Less Conservative Stability Criteria for Neutral Type Neural Networks with Mixed Time-Varying Delays

Abstract

This paper investigates the problem of dependent stability criteria for neutral type neural networks with mixed time-varying delays. Firstly, some new delay-dependent stability results are obtained by employing the more general partitioning approach and generalizing the famous Jensen inequality. Secondly, based on a new type of Lyapunov-Krasovskii functional with the cross terms of variables, less conservative stability criteria are proposed in terms of linear matrix inequalities (LMIs). Furthermore, it is the first time that the idea of second-order convex combination and the property of quadratic convex function applied to the derivation of neutral type neural networks play an important role in reducing the conservatism of the paper. Finally, four numerical examples are given to show the effectiveness and the advantage of the proposed method.

1. Introduction

During the last two decades, delayed neural networks have drawn a great deal of attention because of their extensive applications in various scientific and technical areas, such as pattern recognition, power systems, parallel computing, signal processing, finance, associative memories, mechanics of structures, and other scientific areas [1–30]. It is well known that time delay regarded as a major cause of instability and poor performance often appears in many neural networks. Therefore, the stability analysis for delayed neural networks has been investigated extensively in recent few decades. Generally speaking, studying the dynamical behavior of delayed neural networks can be mainly classified into two types: delay-independent stability and delay-dependent stability. As is known to all, delay-dependent stability criteria are less conservative than delay-independent ones when the size of time delay is small.

On the other hand, due to the complicated dynamic properties of the neural cells in the real world, there exist many neural network models such as distributed networks, chemical reactors, and heat exchanges that cannot characterize the properties of a neural reaction process precisely. It is natural and important that these systems will contain some information about the derivative of the past state to further describe and model the dynamics of the complex neural reactions. This new type of neural networks is called neutral neural networks or neural networks of neutral type. However, many researchers have focused on the global stability of neural networks of neutral type only with constant time delay in recent years, which is very restrictive. Hence, described with neutral functional differential equations with discrete and distributed delays, these neural networks called neutral type neural networks with mixed time-varying delays have a lot of research on space. The differential expression not only defines the derivative term of the current state but also explains the derivative term of the past state. Furthermore, it is necessary to have some information about the derivative of the past state in the systems to characterize the dynamics of such complex neural reactions. Practically, neutral type phenomenon always appears in studies of automatic control, chemical reactors, distributed networks, dynamic process including steam and water pipes, population ecology, heat exchanges, microwave oscillators, systems of turbojet engine, lossless transmission lines, vibrating masses attached to an elastic bar, and so on. For this reason, there has been a growing research interest in the study of delayed neural networks of neutral type in the recent years. Therefore, some less conservative stability criteria for neutral type neural networks with mixed time-varying delays have been reported in recently [25, 31–35]. Many methods have been proposed in these results to reduce the conservatism of the stability criteria, such as model transformation method, free-weighting matrix method, the method of constructing novel Lyapunov-Krasovskii functionals, delay decomposition technique, and weighting-matrix decomposition method. In [36], the authors derived some less conservative stability criteria by considering some useful terms and using free-weighting matrix technique. By considering the relationship between the time-varying delay and its lower and upper bound, the results obtained in [36] were improved in [37]. By constructing a new Lyapunov-Krasovskii functional and using free-weighting matrix method, some more less conservative criteria than those obtained in [37] were proposed in [38]. Further, the problems of stability analysis of neutral type neural networks with discrete and distributed delays have been investigated in [39]. By using a delay-partitioning approach, a new type of Lyapunov-Krasovskii functionals was constructed to obtain some less conservative stability criteria. However, time delay in [39] is not only constant delay, but also the delay-partitioning approach is equational; hence, this method has some limitations.

Motivated by this technique, it is the first attempt to investigate the integral nonuniform partitioning method to be extended for neutral type neural networks with mixed time-varying delays. In the paper, the reduced conservatism of Theorem 6 benefits from the construction of the new Lyapunov-Krasovskii functionals in (17), which contain some integral nonuniform partitioning method and triple-integral terms, which play an important role in the improvement of less conservative results. Secondly, a novel handling method is given to establish the relationship among ∫^𝑡_𝑡−ℎ𝑥̇(𝑠)𝑆₅𝑥̇(𝑠)𝑑𝑠, ∫^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠 and 𝑥(𝑡−ℎ), which play an important role in reducing the conservatism of stability criteria further. Furthermore, compared with previous results by using the first-order convex combination property, our derivation makes full use of the idea of second-order convex combination and the property of quadratic convex function given in the form of a lemma without employing Jensen's inequality. Finally, four numerical examples are given to illustrate the effectiveness and the advantage of the proposed main results.

Notation 1. Notations used in this paper are fairly standard: 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space, 𝑅^𝑛×𝑚 is the set of all 𝑛×𝑚 dimensional matrices; 𝐼 denotes the identity matrix of appropriate dimensions, 𝑇 stands for matrix transposition, the natation 𝑋>0 (resp., 𝑋≥0), for 𝑋∈𝑅^𝑛×𝑛 means that the matrix is real symmetric positive definite (resp., positive semidefinite); diag{𝑟₁,𝑟₂,…,𝑟_𝑛} denotes block diagonal matrix with diagonal elements 𝑟_𝑖, 𝑖=1,2,…,𝑛, the symbol ∗ represents the elements below the main diagonal of a symmetric matrix, and ⟨𝑀⟩_𝑠 is defined as ⟨𝑀⟩_𝑠=(1/2)(𝑀+𝑀^𝑇).

2. Preliminaries

Consider the following neural networks of neutral type with mixed time-varying delays: 𝑧̇(𝑡)=−𝑊₀𝑧(𝑡)+𝑊₁𝑔(𝑧(𝑡))+𝑊₂𝑔(𝑧(𝑡−ℎ(𝑡)))+𝑊₃^𝑡_𝑡−𝑟𝑔^𝑇(𝑧(𝑠))𝑑𝑠+𝑊₄𝑦̇(𝑡−ℎ(𝑡))+𝐽, (1) where 𝑧(𝑡)=[𝑧₁(𝑡),…,𝑧_𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the neural state vector, 𝑔(𝑧(𝑡))=[𝑔(𝑧₁(𝑡)),…,𝑔(𝑧_𝑛(𝑡))]^𝑇 is the neuron activation function, 𝐽=[𝐽₁,…,𝐽_𝑛]^𝑇∈𝑅^𝑛 is an external constant input vector, and 𝑊₀=diag{𝑤₀₁,…,𝑤_0𝑛}>0, 𝑊₁, 𝑊₂, 𝑊₃, and 𝑊₄ are the constant matrices of appropriate dimensions.

Assumption A. The time-varying delay ℎ(𝑡) is continuous and differential function that satisfies 0≤ℎ(𝑡)≤ℎ,ℎ̇(𝑡)≤ℎ_𝐷<1. (2)

Assumption B. For the constants 𝛿⁺_𝑖, 𝛿⁻_𝑖, the nonlinear function 𝑔_𝑖(⋅) in (1) satisfies the following condition: 𝛿⁻_𝑖≤𝑔_𝑖(𝛼)−𝑔_𝑖𝛽𝛼−𝛽≤𝛿⁺_𝑖,∀𝛼,𝛽∈𝑅,𝛼≠𝛽,𝑖=1,2,…,𝑛. (3) Here, we denote Σ=diag{𝛿⁺₁,…,𝛿⁺_𝑛}, Σ=diag{𝛿⁻₁,…,𝛿⁻_𝑛}, Σ=diag{max{|𝛿⁺₁|,|𝛿⁻₁|},…,max{|𝛿⁺_𝑛|,|𝛿⁻_𝑛|}}=diag{𝛿₁,…,𝛿_𝑛}, Σ₁=diag{𝛿⁺₁𝛿⁻₁,…,𝛿⁺_𝑛𝛿⁻_𝑛}, Σ₂=diag{(𝛿⁺₁+𝛿⁻₁)/2,…,(𝛿⁺_𝑛+𝛿⁻_𝑛)/2}.

Assumption C. For given positive scalars 𝜌_𝑖 satisfies: 0=𝜌₀<𝜌₁<⋯<𝜌_𝑙=1(𝑖=1,…,𝑙). (4) It is clear that under Assumption B, the system (1) has one equilibrium point denoted as 𝑧^∗=[𝑧^∗₁,…,𝑧^∗_𝑛]^𝑇. For convenience, we firstly shift the equilibrium point 𝑧^∗ to the origin by letting 𝑥(𝑡)=𝑧(𝑡)−𝑧^∗, 𝑓(𝑥(𝑡))=𝑔(𝑧(𝑡))−𝑔(𝑧^∗); then the system (1) can be transformed into 𝑥̇(𝑡)=−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑊₃^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠+𝑊₄𝑥̇(𝑡−ℎ(𝑡)), (5) where 𝑥(𝑡)=[𝑥₁(𝑡),…,𝑥_𝑛(𝑡)]∈𝑅^𝑛 is the state vector of transformed system, 𝑓(𝑥(𝑡))=[𝑓₁(𝑥₁),…,𝑓_𝑛(𝑥_𝑛)]^𝑇∈𝑅^𝑛. It is easy to check that the transformed neuron activation function 𝑓_𝑖(⋅) satisfies 𝛿⁻_𝑖≤𝑓_𝑖(𝛼)𝛼≤𝛿⁺_𝑖,𝑓_𝑖(0)=0,∀𝛼∈𝑅,𝛼≠0,𝑖=1,2,…,𝑛. (6) The following lemmas are introduced, which will be used in the proof of the main results.

Fact 1 (Boyd et al. [40], (Schur complement)). For a given symmetric matrix 𝑋=𝑋^𝑇=^𝑋11𝑋12_∗𝑋22, where 𝑆₁₁∈𝑅^𝑛×𝑛, the following conditions are equivalent: (1)𝑋<0; (2)𝑋₁₁<0, 𝑋₂₂−𝑋^𝑇₁₂𝑋⁻¹₁₁𝑋₁₂<0;(3)𝑋₂₂<0, 𝑋₁₁−𝑋₁₂𝑋⁻¹₂₂𝑋^𝑇₁₂<0.

Lemma 1 (see [41]). For symmetric matrices 𝑌₀, 𝑌₁, 𝑌₂, and a vector 𝜉_𝑡, Let 𝑓(𝛼)=𝛼²𝜉^𝑇_𝑡𝑌₂𝜉_𝑡+𝛼𝜉^𝑇_𝑡𝑌₁𝜉_𝑡+𝜉^𝑇_𝑡𝑌₀𝜉_𝑡 with 𝑌₂≤0. Then we have 𝑓(𝛼₁)<0 and 𝑓(𝛼₂)<0⇒𝑓(𝛼)<0,  ∀𝛼∈[𝛼₁,𝛼₂].

Lemma 2 (see [42]). Let 𝑊>0, and let 𝜔(𝑠) be an appropriate dimensional vector. Then, we have the following facts for any scalar function 𝛽(𝑠)≥0  ∀𝑠∈[𝑎,𝑏]:(1)−∫^𝑏_𝑎𝜔^𝑇(𝑠)𝑊𝜔(𝑠)𝑑𝑠≤(𝑏−𝑎)𝜉^𝑇_𝑡𝐹₁𝑊⁻¹𝐹₁𝜉_𝑡+2𝜉^𝑇_𝑡𝐹₁∫^𝑏_𝑎𝜔(𝑠)𝑑𝑠; (2)−∫^𝑏_𝑎𝛽(𝑠)𝜔^𝑇(𝑠)𝑊𝜔(𝑠)𝑑𝑠≤∫^𝑏_𝑎𝛽(𝑠)𝑑𝑠𝜉^𝑇_𝑡𝐹₂𝑊⁻¹𝐹₂𝜉_𝑡+2𝜉^𝑇_𝑡𝐹₂∫^𝑏_𝑎𝛽(𝑠)𝜔(𝑠)𝑑𝑠; (3)−∫^𝑏_𝑎𝛽²(𝑠)𝜔^𝑇(𝑠)𝑊𝜔(𝑠)𝑑𝑠≤(𝑏−𝑎)𝜉^𝑇_𝑡𝐹₃𝑊⁻¹𝐹₃𝜉_𝑡+2𝜉^𝑇_𝑡𝐹₃∫^𝑏_𝑎𝛽(𝑠)𝜔(𝑠)𝑑𝑠, where matrices 𝐹_𝑖  (𝑖=1,2,3) and a vector 𝜉_𝑡 independent of the integral variable are appropriate dimensional arbitrary ones.

Lemma 3 (see [43]). For any constant matrix 0<𝑅=𝑅^𝑇∈𝑅^𝑛×𝑛, a scalar 𝑟>0 and a vector function 𝑥∶[0,𝑟]→𝑅^𝑛 such that the integrations concerned are well defined; then −⁰_𝑡−𝑟𝑥^𝑇(𝑠)(𝑠)1𝑅𝑥𝑑𝑠≤−𝑟^𝑡_𝑡−𝑟𝑥(𝑠)𝑑𝑠^𝑇𝑅^𝑡_𝑡−𝑟𝑥(𝑠),−𝑑𝑠_{0−𝑟𝑡𝑡+𝜃}𝑥^𝑇(𝑠)(𝑠)2𝑅𝑥𝑑𝑠𝑑𝜃≤−𝑟²⁰_−𝑟^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑅⁰_−𝑟^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃. (7)

Lemma 4 (see [44]). For any constant matrix 𝑅∈𝑅^𝑛×𝑛, 𝑅=𝑅^𝑇>0, a scalar function ℎ∶=ℎ(𝑡)>0, and a vector-valued function 𝑥̇∶[−ℎ,0]→𝑅^𝑛 such that the following integrations are well defined: −ℎ^𝑡_𝑡−ℎ𝑥̇(𝑠)𝑅𝑥̇(𝑠)𝑥(𝑡)𝑥()𝑑𝑠≤𝑡−ℎ^𝑇𝑥(𝑡)𝑥(),−ℎ−𝑅𝑅𝑅−𝑅𝑡−ℎ₂2⁰_−ℎ^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)≤(𝑡)𝑑𝑠𝑑𝜃ℎ𝑥_{𝑡𝑡−ℎ}𝑥(𝑠)𝑑𝑠^𝑇−𝑅𝑅(𝑡)𝑅−𝑅ℎ𝑥_{𝑡𝑡−ℎ}𝑥(𝑠).𝑑𝑠 (8)

Lemma 5. Let 𝑥(𝑡)∈𝑅^𝑛 has continuous derived function 𝑥̇(𝑡) on interval [0,ℎ]. Then for any matrix 𝑍^𝑛×𝑛>0, scalar ℎ>0, the following inequality holds: −ℎ^𝑡_𝑡−ℎ𝑥̇(𝑠)𝑅𝑥̇(𝑠)2𝑑𝑠≤−ℎ1ℎ^𝑡_𝑡−ℎ𝑥(𝑠)𝑥()𝑑𝑠𝑡−ℎ^𝑇1𝑅−𝑅−𝑅𝑅ℎ^𝑡_𝑡−ℎ𝑥(𝑠)𝑥().𝑑𝑠𝑡−ℎ (9)

Proof. From Lemma 3, we can get 1ℎ^𝑡_𝑡−ℎ^𝜃_𝑡−ℎ𝑥̇(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑅^𝑡_𝑡−ℎ^𝜃_𝑡−ℎ𝑥̇(𝑠)≤𝑑𝑠𝑑𝜃_{𝑡𝑡−ℎ𝜃𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠^𝑇𝑅^𝜃_𝑡−ℎ𝑥̇(𝑠)≤𝑑𝑠𝑑𝜃_{𝑡𝑡−ℎ𝜃𝑡−ℎ}(𝜃−(𝑡−ℎ))𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)≤𝑑𝑠𝑑𝜃_{𝑡𝑡−ℎ𝑡𝑠}(𝜃−(𝑡−ℎ))𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)=𝑑𝜃𝑑𝑠_{𝑡𝑡−ℎ}ℎ²2−(𝑠−𝑡+ℎ)²2𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)≤ℎ𝑑𝑠₂2^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠. (10) Notice that 𝑥̇^𝑡_𝑡−ℎ^𝜃_𝑡−ℎ(𝑠)𝑑𝑠𝑑𝜃^𝑇𝑅^𝑡_𝑡−ℎ^𝜃_𝑡−ℎ𝑥̇(𝑠)=𝑑𝑠𝑑𝜃_{𝑡𝑡−ℎ}𝑥(𝑠)()𝑑𝑠ℎ𝑥𝑡−ℎ^𝑇𝑅−𝑅−𝑅𝑅^{𝑡𝑡−ℎ}𝑥(𝑠)().𝑑𝑠ℎ𝑥𝑡−ℎ (11) Therefore, we get −^𝑡_𝑡−ℎ𝑥̇(𝑠)𝑅𝑥̇(𝑠)2𝑑𝑠≤−ℎ1ℎ^𝑡_𝑡−ℎ𝑥(𝑠)𝑥()𝑑𝑠𝑡−ℎ^𝑇1𝑅−𝑅−𝑅𝑅ℎ^𝑡_𝑡−ℎ𝑥(𝑠)𝑥().𝑑𝑠𝑡−ℎ (12) This completes the proof.

3. Main Results

In this section we will give sufficient conditions under which the system (5) is asymptotically stable.

Theorem 6. For given scalars ℎ>0 and ℎ_𝐷<1, the system (5) with the neuron activation function 𝑓(𝑥(𝑡)) satisfying the condition (6) is asymptotically stable if there exists 𝑃>0,   𝑅^𝑇_𝑖=𝑅_𝑖>0(𝑖=1,2,3),  𝑄^𝑇_𝑖=𝑄_𝑖>0(𝑖=1,2),  𝑆^𝑇_𝑖=𝑆_𝑖>0(𝑖=1,…,8),  𝑍^𝑇_𝑖=𝑍_𝑖>0(𝑖=1,…,𝑙), diagonal matrices 𝐺_𝑖=diag{𝑔_𝑖1,𝑔₂,…,𝑔_𝑖𝑛}>0,  𝐾_𝑖=diag{𝑘_𝑖1,𝑘₂,…,𝑘_𝑖𝑛}>0,  𝐿_𝑖=diag{𝑙_𝑖1,𝑙₂,…,𝑙_𝑖𝑛}>0,  (𝑖=1,2,3) and 𝑀=diag{𝑚_𝑖1,𝑚_𝑖2,…,𝑚_𝑖𝑛}>0,  (𝑖=1,2,3,4),  𝑇_𝑖(𝑖=1,…,5) and 𝐹_𝑖(𝑖=1,…,6) with appropriate dimensions such that the following symmetric linear matrix inequality holds: Ξ₁=Π₀ℎ𝐹^𝑇₁ℎ𝐹^𝑇₂ℎ𝐹^𝑇₃∗−ℎ𝑅₃00∗∗−𝑄₁0ℎ∗∗∗−3𝑄₂Ξ<0,₂=Π₀+ℎΠ₁ℎ𝐹^𝑇₄ℎ𝐹^𝑇₅ℎ𝐹^𝑇₆∗−ℎ𝑅₃00∗∗−𝑄₁0ℎ∗∗∗−3𝑄₂<0, (13)(14) where Π₀=2𝑒₁𝑒₇+𝑒₈𝑃𝑒₂𝑒₁−𝑒₄^𝑇+2𝑒₂0𝑅₁0𝑒₇+𝑒₈^𝑇+2𝑒₂0𝑅₂ℎ𝑒₁𝑒₈^𝑇_𝑠+𝑒₁𝑒₁×𝑅₁+𝑅₂𝑒₁𝑒₁^𝑇−1−ℎ_𝐷𝑒₁𝑒₃𝑅₁𝑒₁𝑒₃^𝑇−𝑒₁𝑒₄𝑅₂𝑒₁𝑒₄^𝑇+ℎ𝑒₁𝑒₂𝑅₃×𝑒₁𝑒₂^𝑇+𝑒₂ℎ²𝑄₁+ℎ³𝑄₂𝑒^𝑇₂+2𝐹^𝑇₁𝑒₇𝑒₃−𝑒₄^𝑇+4𝐹^𝑇₂ℎ𝑒₃−𝑒₇^𝑇+6𝐹^𝑇₃ℎ𝑒₃−𝑒₇^𝑇+2𝐹^𝑇₄𝑒₈𝑒₁−𝑒₃^𝑇−4𝐹^𝑇₅𝑒^𝑇₈−6𝐹^𝑇₆𝑒^𝑇_8𝑠+𝑒₉𝑆₁+𝑆₂𝑒^𝑇₉−1−ℎ_𝐷𝑒₁₀𝑆₁𝑒^𝑇₁₀−𝑒₁₁𝑆₂𝑒^𝑇₁₁−𝑒₆𝑆₄𝑒^𝑇₆+𝑒₂𝑆₃+𝑆₄𝑒^𝑇₂−1−ℎ_𝐷𝑒₅𝑆₃𝑒^𝑇₅+𝑒₂ℎ𝑆₅+ℎ²2𝑆₆𝑒^𝑇₂−2𝑒₁𝑆₆𝑒^𝑇₁+4ℎ𝑒₁𝑆₆𝑒^𝑇₇+4ℎ𝑒₁𝑆₆𝑒^𝑇₈+4ℎ²𝑒₄𝑆₅𝑒^𝑇₇+4ℎ²𝑒₄𝑆₅𝑒^𝑇₈−2ℎ𝑒₄𝑆₅𝑒^𝑇₄−𝑒₇2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₇−𝑒₇4ℎ³𝑆₅+4ℎ²𝑆₆𝑒^𝑇₈−𝑒₈2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₈+𝑒₉𝑟𝑆₇+𝑟²2𝑆₈+^𝑙_𝑖=1𝜌_𝑖−𝜌_𝑖−1𝑟𝑍_𝑖𝑒^𝑇₉−^𝑙_𝑖=11𝜌_𝑖−𝜌_𝑖−1𝑟𝑒_11+𝑖𝑍_𝑖𝑒^𝑇_11+𝑖−1𝑟𝑒₁₂+⋯+𝑒_11+𝑙𝑆₇𝑒₁₂+⋯+𝑒_11+𝑙^𝑇−2𝑟²𝑒_12+𝑙𝑆₈𝑒^𝑇_12+𝑙+𝑒₁Σ𝐾₁−Σ𝐺₁+𝐿₁Σ𝑒^𝑇₂+2𝑒₂𝐺₁−𝐾₁+𝐿₁𝑒^𝑇₉+2𝑒₃Σ𝐾₂−Σ𝐺₂+𝐿₂Σ𝑒^𝑇₅+2𝑒₅𝐺₂−𝐾₂+𝐿₂𝑒^𝑇₁₀+2𝑒₄Σ𝐾₃−Σ𝐺₃+𝐿₃Σ𝑒^𝑇₆+2𝑒₆𝐺₃−𝐾₃+𝐿₃𝑒^𝑇₁₁+𝑒₁Σ𝑀₁Σ−𝑀₂Σ₁𝑒^𝑇₁+2𝑒₁𝑀₂Σ₂𝑒^𝑇₉−𝑒₃𝑀₃Σ₁𝑒^𝑇₃+2𝑒₃𝑀₃Σ₂𝑒^𝑇₁₀−𝑒₉𝑀₁+𝑀₂𝑒^𝑇₉−𝑒₁₀𝑀₃𝑒^𝑇₁₀−𝑒₄𝑀₄Σ₁𝑒^𝑇₄+2𝑒₄𝑀₄Σ₂𝑒^𝑇₁₁+𝑒₂−𝑇₁−𝑇^𝑇₁𝑒^𝑇₂−𝑒₁₁𝑀₄𝑒^𝑇₁₁−2𝑒₁𝑊^𝑇₀𝑇^𝑇₁𝑒^𝑇₂+2𝑒₂𝑇₁𝑊₁𝑒^𝑇₉+2𝑒₂𝑇₁𝑊₂𝑒^𝑇₁₀+2𝑒₂𝑇₁𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇+2𝑒₂𝑇₁𝑊₄𝑒^𝑇₅−2𝑒₁𝑇₂𝑒^𝑇₂+𝑒₁−𝑇₂𝑊₀−𝑊^𝑇₀𝑇^𝑇₂𝑒^𝑇₁+2𝑒₁𝑇₂𝑊₁𝑒^𝑇₉+2𝑒₁𝑇₂𝑊₂𝑒^𝑇₁₀+2𝑒₁𝑇₂𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇+2𝑒₂𝑇₂𝑊₄𝑒^𝑇₅−2𝑒₂𝑇^𝑇₃𝑒^𝑇₉−2𝑒₁𝑊^𝑇₀𝑇^𝑇₃𝑒^𝑇₉+2𝑒₉𝑇₃𝑊₂𝑒^𝑇₁₀+𝑒₉T₃𝑊₁+𝑊^𝑇₁𝑇^𝑇₃𝑒^𝑇₉+2𝑒₉𝑇₃𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇+2𝑒₅𝑊^𝑇₄𝑇^𝑇₃𝑒^𝑇₉−2𝑒₂𝑇^𝑇₄𝑒^𝑇₁₀+𝑒₉𝑊^𝑇₁𝑇^𝑇₄𝑒^𝑇₁₀+𝑒₁₀𝑇₄𝑊₂+𝑊^𝑇₂𝑇^𝑇₄𝑒^𝑇₁₀+2𝑒₁₀𝑇₄𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇+2𝑒₅𝑊^𝑇₄𝑇^𝑇₄𝑒^𝑇₁₀+2𝑒₅𝑊^𝑇₄𝑇^𝑇₃𝑒^𝑇₉−2𝑒₂𝑇^𝑇₅𝑒^𝑇₅−2𝑒₁𝑊^𝑇₀𝑇^𝑇₅𝑒^𝑇₅+𝑒₅𝑇₅𝑊₁𝑒^𝑇₉+𝑒₅𝑇₅𝑊₂𝑒^𝑇₁₀+2𝑒₅𝑇₅𝑊₄+𝑊^𝑇₄𝑇^𝑇₄𝑒^𝑇₅+2𝑒₅𝑇₅𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇−2𝑒₁𝑊^𝑇₀𝑇^𝑇₄𝑒^𝑇₁₀,Π₁=2𝑒₂0𝑅₁𝑒₁𝑒₀^𝑇−4𝐹^𝑇₂𝑒^𝑇₃−6𝐹^𝑇₃𝑒^𝑇₃+4𝐹^𝑇₅𝑒^𝑇₁+6𝐹^𝑇₆𝑒^𝑇_1𝑠, (15) with ⟨⋅⟩_𝑠=12(⋅)+(⋅)^𝑇,𝑒_𝑖=0_{𝑛×(𝑖−1)}𝐼_𝑛×𝑛0_{𝑛×(12+𝑙−𝑖)}^𝑇(𝑖=1,…,12+𝑙). (16)

Proof. Consider a novel augmented Lyapunov-Krasovskii functional for the system (5) as follows: 𝑉𝑥_𝑡=𝑉₁𝑥_𝑡+𝑉₂𝑥_𝑡+𝑉₃𝑥_𝑡+𝑉₄𝑥_𝑡+𝑉₅𝑥_𝑡+𝑉₆𝑥_𝑡+𝑉₇𝑥_𝑡, (17) where 𝑉₁𝑥_𝑡=𝜉^𝑇₁(𝑡)𝑃𝜉₁(𝑡),𝑉₂𝑥_𝑡=^𝑡_{(𝑡)𝑡−ℎ}𝜉^𝑇₂(𝑡,𝑠)𝐸₁𝐸₂𝑅₁𝐸₁𝐸₂^𝑇𝜉₂()+𝑡,𝑠𝑑𝑠_{𝑡𝑡−ℎ}𝜉^𝑇₂(𝑡,𝑠)𝐸₁𝐸₂𝑅₂𝐸₁𝐸₂^𝑇𝜉₂()𝑉𝑡,𝑠𝑑𝑠,₃𝑥_𝑡=^𝑡_𝑡−ℎ(ℎ−𝑡+𝑠)𝜉^𝑇₂(𝑡,𝑠)𝐸₂𝐸₃𝑅₃𝐸₂𝐸₃^𝑇𝜉₂(𝑡,𝑠)𝑑𝑠+^𝑡_𝑡−ℎ(ℎ−𝑡+𝑠)²𝜉^𝑇₂(𝑡,𝑠)𝐸₃𝑄₁𝐸^𝑇₃𝜉₂()+𝑡,𝑠𝑑𝑠_{𝑡𝑡−ℎ}(ℎ−𝑡+𝑠)³𝜉^𝑇₂(𝑡,𝑠)𝐸₃𝑄₂𝐸^𝑇₃𝜉₂()𝑉𝑡,𝑠𝑑𝑠,₄𝑥_𝑡=^𝑡_{(𝑡)𝑡−ℎ}𝑓^𝑇(𝑥(𝑠))𝑆₁𝑓(𝑥(𝑠+))𝑑𝑠_{𝑡𝑡−ℎ}𝑓^𝑇(𝑥(𝑠))𝑆₂𝑓(𝑥(𝑠+))𝑑𝑠_{𝑡(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑆₃𝑥̇(𝑠)𝑑𝑠+^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑆₄𝑥̇(𝑠)𝑉𝑑𝑠,₅𝑥_𝑡=^𝑡_𝑡−ℎ(ℎ−𝑡+𝑠)𝑥̇^𝑇(𝑠)𝑆₅𝑥̇(𝑠)+𝑑𝑠_{0−ℎ0𝜃𝑡𝑡+𝜆}𝑥̇^𝑇(𝑠)𝑆₆𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜆𝑑𝜃,₆𝑥_𝑡=^𝑡_𝑡−𝑟(𝑟−𝑡+𝑠)𝑓^𝑇(𝑥(𝑠))𝑆₇𝑓(𝑥(𝑠+))𝑑𝑠_{0−𝑟0𝜃𝑡𝑡+𝜆}𝑓^𝑇(𝑥(𝑠))𝑆₈𝑓(𝑥(𝑠))𝑑𝑠𝑑𝜆𝑑𝜃+2^𝑙_𝑖=1^{−𝜌𝑖−1𝑟}_{−𝜌𝑖𝑟}^𝑡_𝑡+𝜃𝑓(𝑥(𝑠))𝑍_𝑖𝑓(𝑥(𝑠𝑉))𝑑𝑠𝑑𝜃,₇𝑥_𝑡=2^𝑛_𝑖=1𝑔_1𝑖^𝑥𝑖₀^(𝑡)𝑓_𝑖(𝑠)−𝛿⁻_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑘_1𝑖^𝑥𝑖₀^(𝑡)𝛿⁺_𝑖𝑠−𝑓_𝑖(𝑠)𝑑𝑠+2^𝑛_𝑖=1𝑙_1𝑖^𝑥𝑖₀^(𝑡)𝑓_𝑖(𝑠)+𝛿_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑔_2𝑖^𝑥𝑖₀^{(𝑡−ℎ(𝑡))}𝑓_𝑖(𝑠)−𝛿⁻_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑘_2𝑖^𝑥𝑖₀^{(𝑡−ℎ(𝑡))}𝛿⁺_𝑖𝑠−𝑓_𝑖(𝑠)𝑑𝑠+2^𝑛_𝑖=1𝑙_2𝑖^𝑥𝑖₀^{(𝑡−ℎ(𝑡))}𝑓_𝑖(𝑠)+𝛿_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑔_3𝑖^𝑥𝑖₀^(𝑡−ℎ)𝑓_𝑖(𝑠)−𝛿⁻_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑘_3𝑖^𝑥𝑖₀^(𝑡−ℎ)𝛿⁺_𝑖𝑠−𝑓_𝑖(𝑠)𝑑𝑠+2^𝑛_𝑖=1𝑙_3𝑖^𝑥𝑖₀^(𝑡−ℎ)𝑓_𝑖(𝑠)+𝛿_𝑖𝑠𝑑𝑠, (18)(19)(20)(21)(22)(23) with 𝜉^𝑇₁(𝑡)=𝑥^𝑇(𝑡)^𝑡_𝑡−ℎ𝑥^𝑇(𝑠),𝜉𝑑𝑠_𝑇2(𝑡,𝑠)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠),𝜉^𝑇₃(𝑡)=𝑥^𝑇(𝑡)𝑥̇^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑥^𝑇()𝑥̇𝑡−ℎ_𝑇(𝑡−ℎ(𝑡))𝑥̇^𝑇(),𝜉𝑡−ℎ_𝑇4(𝑡)=∫^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠∫^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑑𝑠𝑓^𝑇(𝑥(𝑡𝑓))_𝑇(𝑥(𝑡−ℎ(𝑡)))𝑓^𝑇(𝑥(,𝜉𝑡−ℎ))_𝑇5(𝑡)=^𝑡_{𝑡−𝜌1𝑟}𝑓^𝑇(𝑥(𝑠))𝑑𝑠⋯^{𝑡−𝜌𝑙−1𝑟}_𝑡−𝑟𝑓^𝑇(𝑥(𝑠,𝜉))𝑑𝑠_𝑇(𝑡)=𝜉^𝑇₃(𝑡)𝜉^𝑇₄(𝑡)𝜉^𝑇₅(𝑡)⁰_−𝑟^𝑡_𝑡+𝜃𝑓^𝑇(𝑥(𝑠,𝐸))𝑑𝑠𝑑𝜃_𝑖=0_{𝑛×(𝑖−1)}𝐼_𝑛×𝑛0_{𝑛×(3−𝑖)}^𝑇(𝑖=1,2,3). (24) The time derivative of  𝑉(𝑥_𝑡) along the trajectory of system (5) is given as 𝑉̇𝑥_𝑡=𝑉̇₁𝑥_𝑡+𝑉̇₂𝑥_𝑡+𝑉̇₃𝑥_𝑡+𝑉̇₄𝑥_𝑡+𝑉̇₅𝑥_𝑡+𝑉̇₆𝑥_𝑡+𝑉̇₇𝑥_𝑡, (25) where 𝑉̇₁𝑥_𝑡=2𝜉^𝑇₁(𝑡)𝑃𝜉̇₁(𝑡)=2𝜉^𝑇(𝑡)𝑒₁𝑒₇+𝑒₈𝑃𝑒₂𝑒₁−𝑒₄^𝑇𝜉(𝑡),𝑉̇₂𝑥_𝑡≤𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)𝑅₁𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)^𝑇+2𝑥̇^𝑇(𝑡)0𝑅₁^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑡)𝑥^𝑇(𝑠)^𝑇𝑑(𝑠)−1−ℎ_𝐷×𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑅₁×𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))^𝑇+𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)𝑅₂×𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)^𝑇+2𝑥̇^𝑇(𝑡)0𝑅₂×^𝑡_𝑡−ℎ𝑥^𝑇(𝑡)𝑥^𝑇(𝑠)^𝑇𝑑(𝑠)−𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ)×𝑅₂𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ)^𝑇=𝜉^𝑇(𝑡)𝑒₁𝑒₁𝑅₁𝑒₁𝑒₁^𝑇+2𝑒₂0×𝑅₁ℎ(𝑡)𝑒₁𝑒₈^𝑇−1−ℎ_𝐷×𝑒₁𝑒₃𝑅₁𝑒₁𝑒₃^𝑇+𝑒₁𝑒₁×𝑅₂𝑒₁𝑒₁^𝑇+2𝑒₂0𝑅₂×ℎ𝑒₁𝑒₇+𝑒₈^𝑇−𝑒₁𝑒₄×𝑅₂𝑒₁𝑒₄^𝑇𝜉(𝑡),𝑉̇₃𝑥_𝑡=ℎ𝜉₂(𝑡,𝑡)𝐸₂𝐸₃𝑅₃𝐸₂𝐸₃^𝑇𝜉₂()+𝑥̇𝑡,𝑡_𝑇(𝑡)ℎ²𝑄₁+ℎ³𝑄₂𝑥̇(𝑡)+𝑉_𝑎(𝑥(𝑡))=𝜉^𝑇(𝑡)ℎ𝑒₁𝑒₂𝑅₃𝑒₁𝑒₂^𝑇+𝑒₂ℎ²𝑄₁+ℎ³𝑄₂𝑒^𝑇₂𝜉(𝑡)+𝑉_𝑎(𝑥(𝑡)). (26) Here, 𝑉_𝑎(𝑥(𝑡)) is the sum of all integral terms expressed as 𝑉_𝑎(𝑥(𝑡))=−^𝑡_𝑡−ℎ𝜂₂(𝑡,𝑠)𝐸₂𝐸₃𝑅₃𝐸₂𝐸₃^𝑇𝜂₂()−𝑡,𝑠𝑑𝑠_{𝑡𝑡−ℎ}2(ℎ−𝑡+𝑠)𝜂₂(𝑡,𝑠)𝐸₃𝑄₁𝐸^𝑇₃𝜂₂()−𝑡,𝑠𝑑𝑠_{𝑡𝑡−ℎ}3(ℎ−𝑡+𝑠)²𝜂₂(𝑡,𝑠)𝐸₃𝑄₂𝐸^𝑇₃𝜂₂(𝑡,𝑠)𝑑𝑠≤−^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝜂₂(𝑡,𝑠)𝐸₂𝐸₃𝑅₃𝐸₂𝐸₃^𝑇𝜂₂()−𝑡,𝑠𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ}2(ℎ−𝑡+𝑠)𝜂₂(𝑡,𝑠)𝐸₃𝑄₁𝐸^𝑇₃𝜂₂()−𝑡,𝑠𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ}3(ℎ−𝑡+𝑠)²𝜂₂(𝑡,𝑠)𝐸₃𝑄₂𝐸^𝑇₃𝜂₂()−𝑡,𝑠𝑑𝑠_{𝑡(𝑡)𝑡−ℎ}𝜂₂(𝑡,𝑠)𝐸₂𝐸₃𝑅₃𝐸₂𝐸₃^𝑇𝜂₂()−𝑡,𝑠𝑑𝑠_{𝑡(𝑡)𝑡−ℎ}2(ℎ(𝑡)−𝑡+𝑠)𝜂₂(𝑡,𝑠)𝐸₃𝑄₁𝐸^𝑇₃𝜂₂()−𝑡,𝑠𝑑𝑠_{𝑡(𝑡)𝑡−ℎ}3(ℎ(𝑡)−𝑡+𝑠)²𝜂₂(𝑡,𝑠)𝐸₃𝑄₂𝐸^𝑇₃×𝜂₂()=𝑉𝑡,𝑠𝑑𝑠_𝑎(𝑥(𝑡)). (27) Apply Lemma 2 to 𝑉_𝑎(𝑥(𝑡)), 𝑉_𝑎(𝑥(𝑡))≤𝜉^𝑇(𝑡)(ℎ−ℎ(𝑡))𝐹^𝑇₁𝑅₃⁻¹𝐹₁+2𝐹^𝑇₁𝑒₇𝑒₃−𝑒₄^𝑇+(ℎ−ℎ(𝑡))²𝐹^𝑇₂𝑄₁⁻¹𝐹₂+4𝐹^𝑇₂(ℎ−ℎ(𝑡))𝑒₃−𝑒₇^𝑇+3(ℎ−ℎ(𝑡))𝐹^𝑇₃𝑄₁⁻¹𝐹₃+6𝐹^𝑇₃(ℎ−ℎ(𝑡))𝑒₃−𝑒₇^𝑇+ℎ(𝑡)𝐹^𝑇₄𝑅₃⁻¹𝐹₄+2𝐹^𝑇₄𝑒₈𝑒₁−𝑒₃^𝑇+ℎ²(𝑡)𝐹^𝑇₅𝑄₁⁻¹𝐹₅+4𝐹^𝑇₅ℎ(𝑡)𝑒₁−𝑒₈^𝑇+3ℎ(𝑡)𝐹^𝑇₆𝑄₂⁻¹𝐹₆+6𝐹^𝑇₆ℎ(𝑡)𝑒₁−𝑒₈^𝑇(𝑡),𝑉̇×𝜉₄𝑥_𝑡≤𝑓^𝑇(𝑥(𝑡))𝑆₁𝑓(𝑥(𝑡−))1−ℎ_𝐷𝑓^𝑇(𝑥(𝑡−ℎ(𝑡)))𝑆₁𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑓^𝑇(𝑥(𝑡))𝑆₂𝑓(𝑥(𝑡))−𝑓^𝑇(𝑥(𝑡−ℎ))𝑆₂𝑓(𝑥(𝑡−ℎ))+𝑥̇^𝑇(𝑡)𝑆₃𝑥̇(𝑡)−1−ℎ_𝐷𝑥̇^𝑇(𝑡−ℎ(𝑡))𝑆₃𝑥̇((𝑡+𝑥̇𝑡−ℎ))_𝑇(𝑡)𝑆₄𝑥̇(𝑡)−𝑥̇^𝑇(𝑡−ℎ)𝑆₄𝑥̇(𝑡−ℎ)=𝜉^𝑇(𝑡)𝑒₉𝑆₁+𝑆₂𝑒^𝑇₉−1−ℎ_𝐷𝑒₁₀𝑆₁𝑒^𝑇₁₀−𝑒₁₁𝑆₂𝑒^𝑇₁₁+𝑒₂𝑆₃+𝑆₄𝑒^𝑇₂−1−ℎ_𝐷𝑒₅𝑆₃𝑒^𝑇₅−𝑒₆𝑆₄𝑒^𝑇₆𝜉(𝑡),𝑉̇₅𝑥_𝑡=𝑥̇^𝑇(𝑡)ℎ𝑆₅+ℎ²2𝑆₆𝑥̇(𝑡)−^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑆₅𝑥̇(𝑠)𝑑𝑠−⁰_−ℎ^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑆₆𝑥̇(𝑠)𝑑𝑠𝑑𝜃. (28)(29) From Lemmas 4 and 5, we have −^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑆₅𝑥̇(𝑠)2𝑑𝑠≤−ℎ1ℎ^𝑡_𝑡−ℎ𝑥(𝑠)𝑥()𝑑𝑠𝑡−ℎ^𝑇𝑆₅−𝑆₅−𝑆₅𝑆₅×1ℎ^𝑡_𝑡−ℎ𝑥(𝑠)𝑥(),−𝑑𝑠𝑡−ℎ_{0−ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝑆₆𝑥̇(𝑠)≤2𝑑𝑠𝑑𝜃ℎ²ℎ𝑥_{𝑡𝑡−ℎ}(𝑡)𝑥(𝑠)𝑑𝑠^𝑇−𝑆₆𝑆₆𝑆₆−𝑆₆ℎ𝑥_{𝑡𝑡−ℎ}(𝑡)𝑥(𝑠).𝑑𝑠 (30)
Then combining (29) and (30), we can obtain that 𝑉̇₅𝑥_𝑡≤𝜉^𝑇(𝑡)𝑒₂ℎ𝑆₅+ℎ²2𝑆₆𝑒^𝑇₂−2𝑒₁𝑆₆𝑒^𝑇₁+4ℎ𝑒₁𝑆₆𝑒^𝑇₇+4ℎ𝑒₁𝑆₆𝑒^𝑇₈+4ℎ²𝑒₄𝑆₅𝑒^𝑇₇+4ℎ²(𝑡)𝑒₄𝑆₅𝑒^𝑇₈−2ℎ𝑒₄𝑆₅𝑒^𝑇₄−𝑒₇2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₇−𝑒₇4ℎ³𝑆₅+4ℎ²𝑆₆𝑒^𝑇₈−𝑒₈2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₈𝜉(𝑡),𝑉̇₆𝑥_𝑡=𝑟𝑓^𝑇(𝑥(𝑡))𝑆₇𝑓(𝑥(𝑡−))_{𝑡𝑡−𝑟}𝑓^𝑇(𝑥(𝑠))𝑆₇𝑓(𝑥(𝑠𝑟))𝑑𝑠+²2𝑓^𝑇(𝑥(𝑡))𝑆₈𝑓(𝑥(𝑡−))_{0−𝑟𝑡𝑡+𝜃}𝑓(𝑥(𝑠))𝑆₈𝑓(𝑥(𝑠+))𝑑𝑠𝑑𝜃_𝑙𝑖=1𝜌_𝑖−𝜌_𝑖−1𝑟𝑓^𝑇(𝑥(𝑡))𝑍_𝑖𝑓(𝑥(𝑡−))_{𝑙𝑖=1𝑡−𝜌𝑖−1𝑟𝑡−𝜌𝑖𝑟}𝑓^𝑇(𝑥(𝑠))𝑍_𝑖𝑓(𝑥(𝑠))𝑑𝑠. (31)(32) From Lemma 3, we get −^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑆₇𝑓^𝑇(𝑥(𝑠1))𝑑𝑠≤−𝑟^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠𝑆₇^𝑡_𝑡−𝑟𝑓(𝑥(𝑠−))𝑑𝑠,_{0−𝑟𝑡𝑡+𝜃}𝑓^𝑇(𝑥(𝑠))𝑆₈𝑓(𝑥(𝑠2))𝑑𝑠𝑑𝜃≤−𝑟²⁰_−𝑟^𝑡_𝑡+𝜃𝑓^𝑇(𝑥(𝑠))𝑑𝑠𝑑𝜃𝑆₈×⁰_−𝑟^𝑡_𝑡+𝜃𝑓(𝑥(𝑠))𝑑𝑠𝑑𝜃,−^{𝑡−𝜌𝑖−1𝑟}_{𝑡−𝜌𝑖𝑟}𝑓^𝑇(𝑥(𝑠))𝑍_𝑖𝑓(𝑥(𝑠1))𝑑𝑠≤−𝜌_𝑖−𝜌_𝑖−1𝑟^{𝑡−𝜌𝑖−1𝑟}_{𝑡−𝜌𝑖𝑟}𝑓^𝑇(𝑥(𝑠))𝑑𝑠𝑍_𝑖×^{𝑡−𝜌𝑖−1𝑟}_{𝑡−𝜌𝑖𝑟}𝑓(𝑥(𝑠))𝑑𝑠. (33) Then combining (32) and (33) we can have that 𝑉̇₆𝑥_𝑡≤𝑟𝑓^𝑇(𝑥(𝑡))𝑆₇𝑓(𝑥(𝑡))+𝑟²2𝑓^𝑇(𝑥(𝑡))𝑆₈𝑓(𝑥(𝑡+))_𝑙𝑖=1𝜌_𝑖−𝜌_𝑖−1𝑟𝑓^𝑇(𝑥(𝑡))𝑍_𝑖𝑓(𝑥(𝑡−1))𝑟^𝑙_𝑖=1^{𝑡−𝜌𝑖−1𝑟}_{𝑡−𝜌𝑖𝑟}𝑓^𝑇(𝑥(𝑠))𝑑𝑠𝑆₇^𝑙_𝑖=1^{𝑡−𝜌𝑖−1𝑟}_{𝑡−𝜌𝑖𝑟}𝑓(𝑥(𝑠−2))𝑑𝑠𝑟²⁰_−𝑟^𝑡_𝑡+𝜃𝑓^𝑇(𝑥(𝑠))𝑑𝑠𝑑𝜃𝑆₈⁰_−𝑟^𝑡_𝑡+𝜃𝑓(𝑥(𝑠−))𝑑𝑠_𝑙𝑖=11𝜌_𝑖−𝜌_𝑖−1𝑟^{𝑡−𝜌𝑖−1𝑟}_{𝑡−𝜌𝑖𝑟}𝑓^𝑇(𝑥(𝑠))𝑑𝑠𝑍_𝑖×^{𝑡−𝜌𝑖−1𝑟}_{𝑡−𝜌𝑖𝑟}𝑓(𝑥(𝑠))𝑑𝑠=𝜉^𝑇(𝑡)𝑒₉𝑟𝑆₇+𝑟²2𝑆₈+^𝑙_𝑖=1𝜌_𝑖−𝜌_𝑖−1𝑟𝑍_𝑖𝑒^𝑇₉−2𝑟²𝑒_12+𝑙𝑆₈𝑒^𝑇_12+𝑙−^𝑙_𝑖=11𝜌_𝑖−𝜌_𝑖−1𝑟𝑒_11+𝑖×𝑍_𝑖𝑒^𝑇_11+𝑖−1𝑟𝑒₁₂+⋯+𝑒_11+𝑙×𝑆₇𝑒₁₂+⋯+𝑒_11+𝑙^𝑇𝜉(𝑡),𝑉̇₇𝑥_𝑡≤2𝑓^𝑇(𝑥(𝑡))𝐺₁−𝐾₁+𝐿₁+𝑥^𝑇(𝑡)Σ𝐾₁−Σ𝐺₁+𝐿₁Σ×𝑥̇(𝑡)+21−ℎ_𝐷×𝑓^𝑇(𝑥(𝑡−ℎ(𝑡)))𝐺₂−𝐾₂+𝐿₂+𝑥^𝑇(𝑡−ℎ(𝑡))Σ𝐾₂−Σ𝐺₂+𝐿₂Σ×𝑥̇((𝑡𝑓𝑡−ℎ))+2^𝑇(𝑥(𝑡−ℎ))𝐺₃−𝐾₃+𝐿₃+𝑥^𝑇(𝑡−ℎ)Σ𝐾₃−Σ𝐺₃+𝐿₃Σ×𝑥̇(𝑡−ℎ)=2𝜉^𝑇(𝑡)𝑒₁Σ𝐾₁−Σ𝐺₁+𝐿₁Σ𝑒^𝑇₂+2𝑒₂𝐺₁−𝐾₁+𝐿₁𝑒^𝑇₉+2𝑒₃Σ𝐾₂−Σ𝐺₂+𝐿₂Σ𝑒^𝑇₅+2𝑒₅𝐺₂−𝐾₂+𝐿₂𝑒^𝑇₁₀+2𝑒₄Σ𝐾₃−Σ𝐺₃+𝐿₃Σ𝑒^𝑇₆+2𝑒₆𝐺₃−𝐾₃+𝐿₃𝑒^𝑇₁₁𝜉(𝑡). (34)(35) From (6), for any positive diagonal matrices 𝑀_𝑖=diag{𝑚_𝑖1,…,min}(𝑖=1,2,3,4) one can easily check 0≤𝑥^𝑇(𝑡)Σ𝑀₁Σ𝑥(𝑡)−𝑓^𝑇(𝑥(𝑡))𝑀₁𝑓(𝑥(𝑡+))−𝑥^𝑇(𝑡)𝑀₂Σ₁𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑀₂Σ₂𝑓(𝑥(𝑡))−𝑓^𝑇(𝑥(𝑡))𝑀₂𝑓(𝑥(𝑡+))−𝑥^𝑇(𝑡−ℎ(𝑡))𝑀₃×Σ₁𝑥(𝑡−ℎ(𝑡))+2𝑥^𝑇(𝑡−ℎ(𝑡))×𝑀₃Σ₂𝑓(𝑥(𝑡−ℎ(𝑡)))−𝑓^𝑇(𝑥(𝑡−ℎ(𝑡)))×𝑀₃𝑓(𝑥((𝑡+𝑡−ℎ)))−𝑥^𝑇(𝑡−ℎ)𝑀₄Σ₁𝑥(𝑡−ℎ)+2𝑥^𝑇(𝑡−ℎ)𝑀₄Σ₂𝑓×(𝑥(𝑡−ℎ))−𝑓^𝑇(𝑥(𝑡−ℎ))𝑀₄𝑓(𝑥(𝑡−ℎ))=𝜉^𝑇(𝑡)𝑒₁Σ𝑀₁Σ−𝑀₂Σ₁𝑒^𝑇₁+2𝑒₁𝑀₂Σ₂𝑒^𝑇₉−𝑒₃𝑀₃Σ₁𝑒^𝑇₃+2𝑒₃𝑀₃Σ₂𝑒^𝑇₁₀−𝑒₉𝑀₁+𝑀₂𝑒^𝑇₉−𝑒₁₀𝑀₃𝑒^𝑇₁₀−𝑒₄𝑀₄Σ₁𝑒^𝑇₄+2𝑒₄𝑀₄Σ₂𝑒^𝑇₁₁−𝑒₁₁𝑀₄𝑒^𝑇₁₁𝜉(𝑡). (36) Furthermore, for arbitrary matrices 𝑇₁, 𝑇₂, 𝑇₃, 𝑇₄, 𝑇₅ with appropriate dimensions, we have 2𝑥̇^𝑇(𝑡)𝑇₁×−𝑥̇(𝑡)−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑊₃^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠+𝑊₄(𝑥̇(𝑡−ℎ(𝑡)))=𝜉^𝑇(𝑡)𝑒₂−𝑇₁−𝑇^𝑇₁𝑒^𝑇₂−2𝑒₁𝑊^𝑇₀𝑇^𝑇₁𝑒^𝑇₂+2𝑒₂𝑇₁𝑊₁𝑒^𝑇₉+2𝑒₂𝑇₁𝑊₂𝑒^𝑇₁₀+2𝑒₂𝑇₁𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇+2𝑒₂𝑇₁𝑊₄𝑒^𝑇₅𝜉(𝑡)=0,2𝑥^𝑇(𝑡)𝑇₂×−𝑥̇(𝑡)−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑊₃^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠+𝑊₄(𝑥̇(𝑡−ℎ(𝑡)))=𝜉^𝑇(𝑡)−2𝑒₁𝑇₂𝑒^𝑇₂+𝑒₁−𝑇₂𝑊₀−𝑊^𝑇₀𝑇^𝑇₂𝑒^𝑇₁+2𝑒₁𝑇₂𝑊₁𝑒^𝑇₉+2𝑒₁𝑇₂𝑊₂𝑒^𝑇₁₀+2𝑒₁𝑇₂𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇+2𝑒₂𝑇₂𝑊₄𝑒^𝑇₅𝜉(𝑡)=0,2𝑓^𝑇(𝑥(𝑡))𝑇₃×−𝑥̇(𝑡)−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑊₃^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠+𝑊₄(𝑥̇(𝑡−ℎ(𝑡)))=𝜉^𝑇(𝑡)−2𝑒₂𝑇^𝑇₃𝑒^𝑇₉−2𝑒₁𝑊^𝑇₀𝑇^𝑇₃𝑒^𝑇₉+𝑒₉𝑇₃𝑊₁+𝑊^𝑇₁𝑇^𝑇₃𝑒^𝑇₉+2𝑒₉𝑇₃𝑊₂𝑒^𝑇₁₀+2𝑒₉𝑇₃𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇+2𝑒₅𝑊^𝑇₄𝑇^𝑇₃𝑒^𝑇₉𝜉(𝑡)=0,2𝑓^𝑇(𝑥(𝑡−ℎ(𝑡)))𝑇₄×−𝑥̇(𝑡)−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑊₃^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠+𝑊₄(𝑥̇(𝑡−ℎ(𝑡)))=𝜉^𝑇(𝑡)−2𝑒₂𝑇^𝑇₄𝑒^𝑇₁₀−2𝑒₁𝑊^𝑇₀𝑇^𝑇₄𝑒^𝑇₁₀+𝑒₉𝑊^𝑇₁𝑇^𝑇₄𝑒^𝑇₁₀+𝑒₁₀𝑇₄𝑊₂+𝑊^𝑇₂𝑇^𝑇₄𝑒^𝑇₁₀+2𝑒₁₀𝑇₄𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇+2𝑒₅𝑊^𝑇₄𝑇^𝑇₄𝑒^𝑇₁₀𝜉(𝑡)𝑥̇=0,_𝑇(𝑡−ℎ(𝑡))𝑇₅×−𝑥̇(𝑡)−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑊₃^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠+𝑊₄(𝑥̇(𝑡−ℎ(𝑡)))=𝜉^𝑇(𝑡)−2𝑒₂𝑇^𝑇₅𝑒^𝑇₅−2𝑒₁𝑊^𝑇₀𝑇^𝑇₅𝑒^𝑇₅+𝑒₅𝑇₅𝑊₁𝑒^𝑇₉+𝑒₅𝑇₅𝑊₂𝑒^𝑇₁₀+2𝑒₅𝑇₅𝑊₃𝑒₁₂+⋯+𝑒_11+𝑙^𝑇+2𝑒₅𝑇₅𝑊₄+𝑊^𝑇₄𝑇^𝑇₄𝑒^𝑇₅𝜉(𝑡)=0. (37)(38)(39)(40)(41)
The combination of (26)–(37) gives that 𝑉𝑥_𝑡≤𝜉^𝑇(𝑡)Π₀+ℎ(𝑡)Π₁+Π_ℎ𝜉(𝑡), (42) where Π₀, Π₁ are defined in (13) and (14), respectively, and Π_ℎ=(ℎ−ℎ(𝑡))𝐹^𝑇₁𝑅₃⁻¹𝐹₁+(ℎ−ℎ(𝑡))²𝐹^𝑇₂𝑄₁⁻¹𝐹₂+3(ℎ−ℎ(𝑡))𝐹^𝑇₃𝑄₁⁻¹𝐹₃+ℎ(𝑡)𝐹^𝑇₄𝑅₃⁻¹𝐹₄+ℎ²(𝑡)𝐹^𝑇₅𝑄₁⁻¹𝐹₅+3ℎ(𝑡)𝐹^𝑇₆𝑄₂⁻¹𝐹₆. (43) Note that the scalar valued function 𝜉^𝑇(𝑡){Π₀+ℎ(𝑡)Π₁+Π_ℎ}𝜉(𝑡) is quadratic function on the scalar ℎ(𝑡) and the coefficient of second order is 𝜉^𝑇(𝑡)[𝐹^𝑇₂𝑄₁⁻¹𝐹₂+𝐹^𝑇₅𝑄₁⁻¹𝐹₅]𝜉(𝑡)≥0 since 𝑄₁>0. This means that the function 𝜉^𝑇(𝑡){Π₀+ℎ(𝑡)Π₁+Π_ℎ}𝜉(𝑡) is a convex quadratic function for ℎ(𝑡). Finally, apply Fact 1 and Lemma 1 in order, then we get Ξ₁<0in(22),Ξ₂()⟺Π<0in23₀+ℎ(𝑡)Π₁+Π_{ℎℎ(𝑡)=0}Π<0,₀+ℎ(𝑡)Π₁+Π_{ℎℎ(𝑡)=ℎ}<0,⟺Π₀+ℎ(𝑡)Π₁+Π_ℎ<0,∀ℎ(𝑡)∈[0,ℎ], (44)(45)(46) which means the asymptotic stability of the system (19). This completes the proof.

Remark 7. In Theorem 6, the augmented vector 𝜉(𝑡) has integrating terms of activation function 𝑓(𝑥(𝑡)) which are ∫^𝑡_{𝑡−𝜌1𝑟}𝑓^𝑇(𝑥(𝑠))𝑑𝑠,…,∫^{𝑡−𝜌𝑙−1𝑟}_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠 and ∫⁰_−𝑟∫^𝑡_𝑡+𝜃𝑓^𝑇(𝑥(𝑠))𝑑𝑠𝑑𝜃. By these terms, more past history of 𝑓(𝑥(𝑡)) can be used, which lead to less conservative results.

Remark 8. Compared with those in previous articles, Ours constructed a new type of Lyapunov-Krasovskii functional which has three differences: (1) an independent augmented variable ∫^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠; (2) the cross terms between entries in 𝜉₁(𝑡), 𝜉₂(𝑡,𝑠), respectively; (3) quadratic terms multiplied by first, second, and third degrees of a scalar function ℎ−𝑡+𝑠 by 1 means the number increase of the integral by 1.

Remark 9. Compared with traditional approach to deal with term like ∫^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑆₅𝑥̇(𝑠)𝑑𝑠, Lemma 5 provides a new handling method. This new handling method can establish the relationship among ∫^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑑𝑠, ∫^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠  and 𝑥(𝑡−ℎ), which may significantly reduce the conservatism of stability criteria.

Remark 10. When 𝑊₃=0, the system (5) reduces to 𝑥̇(𝑡)=−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑊₄𝑥̇(𝑡−ℎ(𝑡)). (47) Similarly, based on Theorem 6, we can obtain the asymptotical stability for system (47) as follows.

Theorem 11. For given scalars ℎ>0 and ℎ_𝐷<1, the system (47) with the neuron activation function 𝑓(𝑥(𝑡)) satisfying the condition (6) is asymptotically stable if there exists 𝑃>0, 𝑅^𝑇_𝑖=𝑅_𝑖>0  (𝑖=1,2,3),  𝑄^𝑇_𝑖=𝑄_𝑖>0(𝑖=1,2), 𝑆^𝑇_𝑖=𝑆_𝑖>0(𝑖=1,…,6),  diagonal matrices 𝐺_𝑖=diag{𝑔_𝑖1,𝑔_𝑖2,…,𝑔_𝑖𝑛}>0, 𝐾_𝑖=diag{𝑘_𝑖1,𝑘_𝑖2,…,𝑘_𝑖𝑛}>0, 𝐿_𝑖=diag{𝑙_𝑖1,𝑙_𝑖2,…,𝑙_𝑖𝑛}>0, (𝑖=1,2,3), and 𝑀=diag{𝑚_𝑖1,𝑚_𝑖2,…,𝑚_𝑖𝑛}>0,(𝑖=1,2,3,4), 𝑇_𝑖(𝑖=1,…,5) and 𝐹_𝑖(𝑖=1,…,6) with appropriate dimensions such that the following symmetric linear matrix inequality holds: Ξ₁=Π₀ℎ𝐹^𝑇₁ℎ𝐹^𝑇₂ℎ𝐹^𝑇₃∗−ℎ𝑅₃00∗∗−𝑄₁0ℎ∗∗∗−3𝑄₂Ξ<0,₂=Π₀+ℎΠ₁ℎ𝐹^𝑇₄ℎ𝐹^𝑇₅ℎ𝐹^𝑇₆∗−ℎ𝑅₃00∗∗−𝑄₁0ℎ∗∗∗−3𝑄₂<0, (48)(49) where Π₀=2𝑒₁̂𝑒₇̂+𝑒₈̂𝑃𝑒₂𝑒₁̂−𝑒₄^𝑇+2𝑒₂̂0𝑅₁0𝑒̂₈^𝑇+2𝑒₂̂0𝑅₂ℎ𝑒₁̂𝑒₇̂+𝑒̂₈^𝑇_𝑠+𝑒₁̂𝑒₁̂×𝑅₁+𝑅₂𝑒₁̂𝑒̂₁^𝑇−1−ℎ_𝐷𝑒₁̂𝑒₃̂×𝑅₁𝑒₁̂𝑒̂₃^𝑇−𝑒₁̂𝑒₄̂𝑅₂𝑒₁̂𝑒̂₄^𝑇+ℎ𝑒₁̂𝑒₂̂𝑅₃×𝑒₁̂𝑒̂₂^𝑇+𝑒₂̂ℎ²𝑄₁+ℎ³𝑄₂𝑒^𝑇₂+2𝐹^𝑇₁𝑒₇̂𝑒₃̂−𝑒̂₄^𝑇+4𝐹^𝑇₂ℎ𝑒₃̂−𝑒̂₇^𝑇+6𝐹^𝑇₃ℎ𝑒₃̂−𝑒̂₇^𝑇+2𝐹^𝑇₄𝑒₈̂𝑒₁̂−𝑒̂₃^𝑇−4𝐹^𝑇₅𝑒^𝑇₈̂−6𝐹^𝑇₆𝑒̂^𝑇_8𝑠+𝑒₉̂𝑆₁+𝑆₂𝑒^𝑇₉̂−1−ℎ_𝐷𝑒̂₁₀𝑆₁𝑒̂^𝑇₁₀−𝑒̂₁₁𝑆₂𝑒̂^𝑇₁₁−𝑒₆̂𝑆₄𝑒^𝑇₆̂+𝑒₂̂𝑆₃+𝑆₄𝑒^𝑇₂̂−1−ℎ_𝐷𝑒₅̂𝑆₃𝑒^𝑇₅+𝑒₂̂ℎ𝑆₅+ℎ²2𝑆₆𝑒^𝑇₂̂−2𝑒₁̂𝑆₆𝑒^𝑇₁̂+4ℎ𝑒₁̂𝑆₆𝑒^𝑇₇̂+4ℎ²𝑒₄̂𝑆₅𝑒^𝑇₇̂+4ℎ²𝑒₄̂𝑆₅𝑒^𝑇₈̂−𝑒₇̂2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₇̂−𝑒₇̂4ℎ³𝑆₅+4ℎ²𝑆₆𝑒^𝑇₈̂−𝑒₈̂2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₈̂+𝑒₁̂Σ𝐾₁−Σ𝐺₁+𝐿₁Σ𝑒^𝑇₂̂+2𝑒₂̂𝐺₁−𝐾₁+𝐿₁𝑒^𝑇₉̂+2𝑒₃̂Σ𝐾₂−Σ𝐺₂+𝐿₂Σ×𝑒^𝑇₅̂+2𝑒₅̂𝐺₂−𝐾₂+𝐿₂𝑒̂^𝑇₁₀+2𝑒₄̂Σ𝐾₃−Σ𝐺₃+𝐿₃Σ𝑒^𝑇₆̂+2𝑒₆̂𝐺₃−𝐾₃+𝐿₃×𝑒̂^𝑇₁₁+𝑒₁Σ𝑀₁Σ−𝑀₂Σ₁𝑒^𝑇₁̂+2𝑒₁̂𝑀₂Σ₂𝑒^𝑇₉̂−𝑒₃̂𝑀₃Σ₁𝑒^𝑇₃̂+2𝑒₃̂𝑀₃Σ₂𝑒̂^𝑇₁₀−𝑒₉̂𝑀₁+𝑀₂𝑒^𝑇₉̂−𝑒̂₁₀𝑀₃𝑒̂^𝑇₁₀−𝑒₄̂𝑀₄Σ₁𝑒^𝑇₄̂+2𝑒₄̂𝑀₄Σ₂𝑒̂^𝑇₁₁+𝑒₂̂−𝑇₁−𝑇^𝑇₁𝑒^𝑇₂̂−𝑒̂₁₁𝑀₄𝑒̂^𝑇₁₁−2𝑒₁̂𝑊^𝑇₀𝑇^𝑇₁𝑒^𝑇₂̂+2𝑒₂̂𝑇₁𝑊₁𝑒^𝑇₉̂+2𝑒₂̂𝑇₁𝑊₂𝑒̂^𝑇₁₀−2𝑒₁𝑇₂𝑒^𝑇₂̂+𝑒₁̂−𝑇₂𝑊₀−𝑊^𝑇₀𝑇^𝑇₂𝑒^𝑇₁̂+2𝑒₁̂𝑇₂𝑊₁𝑒^𝑇₉̂+2𝑒₁̂𝑇₂𝑊₂𝑒̂^𝑇₁₀+2𝑒₂̂𝑇₂𝑊₄𝑒^𝑇₅̂−2𝑒₁̂𝑊^𝑇₀𝑇^𝑇₃𝑒^𝑇₉̂+2𝑒₉̂𝑇₃𝑊₂𝑒̂^𝑇₁₀+𝑒₉̂𝑇₃𝑊₁+𝑊^𝑇₁𝑇^𝑇₃𝑒^𝑇₉̂+2𝑒₅̂𝑊^𝑇₄𝑇^𝑇₃𝑒^𝑇₉̂−2ê₂𝑇^𝑇₄𝑒̂^𝑇₁₀−2𝑒₁̂𝑊^𝑇₀𝑇^𝑇₄𝑒̂^𝑇₁₀+𝑒₉̂𝑊^𝑇₁𝑇^𝑇₄𝑒̂^𝑇₁₀+𝑒̂₁₀𝑇₄𝑊₂+𝑊^𝑇₂𝑇^𝑇₄𝑒̂^𝑇₁₀+2𝑒₅̂𝑊^𝑇₄𝑇^𝑇₃𝑒^𝑇₉̂−2𝑒₂̂𝑇^𝑇₅𝑒^𝑇₅̂−2𝑒₁̂𝑊^𝑇₀𝑇^𝑇₅𝑒^𝑇₅̂+𝑒₅̂𝑇₅𝑊₁𝑒^𝑇₉̂+𝑒₅̂𝑇₅𝑊₂𝑒̂^𝑇₁₀−2ℎ𝑒₄̂𝑆₅𝑒^𝑇₄̂+4ℎ𝑒₁̂𝑆₆𝑒^𝑇₈̂+2𝑒₅̂𝑇₅𝑊₄+𝑊^𝑇₄𝑇^𝑇₄𝑒^𝑇₅̂+2𝑒₅̂𝑊^𝑇₄𝑇^𝑇₄𝑒̂^𝑇₁₀+2𝑒₂̂𝑇₁𝑊₄𝑒^𝑇₅̂−2𝑒₂̂𝑇^𝑇₃𝑒^𝑇₉̂,Π₁=2𝑒₂̂0𝑅₁𝑒₁̂0^𝑇−4𝐹^𝑇₂𝑒^𝑇₃̂−6𝐹^𝑇₃𝑒^𝑇₃̂+4𝐹^𝑇₅𝑒^𝑇₁̂+6𝐹^𝑇₆𝑒̂^𝑇_1𝑠, (50) with ⟨⋅⟩_𝑠=12(⋅)+(⋅)^𝑇,𝑒_𝑖̂=0_{𝑛×(𝑖−1)}𝐼_𝑛×𝑛0_𝑛×(11)^𝑇(𝑖=1,…,11). (51)

Remark 12. When 𝑊₄=0, the system (5) reduces to 𝑥̇(𝑡)=−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑊₃^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠. (52) Similarly, based on Theorem 6, we can obtain the asymptotical stability for system (52) as follows.

Theorem 13. For given scalars ℎ>0 and ℎ_𝐷<1, the system (52) with the neuron activation function 𝑓(𝑥(𝑡)) satisfying the condition (6) is asymptotically stable if there exists 𝑃>0, 𝑅^𝑇_𝑖=𝑅_𝑖>0  (𝑖=1,2,3),  𝑄^𝑇_𝑖=𝑄_𝑖>0(𝑖=1,2),  𝑆^𝑇_𝑖=𝑆_𝑖>0(𝑖=1,2,5…,8),  𝑍^𝑇_𝑖=𝑍_𝑖>0(𝑖=1,…,𝑙), diagonal matrices 𝐺₁=diag{𝑔₁,𝑔₂,…,𝑔_𝑛}>0,  𝐾₁=diag{𝑘₁,𝑘₂,…,𝑘_𝑛}>0,  𝐿₁=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}>0, and 𝑀=diag{𝑚_𝑖1,𝑚_𝑖2,…,𝑚_𝑖𝑛}>0,(𝑖=1,2,3,4), 𝑇_𝑖(𝑖=1,…,4) and 𝐹_𝑖(𝑖=1,…,6) with appropriate dimensions such that the following symmetric linear matrix inequality holds: Ξ₁=Π₀ℎ𝐹^𝑇₁ℎ𝐹^𝑇₂ℎ𝐹^𝑇₃∗−ℎ𝑅₃00∗∗−𝑄₁0ℎ∗∗∗−3𝑄₂Ξ<0,₂=Π₀+ℎΠ₁ℎ𝐹^𝑇₄ℎ𝐹^𝑇₅ℎ𝐹^𝑇₆∗−ℎ𝑅₃00∗∗−𝑄₁0ℎ∗∗∗−3𝑄₂<0, (53)(54) where Π₀=2𝑒₁̃𝑒₅̃+𝑒₆̃𝑃𝑒₂̃𝑒₁̃−𝑒̃₄^𝑇+2𝑒₂̃0𝑅₁0𝑒̃₆^𝑇+2𝑒₂̃0×𝑅₂ℎ𝑒₁̃𝑒₅̃+𝑒̃₆^𝑇_𝑠+𝑒₁̃𝑒₁̃×𝑅₁+𝑅₂𝑒₁̃𝑒̃₁^𝑇−1−ℎ_𝐷𝑒₁̃𝑒₃×𝑅₁𝑒₁̃𝑒̃₃^𝑇−ẽ₁𝑒₄𝑅₂𝑒₁̃𝑒̃₄^𝑇+ℎ𝑒₁̃𝑒₂̃𝑅₃×𝑒₁̃𝑒̃₂^𝑇+𝑒₂̃ℎ²𝑄₁+ℎ³𝑄₂×𝑒^𝑇₂̃+2𝐹^𝑇₁𝑒₅̃𝑒₃̃−𝑒̃₄^𝑇+4𝐹^𝑇₂ℎ𝑒₃̃−𝑒̃₅^𝑇+6𝐹^𝑇₃ℎ𝑒₃̃−𝑒̃₅^𝑇+2𝐹^𝑇₄𝑒₆̃𝑒₁̃−𝑒̃₃^𝑇−4𝐹^𝑇₅𝑒^𝑇₆̃−6𝐹^𝑇₆𝑒̃^𝑇_6𝑠+𝑒₇̃𝑆₁+𝑆₂𝑒^𝑇₇̃−1−ℎ_𝐷𝑒₈̃𝑆₁𝑒^𝑇₈̃−𝑒₉̃𝑆₂𝑒^𝑇₉̃+𝑒₂̃ℎ𝑆₅+ℎ²2𝑆₆×𝑒^𝑇₂̃−2𝑒₁̃𝑆₆𝑒^𝑇₁̃+4ℎ𝑒₁̃𝑆₆𝑒^𝑇₅̃+𝑒₇̃𝑊^𝑇₁𝑇^𝑇₄𝑒^𝑇₈̃+𝑒₈̃𝑇₄𝑊₂+𝑊^𝑇₂𝑇^𝑇₄𝑒^𝑇₈̃+4ℎ𝑒₁̃×𝑆₆𝑒^𝑇₆̃+4ℎ²𝑒₄̃𝑆₅𝑒^𝑇₅̃+4ℎ²𝑒₄̃𝑆₅𝑒^𝑇₆̃−2ℎ𝑒₄̃𝑆₅𝑒^𝑇₄−𝑒₅̃2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₅̃−𝑒₅̃4ℎ³𝑆₅+4ℎ²𝑆₆𝑒^𝑇₆̃−𝑒₆̃2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₆̃+𝑒₇̃𝑟𝑆₇+𝑟²2𝑆₈+^𝑙_𝑖=1𝜌_𝑖−𝜌_𝑖−1𝑟𝑍_𝑖𝑒^𝑇₇̃−^𝑙_𝑖=11𝜌_𝑖−𝜌_𝑖−1𝑟𝑒̃_9+𝑖𝑍_𝑖𝑒̃^𝑇_9+𝑖−1𝑟𝑒̃₁₀+⋯+𝑒̃_9+𝑙×𝑆₇𝑒̃₁₀+⋯+𝑒_9+𝑙^𝑇−2𝑟²𝑒̃_10+𝑙𝑆₈𝑒̃^𝑇_10+𝑙+𝑒₁̃Σ𝐾₁−Σ𝐺₁+𝐿₁Σ𝑒^𝑇₂̃+2𝑒₂̃𝐺₁−𝐾₁+𝐿₁𝑒^𝑇₇+𝑒₁̃Σ𝑀₁Σ−M₂Σ₁𝑒^𝑇₁+2𝑒₁𝑀₂Σ₂𝑒^𝑇₇−𝑒₃𝑀₃Σ₁𝑒^𝑇₃+2𝑒₇𝑇₃𝑊₂𝑒^𝑇₈+2𝑒₃̃𝑀₃Σ₂𝑒^𝑇₈̃−𝑒₇̃𝑀₁+𝑀₂𝑒^𝑇₇̃−𝑒₈̃𝑀₃𝑒^𝑇₈̃−𝑒₄̃𝑀₄Σ₁𝑒^𝑇₄̃+2𝑒₄̃𝑀₄Σ₂𝑒^𝑇₉̃+𝑒₂̃−𝑇₁−𝑇^𝑇₁𝑒^𝑇₂̃−𝑒₉̃𝑀₄𝑒^𝑇₉̃−2𝑒₁̃𝑊^𝑇₀𝑇^𝑇₁𝑒^𝑇₂̃+2𝑒₂𝑇₁𝑊₁𝑒^𝑇₇̃+2𝑒₂̃𝑇₁𝑊₂𝑒^𝑇₈̃+2𝑒₂̃𝑇₁𝑊₃×𝑒̃₁₀+⋯+𝑒̃_9+𝑙^𝑇−2𝑒₁̃×𝑊^𝑇₀𝑇^𝑇₄𝑒^𝑇₈̃−2𝑒₁̃𝑇₂𝑒^𝑇₂+𝑒₁̃−𝑇₂𝑊₀−𝑊^𝑇₀𝑇^𝑇₂𝑒^𝑇₁̃+2𝑒₁̃𝑇₂𝑊₁𝑒^𝑇₇̃+2𝑒₁̃𝑇₂𝑊₂𝑒^𝑇₈−2𝑒₂̃𝑇^𝑇₄𝑒^𝑇₈̃+2𝑒₁̃𝑇₂𝑊₃×𝑒̃₁₀+⋯+𝑒̃_9+𝑙^𝑇−2𝑒₂̃𝑇^𝑇₃𝑒^𝑇₇̃−2𝑒₁𝑊^𝑇₀𝑇^𝑇₃𝑒^𝑇₇̃+𝑒₇̃𝑇₃𝑊₁+𝑊^𝑇₁𝑇^𝑇₃𝑒^𝑇₇̃+2𝑒₇̃𝑇₃𝑊₃𝑒̃₁₀+⋯+𝑒_9+𝑙^𝑇+2𝑒₈̃𝑇₄𝑊₃𝑒̃₁₀+⋯+𝑒̃_9+𝑙^𝑇,Π₁=2𝑒₂̃0𝑅₁𝑒₁̃𝑒̃₀^𝑇−4𝐹^𝑇₂𝑒^𝑇₃̃−6𝐹^𝑇₃𝑒^𝑇₃̃+4𝐹^𝑇₅𝑒^𝑇₁̃+6𝐹^𝑇₆𝑒̃^𝑇_1𝑠, (55) with ⟨⋅⟩_𝑠=12(⋅)+(⋅)^𝑇,𝑒_𝑖̃=0_{𝑛×(𝑖−1)}𝐼_𝑛×𝑛0_{𝑛×(10+𝑙−𝑖)}^𝑇(𝑖=1,…,10+𝑙). (56)

Remark 14. When 𝑊₃=0 and 𝑊₄=0, the system (5) reduces to 𝑥̇(𝑡)=−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡))). (57) Similarly, based on Theorem 6, we can obtain the asymptotical stability for system (57) as follows.

Theorem 15. For given scalars ℎ>0 and ℎ_𝐷<1, the system (57) with the neuron activation function 𝑓(𝑥(𝑡)) satisfying the condition (6) is asymptotically stable if there exists 𝑃>0, 𝑅^𝑇_𝑖=𝑅_𝑖>0  (𝑖=1,2,3), 𝑄^𝑇_𝑖=𝑄_𝑖>0(𝑖=1,2), 𝑆^𝑇_𝑖=𝑆_𝑖>0(𝑖=1,2,5,6), diagonal matrices 𝐺₁=diag{𝑔_𝑖1,𝑔₂,…,𝑔_𝑖𝑛}>0,  𝐾₁=diag{𝑘₁,𝑘₂,…,𝑘_𝑛}>0,  𝐿₁=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}>0,  and 𝑀=diag{𝑚_𝑖1,𝑚_𝑖2,…,𝑚_𝑖𝑛}>0,  (𝑖=1,2,3,4), 𝑇_𝑖(𝑖=1,…,4) and 𝐹_𝑖(𝑖=1,…,6) with appropriate dimensions such that the following symmetric linear matrix inequality holds: Ξ̆₁=Π̆₀ℎ𝐹^𝑇₁ℎ𝐹^𝑇₂ℎ𝐹^𝑇₃∗−ℎ𝑅₃00∗∗−𝑄₁0ℎ∗∗∗−3𝑄₂Ξ̆<0,₂=Π̆₀+ℎΠ̆₁ℎ𝐹^𝑇₄ℎ𝐹^𝑇₅ℎ𝐹^𝑇₆∗−ℎ𝑅₃00∗∗−𝑄₁0ℎ∗∗∗−3𝑄₂<0, (58)(59) where Π̆₀=2𝑒₁̆𝑒₅̆+𝑒₆̆𝑃𝑒₂𝑒₁̆−𝑒₄^𝑇+2𝑒₂̆0𝑅₁0𝑒₆^𝑇+2𝑒₂̆0𝑅₂ℎ𝑒₁̆𝑒₅̆+𝑒̆₆^𝑇_𝑠+𝑒₁̆𝑒₁̂×𝑅₁+𝑅₂𝑒₁̆𝑒̆₁^𝑇−1−ℎ_𝐷𝑒₁̆𝑒₃̆𝑅₁𝑒₁̆𝑒̆₃^𝑇−𝑒₁̆𝑒₄̆𝑅₂𝑒₁̆𝑒̆₄^𝑇+ℎ𝑒₁̆𝑒₂̆𝑅₃×𝑒₁̆𝑒̆₂^𝑇+𝑒₂̆ℎ²𝑄₁+ℎ³𝑄₂𝑒^𝑇₂+2𝐹^𝑇₁𝑒₅̆𝑒₃̆−𝑒̆₄^𝑇+4𝐹^𝑇₂ℎ𝑒₃̆−𝑒̆₅^𝑇+6𝐹^𝑇₃ℎ𝑒₃̆−𝑒̆₅^𝑇+2𝐹^𝑇₄𝑒₆̆𝑒₁̆−𝑒̆₃^𝑇−4𝐹^𝑇₅𝑒^𝑇₆̆−6𝐹^𝑇₆𝑒̆^𝑇_6𝑠+𝑒₇̆𝑆₁+𝑆₂𝑒^𝑇₇̆−1−ℎ_𝐷𝑒₈̆𝑆₁𝑒^𝑇₈̆−𝑒₉̆𝑆₂𝑒^𝑇₉̆+𝑒₂̆ℎ𝑆₅+ℎ²2𝑆₆𝑒^𝑇₂̆−2𝑒₁̆𝑆₆𝑒^𝑇₁̆+2ℎ𝑒₁̆𝑆₆𝑒^𝑇₅̆+2ℎ𝑒₁̆𝑆₆𝑒^𝑇₆̆+2ℎ²𝑒₄̆𝑆₅𝑒^𝑇₅̆+2ℎ²𝑒₄̆𝑆₅𝑒^𝑇₆̆−2ℎ𝑒₄̆𝑆₅𝑒^𝑇₄̆−𝑒₅̆2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₅̆−𝑒₅̆2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₆̆−𝑒₆̆2ℎ³𝑆₅+2ℎ²𝑆₆𝑒^𝑇₆̆+2𝑒₂̆𝐺₁−𝐾₁+𝐿₁𝑒^𝑇₇̆+𝑒₁̆Σ𝐾₁−Σ𝐺₁+𝐿₁Σ𝑒^𝑇₂̆+𝑒₁̆Σ𝑀₁Σ−𝑀₂Σ₁𝑒^𝑇₁̆+2𝑒₁̆𝑀₂Σ₂𝑒^𝑇₇̆−𝑒₃̆𝑀₃Σ₁𝑒^𝑇₃̆+2𝑒₃̆𝑀₃Σ₂𝑒^𝑇₈̂−𝑒₇̆𝑀₁+𝑀₂𝑒^𝑇₇̆−𝑒₈̆𝑀₃𝑒^𝑇₈̆−𝑒₄̆𝑀₄Σ₁𝑒^𝑇₄̆+2𝑒₄̆𝑀₄Σ₂𝑒^𝑇₉̂+𝑒₂̆−𝑇₁−𝑇^𝑇₁𝑒^𝑇₂̆−𝑒₉̆𝑀₄𝑒^𝑇₉̆−2𝑒₁̆𝑊^𝑇₀𝑇^𝑇₁𝑒^𝑇₂̆+2𝑒₂̆𝑇₁𝑊₁𝑒^𝑇₇̆+2𝑒₂̆𝑇₁𝑊₂𝑒^𝑇₈̆−2𝑒₁𝑇₂𝑒^𝑇₂̆+𝑒₁̆−𝑇₂𝑊₀−𝑊^𝑇₀𝑇^𝑇₂𝑒^𝑇₁̆+2𝑒₁̆𝑇₂𝑊₂𝑒^𝑇₈̆−2𝑒₂̆𝑇^𝑇₃𝑒^𝑇₇̆−2𝑒₁̆𝑊^𝑇₀𝑇^𝑇₃𝑒^𝑇₇̆+2𝑒₇̆𝑇₃𝑊₂𝑒^𝑇₈̆+𝑒₇̆𝑇₃𝑊₁+𝑊^𝑇₁𝑇^𝑇₃𝑒^𝑇₇̆−2𝑒₂̆𝑇^𝑇₄𝑒^𝑇₈̆−2𝑒₁̆𝑊^𝑇₀𝑇^𝑇₄𝑒^𝑇₈̆+𝑒₇̆𝑊^𝑇₁𝑇^𝑇₄𝑒^𝑇₈̆+𝑒₈̆𝑇₄𝑊₂+𝑊^𝑇₂𝑇^𝑇₄𝑒^𝑇₈̆+2𝑒₁̆𝑇₂𝑊₁𝑒^𝑇₇̆,Π₁=2𝑒₂̂0𝑅₁𝑒₁̂0^𝑇−4𝐹^𝑇₂𝑒^𝑇₃̂−6𝐹^𝑇₃𝑒^𝑇₃̂+4𝐹^𝑇₅𝑒^𝑇₁̂+6𝐹^𝑇₆𝑒̂^𝑇_1𝑠, (60) with ⟨⋅⟩_𝑠=12(⋅)+(⋅)^𝑇,𝑒_𝑖̂=0_{𝑛×(𝑖−1)}𝐼_𝑛×𝑛0_{𝑛×(9𝑖)}^𝑇(𝑖=1,…,9). (61)

4. Numerical Examples

In this section, four numerical examples are given to show the effectiveness and improvement of the main results proposed in the paper.

Example 1. Consider the following neural networks of neutral type with discrete and distributed delays: 𝑥̇(𝑡)=−𝑊₀𝑥(𝑡)+𝑊₁𝑓(𝑥(𝑡))+𝑊₂𝑓(𝑥(𝑡−ℎ(𝑡)))+𝑊₃^𝑡_𝑡−𝑟𝑓^𝑇(𝑥(𝑠))𝑑𝑠+𝑊₄𝑥̇(𝑡−ℎ(𝑡)), (62) where 𝑊₁=,𝑊−2.5573−1.38131.9574−1.1398−1.0226−0.88450.5045−0.21111.03781.55320.66451.1902−0.38960.7079−0.3398−2.2543₂=,𝑊0.2853−0.07930.46940.5354−0.59551.3352−0.90360.5529−0.1497−0.6065−0.1641−0.2037−0.4348−1.3474−0.6275−2.2543₃=,𝑊0.02650.11570.0578−0.09300.3186−0.1363−0.08590.07420.2037−0.20490.01120.1457−0.3161−0.2469−0.0736−2.2543₄=,𝑊−0.30540.36820.1761−0.0235−0.0546−0.2089−0.07540.26680.45630.00230.14400.6928−0.0115−0.23490.20040.1574₀=diag{1.6305,1.9221,2.5973,1.3775},{},ΣΣ=diag1.0275,0.9960,0.3223,0.2113=diag{0,0,0,0}. (63)

From Table 1, it can be seen that the stability criterion proposed in the paper is less conservative than those obtained by Feng et al. in [32] and Lakshmanan et al. in [39]. Besides, from Tables 1 and 2, the maximum allowable time-delay bounds ℎ will become larger with the values of 𝑙 becoming larger. Moreover, Figures 1 and 2 show that the state vector 𝑥(𝑡) stabilizes to zero asymptotically with different ℎ_𝐷 and initial values. By using the Matlab LMI toolbox, we solve LMIs (13) and (14) for the case ℎ_𝐷=0 and ℎ=4.6531 and obtain 𝑃₁₁=,𝑅0.0048−0.00090.0017−0.0003−0.00090.0072−0.00100.00040.0017−0.00100.0045−0.0009−0.00030.0004−0.00090.0042₂₁₁=,𝑅0.0022−0.00120.0013−0.0004−0.00120.0043−0.00130.00060.0013−0.00130.0021−0.0007−0.00040.0006−0.00070.0015₂₂₂=,𝑅0.0063−0.00060.0011−0.0007−0.00060.0085−0.00150.00060.0011−0.00150.0050−0.0010−0.00070.0006−0.00100.0034₃₁₁=,𝑍0.0030−0.00040.0006−0.0005−0.00040.0041−0.00090.00040.0006−0.00090.0026−0.0006−0.00050.0004−0.00060.0018₁=,𝑍0.0307−0.0024−0.0035−0.0050−0.00240.01810.0044−0.0137−0.00350.00440.0136−0.0027−0.0050−0.0137−0.00270.0302₂=,𝑆0.0022−0.0000−0.00030.0005−0.00000.0017−0.0000−0.0001−0.0003−0.00000.00260.00100.0005−0.00010.00100.0045₁=,𝑆0.00650.00330.0006−0.00230.00330.00580.0017−0.00310.00060.00170.0049−0.0010−0.0023−0.0031−0.00100.0073₂=,𝑇0.0066−0.0003−0.00160.0010−0.00030.00490.0006−0.0003−0.00160.00060.00520.00110.0010−0.00030.00110.0063₁=,𝑇0.0087−0.00390.0032−0.0014−0.00370.0144−0.00170.00210.0030−0.00160.0061−0.0009−0.00070.0012−0.00170.0092₂=.0.0131−0.00080.0044−0.0002−0.00430.0237−0.00740.0003−0.00120.00290.0099−0.0003−0.00160.0014−0.00250.0086 (64)

Table 1 

Maximum allowable time-delay bounds ℎ=𝑟 in Example 1.

Table 2 

Maximum allowable time-delay bounds ℎ=𝑟 for different values ℎ_𝐷 in Example 1.

Figure 1 

The simulation of Example 1 for ℎ=𝑟=4.8652, where the initial value is [−3,−2,2,3]𝑇, ℎ𝐷=0, 𝑙=3.

Figure 2 

The simulation of Example 1 for ℎ=𝑟=4.5312, where the initial value is [−0.8,−0.6,0.4,0.8]𝑇, ℎ𝐷=0.1, 𝑙=3.

Example 2. Using this example, we will show the improvement of our results for (52) by two cases (A) and (B). Consider the following parameters as in [32, 39]:

Case A𝑊₀=,𝑊600050007₁=,𝑊1.2−0.80.60.5−1.50.7−0.8−1.2−1.4₂=,𝑊−1.40.90.5−0.61.20.80.5−0.71.1₃=,−1.80.7−0.80.61.41−0.4−0.61.2Σ=0.50000.50000.5,Σ=.000000000 (65)

Case B𝑊₀=0.9000.8,𝑊₁=,𝑊1−1.7−1.61₂=10.60.50.8,𝑊₃=,0.40.30.10.2Σ=0.25000.25,Σ=.0000 (66)

For the above two Cases A and B of numerical examples, the maximum allowable upper bounds for guaranteeing the asymptotical stability of the corresponding systems obtained from Theorem 13 are listed in Table 3. Table 3 clearly shows that our results have larger improvement over those results of previous literature. In Case A, when ℎ_𝐷=0, ℎ=𝑟, the simulation result is given as Figure 3, which implies that under the given conditions, the state vector 𝑥(𝑡) stabilizes to zero asymptotically.

Table 3 

Maximum allowable time-delay upper bounds ℎ=𝑟 for ℎ_𝐷=0 in Example 2.

Figure 3 

The simulation of Example 2 of Case A for ℎ=𝑟=4.9532, where the initial value is [−0.2,0.5,−0.4]𝑇, ℎ𝐷=0.

Example 3. Consider the following delayed neural network given in (47) with parameters: 𝑊₀=2002,𝑊₁=,𝑊1.60.30.30.5₂=0.20.10.10.2,𝑊₄=,0.15000.15Σ=1001,Σ=.0000 (67)

The corresponding results for maximum allowable upper bounds of the time-varying delay ℎ(𝑡) are given in Table 4. From Table 4, it can be clearly seen that the maximum allowable time-delay bounds ℎ will become small with the values of ℎ_𝐷 becoming large. Furthermore, when ℎ_𝐷=0.1, the state trajectories of the system (47) are shown in Figure 4.

Table 4 

Maximum allowable time delay bounds ℎ for different values ℎ_𝐷 in Example 3.

Figure 4 

The simulation of Example 3 for ℎ=𝑟=9.0123, where the initial value is [−3,3]𝑇, ℎ𝐷=0.1.

Example 4. In this Example, we give four Cases (A)–(C) that show the potential benefits and effectiveness of the developed method for delayed neural networks. Consider the delayed system (57) with the following parameters:

Case A𝑊₁=,𝑊−0.03730.4852−0.33510.2336−1.60330.5988−0.32241.23520.3394−0.0860−0.3824−0.5785−0.13110.3253−0.9534−0.5015₂=,𝑊0.8674−1.2405−0.53250.02200.0474−0.91640.03600.98161.84952.6117−0.37880.8428−2.04130.51791.1734−0.2775₀=diag{1.2769,0.6231,0.9230,0.4480},{},ΣΣ=diag0.1137,0.1279,0.7994,0.2368=diag{0,0,0,0}. (68)

Case B𝑊₁=,𝑊0.05300.04540.09870.2750₂=,𝑊0.23810.93200.03880.5062₀=diag{1.5,0.7},{},ΣΣ=diag0.3,0.8=diag{0,0}. (69)

Case C𝑊₁=11−1−1,𝑊₂=,𝑊0.88111₀={2,2},{},ΣΣ=diag0.4,0.8=diag{0,0}. (70)

From Tables 5, 6, and 7, it is clearly shown that Theorem 15 is less conservative than those in the mentioned literature. In Case A, when ℎ_𝐷=0.1, the simulation result is given as Figure 5, which implies that under the given conditions, the state vector 𝑥(𝑡) stabilizes to zero asymptotically. Hence, this example indicates fully that the method proposed in the paper plays a major role in reducing conservatism.

Table 5 

Maximum allowable upper bounds ℎ for different ℎ_𝐷 in Case A.

Table 6 

Maximum allowable time-delay upper bounds ℎ in Case B.

Table 7 

Maximum allowable time-delay bounds ℎ for different values ℎ_𝐷 in Case C.

Figure 5 

The simulation of Example 4 for ℎ=5.19, where the initial value is [−0.2,0.5,−0.4,0.2]𝑇, ℎ𝐷=0.1.

Finally, we provide the number of decision variables involved in the LIM in each of the Theorems 6–15 in Table 8.

Table 8 

Number of decision variables involved in the stability criteria.

5. Conclusions

This paper is concerned with the stability analysis of neutral type neural networks with mixed time-varying delays. Some improved delay-dependent stability results are established by using a novel approach for the networks. Improved delay-dependent stability criteria in terms of linear matrix inequalities (LMIs) are derived by employing a new type of Lyapunov-Krasovskii functionals with three and four integral terms. Different from previous results by using the first-order convex combination property, our derivation applies the idea of second-order convex combination and the property of quadratic convex function. obtained results are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness and the advantage of the proposed main results.

Acknowledgments

This work was supported by the National Basic Research Program of China (2010CB732501), the National Natural Science Foundation of China (61273015), and the Natural Science Foundation of Hainan province (111002).