Piecewise Convex Technique for the Stability Analysis of Delayed Neural Network

Abstract

On the basis of the fact that the neuron activation function is sector bounded, this paper transforms the researched original delayed neural network into a linear uncertain system. Combined with delay partitioning technique, by using the convex combination between decomposed time delay and positive matrix, this paper constructs a novel Lyapunov function to derive new less conservative stability criteria. The benefit of the method used in this paper is that it can utilize more information on slope of the activations and time delays. To illustrate the effectiveness of the new established stable criteria, one numerical example and an application example are proposed to compare with some recent results.

1. Introduction

As a special class of nonlinear dynamical systems, neural networks (NNs) have attracted considerable attention due to their extensive applications in pattern recognition, signal processing, associative memories, combinatorial optimization, and many other fields. However, time delay is frequently encountered in NNs due to the finite switching speed of amplifier and the inherent communication time of neurons, especially in the artificial neural network. And it is often an important source of instability and oscillations. It has been shown that the existence of time delay can change the topology of neural networks, and then change the dynamic behavior of neural networks, such as oscillation and chaos. Thus, it is significant to introduce time delay into the neural network model. Additionally, stochastic disturbances and parameter uncertainties can also destroy the convergence of a neural network system. This makes the design or performance for the corresponding closed-loop systems become difficult. Therefore, the equilibrium and stability properties of NNs with time delay have been widely considered by many researchers. Up to now, various stability conditions have been obtained, and many excellent papers and monographs have been available (see [1–8]). So far, these obtained stability results are classified into two types: delay independent and delay dependent. Since sufficiently considered the information of time delays, delay-dependent criteria may be less conservative than delay-independent ones when the size of time delay is small, and much attention has been paid to the delay-dependent category [9–12]. In order to utilize more information of time delay, delay interval is always divided into two or many subintervals with the same size [13, 14]. It has been shown that delay partitioning technique is effective, and the more delay subintervals are divided, the less conservatism of stable criterion may be. However, too many delay subintervals must increase the computational burden; how to balance these two contradictions is a very important issue. To solve this problem, weighting delay and convex analysis methods are widely employed [8, 13].

Additionally, as pointed out by Li et al. [15], the choice of an appropriate Lyapunov-Krasovskii functional (LKF) and the utilization of neuron activation function’s information are very important for deriving less conservative stability criteria. Thus, recently, many authors were devoted to propose a new technique to establish less conservative stable results, such as discredited LKF, augmented LKF, free-weighting matrix LKF, weighting delay LKF, and delay-slope dependent LKF.

In view of the previous discussion, one can see that, to reduce criterion’s conservatism, the crucial problem is how to effectively utilize the information of time delays and neuron activation function. Motivated by the preceding discussion, this paper mainly considers the effective utilization of time delay and neuron activation function’s sector bound. By using the convex representation of the neuron activation function’s sector bounds, we first transform the original nonlinear delayed system into a linear uncertain system. Then, a new LKF function is constructed to derive less conservative stable criteria. Different from previous LKF, this new LKF sufficiently employs the convex combination between decomposed time delay and positive matrix. Finally, one numerical example and an application example are presented to illustrate the validity of the main results.

Notation. The notations are used in our paper unless otherwise specified. ‖⋅‖ denotes a vector or a matrix norm; ℛ and ℛ^𝑛 are real and n-dimension real number sets, respectively; diag(⋯) denotes the block diagonal matrix. Real matrix 𝑃>0(<0) denotes that 𝑃 is positive definite (negative definite). 𝑃>𝑄 denotes that 𝑃−𝑄 is positive-definite. Consider 𝑒_𝑖≜[0,…,0,𝐼^11}_𝑖,0,…,0]^𝑇(𝑖=1,2,…,11), where 𝐼 denotes identity matrix.

2. Preliminaries

Consider the following delayed neural networks: 𝑑𝑥(𝑡)(𝑡)(𝑥(𝑡(𝑥((𝑡𝑑𝑡=−𝐶𝑥+𝐴𝑔))+𝐵𝑔𝑡−𝜏)))+𝑢, (1) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇 denotes the neural state vector; 𝑔(𝑥(𝑡))=[𝑔₁(𝑥₁(𝑡)), 𝑔₂(𝑥₂(𝑡)),…,𝑔_𝑛(𝑥_𝑛(𝑡))]^𝑇 denotes the neuron activation function; 𝑢=[𝑢₁,𝑢₂,…,𝑢_𝑛]^𝑇 is external input vector; 𝐶=diag(𝐶₁,𝐶₂,…,𝐶_𝑛) with 𝐶_𝑖>0 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 and 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛 represent the weighting and delayed weighting matrices, respectively; 𝑔(𝑥(𝑡−𝜏(𝑡)))=[𝑔₁(𝑥₁(𝑡−𝜏(𝑡))),𝑔₂(𝑥₂(𝑡−𝜏(𝑡))),…,𝑔_𝑛(𝑥_𝑛(𝑡−𝜏(𝑡)))]^𝑇, 𝜏(𝑡) is time-varying continuous function which satisfies 0≤𝜏_𝑙≤𝜏(𝑡)≤𝜏_𝑢, 𝑏₁≤𝜏̇(𝑡)≤𝑏₂, where 𝜏_𝑙, 𝜏_𝑢, 𝑏₁, and 𝑏₂ are given constants. Additionally, we always assume that each neuron activation function 𝑔_𝑖(⋅) satisfies condition: 𝑙⁻_𝑖≤(𝑔_𝑖(𝑥)−𝑔_𝑖(𝑦))/(𝑥−𝑦)≤𝑙⁺_𝑖, for all 𝑥,𝑦∈ℛ, 𝑥≠𝑦, 𝑖=1,2,…,𝑛, where 𝑙⁻_𝑖 and 𝑙⁺_𝑖 are known constant scalars. As pointed out in [16], under this assumption, system (1) has equilibrium point. Assume that 𝑥^∗=[𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛]^𝑇 is an equilibrium point of system (1), and set 𝑦_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥^∗_𝑖, 𝑓_𝑖(𝑦_𝑖(𝑡))=𝑔_𝑖(𝑦_𝑖(𝑡)+𝑥^∗_𝑖)−𝑔_𝑖(𝑥^∗_𝑖). Then, system (1) can be transformed into the following form: 𝑑𝑦(𝑡)(𝑡)𝑦(𝑡)𝑦((𝑡,𝑑𝑡=−𝐶𝑦+𝐴𝑓+𝐵𝑓𝑡−𝜏)) (2) where 𝑦(𝑡)=[𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)]^𝑇, 𝑓(𝑦(𝑡))=[𝑓₁(𝑦₁(𝑡)),𝑓₂(𝑦₂(𝑡)),…,𝑓_𝑛(𝑦_𝑛(𝑡))]^𝑇. From the previous assumption, for any 𝑦_𝑖(𝑡)∈ℛ, 𝑦_𝑖(𝑡)≠0, function 𝑓_𝑖(⋅) satisfies 𝑙⁻_𝑖≤(𝑓_𝑖(𝑦_𝑖(𝑡)))/(𝑦_𝑖(𝑡))≤𝑙⁺_𝑖, 𝑓_𝑖(0)=0, 𝑖=1,2,…,𝑛.

Notice that the nonlinear function 𝑓_𝑖(⋅)(𝑖=1,2,…,𝑛) can be rewritten as a convex combination form with the sector bounds as follows: 𝑓_𝑖𝑦_𝑖(𝑡)=𝜆_𝑖𝑦_𝑖(𝑡)𝑙⁻_𝑖+1−𝜆_𝑖𝑦_𝑖(𝑡)𝑙⁺_𝑖𝑦_𝑖(𝑡),𝑓_𝑖𝑦_𝑖(𝑡−𝜏(𝑡))=𝜆_𝑖𝑦_𝑖(𝑡−𝜏(𝑡))𝑙⁻_𝑖+1−𝜆_𝑖𝑦_𝑖(𝑡−𝜏(𝑡))𝑙⁺_𝑖𝑦_𝑖(𝑡−𝜏(𝑡)), (3) where 𝜆_𝑖(𝑦_𝑖)=(𝑓_𝑖(𝑦_𝑖(𝑡))−𝑙⁻_𝑖𝑦_𝑖(𝑡))/(𝑙⁺_𝑖−𝑙⁻_𝑖)𝑦_𝑖(𝑡), 𝜆_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡))) = (𝑓_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡)))−𝑙⁻_𝑖𝑦_𝑖(𝑡−𝜏(𝑡)))/(𝑙⁺_𝑖−𝑙⁻_𝑖)𝑦_𝑖(𝑡−𝜏(𝑡)) satisfying 0≤𝜆_𝑖(𝑦_𝑖(𝑡)), 𝜆_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡))) ≤ 1. Namely, 𝑓_𝑖(𝑦(𝑡)) = Λ_𝑖(𝑦_𝑖(𝑡))𝑦_𝑖(𝑡), 𝑓_𝑖(𝑦(𝑡−𝜏(𝑡))) = Λ_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡)))𝑦_𝑖(𝑡−𝜏(𝑡)), where Λ_𝑖(𝑦_𝑖(𝑡)), Λ_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡))), are elements of a convex hull Co{𝑙⁻_𝑖,𝑙⁺_𝑖}.

Set 𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑛), where 𝑙_𝑖=max{|𝑙⁻_𝑖|,|𝑙⁺_𝑖|}. Obviously, −1≤(𝜆_𝑖(𝑦_𝑖(𝑡))𝑙⁻_𝑖+(1−𝜆_𝑖(𝑦_𝑖(𝑡)))𝑙⁺_𝑖)/𝑙_𝑖,(𝜆_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡)))𝑙⁻_𝑖+(1−𝜆_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡))))𝑙⁺_𝑖)/𝑙_𝑖≤1.

Define Δ_𝑖 = (𝜆_𝑖(𝑦_𝑖(𝑡))𝑙⁻_𝑖+(1−𝜆_𝑖(𝑦_𝑖(𝑡)))𝑙⁺_𝑖)/𝑙_𝑖, Δ_𝑖 = (𝜆_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡)))𝑙⁻_𝑖+(1−𝜆_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡))))𝑙⁺_𝑖)/𝑙_𝑖, Δ = diag(Δ₁,Δ₂,…,Δ_𝑛), Δ = diag(Δ₁,Δ₂,…,Δ_𝑛); then nonlinearities 𝑓(𝑦(𝑡)) and 𝑓(𝑦(𝑡−𝜏(𝑡))) can be expressed as 𝑓(𝑦(𝑡))=𝐿Δ𝑦(𝑡), 𝑓(𝑦(t−𝜏(𝑡)))=𝐿Δ𝑦(𝑡−𝜏(𝑡)), where Δ and Δ satisfy Δ^𝑇Δ≤𝐼, Δ^𝑇Δ≤𝐼. And the system (2) can be rewritten as the following delayed uncertain system: 𝑦̇(𝑡)=(−𝐶+𝐴𝐿Δ)𝑦(𝑡)+𝐵𝐿Δ𝑦(𝑡−𝜏(𝑡)). (4)

Remark 1. Different from previous work, in this paper, by using the convex expression 𝑓_𝑖(𝑦(𝑡))=Λ_𝑖(𝑦_𝑖(𝑡))𝑦_𝑖(𝑡), 𝑓_𝑖(𝑦(𝑡−𝜏(𝑡)))=Λ_𝑖(𝑦_𝑖(𝑡−𝜏(𝑡)))𝑦_𝑖(𝑡−𝜏(𝑡)), we transform the original nonlinear system (2) into a linear system with parameter uncertain system (4). As a result, the stability problem of delayed neural network system (1) can be transformed into the robust stability problem of uncertain linear system (4).

Let 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑛), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑛). For further discussion, the following lemmas are needed.

Lemma 2 (see [16]). Let 𝑓₁,𝑓₂,…,𝑓_𝑁∶ℛ^𝑚↦ℛ have positive values in an open subset 𝐷 of ℛ^𝑚. Then, the reciprocally convex combination of 𝑓_𝑖 over 𝐷 satisfies lim_{{𝛼𝑖∣𝛼𝑖>0,∑𝑖𝛼𝑖=1}𝑖}1𝛼_𝑖𝑓_𝑖(𝑡)=_𝑖𝑓_𝑖(𝑡)+max_{𝑔𝑖,𝑗(𝑡)𝑖≠𝑗}𝑔_𝑖,𝑗(𝑡) (5) subject to 𝑔_𝑖,𝑗∶ℛ^𝑚↦ℛ,𝑔_𝑗,𝑖(𝑡)≜𝑔_𝑖,𝑗(𝑡),𝑓_𝑖(𝑡)𝑔_𝑖,𝑗(𝑡)𝑔_𝑗,𝑖(𝑡)𝑓_𝑗(𝑡)≥0. (6)

Lemma 3 (see [17]). Given the symmetric matrix 𝑃₁ and any real matrices 𝑃₂, 𝑃₃ of appropriate dimensions, then 𝑃₁+𝑃₂Δ𝑃₃+𝑃^𝑇₃Δ^𝑇𝑃^𝑇₂<0 (7) for all Δ∈Θ satisfying Δ^𝑇Δ≤𝐼 if and only if there exists 𝑆∈𝑆_Δ such that 𝑃₁+𝑃^𝑇₃𝑆𝑃₃𝑃₂𝑃^𝑇₂−𝑆<0, (8) where 𝑆_Δ=∶{diag(𝑠₁𝐼,…,𝑠_𝑘𝐼,𝑆₁,…,𝑆_𝑙)∶𝑆_𝑖>0,𝑘,𝑙∈𝑁}.

Lemma 4 ([18], Jensen inequality). Consider two scalars 𝑎<𝑏 and a positive definite matrix 𝑅∈ℛ^𝑛×𝑛. For any conditions function 𝜔∶[𝑎,𝑏]→ℛ^𝑛 and any strictly positive condition 𝑓∶[𝑎,𝑏]→ℛ, the following inequality holds: ^𝑏_𝑎𝑤^𝑇(𝑠)𝑓(𝑠)(𝑠)≥𝑅𝑤𝑑𝑠_𝑏𝑎𝑤(𝑠)𝑑𝑠^𝑇𝑏_𝑎𝑓(𝑠)⁻¹𝑑s⁻¹𝑅^𝑏_𝑎𝑤(𝑠)𝑑𝑠. (9)

Lemma 5 (see [19]). For any constant matrix 𝑍∈ℛ^𝑛×𝑛, 𝑍=𝑍^𝑇>0, scalars ℎ₂>ℎ₁>0, such that the following integrations are well defined; then −12ℎ²₂−ℎ²₁^−ℎ1_−ℎ2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑍𝑥(𝑠)𝑑𝑠𝑑𝜃≤−^−ℎ1_−ℎ2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑑𝑠𝑑𝜃𝑍^−ℎ1_−ℎ2^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃. (10)

3. Stability Analysis

Set 𝜏=(𝜏_𝑢+𝜏_𝑙)/2, 𝜏=(𝜏_𝑢−𝜏_𝑙)/2, and 𝑄₁, 𝑄₂, 𝑄₃, 𝑃₁>𝑃₂, 𝑃₃>𝑃₄, 𝑃₅>𝑃₆ are positive matrices. Let 𝑄(𝜏(𝑡=𝜏(𝑡)))−𝜏_𝑙𝜏𝑄₁+𝜏−𝜏(𝑡)𝜏𝑄₂∶𝜏_𝑙≤𝜏(𝑡)≤𝜏(𝑡)−𝜏,𝜏𝜏𝑄₃+𝜏_𝑢−𝜏(𝑡)𝜏𝑄₁∶𝜏<𝜏(𝑡)≤𝜏_𝑢,𝑃(𝑠)=𝑠−𝑡+𝜏_𝑢𝜏𝑃₁+𝑡−𝜏−𝑠𝜏𝑃₂∶𝑡−𝜏_𝑢≤𝑠<𝑡−𝜏,𝑠−𝑡+𝜏𝜏𝑃₃+𝑡−𝜏_𝑙−𝑠𝜏𝑃₄∶𝑡−𝜏≤𝑠<𝑡−𝜏_𝑙,𝑠−𝑡+𝜏_𝑙𝜏_𝑙𝑃₅+𝑡−𝑠𝜏_𝑙𝑃₆∶𝑡−𝜏_𝑙≤𝑠<𝑡. (11) Consider a new class of Lyapunov functional candidate as follows: 𝑉𝑦(𝑡)=𝑉₁𝑦(𝑡)+𝑉₂𝑦(𝑡), (12) where 𝑉₁𝑦(𝑡)=𝑦^𝑇(𝑡)𝑄(𝜏(𝑡))𝑦(𝑡),𝑉₂𝑦(𝑡)=^𝜏𝑢_𝜏𝑙^𝑡_𝑡−𝜃𝑦̇^𝑇(𝑠)𝑃(𝑠)𝑦̇(𝑠)𝑑𝑠𝑑𝜃. (13) Define 𝑉(𝑡,𝑥_𝑡,𝑥̇_𝑡)=limsup_𝑠→0+(1/𝑠)[𝑉(𝑡+𝑠,𝑥_𝑡+𝑠,𝑥̇_𝑡+𝑠)−𝑉(𝑡,𝑥_𝑡,𝑥̇_𝑡)]; we can obtain 𝑉̇₁(𝑦(𝑡))=𝑦^𝑇(𝑡)𝑄̇(𝜏(𝑡)𝑦(𝑡)) + 2𝑦̇^𝑇(𝑡)[𝜒(𝜏(𝑡))[((𝜏(𝑡)−𝜏_𝑙)/𝜏)𝑄₁ + ((𝜏−𝜏(𝑡))/𝜏)𝑄₂] + (1−𝜒(𝜏(𝑡)))[((𝜏(𝑡)−𝜏)/𝜏)𝑄₃ + ((𝜏_𝑢−𝜏(𝑡))/𝜏)𝑄₁]]𝑦(𝑡), where 𝑄̇(𝜏(𝑡))_|𝜏≠𝜏 = 𝜏̇(𝑡)[𝜒(𝜏(𝑡))((𝑄₁−𝑄₂)/𝜏)+(1−𝜒(𝜏(𝑡)))((𝑄₃−𝑄₁)/𝜏)], 𝜒(𝜏(𝑡=(𝑡)∈𝜏))1∶if𝜏_𝑙,𝜏0∶otherwise. (14)

Notice that 𝑉₂𝑦(𝑡)=^𝑡_{𝑡−𝜏𝑙}^𝜏𝑢_𝜏𝑙𝑦̇^𝑇(𝑠)𝑃(𝑠)𝑦̇(𝑠)+𝑑𝜃𝑑𝑠_{𝑡−𝜏𝑙𝑡−𝜏𝑢𝜏𝑢𝑡−𝑠}𝑦̇^𝑇(𝑠)𝑃(𝑠)𝑦̇(𝑠)𝑑𝜃𝑑𝑠=2𝜏^𝑡_{𝑡−𝜏𝑙}𝑦̇^𝑇(𝑠)𝑠−𝑡+𝜏_𝑙𝜏_𝑙𝑃₅+𝑡−𝑠𝜏_𝑙𝑃₆𝑦̇(𝑠)+𝑑𝑠_{𝑡−𝜏𝑡−𝜏𝑢}𝜏_𝑢+𝑠−𝑡𝑦̇^𝑇(𝑠)×𝑠−𝑡+𝜏_𝑢𝜏𝑃₁+𝑡−𝜏−𝑠𝜏𝑃₂𝑦̇(𝑠)+𝑑𝑠_{𝑡−𝜏𝑙𝑡−𝜏}𝜏_𝑢+𝑠−𝑡𝑦̇^𝑇(𝑠)×𝑠−𝑡+𝜏𝜏𝑃₃+𝑡−𝜏_𝑙−𝑠𝜏𝑃₄𝑦̇(𝑠)𝑑𝑠. (15)

Set 𝜉^𝑇(𝑡) = [𝑦^𝑇(𝑡), 𝑦^𝑇(𝑡−𝜏(𝑡)), 𝑦^𝑇(𝑡−𝜏_𝑙), 𝑦^𝑇(𝑡−𝜏_𝑢), 𝑦^𝑇(𝑡−𝜏), 𝑦̇^𝑇(𝑡), 𝑦̇^𝑇(𝑡−𝜏_𝑙), 𝑦̇^𝑇(𝑡−𝜏), ∫^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑦^𝑇(𝑠)𝑑𝑠, ∫^{𝑡−𝜏𝑙}_{𝑡−𝜏(𝑡)}𝑦^𝑇(𝑠)𝑑𝑠, ∫^𝑡−𝜏_{𝑡−𝜏𝑢}𝑦^𝑇(𝑠)𝑑𝑠, ∫^{𝑡−𝜏𝑙}_𝑡−𝜏𝑦^𝑇(𝑠)𝑑𝑠]. It yields that 𝑉̇₂𝑦(𝑡)=𝜉^𝑇(𝑡)2𝜏𝑒^𝑇₆𝑃₅𝑒₆−𝑒^𝑇₇2𝜏𝑃₆−2𝜏𝑃₃𝑒₇−𝑒^𝑇₈𝜏𝑃₄−𝜏𝑃₁𝑒₈𝜉(𝑡)−^𝑡−𝜏_{𝑡−𝜏𝑢}𝑦̇^𝑇(𝑠)𝑠−𝑡+𝜏_𝑢𝜏𝑃₁+𝑡−𝜏−𝑠𝜏𝑃₂𝑦̇(𝑠)−1𝑑𝑠𝜏^𝑡−𝜏_{𝑡−𝜏𝑢}𝜏_𝑢+𝑠−𝑡𝑦̇^𝑇(𝑠)𝑃₁−𝑃₂𝑦̇(𝑠)−𝑑𝑠_{𝑡−𝜏𝑙𝑡−𝜏}𝑦̇^𝑇(𝑠)𝑠−𝑡+𝜏𝜏𝑃₃+𝑡−𝜏_𝑙−𝑠𝜏𝑃₄𝑦̇(𝑠)−1𝑑𝑠𝜏^{𝑡−𝜏𝑙}_𝑡−𝜏𝜏_𝑢+𝑠−𝑡𝑦̇^𝑇(𝑠)𝑃₃−𝑃₄𝑦̇(𝑠)−𝑑𝑠2𝜏𝜏_𝑙^𝑡_{𝑡−𝜏𝑙}𝑦̇^𝑇(𝑠)𝑃₅−𝑃₆𝑦̇(𝑠)𝑑𝑠≜𝜉^𝑇(𝑡)2𝜏𝑒^𝑇₆𝑃₅𝑒₆−𝑒^𝑇₇2𝜏𝑃₆−2𝜏𝑃₃𝑒₆−𝑒^𝑇₈𝜏𝑃₄−𝜏𝑃₁𝑒₈𝜉(𝑡)+𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅. (16)

We start with the case 𝜏(𝑡)∈[𝜏_𝑙,𝜏], where 𝜒(𝜏(𝑡))=1.

Therefore 𝑉̇₁𝑦(𝑡)=𝜉^𝑇(𝑡)𝜏̇(𝑡)𝜏𝑒^𝑇₁𝑄₁−𝑄₂𝑒^𝑇₁+2𝑒^𝑇₆𝜏(𝑡)−𝜏_𝑙𝜏𝑄₁+𝜏−𝜏(𝑡)𝜏𝑄₂𝑒₁×𝜉(𝑡). (17)

Notice that 𝐼₁=−(1/𝜏)[∫^𝑡−𝜏_{𝑡−𝜏𝑢}∫^𝜏𝑢_𝑡−𝑠𝑦̇^𝑇(𝑠)𝑃₁𝑦(̇𝑠)𝑑𝜃𝑑𝑠+∫^𝑡−𝜏_{𝑡−𝜏𝑢}∫^−𝜏_𝑠−𝑡𝑦̇^𝑇(𝑠)𝑃₂𝑦(̇𝑠)𝑑𝜃𝑑𝑠]=−(1/𝜏)[∫^𝜏𝑢_𝜏∫^𝑡−𝜏_𝑡−𝜃𝑦̇^𝑇(𝑠)𝑃₁𝑦(̇𝑠)𝑑𝑠𝑑𝜃+∫^−𝜏_−𝜏𝑢∫^𝑡+𝜃_{𝑡−𝜏𝑢}𝑦̇^𝑇(𝑠)𝑃₂𝑦(̇𝑠)𝑑𝜃𝑑𝑠]. 𝐼₂ = −(1/𝜏)∫^𝑡−𝜏_{𝑡−𝜏𝑢}∫^𝜏𝑢_𝑡−𝑠𝑦̇^𝑇(𝑠) [𝑃₁−𝑃₂]𝑦(̇𝑠)𝑑𝜃𝑑𝑠 = −(1/𝜏)∫^𝜏𝑢_𝜏∫^𝑡−𝜏_𝑡−𝜃𝑦̇^𝑇(𝑠) [𝑃₁−𝑃₂]𝑦(̇𝑠)𝑑𝑠𝑑𝜃, ∫^𝑡−𝜏_{𝑡−𝜏𝑢}(𝑠−𝑡+𝜏_𝑢) 𝑦(̇𝑠)𝑑𝑠 = (𝜏_𝑢−𝜏)𝑦(𝑡−𝜏)−∫^𝑡−𝜏_{𝑡−𝜏𝑢}𝑦(𝑠)𝑑𝑠, ∫^𝑡−𝜏_{𝑡−𝜏𝑢}(𝑡−𝜏−𝑠)𝑦(̇𝑠)𝑑𝑠 = −(𝜏_𝑢−𝜏)𝑦(𝑡−𝜏_𝑢) + ∫^𝑡−𝜏_{𝑡−𝜏𝑢}𝑦(𝑠)𝑑𝑠; from Lemma 5, it yields that 𝐼₁+𝐼₂4≤−𝜏𝜏²_𝑢−𝜏²𝜉^𝑇(𝑡)𝜏_𝑢−𝜏𝑒^𝑇₅−𝑒^𝑇₁₁×𝑃₁𝜏_𝑢−𝜏𝑒₅−𝑒₁₁𝜉(𝑡)−2𝜏𝜏²_𝑢−𝜏²𝜉^𝑇(𝑡)𝑒^𝑇₁₁−𝜏_𝑢−𝜏𝑒^𝑇₄×𝑃₂𝑒₁₁−𝜏_𝑢−𝜏𝑒₄𝜉(𝑡)+2𝜏𝜏²_𝑢−𝜏²𝜉^𝑇(𝑡)𝜏_𝑢−𝜏𝑒^𝑇₅−𝑒^𝑇₁₁×𝑃₂𝜏_𝑢−𝜏𝑒₅−𝑒₁₁𝜉(𝑡). (18) Notice that 𝐼₃1=−𝜏^{𝑡−𝜏(𝑡)}_𝑡−𝜏^𝜏_𝑡−𝑠𝑦̇^𝑇(𝑠)𝑃₃𝑦̇(𝑠)+𝑑𝜃𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−𝜏−𝜏𝑙𝑠−𝑡}𝑦̇^𝑇(𝑠)𝑃₄𝑦̇(𝑠)−1𝑑𝜃𝑑𝑠𝜏^{𝑡−𝜏𝑙}_{(𝑡)𝑡−𝜏}^𝜏_𝑡−𝑠𝑦̇^𝑇(𝑠)𝑃₃𝑦̇(𝑠)+𝑑𝜃𝑑𝑠_{𝑡−𝜏𝑙(𝑡)𝑡−𝜏−𝜏𝑙𝑠−𝑡}𝑦̇^𝑇(𝑠)𝑃₄𝑦̇(𝑠)1𝑑𝜃𝑑𝑠=−𝜏^𝜏_𝜏(𝑡)^{𝑡−𝜏(𝑡)}_𝑡−𝜃𝑦̇^𝑇(𝑠)𝑃₃𝑦̇(𝑠)+𝑑𝑠𝑑𝜃_{−−𝜏(𝑡)𝜏𝑡+𝜃𝑡−𝜏}𝑦̇^𝑇(𝑠)𝑃₄𝑦̇(𝑠)−1𝑑𝑠𝑑𝜃𝜏_𝜏^𝜏(𝑡)_𝑙^{𝑡−𝜏𝑙}_𝑡−𝜃𝑦̇^𝑇(𝑠)𝑃₃𝑦̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝜏𝑙(𝑡)−𝜏(𝑡)𝑡+𝜃𝑡−𝜏}𝑦̇^𝑇(𝑠)𝑃₄𝑦̇(𝑠)𝑑𝑠𝑑𝜃. (19)

Thus, from Lemma 5, it yields that 𝐼₃2≤−𝜏𝜏²−𝜏(𝑡)²𝜉^𝑇(𝑡)(𝜏−𝜏(𝑡))𝑒^𝑇₂−𝑒^𝑇₉×𝑃₃(𝜏−𝜏(𝑡))𝑒₂−𝑒₉𝜉(𝑡)−2𝜏−𝜏(𝑡)²𝜉^𝑇(𝑡)−𝜏−𝜏_𝑙𝑒^𝑇₅+𝑒^𝑇₉×𝑃₄−𝜏−𝜏_𝑙𝑒₅+𝑒₉𝜉(𝑡)−2𝜏𝜏(𝑡)²−𝜏²_𝑙𝜉^𝑇(𝑡)𝜏−𝜏_𝑙𝑒^𝑇₃−𝑒^𝑇₁₀×𝑃₃𝜏−𝜏_𝑙𝑒₃−𝑒₁₀𝜉(𝑡)−2𝜏𝜏(𝑡)²−𝜏²_𝑙𝜉^𝑇(𝑡)−𝜏(𝑡)−𝜏_𝑙𝑒^𝑇₂+𝑒^𝑇₁₀×𝑃₄−𝜏(𝑡)−𝜏_𝑙𝑒₂+𝑒₁₀𝜉(𝑡). (20) Set 𝐸₁=^{(𝜏−𝜏(𝑡))𝑒𝑇2−𝑒𝑇9−(𝜏−𝜏𝑙)𝑒𝑇5+𝑒𝑇9}_{(𝜏−𝜏𝑙)𝑒𝑇3−𝑒𝑇10−(𝜏(𝑡)−𝜏𝑙)𝑒𝑇2+𝑒𝑇10}, 𝑃=1/(𝜏²(𝜏+𝜏_𝑙))^{𝑃3𝑆1,2𝑆1,3𝑆1,4∗𝑃4𝑆2,2𝑆2,3}_{∗∗𝑃3𝑆3,3∗∗∗𝑃4}, where 𝑆_𝑖,𝑗(1≤𝑖≤𝑗≤3) are arbitrary matrices. If 𝑃>0, since (𝜏²−𝜏(𝑡)²)/(𝜏(𝜏+𝜏_𝑙))+(𝜏(𝑡)²−𝜏²_𝑙)/(𝜏(𝜏+𝜏_𝑙))=1, from Lemma 2, one can obtain 𝐼₃≤−𝜉^𝑇(𝑡)𝐸^𝑇₁𝑃𝐸₁𝜉(𝑡). (21)

Note that inequality (21) holds for all 𝜏(𝑡)∈[𝜏_𝑙,𝜏]. When 𝜏(𝑡)=𝜏_𝑙 and 𝜏, inequality (21) still holds; it yields that𝐼₃≤−𝜉^𝑇(𝑡)𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾𝜉(𝑡),𝐼₃≤−𝜉^𝑇(𝑡)𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾𝜉(𝑡), (22a)(22b)where 𝐸₁⁽¹⁾=𝜏−𝜏_𝑙𝑒^𝑇₃−𝑒^𝑇₉−𝜏−𝜏_𝑙𝑒^𝑇₅+𝑒^𝑇₉𝜏−𝜏_𝑙𝑒^𝑇₃−𝑒^𝑇₁₀𝑒^𝑇₁₀,𝐸₁⁽²⁾=−𝑒^𝑇₉−𝜏−𝜏_𝑙𝑒^𝑇₅+𝑒^𝑇₉𝜏−𝜏_𝑙𝑒^𝑇₃−𝑒^𝑇₁₀−𝜏−𝜏_𝑙𝑒^𝑇₅+𝑒^𝑇₁₀. (23) Using the fact that 𝐼₄ = −(1/𝜏)[∫^{𝑡−𝜏(𝑡)}_𝑡−𝜏(𝜏_𝑢+𝑠−𝑡)𝑦̇^𝑇(𝑠) [𝑃₃−𝑃₄]𝑦(̇𝑠)𝑑𝑠+∫^{𝑡−𝜏𝑙}_{𝑡−𝜏(𝑡)} (𝜏_𝑢+𝑠−𝑡)𝑦̇^𝑇(𝑠)[𝑃₃−𝑃₄]𝑦(̇𝑠)𝑑𝑠] and 𝜏_𝑢+𝑠−𝑡 is a strictly positive continuous function on [𝑡−𝜏,𝑡−𝜏_𝑙], by Lemma 4, one can obtain 𝐼₄1≤−𝜏𝜉ln2^𝑇(𝑡)×𝑒^𝑇₂−𝑒^𝑇₅𝑃₃−𝑃₄𝑒₂−𝑒₅+𝑒^𝑇₃−𝑒^𝑇₂𝑃₃−𝑃₄𝑒₃−𝑒₂𝜉(𝑡).𝐼₅≤−2𝜏𝜏²_𝑙𝜉^𝑇(𝑡)𝑒^𝑇₁−𝑒^𝑇₃𝑃₅−𝑃₆𝑒₁−𝑒₃𝜉(𝑡). (24)

Furthermore, from (4), for arbitrary matrices 𝑀₁ and 𝑀₂ of appropriate dimensions, the following equalities hold:2𝜉^𝑇(𝑡)𝑒^𝑇₁𝑀₁−𝐶𝑒₁−𝑒₆𝜉(𝑡)+𝜉^𝑇(𝑡)𝑒^𝑇₁𝑀₁𝐴𝐿Δ𝑒₁+𝐵𝐿Δ𝑒₂𝜉(𝑡)=0,2𝜉^𝑇(𝑡)𝑒^𝑇₆𝑀₂−𝐶𝑒₁−𝑒₆𝜉(𝑡)+𝜉^𝑇(𝑡)𝑒^𝑇₆𝑀₂𝐴𝐿Δ𝑒₁+𝐵𝐿Δ𝑒₂𝜉(𝑡)=0. (25a)(25b)

Define𝐸₂⁽¹⁾=2𝑒^𝑇₆𝑄₂𝑒₁,𝐸₂⁽²⁾=2𝑒^𝑇₆𝑄₁𝑒₁,𝐸₂⁽³⁾=2𝑒^𝑇₆𝜏(𝑡)−𝜏_𝑙𝜏𝑄₁+𝜏−𝜏(𝑡)𝜏𝑄₂𝑒₁,𝐸₃⁽¹⁾=𝑏₁𝜏𝑒^𝑇₁𝑄₁−𝑄₂𝑒₁,𝐸₃⁽²⁾=𝑏₂𝜏𝑒^𝑇₁𝑄₁−𝑄₂𝑒₁,𝐸₃⁽³⁾=𝜏̇(𝑡)𝜏𝑒^𝑇₁𝑄₁−𝑄₂𝑒₁. (26a)(26b)Therefore 𝐸¹=2𝜏𝑒^𝑇₆𝑃₅𝑒₆−𝑒^𝑇₇2𝜏𝑃₆−2𝜏𝑃₃𝑒₆−𝑒^𝑇₈𝜏𝑃₄−𝜏𝑃₁𝑒₈−4𝜏𝜏²_𝑢−𝜏²𝜏_𝑢−𝜏𝑒^𝑇₅−𝑒^𝑇₁₁×𝑃₁𝜏_𝑢−𝜏𝑒₅−𝑒₁₁−2𝜏𝜏²_𝑢−𝜏²𝑒^𝑇₁₁−𝜏_𝑢−𝜏𝑒^𝑇₄×𝑃₂𝑒₁₁−𝜏_𝑢−𝜏𝑒₄+2𝜏𝜏²_𝑢−𝜏²𝜏_𝑢−𝜏𝑒^𝑇₅−𝑒^𝑇₁₁×𝑃₂𝜏_𝑢−𝜏𝑒₅−𝑒₁₁−1𝜏𝑒ln2^𝑇2−𝑒^𝑇₅𝑃₃−𝑃₄𝑒₂−𝑒₅+𝑒^𝑇₃−𝑒^𝑇₂𝑃₃−𝑃₄𝑒₃−𝑒₂−2𝜏𝜏²_𝑙𝑒^𝑇₁−𝑒^𝑇₃𝑃₆−𝑃₅𝑒₁−𝑒₃+𝑒^𝑇₁𝑀₁−𝐶𝑒₁−𝑒₆+𝑒^𝑇₆𝑀₂−𝐶𝑒₁−𝑒₆+−𝐶𝑒₁−𝑒₆^𝑇𝑀^𝑇₁𝑒₁+−𝐶𝑒₁−𝑒₆^𝑇𝑀^𝑇₂𝑒₆. (27)Moreover Φ=𝑒^𝑇₁,𝑒^𝑇₂,𝑒^𝑇₁,𝑒^𝑇₂^𝑇,Δ=diagΔ,Δ,Δ,Δ,Ψ=𝑒^𝑇₁𝑀₁𝐴𝐿,𝑒^𝑇₁𝑀₁𝐵𝐿,𝑒^𝑇₆𝑀₂𝐴𝐿,𝑒^𝑇₆𝑀₂𝐵𝐿. (28a)(28b) From (16)–(28b), we get that, along (4), 𝑉̇𝑦(𝑡)≤𝛼₁‖𝑥(𝑡)‖², (29) for some scalar 𝛼₁>0 if Π₁=𝐸¹+𝐸₂⁽³⁾+𝐸₃⁽³⁾−𝐸₁^𝑇𝑃𝐸₁+ΨΔΦ+Φ^𝑇ΔΨ^𝑇<0. (30)

Inequality (30) leads, for 𝜏(𝑡)=𝜏_𝑙,𝜏, to the following two inequalities:Π₁⁽¹⁾=𝐸¹+𝐸₂⁽¹⁾+𝐸₃⁽³⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₂₍₁₎=𝐸¹+𝐸₂⁽²⁾+𝐸₃⁽³⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇<0. (31a)(31b)

Inequalities (31a) and (31b) imply (30), because Π₁=((𝜏(𝑡)−𝜏_𝑙)/𝜏)Π₂⁽¹⁾+((𝜏−𝜏(𝑡))/𝜏)Π₁⁽¹⁾ is convex in 𝜏(𝑡)∈[𝜏_𝑙,𝜏]. At the same time, inequalities (31a) and (31b) lead, for 𝜏̇(𝑡)=𝑏₁,𝑏₂, to the following inequalities: Π₍₁₎≜𝐸¹+𝐸₂⁽¹⁾+𝐸₃⁽¹⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₍₂₎≜𝐸¹+𝐸₂⁽¹⁾+𝐸₃⁽²⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₍₃₎≜𝐸¹+𝐸₂⁽²⁾+𝐸₃⁽¹⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₍₄₎≜𝐸¹+𝐸₂⁽²⁾+𝐸₃⁽²⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇<0. (32a)(32b)(32c)(32d)

Obviously, inequalities (32a)–(32d) imply (31a) and (31b) since Π₁⁽¹⁾ = ((𝑏₂−𝜏̇)/(𝑏₂−𝑏₁))Π₍₂₎ + ((𝜏̇−𝑏₁)/(𝑏₂−𝑏₁))Π₍₁₎, and Π₂⁽¹⁾ = ((𝑏₂−𝜏̇)/(𝑏₂−𝑏₁))Π₍₄₎ + ((𝜏̇−𝑏₁)/(𝑏₂−𝑏₁))Π₍₃₎ are convex in 𝜏̇(𝑡)∈[𝑏₁,𝑏₂].

Similarly, there exists scalar 𝛼₂>0 such that 𝑉̇𝑦(𝑡)≤𝛼₂‖𝑥(𝑡)‖², (33) if Π₍₁₎≜𝐸¹+𝐸₂⁽¹⁾+𝐸₃⁽¹⁾−𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇<0,Π₍₂₎≜𝐸¹+𝐸₂⁽¹⁾+𝐸₃⁽²⁾−𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇<0,Π₍₃₎≜𝐸¹+𝐸₂⁽²⁾+𝐸₃⁽¹⁾−𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇<0,Π₍₄₎≜𝐸¹+𝐸₂⁽²⁾+𝐸₃⁽²⁾−𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇<0. (34a)(34b)(34c)(34d)

Similarly, set 𝜉̃^𝑇(𝑡) = [𝑦^𝑇(𝑡), 𝑦^𝑇(𝑡−𝜏(𝑡)), 𝑦^𝑇(𝑡−𝜏_𝑙), 𝑦^𝑇(𝑡−𝜏_𝑢), 𝑦^𝑇(𝑡−𝜏), 𝑦̇^𝑇(𝑡), 𝑦̇^𝑇(𝑡−𝜏_𝑙), 𝑦̇^𝑇(𝑡−𝜏), ∫^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑢}𝑦^𝑇(𝑠)𝑑𝑠, ∫^𝑡−𝜏_{𝑡−𝜏(𝑡)}𝑦^𝑇(𝑠)𝑑𝑠, ∫^𝑡−𝜏_{𝑡−𝜏𝑢}𝑦^𝑇(𝑠)𝑑𝑠, ∫^{𝑡−𝜏𝑙}_𝑡−𝜏𝑦^𝑇(𝑠)𝑑𝑠].

If 𝜏(𝑡)∈(𝜏,𝜏_𝑢], then, 𝜒(𝜏(𝑡))=0.

Therefore 𝑉̇₁𝑦(𝑡)=𝜉̃^𝑇(𝑡)𝜏̇(𝑡)𝜏𝑒^𝑇₁𝑄₃−𝑄₁𝑒₁+2𝑒^𝑇₆𝜏(𝑡)−𝜏𝜏𝑄₃+𝜏_𝑢−𝜏(𝑡)𝜏𝑄₁𝑒₁×𝜉̃(𝑡). (35) Moreover 𝐼₁1=−𝜏^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑢}^𝜏𝑢_𝑡−𝑠𝑦̇^𝑇(𝑠)𝑃₁𝑦̇(𝑠)−1𝑑𝜃𝑑𝑠𝜏^𝑡−𝜏_{(𝑡)𝑡−𝜏}^𝜏𝑢_𝑡−𝑠𝑦̇^𝑇(𝑠)𝑃₁𝑦̇(𝑠)−1𝑑𝜃𝑑𝑠𝜏^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑢}^−𝜏_𝑠−𝑡𝑦̇^𝑇(𝑠)𝑃₂𝑦̇(𝑠)−1𝑑𝜃𝑑𝑠𝜏^𝑡−𝜏_{𝑡−𝜏(𝑡)}^−𝜏_𝑠−𝑡𝑦̇^𝑇(𝑠)𝑃₂𝑦̇(𝑠)𝐼𝑑𝜃𝑑𝑠.₂1=−𝜏^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑢}^𝜏𝑢_𝑡−𝑠𝑦̇^𝑇(𝑠)𝑃₁−𝑃₂𝑦̇(𝑠)−1𝑑𝜃𝑑𝑠𝜏^𝑡−𝜏_{𝑡−𝜏(𝑡)}^𝜏𝑢_𝑡−𝑠𝑦̇^𝑇(𝑠)𝑃₁−𝑃₂𝑦̇(𝑠)𝑑𝜃𝑑𝑠. (36)

Set 𝐸^𝑇₁=𝜏_𝑢−𝜏(𝑡)𝑒^𝑇₂−𝑒^𝑇₉(𝜏(𝑡)−𝜏)𝑒^𝑇₅−𝑒^𝑇₁₀−𝜏_𝑢−𝜏(𝑡)𝑒^𝑇₄+𝑒^𝑇₉−(𝜏(𝑡)−𝜏)𝑒^𝑇₂+𝑒^𝑇₁₀𝜏_𝑢−𝜏(𝑡)𝑒^𝑇₂−𝑒^𝑇₉(𝜏(𝑡)−𝜏)𝑒^𝑇₅−𝑒^𝑇₁₀,2𝑃=3𝜏²𝜏_𝑢+𝜏𝑃₁𝑆_1,2𝑆_1,3𝑆_1,4𝑆_1,5𝑆_1,6∗𝑃₁𝑆_2,2𝑆_2,3𝑆_2,4𝑆_2,5∗∗𝑃₂𝑆_3,3𝑆_3,4𝑆_3,5∗∗∗𝑃₂𝑆_4,4𝑆_4,5∗∗∗∗𝑃₁−𝑃₂𝑆_5,5∗∗∗∗∗𝑃₁−𝑃₂, (37) where 𝑆_𝑖,𝑗(1≤𝑖≤𝑗≤5) are arbitrary matrices. If 𝑃>0, from Lemmas 5 and 2, similar to the proof of (19)–(21), one can obtain 𝐼₁+𝐼₂≤−𝜉̃^𝑇(𝑡)𝐸^𝑇₁𝑃𝐸₁𝜉̃(𝑡). (38) When 𝜏(𝑡)=𝜏 and 𝜏_𝑢, inequality (38) still holds; it yields that 𝐼₁+𝐼₂≤−𝜉̃^𝑇(𝑡)𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾𝜉̃(𝑡),𝐼₁+𝐼₂≤−𝜉̃^𝑇(𝑡)𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾𝜉̃(𝑡), (39a)(39b)where 𝐸₁⁽¹⁾=𝜏_𝑢−𝜏𝑒^𝑇₅−𝑒^𝑇₉−𝑒^𝑇₁₀−𝜏_𝑢−𝜏𝑒^𝑇₄+𝑒^𝑇₉𝑒^𝑇₁₀𝜏_𝑢−𝜏𝑒^𝑇₅−𝑒^𝑇₉−𝑒^𝑇₁₀,𝐸₁⁽²⁾=−𝑒^𝑇₉𝜏_𝑢−𝜏𝑒^𝑇₅−𝑒^𝑇₁₀𝑒^𝑇₉−𝜏_𝑢−𝜏𝑒^𝑇₄+𝑒^𝑇₁₀−𝑒^𝑇₉𝜏_𝑢−𝜏𝑒^𝑇₅−𝑒^𝑇₁₀. (40) One has 𝐼₃2≤−𝜏𝜏²−𝜏²_𝑙𝜉̃^𝑇(𝑡)𝜏−𝜏_𝑙𝑒^𝑇₃−𝑒^𝑇₁₁×𝑃₃𝜏−𝜏_𝑙𝑒₃−𝑒₁₁𝜉̃(𝑡)−2𝜏𝜏²−𝜏²_𝑙𝜉̃^𝑇(𝑡)−𝜏−𝜏_𝑙𝑒^𝑇₄−𝑒^𝑇₁₁×𝑃₄−𝜏−𝜏_𝑙𝑒₄−𝑒₁₁𝜉̃(𝑡).𝐼₄1≤−𝜏𝜉̃ln2^𝑇(𝑡)𝑒^𝑇₃−𝑒^𝑇₅𝑃₃−𝑃₄𝑒₃−𝑒₅𝜉̃(𝑡).𝐼₅≤−2𝜏𝜏²_𝑙𝜉̃^𝑇(𝑡)𝑒^𝑇₁−𝑒^𝑇₃𝑃₅−𝑃₆𝑒₁−𝑒₃𝜉̃(𝑡). (41)

Furthermore, from (4), for arbitrary matrices 𝑁₁ and 𝑁₂ of appropriate dimensions, the following equalities hold:2𝜉̃^𝑇(𝑡)𝑒^𝑇₁𝑁₁−𝐶𝑒₁−𝑒₆𝜉̃(𝑡)+𝜉̃^𝑇(𝑡)𝑒^𝑇₁𝑁₁𝐴𝐿Δ𝑒₁+𝐵𝐿Δ𝑒₂𝜉̃(𝑡)2𝜉̃=0,_𝑇(𝑡)𝑒^𝑇₆𝑁₂−𝐶𝑒₁−𝑒₆𝜉̃(𝑡)+𝜉̃^𝑇(𝑡)𝑒^𝑇₆𝑁₂𝐴𝐿Δ𝑒₁+𝐵𝐿Δ𝑒₂𝜉̃(𝑡)=0. (42a)(42b)

Define 𝐸₂⁽¹⁾=2𝑒^𝑇₆𝑄₁𝑒₁,𝐸₂⁽²⁾=2𝑒^𝑇₆𝑄₃𝑒₁,𝐸₂⁽³⁾=2𝑒^𝑇₆𝜏(𝑡)−𝜏𝜏𝑄₃+𝜏_𝑢−𝜏(𝑡)𝜏𝑄₁,𝐸₃⁽¹⁾=𝑏₁𝜏𝑒^𝑇₁𝑄₃−𝑄₁𝑒₁,𝐸₃⁽²⁾=𝑏₂𝜏𝑒^𝑇₁𝑄₃−𝑄₁𝑒₁,𝐸₃⁽³⁾=𝜏̇(𝑡)𝜏𝑒^𝑇₁𝑄₃−𝑄₁𝑒₁. (43a)(43b)Therefore 𝐸¹=2𝜏𝑒^𝑇₆𝑃₅𝑒₆−𝑒^𝑇₇2𝜏𝑃₆−2𝜏𝑃₃𝑒₆−𝑒^𝑇₈𝜏𝑃₄−𝜏𝑃₁𝑒₈−2𝜏𝜏²−𝜏²_𝑙𝜏−𝜏_𝑙𝑒^𝑇₃−𝑒^𝑇₁₁𝑃₃𝜏−𝜏_𝑙𝑒₃−𝑒₁₁−2𝜏𝜏²−𝜏²_𝑙−𝜏−𝜏_𝑙𝑒^𝑇₄−𝑒^𝑇₁₁𝑃₄−𝜏−𝜏_𝑙𝑒₄−𝑒₁₁−1𝜏𝑒ln2^𝑇3−𝑒^𝑇₅𝑃₃−𝑃₄𝑒₃−𝑒₅−2𝜏𝜏²_𝑙𝑒^𝑇₁−𝑒^𝑇₃𝑃₅−𝑃₆𝑒₁−𝑒₃+𝑒^𝑇₁𝑁₁−𝐶𝑒₁−𝑒₆+𝑒^𝑇₆𝑁₂−𝐶𝑒₁−𝑒₆+−𝐶𝑒₁−𝑒₆^𝑇𝑁^𝑇₁𝑒₁+−𝐶𝑒₁−𝑒₆^𝑇𝑁^𝑇₂𝑒₆. (44)Moreover Φ=𝑒^𝑇₁,𝑒^𝑇₂,𝑒^𝑇₁,𝑒^𝑇₂^𝑇,Δ=diagΔ,Δ,Δ,Δ,Ψ=𝑒^𝑇₁𝑁₁𝐴𝐿,𝑒^𝑇₁𝑁₁𝐵𝐿,𝑒^𝑇₆𝑁₂𝐴𝐿,𝑒^𝑇₆𝑁₂𝐵𝐿. (45a)(45b)From (35)–(45b), similar to (30)–(32d), there exist 𝛼₃ and 𝛼₄ such that 𝑉̇𝑦(𝑡)≤𝛼₃‖𝑥(𝑡)‖²,or𝑉̇𝑦(𝑡)≤𝛼₄‖𝑥(𝑡)‖², (46) ifΠ₍₁₎≜𝐸¹+𝐸₂⁽¹⁾+𝐸₃⁽¹⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₍₂₎≜𝐸¹+𝐸₂⁽¹⁾+𝐸₃⁽²⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₍₃₎≜𝐸¹+𝐸₂⁽²⁾+𝐸₃⁽¹⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₍₄₎≜𝐸¹+𝐸₂⁽²⁾+𝐸₃⁽²⁾−𝐸₁^(1)𝑇𝑃𝐸₁⁽¹⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇<0, (47a)(47b)(47c)(47d)or Π₍₁₎≜𝐸¹+𝐸₂⁽¹⁾+𝐸₃⁽¹⁾−𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₍₂₎≜𝐸¹+𝐸₂⁽¹⁾+𝐸₃⁽²⁾−𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₍₃₎≜𝐸¹+𝐸₂⁽²⁾+𝐸₃⁽¹⁾−𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇Π<0,₍₄₎≜𝐸¹+𝐸₂⁽²⁾+𝐸₃⁽²⁾−𝐸₁^(2)𝑇𝑃𝐸₁⁽²⁾+ΨΔΦ+Φ^𝑇ΔΨ^𝑇<0. (48a)(48b)(48c)(48d)

Theorem 6. For given scalars 𝜏_𝑙≥0, 𝜏_𝑢>0, 𝑏₁, 𝑏₂, 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑛), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑛), system (1) is globally asymptotically stable if there exist 𝑆^𝑖, 𝑆^𝑗∈𝑆_Δ(𝑖,𝑗=1,2,3,4) of appropriate dimensions such that the following conditions hold: Π_(𝑖)+Ψ^𝑇𝑆^𝑖ΦΨΦ_𝑇−𝑆^𝑖Π<0,^(𝑗)+Ψ^𝑇𝑆^𝑗ΨΦΦ^𝑇−𝑆^𝑗<0, (49) where 𝑆_Δ=∶{diag(𝑠₁𝐼,…,𝑠_𝑘𝐼,𝑆₁,…,𝑆_𝑙)∶𝑆_𝑖>0,𝑘,𝑙∈𝑁}.

Proof. By Lemma 3, the conditions in Theorem 6 are equivalent to (32a)–(32d) or (47a)–(47d). In view of the previous analysis from (30) to (32d) and (46) to (47d), one can obtain that there exist 𝛼₁>0and𝛼₃>0 such that 𝑉̇(𝑦(𝑡))≤−min(𝛼₁,𝛼₃)‖𝑥(𝑡)‖². By Lyapunov stable theory, system (1) is globally asymptotically stable, which completes the proof.

Remark 7. Different from previous work, the LKF function in this paper is constructed by using the convex combination between decomposed time delay and positive matrix, which may reduce the conservatism of criterion.

Remark 8. From the proof of Theorem 6, one can see that, by using the different combinations among Π_(𝑖), Π_(𝑖), and Π_(𝑖), Π_(𝑖), we can establish different stable criteria as follows.

Corollary 9. For given scalars 𝜏_𝑙≥0, 𝜏_𝑢>0, 𝑏₁, 𝑏₂, 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑛), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑛), system (1) is globally asymptotically stable if there exist 𝑆^𝑖, 𝑆^𝑗∈𝑆_Δ  (𝑖,𝑗=1,2,3,4) of appropriate dimensions such that the following conditions hold: Π_(𝑖)+Ψ^𝑇𝑆^𝑖ΦΨΦ_𝑇−𝑆^𝑖Π<0,^(𝑗)+Ψ^𝑇𝑆^𝑗ΨΦΦ^𝑇−𝑆^𝑗<0. (50)

Corollary 10. For given scalars 𝜏_𝑙≥0, 𝜏_𝑢>0, 𝑏₁, 𝑏₂, 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑛), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑛), system (1) is globally asymptotically stable if there exist 𝑆^𝑖, 𝑆^𝑗∈𝑆_Δ  (𝑖,𝑗=1,2,3,4) of appropriate dimensions such that the following conditions hold: Π_(𝑖)+Ψ^𝑇𝑆^𝑖ΦΨΦ_𝑇−𝑆^𝑖Π<0,^(𝑗)+Ψ^𝑇𝑆^𝑗ΨΦΦ^𝑇−𝑆^𝑗<0. (51)

Corollary 11. For given scalars 𝜏_𝑙≥0, 𝜏_𝑢>0, 𝑏₁, 𝑏₂, 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑛), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑛), system (1) is globally asymptotically stable if there exist 𝑆^𝑖, 𝑆^𝑗∈𝑆_Δ  (𝑖,𝑗=1,2,3,4) of appropriate dimensions such that the following conditions hold: Π_(𝑖)+Ψ^𝑇𝑆^𝑖ΦΨΦ_𝑇−𝑆^𝑖Π<0,^(𝑗)+Ψ^𝑇𝑆^𝑗ΨΦΦ^𝑇−𝑆^𝑗<0. (52)

Remark 12. Since the existence of items Ψ^𝑇𝑆^𝑖Ψ and Ψ^𝑇𝑆^𝑖Ψ, the results established in Theorem 6 and Corollaries 9–11 are not LMI criteria. In order to use LMI toolbox in computing software, by using the lemma derived in [20], we further establish the following more practicable stable rules.

Theorem 13. For given scalars 𝜏_𝑙≥0, 𝜏_𝑢>0, 𝑏₁, 𝑏₂, 𝐿⁻=diag(𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑛), 𝐿⁺=diag(𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑛), system (1) is globally asymptotically stable if there exist positive constants 𝛿_𝑖, 𝛿_𝑗>0  (𝑖,𝑗=1,2,3,4) such that the following conditions hold: 𝛿_𝑖Π_(𝑖)Ψ𝛿_𝑖Φ^𝑇𝛿∗−𝐼0∗∗−𝐼<0,^𝑗Π_(𝑗)Ψ𝛿_𝑗Φ^𝑇∗−𝐼0∗∗−𝐼<0. (53)

Remark 14. Similar to the analysis of Remark 12, the related practicable stable can also be established by using Corollaries 9–11, since the expressions are similar to Theorem 13, which are omitted here.

4. Illustrative Examples

In this section, two numerical examples are given to illustrate the effectiveness of the proposed method.

4.1. Numerical Examples

Example 1. Consider the delayed neural networks (1) with parameters given as (),,,𝐶=diag1.2769,0.6231,0.9230,0.4480𝐴=−0.03730.4852−0.33510.2336−1.60330.5988−0.32241.23520.3394−0.0860−0.3824−0.5785−0.13110.3253−0.9534−0.5015𝐵=0.8674−1.2405−0.53250.02200.0474−0.91640.03600.98161.84952.6117−0.37880.8428−2.04130.51791.1734−0.2775 (54)𝑔₁(𝑠)=tanh(−0.1137𝑠), 𝑔₂(𝑠)=tanh(−0.1279𝑠), 𝑔₃(𝑠)=tanh(−0.7994𝑠), 𝑔₄(𝑠)=tanh(−0.2368𝑠). Obviously, 𝑙⁻₁=𝑙⁻₂=𝑙⁻₃=𝑙⁻₄=0, 𝑙⁺₁=0.1137, 𝑙⁺₂=0.1279, 𝑙⁺₃=0.7994, 𝑙⁺₄=0.2368, 𝐿=diag(0,0,0,0).

Similar to [15], our purpose is to estimate the allowable upper bounds delay 𝜏_𝑢 under 𝜏_𝑙=0 such that the system (1) is globally asymptotically stable. For this example, when 𝜏̇(𝑡)=0, the maximum allowable delay bound 𝜏_𝑢 is 1.4224 in [9], 1.9321 in [10], 3.5841 in [11], 3.6156 in [13], and 3.7327 in [12]. Recently, by using delay-scope-dependent method, Li et al. improved the previous results further in [15] and gave out the maximum allowable delay bound 𝜏_𝑢 as 3.8363. Applying Theorem 13 in this paper, the maximum allowable delay bound is 3.9221 with 𝛿_𝑖=𝛿_𝑗=0.1, which means that, for this example, the result obtained in this paper is less conservative that those established in [9–13, 15]. Additionally, for this example, the computed variables in [12, 15, 21] are 130, 198, and 86, respectively. In Theorem 13, the computed variables are 150, which is less computationally demanding than in [21], but heavier than in [12, 15]. If 𝑄₁, 𝑄₂, 𝑄₃, 𝑃₁, 𝑃₂, 𝑃₃, 𝑃₄, 𝑃₅, and 𝑃₆ are all diagonal matrices, Theorem 13 established in this paper still holds. In this case, the computed variables in Theorem 13 are 96, which is less computationally demanding than in [12, 21], but heavier than in [15]. For the given initial value [6,7,−5−8], when 𝜏_𝑢=3.9221, the simulation result can be seen in Figure 1. Simulation result shows that, for the given parameters in Example 1, system (1) is asymptotically stable.

Figure 1 

The state variables of system (1) in Example 1.

4.2. An Application Example

Example 2. Consider the continuous pH neutralization of an acid stream by a highly concentrated basic stream, which can be expressed in the following form [13]: 𝑣𝑦̇(𝑡)=−𝑎𝑦(𝑡)−𝑢(𝑡),pH=𝑤₂tanh𝑤₁𝑦(𝑡), (55) where 𝑣 is the volume of the mixing tank, 𝑦(𝑡) is the strong acid, 𝑎 is the acid flow rate, 𝑢(𝑡) is the manipulated variable representing the base flow rate, pH is the measured output signal, and 𝑤₂ and 𝑤₁ are some constants.

The purpose of this application is to find the maximum allowable upper bound of delay 𝜏 for a feedback gain 𝐾 with output feedback controller 𝑢=−𝐾×pH such that the closed-loop system is asymptotically stable. In order to do this, we can rewrite system (55) in the following form: ̇𝑦(𝑡)=−𝐶𝑦(𝑡)+𝐴𝑓𝑦(𝑡)+𝐵𝑓𝑦(𝑡−𝜏), (56) where 𝑦(𝑡)=𝑤₁𝑦(𝑡), 𝑓(𝑦(𝑡))=tanh(𝑦(𝑡)), 𝐶=𝑎/𝑣, 𝐴=𝐾𝑤₁w₂/𝑣, 𝐵=𝐾𝑤₁𝑤₂𝑤₃/𝑣. For this application problem, [16, 17] gave out the maximum allowable upper bound of delay 𝜏 as 17.4956 when the parameters are given as 𝑎=5.8154, 𝑣=1500.3732, 𝑤₁=28.9860, 𝑤₂=−3.8500, and 𝑤₃=2.56, and the feedback gain 𝐾 is selected as 𝐾=0.5022. Recently, by employing weighting-delay method, the maximum allowable upper bound of delay 𝜏 is improved to 18.2871 in [13]. Meanwhile, by using Theorem 13, the maximum allowable upper bound of delay 𝜏 is 18.7436. Namely, a little better result can be obtained by using our criteria. For a given initial value 𝑦₀=0.5, when 𝜏_𝑢=18.7436, the simulation result can be seen in Figure 2. Simulation result shows that, for the given parameters in Example 2, system (55) is asymptotically stable.

Figure 2 

The state variables of system (55) in Example 2.

5. Conclusions

Combined with delay partitioning technique, by using the convex combination between decomposed time delay and positive matrix, this paper researches the stability problem of a class of delayed neural networks with interval time-varying delays. The benefit of the method used in this paper is that it can utilize more information on the slope of activations and time delays. Illustrative examples show that the new criteria derived in this paper are less conservative than some previous results obtained in the references cited therein.

Acknowledgments

This work was supported by China Postdoctoral Science Foundation Grant (2012 M521718) and Soft Science Research Project in Guizhou Province ([2011]LKC2004).