the equilibrium point of the concerned networks is stable or not due to the fact that the application of these networks is heavily dependent on the dynamic behavior of the equilibrium points. Methodologically, the delay-derivative dependent [10], the weighting-delay based [11], delay-slope dependent [12], and delay-partitioning [13] analyses were taken by the use of the information for time-delay. Also, Faydasicok and Arik [14, 15] addressed the stability of neural networks with multiple time delays. The asymptotic stability analysis was dealt with in the aforesaid works, while, in [16–19], the exponential stability analysis was studied. In addition to this, the stochastic perturbation condition was considered in [20]. Furthermore, more attentions have been received in delay-dependent stability analysis since time delays encountered in neural networks are usually not very big [21].

One of the hot issues in delay-dependent stability analysis for dynamic systems such as linear systems, neutral systems, neural networks, and so on is to reduce the conservatism of stability criteria or to enhance the feasible region of the concerned criteria. To check the enhancement of the feasible region of stability criteria, maximum delay bounds for guaranteeing the asymptotic stability of the concerned systems are compared with the other methods. The Jensen inequality [22], Park’s inequality [23], model transformation [24, 25], free-weighting matrices techniques [26, 27], convex combination technique [28], and reciprocally convex optimization [29] are well recognized and have been utilized in many fields as tools of reducing the conservatism of stability and stabilization criteria. Very recently, Seuret and Gouaisbaut [30] proposed the Wirtinger-based integral inequality and the advantages of the proposed integral inequality were shown via the comparison of maximum delay bounds for various systems such as systems with constant and known delay, systems with a time-varying delay, and sampled-data systems.

Another remarkable approach to reduce the conservatism of stability criteria is the delay-partitioning idea which was firstly proposed by Gu [22]. The advantage of this method provides larger delay bounds when the delay-partitioning number increases. The idea of the work [22] has been utilized in stability analysis of neural networks with time delays by many researchers [4, 6, 7, 9–11, 13, 21]. In [9], with the recent techniques such as free-weighting matrices techniques and reciprocally convex optimization, a delay-partitioning approach was presented by considering the time-varying delay at each subinterval. Zhang et al. [11] proposed a new delay-partitioning stability analysis by introducing a tuning parameter adjusting delay interval. Zhang et al. [21] proposed new Lyapunov-Krasovskii functionals and delay-partitioning method to investigate the problem of delay-dependent stability analysis for neural networks with two additive time-varying delays and some discussions about the recent works were described. However, when the delay-partitioning number increases, the computational burden and time-consumption increase while the increasing rate of maximum delay bounds is decreased.

In addition to the techniques mentioned above, the choice of Lyapunov-Krasovskii functional and augmented state vectors also play important roles to enhance the feasible region of stability criteria. Since the introduction of the triple integral Lyapunov-Krasovskii functional [32, 33], some new results in stability analysis of neural networks with time-varying delays have been presented in [4, 19]. Furthermore, in the authors’ previous works [5, 8] with the addition of zero equalities [34], it was shown that larger delay bounds can be obtained by constructing newly augmented Lyapunov-Krasovskii functional and introducing some new methods in the activation function condition. However, the application of the Wirtinger-based integral inequality to the terms obtained by calculating the augmented Lyapunov-Krasovskii functional has not been fully investigated yet, and thus there is room for further improvements on the reduction of conservatism.

With motivation mentioned above, in this paper, two improved delay-dependent stability criteria for neural networks with time-varying delays will be proposed. First, in Theorem 6, by constructing a suitable augmented Lyapunov-Krasovskii functional, an improved stability condition such that the considered neural networks are asymptotically stable is derived in terms of linear matrix inequalities (LMIs) by applying Wirtinger-based integral inequality to the augmented quadratic integral term and utilizing the zero equalities [34]. Second, based on the results of Theorem 6 and motivated by the works [35–37], a further improved stability criterion will be proposed in Theorem 9 by ensuring the positiveness of the Lyapunov-Krasovskii functional and utilizing the Wirtinger-based integral inequality. Through three numerical examples utilized in many previous works to check the conservatism of stability criteria, it will be shown that the proposed stability criteria can provide larger delay bounds than the recent existing results. By extension, the developed methods can be applied into the networked control [38, 39], the filtering problem [40, 41], the uncertain systems of neutral type [42], and so on.

Notation. ℝ^𝑛 is the 𝑛-dimensional Euclidean space, and ℝ^𝑚×𝑛 means the set of all 𝑚×𝑛 real matrices. For symmetric matrices 𝑋 and 𝑌, 𝑋>𝑌 (resp., 𝑋≥𝑌) means that the matrix 𝑋−𝑌 is positive definite (resp., nonnegative). 𝑋^⟂ denotes a basis for the null-space of 𝑋. 𝐼_𝑛, 0_𝑛, and 0_𝑚⋅𝑛 denote 𝑛×𝑛 identity matrix and 𝑛×𝑛 and 𝑚×𝑛 zero matrices, respectively. ‖⋅‖ refers to the Euclidean vector norm or the induced matrix norm. diag{⋯} denotes the block diagonal matrix. For square matrix 𝑆, Sym{𝑆} means the sum of 𝑆 and its symmetric matrix 𝑆^𝑇; that is, Sym{𝑆}=𝑆+𝑆^𝑇. ⋆ represents the elements below the main diagonal of a symmetric matrix. 𝑋_[𝑓(𝑡)]∈ℝ^𝑚×𝑛 means that the elements of matrix 𝑋_[𝑓(𝑡)] include the scalar value of 𝑓(𝑡).

2. Problem Statement and Preliminaries

Consider the following neural networks with discrete time-varying delays: 𝑦̇(𝑡)=−𝐴𝑦(𝑡)+𝑊₀𝑔𝑦(𝑡)+𝑊₁𝑔𝑦(𝑡−ℎ(𝑡))+𝑏, (1) where 𝑦(𝑡)=[𝑦₁(𝑡),…,𝑦_𝑛(𝑡)]^𝑇∈ℝ^𝑛 is the neuron state vector, 𝑛 denotes the number of neurons in a neural network, 𝑔(𝑦(𝑡))=[𝑔₁(𝑦₁(𝑡)),…,𝑔_𝑛(𝑦_𝑛(𝑡))]^𝑇∈ℝ^𝑛 denotes the neuron activation function, 𝑔(𝑦(𝑡−ℎ(𝑡)))=[𝑔₁(𝑦₁(𝑡−ℎ(𝑡))),…,𝑔_𝑛(𝑦_𝑛(𝑡−ℎ(𝑡)))]^𝑇∈ℝ^𝑛, 𝐴=diag{𝑎_𝑖}∈ℝ^𝑛×𝑛 is a positive diagonal matrix, 𝑊₀=(𝑤⁰_𝑖𝑗)_𝑛×𝑛∈ℝ^𝑛×𝑛 and 𝑊₁=(𝑤¹_𝑖𝑗)_𝑛×𝑛∈ℝ^𝑛×𝑛 are the interconnection matrices representing the weight coefficients of the neurons, and 𝑏=[𝑏₁,𝑏₂,…,𝑏_𝑛]^𝑇∈ℝ^𝑛 represents a constant input vector.

The delay, ℎ(𝑡), is a time-varying continuous function satisfying 0≤ℎ(𝑡)≤ℎ_𝑈,ℎ̇(𝑡)≤ℎ_𝐷, (2) where ℎ_𝑈 is a known positive scalar and ℎ_𝐷 is any constant one.

The neuron activation functions satisfy the following assumption.

Assumption 1. The neuron activation functions 𝑔_𝑖(⋅), 𝑖=1,…,𝑛, with 𝑔_𝑖(0)=0 are continuous, bounded, and they satisfy 𝑘⁻_𝑖≤𝑔_𝑖(𝑢)−𝑔_𝑖(𝑣)𝑢−𝑣≤𝑘⁺_𝑖,𝑢,𝑣∈ℝ,𝑢≠𝑣,𝑖=1,…,𝑛, (3) where 𝑘⁺_𝑖 and 𝑘⁻_𝑖 are constants.

Remark 2. In Assumption 1, 𝑘⁺_𝑖 and 𝑘⁻_𝑖 can be allowed to be positive, negative, or zero. As mentioned in [12], Assumption 1 describes the class of globally Lipschitz continuous and monotone nondecreasing activation when 𝑘⁻_𝑖=0 and 𝑘⁺_𝑖>0. And the class of globally Lipschitz continuous and monotone increasing activation functions can be described when 𝑘⁺_𝑖>𝑘⁻_𝑖>0.

For simplicity, in stability analysis of the neural networks (1), the equilibrium point 𝑦^∗=[𝑦^∗₁,…,𝑦^∗_𝑛]^𝑇 whose uniqueness has been reported in [17] is shifted to the origin by utilizing the transformation 𝑥(⋅)=𝑦(⋅)−𝑦^∗, which leads the system (1) to the following form: 𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝑊₀𝑓(𝑥(𝑡))+𝑊₁𝑓(𝑥(𝑡−ℎ(𝑡))), (4) where 𝑥(𝑡)=[𝑥₁(𝑡),…,𝑥_𝑛(𝑡)]^𝑇∈ℝ^𝑛 is the state vector of the transformed system, 𝑓(𝑥(𝑡))=[𝑓₁(𝑥₁(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡))]^𝑇, and 𝑓_𝑗(𝑥_𝑗(𝑡))=𝑔_𝑗(𝑥_𝑗(𝑡)+𝑦^∗_𝑗)−𝑔_𝑗(𝑦^∗_𝑗) with 𝑓_𝑗(0)=0(𝑗=1,…,𝑛).

It should be noted that the activation functions 𝑓_𝑖(⋅)  (𝑖=1,…,𝑛) satisfy the following condition [16]: 𝑘⁻_𝑖≤𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)𝑢−𝑣≤𝑘⁺_𝑖,𝑢,𝑣∈ℝ,𝑢≠𝑣,𝑖=1,…,𝑛;𝑘⁻_𝑖≤𝑓_𝑖(𝑢)𝑢≤𝑘⁺_𝑖,∀𝑢≠0,𝑖=1,…,𝑛. (5)(6) The objective of this paper is to investigate the delay-dependent stability analysis of system (4) which will be introduced in Section 3.

Before deriving our main results, the following lemmas will be utilized in deriving the main results.

Lemma 3 (see [30]). Consider a given matrix 𝑅>0. Then, for all continuous function 𝜔 in [𝑎,𝑏]→ℝ^𝑛, the following inequality holds: ^𝑏_𝑎𝜔^𝑇(𝑠)(𝑠)≥1𝑅𝜔𝑑𝑠𝑏−𝑎^𝑏_𝑎𝜔(𝑠)𝑑𝑠^𝑇𝑏_𝑎𝜔(𝑠)+3𝑑𝑠𝑏−𝑎^𝑏_𝑎𝜔(𝑠)2𝑑𝑠−𝑏−𝑎^𝑏_𝑎^𝑠_𝑎𝜔(𝑢)𝑑𝑢𝑑𝑠^𝑇×𝑅^𝑏_𝑎𝜔(𝑠)2𝑑𝑠−𝑏−𝑎^𝑏_𝑎^𝑠_𝑎𝜔(𝑢)𝑑𝑢𝑑𝑠. (7)

Lemma 4 (see [43]). Let 𝜁∈ℝ^𝑛, Φ=Φ^𝑇∈ℝ^𝑛×𝑛, and 𝐵∈ℝ^𝑚×𝑛 such that 𝑟𝑎𝑛𝑘(𝐵)<𝑛. Then, the following statements are equivalent:(1)𝜁^𝑇Φ𝜁<0, 𝐵𝜁=0, 𝜁≠0,(2)(𝐵^⟂)^𝑇Φ𝐵^⟂<0, where 𝐵^⟂ is a right orthogonal complement of 𝐵.

Lemma 5 (see [44]). For the symmetric appropriately dimensional matrices Ω>0, Ξ, matrix Λ, the following two statements are equivalent:(1)Ξ−Λ^𝑇ΩΛ<0,(2)there exists a matrix of appropriate dimension Ψ such that Ξ+Λ^𝑇Ψ+Ψ^𝑇ΛΨ^𝑇Ψ−Ω<0. (8)

3. Main Results

In this section, two new stability criteria for system (4) will be proposed. For the sake of simplicity of matrix and vector representation, 𝑒_𝑖(𝑖=1,2,…,14)∈ℝ^14𝑛×𝑛 which will be used in Theorems 6 and 9 are defined as block entry matrices (e.g., 𝑒₃=[0_𝑛⋅2𝑛,𝐼_𝑛,0_{𝑛⋅11𝑛}]^𝑇). The other notations for some vectors and matrices are defined as follows: 𝜁(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ(𝑡)),𝑥^𝑇𝑡−ℎ_𝑈,𝑥̇^𝑇(𝑡),𝑥̇^𝑇𝑡−ℎ_𝑈,1ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)1𝑑𝑠,ℎ_𝑈−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥^𝑇(𝑠)𝑓𝑑𝑠,_𝑇(𝑥(𝑡)),𝑓^𝑇(𝑥(𝑡−ℎ(𝑡))),𝑓^𝑇𝑥𝑡−ℎ_𝑈,_{𝑡(𝑡)𝑡−ℎ}𝑓^𝑇(𝑥(𝑠))𝑑𝑠,^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑓^𝑇(𝑥(𝑠1))𝑑𝑠,ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}^𝑠_{𝑡−ℎ(𝑡)}𝑥(𝑢)1𝑑𝑢𝑑𝑠,ℎ_𝑈−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}^𝑠_{𝑡−ℎ𝑈}𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠^𝑇,Γ=−𝐴,0_𝑛⋅2𝑛,−𝐼_𝑛,0_𝑛⋅3𝑛,𝑊₀,𝑊₁,0_𝑛⋅5𝑛,Ξ_1[ℎ(𝑡)]=Sym𝑒₁,𝑒₃,ℎ(𝑡)𝑒₆+(ℎ−ℎ(𝑡))𝑒₇,𝑒₁₁+𝑒₁₂×ℛ𝑒₄,𝑒₅,𝑒₁−𝑒₃,𝑒₈−𝑒₁₀^𝑇,Ξ₂=𝑒₁,𝑒₄,𝑒₈𝒩𝑒₁,𝑒₄,𝑒₈^𝑇−𝑒₃,𝑒₅,𝑒₁₀𝒩𝑒₃,𝑒₅,𝑒₁₀^𝑇+Sym𝑒₈−𝐾_𝑚𝑒₁Λ₁𝑒^𝑇₄+Sym𝐾_𝑝𝑒₁−𝑒₈Δ₁𝑒^𝑇₄+Sym𝑒₁₀−𝐾_𝑚𝑒₃Λ₂𝑒^𝑇₅+Sym𝐾_𝑝𝑒₃−𝑒₁₀Δ₂𝑒^𝑇₅,Ξ₃=𝑒₁,𝑒₈,0_{14𝑛⋅𝑛}𝒢𝑒₁,𝑒₈,0_{14𝑛⋅𝑛}−1−ℎ_𝐷𝑒₂,𝑒₉,𝑒₁−𝑒₂𝒢𝑒₂,𝑒₉,𝑒₁−𝑒₂^𝑇+Sym𝑒₁₁𝐺₂₃𝑒^𝑇₄,Ξ₄=ℎ²_𝑈𝑒₄,𝑒₁𝒬₁𝑒₄,𝑒₁^𝑇,Ξ₅=ℎ²_𝑈𝑒₈𝑄₂𝑒^𝑇₈−𝑒₁₁,𝑒₁₂𝑄₂𝑆₂𝑆^𝑇₂𝑄₂𝑒₁₁,𝑒₁₂^𝑇,Ξ₆=ℎ_𝑈𝑒₁𝑃₁𝑒^𝑇₁−𝑒₂𝑃₁𝑒^𝑇₂+𝑒₂𝑃₂𝑒^𝑇₂−𝑒₃𝑃₂𝑒^𝑇₃,Υ_[ℎ(𝑡)]=Sym𝑒₆ℎ(𝑡)𝐺₁₃𝑒^𝑇₄+𝑒₁−𝑒₆ℎ(𝑡)𝐺₃₃𝑒^𝑇₄,Π_[ℎ(𝑡)]=𝑒^𝑇₁−𝑒^𝑇₂ℎ(𝑡)𝑒^𝑇₆𝑒^𝑇₁+𝑒^𝑇₂−2𝑒^𝑇₆ℎ(𝑡)𝑒^𝑇₆−2𝑒^𝑇₁₃𝑒^𝑇₂−𝑒^𝑇₃ℎ_𝑈−ℎ(𝑡)𝑒^𝑇₇𝑒^𝑇₂+𝑒^𝑇₃−2𝑒^𝑇₇ℎ_𝑈−ℎ(𝑡)𝑒^𝑇₇−2𝑒^𝑇₁₄,Θ₁=−Sym𝑒₈−𝑒₁𝐾_𝑚𝐻₁𝑒₈−𝑒₁𝐾_𝑝^𝑇−Sym𝑒₉−𝑒₂𝐾_𝑚𝐻₂𝑒₉−𝑒₂𝐾_𝑝^𝑇−Sym𝑒₁₀−𝑒₃𝐾_𝑚𝐻₃𝑒₁₀−𝑒₃𝐾_𝑝^𝑇,Θ₂=−Sym𝑒₈−𝑒₉−𝑒₁−𝑒₂𝐾_𝑚×𝐻₄𝑒₈−𝑒₉−𝑒₁−𝑒₂𝐾_𝑝^𝑇−Sym𝑒₉−𝑒₁₀−𝑒₂−𝑒₃𝐾_𝑚×𝐻₅𝑒₉−𝑒₁₀−𝑒₂−𝑒₃𝐾_𝑝^𝑇,𝒬_Aug1=𝒬₁+0_𝑛𝑃₁𝑃₁0_𝑛,𝒬_Aug2=𝒬₂+0_𝑛𝑃₂𝑃₂0_𝑛,𝒬Ω=^Aug10_2𝑛0_2𝑛3𝒬_Aug1𝒮₁𝒮^𝑇₁𝒬_Aug20_2𝑛0_2𝑛3𝒬_Aug2,Σ_[ℎ(𝑡)]=Ξ_1[ℎ(𝑡)]+Ξ₂+Ξ₃+Ξ₄+Ξ₅+Ξ₆+Υ_[ℎ(𝑡)]+Θ₁+Θ₂,Φ_[ℎ(𝑡)]=Γ^⟂𝑇Σ_[ℎ(𝑡)]Γ^⟂+SymΓ^⟂𝑇Π^𝑇_[ℎ(𝑡)]Ψ. (9) Now, the following theorem is given by the main result.

Theorem 6. For given scalars ℎ_𝑈>0, ℎ_𝐷, and diagonal matrices 𝐾_𝑝=diag{𝑘⁺₁,…,𝑘⁺_𝑛} and 𝐾_𝑚=diag{𝑘⁻₁,…,𝑘⁻_𝑛}, system (4) is asymptotically stable for 0≤ℎ(𝑡)≤ℎ_𝑈 and ℎ̇(𝑡)≤ℎ_𝐷, if there exist positive diagonal matrices Λ_𝑖=diag{𝜆_1𝑖,…,𝜆_𝑛𝑖}(𝑖=1,2), Δ_𝑖=diag{𝛿_1𝑖,…,𝛿_𝑛𝑖}(𝑖=1,2), and 𝐻_𝑖=diag{ℎ_1𝑖,…,ℎ_𝑛𝑖}(𝑖=1,…,5), positive definite matrices ℛ∈ℝ^4𝑛×4𝑛, 𝒩∈ℝ^3𝑛×3𝑛, 𝒢=[𝐺_𝑖𝑗]∈ℝ^3𝑛×3𝑛, 𝒬₁∈ℝ^2𝑛×2𝑛, and 𝑄₂∈ℝ^𝑛×𝑛, any matrices 𝒮₁∈ℝ^4𝑛×4𝑛, 𝑆₂∈ℝ^𝑛×𝑛, and Ψ∈ℝ^{8𝑛×13𝑛}, and any symmetric matrices 𝑃_𝑖∈ℝ^𝑛×𝑛(𝑖=1,2) satisfying the following LMIs: Φ_{[ℎ(𝑡)=0]}Ψ^𝑇Ψ−Ω<0,Φ_{[ℎ(𝑡)=ℎ𝑈]}Ψ^𝑇Ψ−Ω<0,𝑄₂𝑆₂𝑆^𝑇₂𝑄₂>0, (10)(11)(12) where Φ_[ℎ(𝑡)], Ψ, Ω, and other notations are defined in (9).

Proof. Let us consider the following candidate for the appropriate Lyapunov-Krasovskii functional: 𝑉(𝑡)=⁵_𝑖=1𝑉_𝑖(𝑡), (13) where 𝑉₁(𝑡)=𝑥(𝑡)𝑥𝑡−ℎ_{𝑈𝑡𝑡−ℎ𝑈}𝑥(𝑠)𝑑𝑠_{𝑡𝑡−ℎ𝑈}𝑓(𝑥(𝑠))𝑑𝑠^𝑇ℛ𝑥(𝑡)𝑥𝑡−ℎ_{𝑈𝑡𝑡−ℎ𝑈}𝑥(𝑠)𝑑𝑠_{𝑡𝑡−ℎ𝑈}𝑓(𝑥(𝑠,𝑉))𝑑𝑠₂(𝑡)=^𝑡_{𝑡−ℎ𝑈}𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠))^𝑇𝒩𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠))𝑑𝑠+2^𝑛_𝑖=1𝜆_𝑖1^𝑥𝑖₀^(𝑡)𝑓_𝑖(𝑠)−𝑘⁻_𝑖𝑠𝑑𝑠+𝛿_𝑖1^𝑥𝑖₀^(𝑡)𝑘⁺_𝑖𝑠−𝑓_𝑖(𝑠)𝑑𝑠+2^𝑛_𝑖=1𝜆_𝑖2^{𝑥𝑖(𝑡−ℎ𝑈)}₀𝑓_𝑖(𝑠)−𝑘⁻_𝑖𝑠𝑑𝑠+𝛿_𝑖2^{𝑥𝑖(𝑡−ℎ𝑈)}₀𝑘⁺_𝑖𝑠−𝑓_𝑖(𝑠)𝑉𝑑𝑠₃(𝑡)=^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑓(𝑥(𝑠))_𝑡𝑠𝑥̇(𝑢)𝑑𝑢^𝑇𝒢𝑥(𝑠)𝑓(𝑥(𝑠))_𝑡𝑠𝑥̇(𝑢),𝑉𝑑𝑢₄(𝑡)=ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}^𝑡_𝑠𝑥̇(𝑢)𝑥(𝑢)^𝑇𝒬₁𝑥̇(𝑢)𝑥(𝑢)𝑉𝑑𝑢𝑑𝑠,₅(𝑡)=ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}^𝑡_𝑠𝑓^𝑇(𝑥(𝑢))𝑄₂𝑓(𝑥(𝑢))𝑑𝑢𝑑𝑠. (14) It should be noted that𝑥(𝑡)𝑥𝑡−ℎ_{𝑈𝑡𝑡−ℎ𝑈}𝑥(𝑠)𝑑𝑠_{𝑡𝑡−ℎ𝑈}𝑓(𝑥(𝑠=𝑥(𝑡)𝑥))𝑑𝑠𝑡−ℎ_𝑈ℎ(𝑡1)ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠+ℎ_𝑈(𝑡1−ℎ)ℎ_𝑈−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥(𝑠)𝑑𝑠_{𝑡𝑡−ℎ(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠+^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑓(𝑥(𝑠=𝑒))𝑑𝑠₁,𝑒₃,ℎ(𝑡)𝑒₆+ℎ_𝑈−ℎ(𝑡)𝑒₇,𝑒₁₁+𝑒₁₂^𝑇𝜁(𝑡),𝑥̇(𝑡)𝑥̇𝑡−ℎ_𝑈𝑥(𝑡)−𝑥𝑡−ℎ_𝑈𝑓(𝑥(𝑡))−𝑓𝑥𝑡−ℎ_𝑈=𝑒₄,𝑒₅,𝑒₁−𝑒₃,𝑒₈−𝑒₁₀^𝑇𝜁(𝑡). (15)From (15), 𝑉̇₁(𝑡) can be represented as 𝑉̇₁(𝑡)=𝜁^𝑇(𝑡)Ξ_1[ℎ(𝑡)]𝜁(𝑡). (16) The result of the time-derivative of 𝑉₂(𝑡) is as follows: 𝑉̇₂(𝑡)=𝑥(𝑡)𝑥̇(𝑡)𝑓(𝑥(𝑡))^𝑇𝒩𝑥(𝑡)𝑥̇(𝑡)𝑓(𝑥(𝑡−𝑥))𝑡−ℎ_𝑈𝑥̇𝑡−ℎ_𝑈𝑓𝑥𝑡−ℎ_𝑈^𝑇𝒩𝑥𝑡−ℎ_𝑈𝑥̇𝑡−ℎ_𝑈𝑓𝑥𝑡−ℎ_𝑈+2𝑓(𝑥(𝑡))−𝐾_𝑚𝑥(𝑡)^𝑇Σ₁𝑥̇(𝑡)+2𝐾_𝑝𝑥(𝑡)−𝑓(𝑥(𝑡))^𝑇Δ₁𝑥̇(𝑡)+2𝑓𝑥𝑡−ℎ_𝑈−𝐾_𝑚𝑥𝑡−ℎ_𝑈^𝑇Σ₂𝑥̇𝑡−ℎ_𝑈+2𝐾_𝑝𝑥𝑡−ℎ_𝑈−𝑓𝑥𝑡−ℎ_𝑈^𝑇Δ₂𝑥̇𝑡−ℎ_𝑈=𝜁^𝑇(𝑡)Ξ₂𝜁(𝑡). (17) By calculating 𝑉̇₃(𝑡), it follows that 𝑉̇₃(𝑡)=𝑑𝑑𝑡^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑓(𝑥(𝑠))_𝑡𝑠𝑥̇(𝑢)𝑑𝑢^𝑇𝒢𝑥(𝑠)𝑓(𝑥(𝑠))_𝑡𝑠𝑥̇(𝑢)=𝑥(𝑠)𝑓(𝑥(𝑠𝑑𝑢𝑑𝑠))_𝑡𝑠𝑥̇(𝑢)𝑑𝑢^𝑇𝒢𝑥(𝑠)𝑓(𝑥(𝑠))_𝑡𝑠𝑥̇(𝑢)𝑑𝑢_𝑠=𝑡×𝑑(𝑡)−𝑥(𝑠)𝑓(𝑥(𝑠𝑑𝑡))_𝑡𝑠𝑥̇(𝑢)𝑑𝑢^𝑇𝒢𝑥(𝑠)𝑓(𝑥(𝑠))_𝑡𝑠𝑥̇(𝑢)𝑑𝑢_{𝑠=𝑡−ℎ(𝑡)}×𝑑((𝑡+𝑑𝑡𝑡−ℎ))_{𝑡𝑡−ℎ(𝑡)}𝑑𝑥(𝑠)𝑓(𝑥(𝑠𝑑𝑡))_𝑡𝑠𝑥̇(𝑢)𝑑𝑢^𝑇𝒢𝑥(𝑠)𝑓(𝑥(𝑠))_𝑡𝑠𝑥̇(𝑢)≤𝑥(𝑡)𝑓(𝑥(𝑡0𝑑𝑢𝑑𝑠))_𝑛^𝑇𝒢𝑥(𝑡)𝑓(𝑥(𝑡0))_𝑛−1−ℎ_𝐷𝑥((𝑡𝑓(𝑥((𝑡𝑥(𝑡)((𝑡𝑡−ℎ))𝑡−ℎ)))−𝑥𝑡−ℎ))^𝑇𝒢𝑥((𝑡𝑓(𝑥((𝑡𝑥(𝑡)((𝑡+𝑡−ℎ))𝑡−ℎ)))−𝑥𝑡−ℎ))_{𝑡𝑡−ℎ(𝑡)}2𝑥(𝑠)𝑓(𝑥(𝑠))_𝑡𝑠𝑥̇(𝑢)𝑑𝑢^𝑇𝒢0_𝑛0_𝑛𝑥̇(𝑡)𝑑𝑠=𝜁^𝑇(𝑡)𝑒₁,𝑒₈,0_{14𝑛⋅𝑛}𝒢𝑒₁,𝑒₈,0_{14𝑛⋅𝑛}^𝑇−1−ℎ_𝐷𝑒₂,𝑒₉,𝑒₁−𝑒₂𝒢𝑒₂,𝑒₉,𝑒₁−𝑒₂^𝑇𝜁(𝑡)1+2ℎ(𝑡)^𝑡_{(𝑡)𝑡−ℎ}𝑥(𝑠)𝑑𝑠ℎ(𝑡)𝐺₁₃𝑥̇(𝑡)+2^𝑡_{(𝑡)𝑡−ℎ}𝑓(𝑥(𝑠))𝑑𝑠𝐺₂₃𝑥̇(𝑡)𝑥(𝑡)−1+2ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠^𝑇ℎ(𝑡)𝐺₃₃𝑥̇(𝑡)=𝜁^𝑇(𝑡)Ξ₃+Υ_[ℎ(𝑡)]𝜁(𝑡). (18) Inspired by the work of [34], the following two zero equalities with any symmetric matrices 𝑃_𝑖(𝑖=1,2) are considered: 0=ℎ_𝑈𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)−𝑥^𝑇(𝑡−ℎ(𝑡))𝑃₁𝑥(𝑡−ℎ(𝑡))−2^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₁𝑥̇(𝑠)𝑑𝑠,0=ℎ_𝑈𝑥^𝑇(𝑡−ℎ(𝑡))𝑃₂𝑥(𝑡−ℎ(𝑡))−𝑥^𝑇𝑡−ℎ_𝑈𝑃₂𝑥𝑡−ℎ_𝑈−2^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥^𝑇(𝑠)𝑃₂𝑥̇(𝑠)𝑑𝑠. (19) Summing the two zero equalities presented at (19) leads to 0=𝜁^𝑇(𝑡)Ξ₆𝜁(𝑡)−2ℎ_𝑈^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₁𝑥̇(𝑠)𝑑𝑠−2ℎ_𝑈^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥^𝑇(𝑠)𝑃₂𝑥̇(𝑠)𝑑𝑠. (20) The result of the 𝑉̇₄(𝑡) is given by 𝑉̇₄(𝑡)=ℎ²_𝑈𝑥̇(𝑡)𝑥(𝑡)^𝑇𝒬₁𝑥̇(𝑡)𝑥(𝑡)−ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}𝑥̇(𝑠)𝑥(𝑠)^𝑇𝒬₁𝑥̇(𝑠)𝑥(𝑠)𝑑𝑠. (21) The integral term −ℎ_𝑈∫^𝑡_{𝑡−ℎ𝑈}^̇𝑥(𝑠)_𝑥(𝑠)^𝑇𝒬₁^̇𝑥(𝑠)_𝑥(𝑠)𝑑𝑠 with the addition of the two integral terms at (20) can be described as −ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}𝑥̇(𝑠)𝑥(𝑠)^𝑇𝒬₁𝑥̇(𝑠)𝑥(𝑠)𝑑𝑠−2ℎ_𝑈^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₁𝑥̇(𝑠)𝑑𝑠−2ℎ_𝑈^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥^𝑇(𝑠)𝑃₂𝑥̇(𝑠)𝑑𝑠=−ℎ_𝑈^𝑡_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑥(𝑠)^𝑇𝒬₁+0_𝑛𝑃₁𝑃₁0_𝑛𝒬Aug1𝑥̇(𝑠)𝑥(𝑠)𝑑𝑠−ℎ_𝑈^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥̇(𝑠)𝑥(𝑠)^𝑇𝒬₁+0_𝑛𝑃₂𝑃₂0_𝑛𝒬Aug2𝑥̇(𝑠)𝑥(𝑠)𝑑𝑠. (22) Here, by applying Lemma 3, we have −ℎ_𝑈^𝑡_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑥(𝑠)^𝑇𝒬_Aug1𝑥̇(𝑠)𝑥(𝑠)ℎ𝑑𝑠≤−^𝑈ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑥(𝑠)𝑑𝑠^𝑇𝒬_Aug1^𝑡_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑥(𝑠)−𝑑𝑠3ℎ_𝑈ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑥(𝑠)−2𝑑𝑠ℎ(𝑡)^𝑡_{(𝑡)𝑡−ℎ}^𝑠_{(𝑡)𝑡−ℎ}𝑥̇(𝑢)𝑥(𝑢)𝑑𝑢𝑑𝑠^𝑇×𝒬_Aug1^𝑡_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑥(𝑠)−2𝑑𝑠ℎ(𝑡)^𝑡_{(𝑡)𝑡−ℎ}^𝑠_{(𝑡)𝑡−ℎ}𝑥̇(𝑢)𝑥(𝑢)ℎ𝑑𝑢𝑑𝑠=−^𝑈ℎ(𝑡)((𝑡1𝑥(𝑡)−𝑥𝑡−ℎ))ℎ(𝑡)ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠^𝑇×𝒬_Aug1𝑥(𝑡)((𝑡ℎ(𝑡)1−𝑥𝑡−ℎ))ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)−𝑑𝑠3ℎ_𝑈ℎ(𝑡)𝑥(𝑡)((𝑡ℎ(𝑡)1+𝑥𝑡−ℎ))ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)1𝑑𝑠−2ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)1𝑑𝑠ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}^𝑠_{𝑡−ℎ(𝑡)}𝑥(𝑢)𝑑𝑢𝑑𝑠^𝑇×𝒬_{𝐴𝑢𝑔1}𝑥(𝑡)((𝑡ℎ(𝑡)1+𝑥𝑡−ℎ))ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)1𝑑𝑠−2ℎ(𝑡)^𝑡_{(𝑡)𝑡−ℎ}𝑥(𝑠)1𝑑𝑠ℎ(𝑡)^𝑡_{(𝑡)𝑡−ℎ}^𝑠_{(𝑡)𝑡−ℎ}𝑥(𝑢)ℎ𝑑𝑢𝑑𝑠=−^𝑈ℎ(𝑡)𝜁^𝑇(𝑡)$_1[ℎ(𝑡)]𝜁(𝑡), (23) where $_1[ℎ(𝑡)]=𝑒₁−𝑒₂,ℎ(𝑡)𝑒₆,𝑒₁+𝑒₂−2𝑒₆,ℎ(𝑡)𝑒₆−𝑒₁₃×𝒬_Aug10_2𝑛0_2𝑛3𝒬_Aug1×𝑒₁−𝑒₂,ℎ(𝑡)𝑒₆,𝑒₁+𝑒₂−2𝑒₆,ℎ(𝑡)𝑒₆−𝑒₁₃^𝑇. (24) With the similar process presented in (23), it can be obtained that −ℎ_𝑈^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥̇(𝑠)𝑥(𝑠)^𝑇𝒬_Aug2𝑥̇(𝑠)𝑥(𝑠)ℎ𝑑𝑠≤−^𝑈ℎ_𝑈(𝑡)𝜁−ℎ^𝑇(𝑡)$_2[ℎ(𝑡)]𝜁(𝑡), (25) where $_2[ℎ(𝑡)]=𝑒₂−𝑒₃,(ℎ−ℎ(𝑡))𝑒₇,𝑒₂+𝑒₃−2𝑒₇,ℎ_𝑈−ℎ(𝑡)𝑒₇−2𝑒₁₄𝒬_Aug20_2𝑛0_2𝑛3𝒬_Aug2×𝑒₂−𝑒₃,(ℎ−ℎ(𝑡))𝑒₇,𝑒₂+𝑒₃−2𝑒₇,ℎ_𝑈−ℎ(𝑡)𝑒₇−2𝑒₁₄^𝑇. (26) With (23)–(26) and utilizing reciprocally convex optimization [29], if Ω>0 which is defined in (9), an upper bound of the term (22) can be obtained as −ℎ_𝑈^𝑡_{𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑥(𝑠)^𝑇𝒬_Aug1𝑥̇(𝑠)𝑥(𝑠)𝑑𝑠−ℎ_𝑈^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥̇(𝑠)𝑥(𝑠)^𝑇𝒬_Aug2𝑥̇(𝑠)𝑥(𝑠)ℎ𝑑𝑠≤−^𝑈ℎ(𝑡)𝜁^𝑇(𝑡)$_1[ℎ(𝑡)]𝜁(𝑡)−ℎ_𝑈ℎ_𝑈(𝑡)𝜁−ℎ^𝑇(𝑡)$_2[ℎ(𝑡)]𝜁(𝑡)≤−𝜁^𝑇(𝑡)Π^𝑇_[ℎ(𝑡)]ΩΠ_[ℎ(𝑡)]𝜁(𝑡), (27) where Π_[ℎ(𝑡)] and Ω are defined in (9).
Thus, from (21) to (27), the following inequality holds: 𝑉̇₄(𝑡)−2ℎ_𝑈^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑃₁𝑥̇(𝑠)𝑑𝑠−2ℎ_𝑈^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥^𝑇(𝑠)𝑃₂𝑥̇(𝑠)𝑑𝑠≤𝜁^𝑇(𝑡)Ξ₄−Π^𝑇_[ℎ(𝑡)]ΩΠ_[ℎ(𝑡)]𝜁(𝑡). (28) By utilizing Jensen’s inequality [22] and reciprocally convex optimization [29], if the inequality (12) holds, then the estimation of 𝑉̇₅(𝑡) can be 𝑉̇₅(𝑡)=ℎ²_𝑈𝑓^𝑇(𝑥(𝑡))𝑄₂𝑓(𝑥(𝑡))−ℎ_𝑈^𝑡_{𝑡−ℎ(𝑡)}𝑓^𝑇(𝑥(𝑠))𝑄₂𝑓(𝑥(𝑠))𝑑𝑠−ℎ_𝑈^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑓^𝑇(𝑥(𝑠))𝑄₂𝑓(𝑥(𝑠))𝑑𝑠≤ℎ²_𝑈𝑓^𝑇(𝑥(𝑡))𝑄₂𝑓(𝑥(𝑡−ℎ))_𝑈ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠^𝑇𝑄₂^𝑡_{𝑡−ℎ(𝑡)}𝑓(𝑥(𝑠−ℎ))𝑑𝑠_𝑈ℎ_𝑈−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑓(𝑥(𝑠))𝑑𝑠^𝑇𝑄₂^𝑡_{𝑡−ℎ(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠≤ℎ²_𝑈𝑓^𝑇(𝑥(𝑡))𝑄₂𝑓(𝑥(𝑡−))_{𝑡𝑡−ℎ(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑈}𝑓(𝑥(𝑠))𝑑𝑠^𝑇𝑄₂𝑆₂𝑆^𝑇₂𝑄₂^{𝑡𝑡−ℎ(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ𝑈}𝑓(𝑥(𝑠))𝑑𝑠=𝜁^𝑇(𝑡)Ξ₅𝜁(𝑡). (29) From (6), for any positive diagonal matrices 𝐻_𝑖=diag{ℎ_1𝑖,…,ℎ_𝑛𝑖}  (𝑖=1,…,3), the following inequality holds: 0≤−2^𝑛_𝑖=1ℎ_𝑖1𝑓_𝑖𝑥_𝑖(𝑡)−𝑘⁻_𝑖𝑥_𝑖(𝑡)𝑓_𝑖𝑥_𝑖(𝑡)−𝑘⁺_𝑖𝑥_𝑖(𝑡)−2^𝑛_𝑖=1ℎ_𝑖2𝑓_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−𝑘⁻_𝑖𝑥_𝑖((𝑡×𝑓𝑡−ℎ))_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−𝑘⁺_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−2^𝑛_𝑖=1ℎ_𝑖3𝑓_𝑖𝑥_𝑖𝑡−ℎ_𝑈−𝑘⁻_𝑖𝑥_𝑖𝑡−ℎ_𝑈×𝑓_𝑖𝑥_𝑖𝑡−ℎ_𝑈−𝑘⁺_𝑖𝑥_𝑖𝑡−ℎ_𝑈=𝜁^𝑇(𝑡)Θ₁𝜁(𝑡), (30) where Θ is defined in (9).
Inspired by the authors’ work of [5], from (5), the following conditions hold: 𝑘⁻_𝑖≤𝑓_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡−ℎ(𝑡))≤𝑘⁺_𝑖,𝑘⁻_𝑖≤𝑓_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−𝑓_𝑗𝑥_𝑖𝑡−ℎ_𝑈𝑥_𝑖(𝑡−ℎ(𝑡))−𝑥_𝑖𝑡−ℎ_𝑈≤𝑘⁺_𝑖,𝑖=1,…,𝑛. (31) Therefore, for any positive diagonal matrices 𝐻_𝑖=diag{ℎ_1𝑖,…,ℎ_𝑛𝑖}  (𝑖=4,5), the following inequality holds: 0≤−2^𝑛_𝑖=1ℎ_𝑖4𝑓_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−𝑘⁻_𝑖𝑥_𝑖(𝑡)−𝑥_𝑖((𝑡×𝑓𝑡−ℎ))_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−𝑘⁺_𝑖𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡−ℎ(𝑡))−2^𝑛_𝑖=1ℎ_𝑖5𝑓_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−𝑓_𝑖𝑥_𝑖𝑡−ℎ_𝑈−𝑘⁻_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−𝑥_𝑖𝑡−ℎ_𝑈×𝑓_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−𝑓_𝑖𝑥_𝑖𝑡−ℎ_𝑈−𝑘⁺_𝑖𝑥_𝑖(𝑡−ℎ(𝑡))−𝑥_𝑖𝑡−ℎ_𝑈=𝜁^𝑇(𝑡)Θ₂𝜁(𝑡). (32) From (13) to (32) and by application of the 𝑆-procedure [45], an upper bound of 𝑉̇(𝑡)=∑⁵_𝑖=1𝑉_𝑖(𝑡) with the addition of (20) can be written as 𝑉̇(𝑡)+𝜁^𝑇(𝑡)Ξ₆𝜁(𝑡)−2ℎ_𝑈^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑃₁𝑥̇(𝑠)𝑑𝑠−2ℎ_𝑈^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥^𝑇(𝑠)𝑃₂𝑥̇(𝑠)𝑑𝑠≤𝜁^𝑇(𝑡)Ξ_1[ℎ(𝑡)]+²_𝑖=6Ξ_𝑖+Υ_[ℎ(𝑡)]+Θ₁+Θ_{2Σ[ℎ(𝑡)]}−Π^𝑇_[ℎ(𝑡)]ΩΠ_[ℎ(𝑡)]𝜁(𝑡). (33) By Lemma 4, 𝜁^𝑇(𝑡)(Σ_[ℎ(𝑡)]−Π^𝑇_[ℎ(𝑡)]ΩΠ_[ℎ(𝑡)])𝜁(𝑡)<0 with 0=Γ𝜁(𝑡) is equivalent to Γ^⟂𝑇Σ_[ℎ(𝑡)]−Π^𝑇_[ℎ(𝑡)]ΩΠ_[ℎ(𝑡)]Γ^⟂<0. (34) Then, by Lemma 5, condition (34) is equivalent to the following inequality with any matrix Ψ∈ℝ^{8𝑛×13𝑛}: Γ^⟂𝑇Σ_[ℎ(𝑡)]Γ^⟂+SymΓ^⟂𝑇Π^𝑇_[ℎ(𝑡)]Ψ_{Φ[ℎ(𝑡)]}Ψ^𝑇Ψ−Ω<0. (35) The above condition is affinely dependent on ℎ(𝑡). Therefore, if inequalities (10) and (11) hold, then inequality (35) is satisfied, which means that system (4) is asymptotically stable for 0≤ℎ(𝑡)≤ℎ_𝑈 and ℎ̇(𝑡)≤ℎ_𝐷. It should be noted that Ω>0 holds if inequalities (11) and (12) are feasible. This completes our proof.

Remark 7. As mentioned in the Introduction section, Lemma 3 was first introduced in [30] to reduce the conservatism of delay-dependent stability criteria. In [30], an upper bound of the integral form of −∫^𝑡_{𝑡−ℎ𝑈}𝑥̇^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠(𝑄>0) was obtained by Lemma 3 with augmented vectors (1/ℎ(𝑡))∫^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠 and (1/(ℎ_𝑈−ℎ(𝑡)))∫^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}𝑥(𝑠)𝑑𝑠. However, unlike the presented method in [30], the augmented integral term of −∫^𝑡_{𝑡−ℎ𝑈}^̇𝑥(𝑠)_𝑥(𝑠)^𝑇𝒬₁^̇𝑥(𝑠)_𝑥(𝑠)𝑑𝑠(𝒬₁>0) was estimated in (23)–(26) with the consideration of the two integral terms obtained by zero equality (20). Thus, by utilizing the newly introduced state vectors such as (1/ℎ(𝑡))∫^𝑡_{𝑡−ℎ(𝑡)}∫^𝑠_{𝑡−ℎ(𝑡)}𝑥(𝑢)𝑑𝑢𝑑𝑠 and (1/(ℎ_𝑈−ℎ(𝑡)))∫^{𝑡−ℎ(𝑡)}_{𝑡−ℎ𝑈}∫^𝑠_{𝑡−ℎ𝑈}𝑥(𝑢)𝑑𝑢𝑑𝑠, more relaxed conditions can be expected since more information about the past history of states and some new cross-terms which may play roles to reduce the conservatism of stability condition were considered in Theorem 6. In the authors’ future works, this method will be extended to various problems such as state estimation, 𝐻_∞ performance analysis, filtering, synchronization between two chaotic systems, and stability and stabilization of other dynamic systems which are receiving much attention in the control society.

Remark 8. Another novelty in Theorem 6 is 𝑉₃(𝑡) introduced in (13). In many works dealing with stability of neural networks with time-varying delays, the proposed Lyapunov-Krasovskii functionals having information of time-varying delays have been of the form of ∫^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+∫^𝑡_{𝑡−ℎ(𝑡)}𝑓^𝑇(𝑥(𝑠))𝑄₂𝑥(𝑠)𝑑𝑠(𝑄_𝑖>0) or ∫^𝑡_{𝑡−ℎ(𝑡)}^𝑥(𝑠)_{𝑓(𝑥(𝑠))}^𝑇𝑄^𝑥(𝑠)_{𝑓(𝑥(𝑠))}𝑑𝑠(𝑄>0). However, the proposed Lyapunov-Krasovskii functional is ∫^𝑡_{𝑡−ℎ(𝑡)∫}^{𝑥(𝑠)𝑓(𝑥(𝑠))}_{𝑡𝑠̇𝑥(𝑢)𝑑𝑢}^𝑇𝒢_∫^{𝑥(𝑠)𝑓(𝑥(𝑠))}_{𝑡𝑠̇𝑥(𝑢)𝑑𝑢}𝑑𝑠 which is different from the previous works. Thus, the results of time-derivative of the proposed 𝑉₃(𝑡) contain some new cross-terms such as 21ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠ℎ(𝑡)𝐺₁₃𝑥̇(𝑡),2^𝑡_{𝑡−ℎ(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠𝐺₂₃𝑥̇(𝑡),2𝑥(𝑡)−1ℎ(𝑡)^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠^𝑇ℎ(𝑡)𝐺₃₃𝑥̇(𝑡), (36) which were presented in (18) and does not be used in existing works.

In the proposed Theorem 6, the positiveness of 𝑉(𝑡) is included such as ℛ>0, 𝒩>0, Σ_𝑖>0  (𝑖=1,2), Δ_𝑖>0  (𝑖=1,2), 𝒢>0, 𝒬₁>0, and 𝑄₂>0. These conditions guarantee the positiveness of each 𝑉_𝑖(𝑡)  (𝑖=1,…,5). However, as mentioned in [35–37], by incorporating some functional of 𝑉(𝑡), the positiveness of 𝑉(𝑡) can be relaxed which will be introduced in Theorem 9. For the sake of simplicity of matrix and vector representation in Theorem 9, 𝑒_𝑖̃(𝑖=1,…,6)∈ℝ^6𝑛×𝑛 which will be used are defined as block entry matrices (e.g., 𝑒₃̃=[0_𝑛⋅2𝑛,𝐼_𝑛,0_𝑛⋅3𝑛]^𝑇). Assume that 𝒩>0, Σ_𝑖>0  (𝑖=1,2), Δ_𝑖>0  (𝑖=1,2), 𝒢>0, 𝒬₁>0, and 𝑄₂>0. Then, 𝑉(𝑡) has a lower bound as follows: 𝑉(𝑡)>𝑥(𝑡)𝑥𝑡−ℎ_{𝑈𝑡𝑡−ℎ𝑈}𝑥(𝑠)𝑑𝑠_{𝑡𝑡−ℎ𝑈}𝑓(𝑥(𝑠))𝑑𝑠^𝑇ℛ𝑥(𝑡)𝑥𝑡−ℎ_{𝑈𝑡𝑡−ℎ𝑈}𝑥(𝑠)𝑑𝑠_{𝑡𝑡−ℎ𝑈}𝑓(𝑥(𝑠+))𝑑𝑠_{𝑡𝑡−ℎ𝑈}𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠))^𝑇𝒩𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠))𝑑𝑠. (37) By Lemma 3, the lower bound of the second integral term at the right side of the inequality (35) can be obtained as _{𝑡𝑡−ℎ𝑈}𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠))^𝑇𝒩𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠≥1))𝑑𝑠ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠))𝑑𝑠^𝑇𝒩^𝑡_{𝑡−ℎ𝑈}𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠+3))𝑑𝑠ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠2))𝑑𝑠−ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}^𝑠_{𝑡−ℎ𝑈}𝑥(𝑢)𝑥̇(𝑢)𝑓(𝑥(𝑢𝑠))𝑑𝑢𝑑𝑠^𝑇×𝒩^𝑡_{𝑡−ℎ𝑈}𝑥(𝑠)𝑥̇(𝑠)𝑓(𝑥(𝑠2))𝑑𝑠−ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}^𝑠_{𝑡−ℎ𝑈}𝑥(𝑢)𝑥̇(𝑢)𝑓(𝑥(=1𝑢𝑠))𝑑𝑢𝑑𝑠ℎ_𝑈^{𝑡𝑡−ℎ𝑈}𝑥(𝑠)𝑥(𝑡)𝑑𝑠−𝑥𝑡−ℎ_{𝑈𝑡𝑡−ℎ𝑈}𝑓(𝑥(𝑠))𝑑𝑠^𝑇𝒩^{𝑡𝑡−ℎ𝑈}𝑥(𝑠)𝑥(𝑡)𝑑𝑠−𝑥𝑡−ℎ_{𝑈𝑡𝑡−ℎ𝑈}𝑓(𝑥(𝑠+3))𝑑𝑠ℎ_𝑈^{𝑡𝑡−ℎ𝑈}𝑥(𝑠)2𝑑𝑠−ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}^𝑠_{𝑡−ℎ𝑈}𝑥(𝑢)𝑥(𝑡)𝑑𝑢𝑑𝑠+𝑥𝑡−ℎ_𝑈−2ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}𝑥(𝑠)𝑑𝑠_{𝑡𝑡−ℎ𝑈}𝑓(𝑥(𝑠2))𝑑𝑠−ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}^𝑠_{𝑡−ℎ𝑈}𝑓(𝑥(𝑢))𝑑𝑢𝑑𝑠^𝑇×𝒩^{𝑡𝑡−ℎ𝑈}𝑥(𝑠)2𝑑𝑠−ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}^𝑠_{𝑡−ℎ𝑈}𝑥(𝑢)𝑥(𝑡)𝑑𝑢𝑑𝑠+𝑥𝑡−ℎ_𝑈−2ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}𝑥(𝑠)𝑑𝑠_{𝑡𝑡−ℎ𝑈}𝑓(𝑥(𝑠2))𝑑𝑠−ℎ_𝑈^𝑡_{𝑡−ℎ𝑈}^𝑠_{𝑡−ℎ𝑈}𝑓(𝑥(𝑢))𝑑𝑢𝑑𝑠=𝛼^𝑇(𝑡)Π𝛼(𝑡), (38)where 𝛼(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇𝑡−ℎ_𝑈,^𝑡_{𝑡−ℎ𝑈}𝑥^𝑇(𝑠)𝑑𝑠,_{𝑡𝑡−ℎ𝑈}𝑓^𝑇(𝑥(𝑠))𝑑𝑠,^𝑡_{𝑡−ℎ𝑈}^𝑠_{𝑡−ℎ𝑈}𝑥^𝑇(𝑢)𝑑𝑢𝑑𝑠,_{𝑡𝑡−ℎ𝑈𝑠𝑡−ℎ𝑈}𝑓^𝑇(𝑥(𝑢))𝑑𝑢𝑑𝑠^𝑇,1Π=ℎ_𝑈𝑒₃̃,𝑒₁̃−𝑒₂̃,𝑒₅̃𝒩𝑒₃̃,𝑒₁̃−𝑒₂̃,𝑒̃₅^𝑇+3ℎ𝑒₃̃−2ℎ_𝑈𝑒₅̃,𝑒₁̃+𝑒₂̃−2ℎ𝑒₃̃,𝑒₅̃−2ℎ_𝑈𝑒₆̃×𝒩𝑒₃̃−2ℎ_𝑈𝑒₅̃,𝑒₁̃+𝑒₂̃−2ℎ𝑒₃̃,𝑒₅̃−2ℎ_𝑈𝑒̃₆^𝑇. (39)(40) Therefore, if the following inequality holds 𝑒₁̃,𝑒₂̃,𝑒₃̃,𝑒₄̃ℛ𝑒₁̃,𝑒₂̃,𝑒₃̃,𝑒̃₄^𝑇+Π>0, (41) then the lower bound of 𝑉(𝑡) can be guaranteed to be of the positiveness. Thus, by deleting the positiveness of the matrix ℛ and adding the inequality (40) into the stability condition of Theorem 6, we have the following theorem.

Theorem 9. For given scalars ℎ_𝑈>0, ℎ_𝐷, and diagonal matrices 𝐾_𝑝=diag{𝑘⁺₁,…,𝑘⁺_𝑛} and 𝐾_𝑚=diag{𝑘⁻₁,…,𝑘⁻_𝑛}, system (4) is asymptotically stable for 0≤ℎ(𝑡)≤ℎ_𝑈 and ℎ̇(𝑡)≤ℎ_𝐷, if there exist positive diagonal matrices Λ_𝑖=diag{𝜆_1𝑖,…,𝜆_𝑛𝑖}  (𝑖=1,2), Δ_𝑖=diag{𝛿_1𝑖,…,𝛿_𝑛𝑖}(𝑖=1,2), and 𝐻_𝑖=diag{ℎ_1𝑖,…,ℎ_𝑛𝑖}  (𝑖=1,…,5), positive definite matrices 𝒩∈ℝ^3𝑛×3𝑛, 𝒢=[𝐺_𝑖𝑗]∈ℝ^3𝑛×3𝑛, 𝒬₁∈ℝ^2𝑛×2𝑛, and 𝑄₂∈ℝ^𝑛×𝑛, any matrices 𝒮₁∈ℝ^4𝑛×4𝑛, 𝑆₂∈ℝ^𝑛×𝑛, and Ψ∈ℝ^{8𝑛×13𝑛}, and any symmetric matrices ℛ∈ℝ^4𝑛×4𝑛, 𝑃_𝑖∈ℝ^𝑛×𝑛  (𝑖=1,2) satisfying the LMIs (10)–(12) and (41).

Remark 10. Unlike Theorem 6, Theorem 9 does not require the positiveness condition of ℛ>0. Instead, the condition (41) was added. It should be noted that the positive definiteness of a chosen Lyapunov-Krasovskii functional does not necessarily require all the involved symmetric matrices in the Lyapunov-Krasovskii functional to be positive definite [36]. As presented in (39), by taking the states ∫^𝑡_{𝑡−ℎ𝑈}∫^𝑠_{𝑡−ℎ𝑈}𝑥(𝑢)𝑑𝑢𝑑𝑠 and ∫^𝑡_{𝑡−ℎ𝑈}∫^𝑠_{𝑡−ℎ𝑈}𝑓(𝑥(𝑢))𝑑𝑢𝑑𝑠 as elements of the augmented vector 𝛼(𝑡), Lemma 3 was applied to obtain a lower bound of 𝑉(𝑡) in (38) for the first time. Through three numerical examples, it will be shown that larger delay bounds comparing with the results of Theorem 6 can be obtained.

Remark 11. When information about the upper bound of ℎ̇(𝑡) is unknown, then Theorems 6 and 9 can provide delay-dependent stability criteria for 0≤ℎ(𝑡)≤ℎ_𝑈 by choosing 𝒢=0_3𝑛.

4. Numerical Examples

In this section, three numerical examples are introduced to show the improvements of the proposed methods. In the examples, MATLAB, YALMIP, and SeDuMi 1.3 are used to solve LMI problems.

Example 1. Consider the neural networks (4) with the parameters as follows: 𝐴=2002,𝑊₀=,𝑊11−1−1₁=0.88111,𝐾_𝑝{},𝐾=diag0.4,0.8_𝑚=diag{0,0}. (42)

When ℎ_𝐷 is 0.8, 0.9, and unknown, maximum delay bounds for guaranteeing the asymptotic stability are listed in Table 1 which conducts the comparison of the previous results [4, 5, 9, 10, 13, 19]. In spite of not employing delay-partitioning approach, the obtained delay bounds by applying Theorem 6 are larger than those of [4, 9, 10, 13, 19] which divided the delay interval into two subintervals. Furthermore, comparing with the results of Theorem 6, it can be confirmed that Theorem 9 enhances the feasible region of stability criteria which shows the effectiveness of the lower bound of Lyapunov-Krasovskii functional as mentioned in Remark 10.

Table 1 

Delay bounds ℎ_𝑈 with different ℎ_𝐷 (Example 1).

Example 2. Consider the neural networks (4) where ,𝑊𝐴=1.276900000.623100000.923000000.4480₀=,𝑊−0.03730.4852−0.33510.2336−1.60330.5988−0.32241.23520.3394−0.0860−0.3824−0.5785−0.13110.3253−0.9534−0.5015₁=,𝐾0.8674−1.2405−0.53250.02200.0474−0.91640.03600.98161.84952.6117−0.37880.8428−2.04130.51791.1734−0.2775_𝑝{},𝐾=diag0.1137,0.1279,0.7994,0.2368_𝑚=diag{0,0,0,0}. (43)

In Table 2, maximum delay bounds for guaranteeing the asymptotic stability of system (43) are listed for various conditions of ℎ̇(𝑡). As presented in Table 2, one can see that Theorem 6 effectively reduces the conservatism of stability criteria compared with the previous results [5–7, 9, 11, 13]. Also, it can be confirmed that Theorem 9 slightly enhances the feasible region of Theorem 6.

Table 2 

Delay bounds ℎ_𝑈 with different ℎ_𝐷 (Example 2).

Example 3. Consider the neural networks (4) with the following parameters: 𝐴=1.5000.7,𝑊₀=,𝑊0.05030.04540.09870.2075₁=,𝐾0.23810.93200.03880.5062_𝑝=diag{0.3,0.8},𝐾_𝑚=diag{0,0}. (44)

For this example, one can see that maximum delay bounds are much larger than those of [4, 7, 8] in Table 3 which shows maximum delay bounds when ℎ̇(𝑡) is 0.4, 0.45, 0.5, and 0.55. It can be also confirmed that the feasible region of Theorem 9 is effectively enhanced compared with those of Theorem 6, which also supports the effectiveness of the statements in Remark 10. To explain the contribution of 𝑉₃(𝑡) mentioned in Remark 8, let us refer Corollary 1 when 𝑉₃(𝑡) in Theorem 6 is replaced by the traditional Lyapunov-Krasovskii functional ∫^𝑡_{𝑡−ℎ(𝑡)}^𝑥(𝑠)_{𝑓(𝑥(𝑠))}^𝑇𝑄^𝑥(𝑠)_{𝑓(𝑥(𝑠))}𝑑𝑠(𝑄>0). From the results of Corollary 1 listed in Table 3, one can confirm that the results of Corollary 1 is smaller than those of Theorem 6. This means that the proposed 𝑉₃(𝑡) in Theorem 6 is effective in reducing the conservatism of stability condition.

Table 3 

Delay bounds ℎ_𝑈 with different ℎ_𝐷 (Example 3).

5. Conclusion

In this paper, two improved delay-dependent stability criteria for neural networks with time-varying delays have been proposed by the use of the Lyapunov stability theorem and LMI framework. In Theorem 6, by constructing the suitable augmented Lyapunov-Krasovskii functional and utilizing some novel Lyapunov-Krasovskii functionals and techniques mentioned in Remarks 7 and 8, a delay-dependent sufficient condition for asymptotic stability of the concerned network was derived. Based on the results of Theorem 6 and by taking lower bound of Lyapunov-Krasovskii functional and utilizing the property of its positiveness with the newly constructed augmented vectors, the further improved stability condition was derived in Theorem 9. Via three numerical examples dealt with in many previous works to check the conservatism of stability criteria, the improvements of the feasible region of the three proposed stability criteria have been successfully verified. Moreover, in [46], the triple integral forms of Lyapunov-Krasovskii functional were shown effectiveness in reducing the conservatism of stability sufficient conditions. Thus, by grafting such an approach onto the proposed idea of this paper, further improved results will be investigated in the near future.

Conflict of Interests

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

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2008-0062611), and by a Grant of the Korea Healthcare Technology R&D Project, Ministry of Health & Welfare, Republic of Korea (A100054).