New Delay-Dependent Robust Exponential Stability Criteria of LPD Neutral Systems with Mixed Time-Varying Delays and Nonlinear Perturbations

Abstract

This paper is concerned with the problem of robust exponential stability for linear parameter-dependent (LPD) neutral systems with mixed time-varying delays and nonlinear perturbations. Based on a new parameter-dependent Lyapunov-Krasovskii functional, Leibniz-Newton formula, decomposition technique of coefficient matrix, free-weighting matrices, Cauchy’s inequality, modified version of Jensen’s inequality, model transformation, and linear matrix inequality technique, new delay-dependent robust exponential stability criteria are established in terms of linear matrix inequalities (LMIs). Numerical examples are given to show the effectiveness and less conservativeness of the proposed methods.

1. Introduction

Over the past decades, the problem of stability for neutral differential systems, which have delays in both their state and the derivatives of their states, has been widely investigated by many researchers, especially in the last decade. It is well known that nonlinearities, as time delays, may cause instability and poor performance of practical systems such as engineering, biology, and economics [1]. The problems of various stability and stabilization for dynamical systems with or without state delays and nonlinear perturbations have been intensively studied in the past years by many researchers of mathematics and control communities [1–35]. Stability criteria for dynamical systems with time delay are generally divided into two classes: delay-independent one and delay-dependent one. Delay-independent stability criteria tend to be more conservative, especially for small size delay; such criteria do not give any information on the size of the delay. On the other hand, delay-dependent stability criteria are concerned with the size of the delay and usually provide a maximal delay size.

Recently, many researchers have studied the stability problem for neutral systems with time-varying delays and nonlinear perturbations have appeared [29, 31]. Furthermore, the convergence rates are essential for the practical system; then the exponential stability analysis of time delay systems has been favorably approved in the past decades; see, for example, [3, 9, 10, 14, 18–21, 25–28].

In addition, many researchers have paid attention to the problem of stability for linear systems with polytope uncertainties. The linear systems with polytopic-type uncertainties are called linear parameter-dependent (LPD) systems. That is, the uncertain state matrices are in the polytope consisting of all convex combination of known matrices. Most of sufficient (or necessary and sufficient) conditions have been obtained via Lyapunov-Krasovskii theory approaches in which parameter-dependent Lyapunov-Krasovskii functional has been employed. These conditions are always expressed in terms of linear matrix inequalities (LMIs). The results have been obtained for robust stability for LPD systems in which time-delay occurs in state variable; for example, [17, 18] presented sufficient conditions for robust stability of LPD discrete-time systems with delays. Moreover, robust stability of LPD continuous-time systems with delays was studied in [6, 19, 22, 30].

In consequence, it is important and interesting to study the problem of robust exponential stability for neutral systems with parametric uncertainties. This paper investigates the robust exponential stability analysis for LPD neutral systems with mixed time-varying delays and nonlinear perturbations. Based on combination of Leibniz-Newton formula, free-weighting matrices, Cauchy’s inequality, modified version of Jensen's inequality, decomposition technique of coefficient matrix, the use of suitable parameter-dependent Lyapunov-Krasovskii functional, model transformation, and linear matrix inequality technique, new delay-dependent robust exponential stability criteria for these systems will be obtained in terms of LMIs. Finally, numerical examples will be given to show the effectiveness of the obtained results.

2. Problem Formulation and Preliminaries

We introduce some notations, a definition, and lemmas that will be used throughout the paper. 𝑅⁺ denotes the set of all real nonnegative numbers; 𝑅^𝑛 denotes the 𝑛-dimensional space with the vector norm ‖⋅‖; ‖𝑥‖ denotes the Euclidean vector norm of 𝑥∈𝑅^𝑛; 𝑅^𝑛×𝑟 denotes the set of 𝑛×𝑟 real matrices; 𝐴^𝑇 denotes the transpose of the matrix 𝐴; 𝐴 is symmetric if 𝐴=𝐴^𝑇; 𝐼 denotes the identity matrix; 𝜆(𝐴) denotes the set of all eigenvalues of 𝐴; 𝜆_max(𝐴)= max{Re𝜆∶𝜆∈𝜆(𝐴)}; 𝜆_min(𝐴)= min{Re𝜆∶𝜆∈𝜆(𝐴)}; 𝜆_max(𝐴(𝛼))= max{𝜆_max(𝐴_𝑖)∶𝑖=1,2,…,𝑁}; 𝜆_min(𝐴(𝛼))= min{𝜆_min(𝐴_𝑖)∶𝑖=1,2,…,𝑁}; 𝐶([−𝑏,0],𝑅^𝑛) denotes the space of all continuous vector functions mapping [−𝑏,0] into 𝑅^𝑛, where 𝑏=max{ℎ,𝑟}, ℎ,𝑟∈𝑅⁺; ∗ represents the elements below the main diagonal of a symmetric matrix.

Consider the system described by the following state equations of the form: 𝑥̇(𝑡)(𝛼)𝑥(𝑡)(𝛼)𝑥((𝑡(𝛼)𝑥̇((𝑡((𝑡(((𝑡(𝑥̇((𝑡𝑥=𝐴+𝐵𝑡−ℎ))+𝐶𝑡−𝑟))+𝑓𝑡,𝑥))+𝑔𝑡,𝑥𝑡−ℎ)))+𝑤𝑡,𝑡−𝑟))),𝑡>0;𝑡+𝑡₀=𝜙(𝑡),𝑥̇𝑡+𝑡₀=𝜓(𝑡),𝑡∈[−𝑏,0], (1) where 𝑥(𝑡)∈𝑅^𝑛 is the state variable and 𝐴(𝛼), 𝐵(𝛼), 𝐶(𝛼)∈𝑅^𝑛×𝑛 are uncertain matrices belonging to the polytope 𝐴(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐴_𝑖,𝐵(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐵_𝑖,𝐶(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐶_𝑖,_𝑁𝑖=1𝛼_𝑖=1,𝛼_𝑖≥0,𝐴_𝑖,𝐵_𝑖,𝐶_𝑖∈𝑅^𝑛×𝑛,𝑖=1,…,𝑁. (2)ℎ(𝑡) and 𝑟(𝑡) are discrete and neutral time-varying delays, respectively, 0≤ℎ(𝑡)≤ℎ,ℎ̇(𝑡)≤ℎ_𝑑,0≤𝑟(𝑡)≤𝑟,𝑟̇(𝑡)≤𝑟_𝑑, (3)(4) where ℎ, 𝑟, ℎ_𝑑, and 𝑟_𝑑 are given positive real constants. Consider the initial functions 𝜙(𝑡), 𝜓(𝑡)∈ 𝐶([−𝑏,0],𝑅^𝑛) with the norm ‖𝜙‖=sup_{𝑡∈[−𝑏,0]}‖𝜙(𝑡)‖ and ‖𝜓‖=sup_{𝑡∈[−𝑏,0]}‖𝜓(𝑡)‖. The uncertainties 𝑓(𝑡,𝑥(𝑡)), 𝑔(𝑡,𝑥(𝑡−ℎ(𝑡))), and 𝑤(𝑡,𝑥̇(𝑡−𝑟(𝑡))) are the nonlinear perturbations with respect to current state 𝑥(𝑡), discrete delayed state 𝑥(𝑡−ℎ(𝑡)), and neutral delayed state 𝑥̇(𝑡−𝑟(𝑡)), respectively, and are bounded in magnitude: 𝑓^𝑇(𝑡,𝑥(𝑡))𝑓(𝑡,𝑥(𝑡))≤𝜂²𝑥^𝑇(𝑡)𝑥(𝑡),𝑔^𝑇(𝑡,𝑥(𝑡−ℎ(𝑡)))𝑔(𝑡,𝑥(𝑡−ℎ(𝑡)))≤𝜌²𝑥^𝑇((𝑡𝑥((𝑡,𝑤𝑡−ℎ))𝑡−ℎ))_𝑇(𝑡,𝑥̇(𝑡−𝑟(𝑡)))𝑤(𝑡,𝑥̇(𝑡−𝑟(𝑡)))≤𝛾²𝑥̇^𝑇(𝑡−𝑟(𝑡))𝑥̇(𝑡−𝑟(𝑡)), (5) where 𝜂,𝜌, and 𝛾 are given positive real constants.

In order to improve the bound of the discrete time-varying delayed ℎ(𝑡) in system (1), let us decompose the constant matrix 𝐵(𝛼) as 𝐵(𝛼)=𝐵₁(𝛼)+𝐵₂(𝛼), (6) where 𝐵₁(𝛼)=∑^𝑁_𝑖=1𝛼_𝑖𝐵¹_𝑖, 𝐵₂(𝛼)=∑^𝑁_𝑖=1𝛼_𝑖𝐵²_𝑖, and ∑^𝑁_𝑖=1𝛼_𝑖=1, 𝛼_𝑖≥0 with 𝐵¹_𝑖,𝐵²_𝑖∈𝑅^𝑛×𝑛, 𝑖=1,…,𝑁 being real constant matrices. By Leibniz-Newton formula, we have 0=𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))−^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠. (7)

By utilizing the following zero equation, we obtain 0=𝐺𝑥(𝑡)−𝐺𝑥𝑡−𝛽ℎ(𝑡)−𝐺^𝑡_{(𝑡)𝑡−𝛽ℎ}𝑥̇(𝑠)𝑑𝑠, (8) where 𝛽 is a given positive real constant and 𝐺∈𝑅^𝑛×𝑛 will be chosen to guarantee the robust exponential stability of system (1). By (6), (7), and (8), system (1) can be represented by the form 𝑥̇(𝑡)=𝐴(𝛼)+𝐵₁(𝛼)+𝐺𝑥(𝑡)+𝐵₂(𝛼)𝑥((𝑡(𝑡)((𝑡(((𝑡(𝑥̇((𝑡(𝛼)𝑥̇((𝑡𝑡−ℎ))−𝐺𝑥𝑡−𝛽ℎ+𝑓𝑡,𝑥))+𝑔𝑡,𝑥𝑡−ℎ)))+𝑤𝑡,𝑡−𝑟)))+𝐶𝑡−𝑟))−𝐵₁(𝛼)^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠−𝐺^𝑡_{(𝑡)𝑡−𝛽ℎ}𝑥̇(𝑠)𝑑𝑠. (9)

Definition 1. The system (1) is robustly exponentially stable, if there exist positive real constants 𝑘 and 𝑀 such that, for each 𝜙(𝑡),𝜓(𝑡)∈𝐶([−𝑏,0],𝑅^𝑛), the solution 𝑥(𝑡,𝜙,𝜓) of the system (1) satisfies 𝑥𝑡,𝜙,𝜓≤𝑀max𝜙,𝜓𝑒^−𝑘𝑡,∀𝑡∈𝑅⁺. (10)

Lemma 2 (Cauchy inequality). For any constant symmetric positive definite matrix 𝑃∈𝑅^𝑛×𝑛 and 𝑎,𝑏∈𝑅^𝑛, one has ±2𝑎^𝑇𝑏≤𝑎^𝑇𝑃𝑎+𝑏^𝑇𝑃⁻¹𝑏. (11)

Lemma 3 (see [15]). The following inequality holds for any 𝑎∈𝑅^𝑛, 𝑏∈𝑅^𝑚, 𝑁,𝑌∈𝑅^𝑛×𝑚, 𝑋∈𝑅^𝑛×𝑛, and 𝑍∈𝑅^𝑚×𝑚: −2𝑎^𝑇𝑎𝑏𝑁𝑏≤^𝑇𝑎𝑏,𝑋𝑌−𝑁∗𝑍 (12) where ^𝑋𝑌_∗𝑍≥0.

Lemma 4. For any constant symmetric positive definite matrix 𝑄∈𝑅^𝑛×𝑛 and ℎ(𝑡) which is discrete time-varying delays with (3), vector function 𝜔∶[−ℎ,0]→𝑅^𝑛 such that the integrations concerned are well defined; then ℎ⁰_−ℎ𝜔^𝑇(𝑠)𝑄𝜔(𝑠)𝑑𝑠≥⁰_(𝑡)−ℎ𝜔^𝑇(𝑠)𝑑𝑠𝑄⁰_(𝑡)−ℎ𝜔(𝑠)𝑑𝑠. (13)

Proof. From Lemma 2, it is easy to see that ℎ⁰_−ℎ𝜔^𝑇(𝑠)𝑄𝜔(𝑠)𝑑𝑠=ℎ⁰_(𝑡)−ℎ𝜔^𝑇(𝑠)𝑄𝜔(𝑠)𝑑𝑠+ℎ^−ℎ(𝑡)_−ℎ𝜔^𝑇(𝑠)(𝑠)(𝑡)𝑄𝜔𝑑𝑠≥ℎ⁰_(𝑡)−ℎ𝜔^𝑇(𝑠)(𝑠)=1𝑄𝜔𝑑𝑠2⁰_(𝑡)−ℎ𝜔^𝑇(𝑠)𝑄𝜔(𝑠)+𝜔^𝑇(𝜉)(𝜉≥1𝑄𝜔)𝑑𝑠𝑑𝜉2⁰_(𝑡)−ℎ2𝜔^𝑇(𝑠)𝑄^1/2𝑇×𝑄^1/2𝜔(𝜉)𝑑𝑠𝑑𝜉=⁰_(𝑡)−ℎ𝜔^𝑇(𝑠)𝑑𝑠𝑄⁰_(𝑡)−ℎ𝜔(𝑠)𝑑𝑠. (14)

Lemma 5. Let 𝑥(𝑡)∈𝑅^𝑛 be a vector-valued function with first order continuous derivative entries. Then, the following integral inequality holds for any matrices 𝑀_𝑖∈𝑅^𝑛×𝑛, 𝑖=1,2,…,5, and ℎ(𝑡) is discrete time-varying delays with (3) and symmetric positive definite matrix 𝑋∈𝑅^𝑛×𝑛: −^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)𝑥(𝑡)𝑥((𝑡𝑑𝑠≤𝑡−ℎ))^𝑇×𝑀₁+𝑀^𝑇₁−𝑀^𝑇₁+𝑀₂∗−𝑀₂−𝑀^𝑇₂×𝑥(𝑡)𝑥((𝑡𝑥(𝑡)𝑥((𝑡𝑡−ℎ))+ℎ𝑡−ℎ))^𝑇𝑀₃𝑀₄∗𝑀₅×𝑥(𝑡)𝑥((𝑡,𝑡−ℎ)) (15) where 𝑋𝑀₁𝑀₂∗𝑀₃𝑀₄∗∗𝑀₅≥0. (16)

Proof. From the Leibniz-Newton formula, one has 0=𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))−^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑡)𝑑𝑠. (17) Therefore, for any 𝐻₁, 𝐻₂∈𝑅^𝑛×𝑛, the following equation is true: 0=2𝑥^𝑇(𝑡)−𝑥^𝑇(𝑡−ℎ(𝑡))−^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑡)×𝐻𝑑𝑠₁𝑥(𝑡)+𝐻₂𝑥(𝑡−ℎ(𝑡))=2𝑥^𝑇(𝑡)𝐻₁𝑥(𝑡)+2𝑥^𝑇(𝑡)𝐻₂𝑥(𝑡−ℎ(𝑡))−2𝑥^𝑇(𝑡−ℎ(𝑡))𝐻₁𝑥(𝑡)−2𝑥^𝑇(𝑡−ℎ(𝑡))𝐻^𝑇₂𝑥(𝑡−ℎ(𝑡))−2^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝐻₁𝑥(𝑡)−2^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝐻₂𝑥((𝑡=𝑥(𝑡)𝑥((𝑡𝑡−ℎ))𝑡−ℎ))^𝑇𝐻₁+𝐻^𝑇₁−𝐻^𝑇₁+𝐻₂∗−𝐻₂−𝐻^𝑇₂×𝑥(𝑡)𝑥(𝑡−ℎ(𝑡))−2^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐻₁𝐻₂𝑥(𝑡)𝑥(𝑡−ℎ(𝑡))𝑑𝑠. (18)
Using Lemma 3 with 𝑎=𝑥̇(𝑠), 𝑏=^𝑥(𝑡)_{𝑥(𝑡−ℎ(𝑡))}, 𝑁=[𝐻₁𝐻₂], 𝑌=[𝑀₁𝑀₂], and 𝑍=^𝑀3𝑀4_∗𝑀5, we obtain −2^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐻₁𝐻₂𝑥(𝑡)𝑥((𝑡≤𝑡−ℎ))𝑑𝑠_{𝑡𝑡−ℎ(𝑡)}𝑥̇(𝑠)𝑥(𝑡)𝑥(𝑡−ℎ(𝑡))^𝑇𝑋𝑀₁−𝐻₁𝑀₂−𝐻₂∗𝑀₃𝑀₄∗∗𝑀₅×𝑥̇(𝑠)𝑥(𝑡)𝑥((𝑡=𝑡−ℎ))𝑑𝑠_{𝑡(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)+𝑥(𝑡)𝑥((𝑡𝑑𝑠𝑡−ℎ))^𝑇×𝑀₁+𝑀^𝑇₁−𝐻₁−𝐻^𝑇₁−𝑀^𝑇₁+𝑀₂+𝐻^𝑇₁−𝐻₂∗−𝑀₂−𝑀^𝑇₂+𝐻₂+𝐻^𝑇₂×𝑥(𝑡)𝑥((𝑡(𝑡)𝑥(𝑡)𝑡−ℎ))𝑥((𝑡+ℎ𝑡−ℎ))^𝑇𝑀₃𝑀₄∗𝑀₅×𝑥(𝑡)𝑥(𝑡−ℎ(𝑡))≤^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)𝑥(𝑡)𝑥((𝑡𝑑𝑠+𝑡−ℎ))^𝑇×𝑀₁+𝑀^𝑇₁−𝑀^𝑇₁+𝑀₂∗−𝑀₂−𝑀^𝑇₂−𝐻₁+𝐻^𝑇₁−𝐻^𝑇₁+𝐻₂∗−𝐻₂−𝐻^𝑇₂𝑥(𝑡)𝑥((𝑡𝑥(𝑡)𝑥((𝑡𝑡−ℎ))+ℎ𝑡−ℎ))^𝑇𝑀₃𝑀₄∗𝑀₅𝑥(𝑡)𝑥((𝑡.𝑡−ℎ)) (19)
Substituting (19) into (18), we obtain −^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)𝑥(𝑡)𝑥((𝑡𝑑𝑠≤𝑡−ℎ))^𝑇×𝐻₁+𝐻^𝑇₁−𝐻^𝑇₁+𝐻₂∗−𝐻₂−𝐻^𝑇₂×𝑥(𝑡)𝑥((𝑡+𝑥(𝑡)𝑥((𝑡𝑡−ℎ))𝑡−ℎ))^𝑇×𝑀₁+𝑀^𝑇₁−𝑀^𝑇₁+𝑀₂∗−𝑀₂−𝑀^𝑇₂−𝐻₁+𝐻^𝑇₁−𝐻^𝑇₁+𝐻₂∗−𝐻₂−𝐻^𝑇₂×𝑥(𝑡)𝑥((𝑡𝑥(𝑡)𝑥((𝑡𝑡−ℎ))+ℎ𝑡−ℎ))^𝑇𝑀₃𝑀₄∗𝑀₅×𝑥(𝑡)𝑥((𝑡=𝑡−ℎ))𝑥(𝑡)𝑥(𝑡−ℎ(𝑡))^𝑇×𝑀₁+𝑀^𝑇₁−𝑀^𝑇₁+𝑀₂∗−𝑀₂−𝑀^𝑇₂×𝑥(𝑡)𝑥((𝑡𝑥(𝑡)𝑥((𝑡𝑡−ℎ))+ℎ𝑡−ℎ))^𝑇𝑀₃𝑀₄∗𝑀₅×𝑥(𝑡)𝑥((𝑡.𝑡−ℎ)) (20)
From (3), it is clear that −^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)𝑑𝑠≤−^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)𝑑𝑠. (21)
From (20) and (21), the integral inequality becomes −^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑋𝑥̇(𝑠)𝑥(𝑡)𝑥((𝑡𝑑𝑠≤𝑡−ℎ))^𝑇×𝑀₁+𝑀^𝑇₁−𝑀^𝑇₁+𝑀₂∗−𝑀₂−𝑀^𝑇₂×𝑥(𝑡)𝑥((𝑡𝑥(𝑡)𝑥((𝑡𝑡−ℎ))+ℎ𝑡−ℎ))^𝑇𝑀₃𝑀₄∗𝑀₅×𝑥(𝑡)𝑥((𝑡.𝑡−ℎ)) (22) The proof of the theorem is complete.

Remark 6. In Lemma 4 and Lemma 5, we have modified the method from [8, 33], respectively.

3. Main results

3.1. Robust Exponential Stability Criteria

In this section, robust exponential stability criteria dependent on mixed time-varying delays of LPD neutral delayed system (1) with nonlinear perturbations via linear matrix inequality (LMI) approach will be presented. We introduce the following notations for later use: ∑^{𝑃𝑗(𝛼)=𝑁𝑖=1𝛼𝑖𝑃𝑗𝑖,𝑍1(𝛼)=𝑁𝑖=1𝛼𝑖𝑍1𝑖,𝑅𝑝(𝛼)=𝑁𝑖=1𝛼𝑖𝑅𝑝𝑖,𝑁𝑗(𝛼)=𝑁𝑖=1𝛼𝑖𝑁𝑗𝑖,𝑂𝑗(𝛼)=𝑁𝑖=1𝛼𝑖𝑂𝑗𝑖,𝑊𝑗(𝛼)=𝑁𝑖=1𝛼𝑖𝑊𝑗𝑖,𝑀𝑗(𝛼)=𝑁𝑖=1𝛼𝑖𝑀𝑗𝑖,𝑁𝑖=1𝛼𝑖=1,𝛼𝑖𝑃≥0,𝑗𝑖,𝑍1𝑖,𝑅𝑝𝑖,𝑊𝑗𝑖,𝑁𝑗𝑖,𝑂𝑗𝑖,𝑀𝑗𝑖∈𝑅𝑛×𝑛,𝑗=1,2,…,10,𝑝=1,2,…,6,𝑖=1,2,…,𝑁;}_𝑖,𝑗=Σ^1,1_𝑖,𝑗Σ^1,2_𝑖,𝑗Σ^1,3_𝑖,𝑗Σ^1,4_𝑖,𝑗Σ^1,5_𝑖,𝑗Σ^1,6_𝑖,𝑗Σ^1,7_𝑖,𝑗Σ^1,8_𝑖,𝑗Σ^1,9_𝑖,𝑗Σ^1,10_𝑖,𝑗∗Σ^2,2_𝑖,𝑗Σ^2,3_𝑖,𝑗Σ^2,4_𝑖,𝑗Σ^2,5_𝑖,𝑗Σ^2,6_𝑖,𝑗Σ^2,7_𝑖,𝑗Σ^2,8_𝑖,𝑗Σ^2,9_𝑖,𝑗Σ^2,10_𝑖,𝑗∗∗Σ_𝑖^3,3Σ^3,4_𝑖,𝑗Σ_𝑖^3,5Σ_𝑖^3,6Σ^3,7_𝑖,𝑗Σ_𝑖^3,8Σ_𝑖^3,9Σ_𝑖^3,10∗∗∗Σ^4,4_𝑖,𝑗Σ^4,5_𝑖,𝑗Σ^4,6_𝑖,𝑗Σ^4,7_𝑖,𝑗Σ_𝑖^4,8Σ_𝑖^4,9Σ_𝑖^4,10∗∗∗∗Σ_𝑖^5,5Σ_𝑖^5,6Σ^5,7_𝑖,𝑗Σ_𝑖^5,8Σ_𝑖^5,9Σ_𝑖^5,10∗∗∗∗∗Σ_𝑖^6,6Σ^6,7_𝑖,𝑗Σ_𝑖^6,8Σ_𝑖^6,9Σ_𝑖^6,10∗∗∗∗∗∗Σ^7,7_𝑖,𝑗Σ^7,8_𝑖,𝑗Σ^7,9_𝑖,𝑗Σ^7,10_𝑖,𝑗∗∗∗∗∗∗∗Σ_𝑖^8,8Σ_𝑖^8,9Σ_𝑖^8,10∗∗∗∗∗∗∗∗Σ_𝑖^9,9Σ_𝑖^9,10∗∗∗∗∗∗∗∗∗Σ_𝑖^10,10, (23) where Σ^1,1_𝑖,𝑗=𝐴^𝑇_𝑖𝑃¹_𝑗+𝐵¹_𝑖^𝑇𝑃¹_𝑗+𝑃¹_𝑖𝐴_𝑗+𝑃¹_𝑖𝐵¹_𝑗+𝑍¹_𝑖+𝑍¹_𝑖^𝑇+𝑃³_𝑖+𝑃⁴_𝑖+ℎ𝑀¹_𝑖^𝑇+ℎ𝑀¹_𝑖−𝑒^−2𝑘ℎ𝑃⁹_𝑖−𝑒^{−2𝑘𝛽ℎ}𝑃_𝑖¹⁰+ℎ²𝑀³_𝑖+𝛽ℎ𝑀⁶_𝑖+𝛽ℎ𝑀⁶_𝑖^𝑇+𝛽²ℎ²𝑀⁸_𝑖+𝑁¹_𝑖^𝑇+𝑁¹_𝑖+𝑂¹_𝑖^𝑇+𝑂¹_𝑖+𝐴^𝑇_𝑖𝑊¹_𝑗+𝑊¹_𝑖^𝑇𝐴_𝑗+𝐵¹_𝑖^𝑇𝑊¹_𝑗+𝑊¹_𝑖^𝑇𝐵¹_𝑗+𝜖₁𝜂²𝐼+2𝑘𝑃¹_𝑖,Σ^1,2_𝑖,𝑗=𝑃¹_𝑖𝐵²_𝑗−ℎ𝑀¹_𝑖^𝑇+ℎ𝑀²_𝑖+ℎ²𝑀⁴_𝑖+𝑒^−2𝑘ℎ𝑃⁹_𝑖+ℎ𝑅²_𝑖−𝑁¹_𝑖^𝑇+𝑁²_𝑖+𝑂²_𝑖+𝑊¹_𝑖^𝑇𝐵²_𝑗+𝐴^𝑇_𝑖𝑊²_𝑗+𝐵¹_𝑖^𝑇𝑊²_𝑗,Σ^1,3_𝑖,𝑗=−𝑍¹_𝑖−𝛽ℎ𝑀⁶_𝑖^𝑇+𝛽ℎ𝑀⁷_𝑖+𝛽ℎ²𝑀⁹_𝑖+𝛽ℎ𝑅⁵_𝑖+𝑒^{−2𝑘𝛽ℎ}𝑃_𝑖¹⁰+𝑁³_𝑖−𝑂¹_𝑖^𝑇+𝑂³_𝑖+𝐴^𝑇_𝑖𝑊³_𝑗+𝐵¹_𝑖^𝑇𝑊³_𝑗,Σ^1,4_𝑖,𝑗=−𝑃¹_𝑖𝐵¹_𝑗−𝑁¹_𝑖^𝑇+𝑁⁴_𝑖+𝑂⁴_𝑖−𝑊¹_𝑖^𝑇𝐵¹_𝑗+𝐴^𝑇_𝑖𝑊⁴_𝑗+𝐵¹_𝑖^𝑇𝑊⁴_𝑗,Σ^1,5_𝑖,𝑗=−𝑍¹_𝑖+𝑁⁵_𝑖−𝑂¹_𝑖^𝑇+𝑂⁵_𝑖+𝐴^𝑇_𝑖𝑊⁵_𝑗+𝐵¹_𝑖^𝑇𝑊⁵_𝑗,Σ^1,6_𝑖,𝑗=𝑁⁶_𝑖+𝑂⁶_𝑖−𝑊¹_𝑖^𝑇+𝐴^𝑇_𝑖𝑊⁶_𝑗+𝐵¹_𝑖^𝑇𝑊⁶_𝑗,Σ^1,7_𝑖,𝑗=𝑃¹_𝑖𝐶_𝑗+𝑁⁷_𝑖+𝑂⁷_𝑖+𝑊¹_𝑖^𝑇𝐶_𝑗+𝐴^𝑇_𝑖𝑊⁷_𝑗+𝐵¹_𝑖^𝑇𝑊⁷_𝑗,Σ^1,8_𝑖,𝑗=𝑃¹_𝑖+𝑁⁸_𝑖+𝑂⁸_𝑖+𝑊¹_𝑖^𝑇+𝐴^𝑇_𝑖𝑊⁸_𝑗+𝐵¹_𝑖^𝑇𝑊⁸_𝑗,Σ^1,9_𝑖,𝑗=𝑃¹_𝑖+𝑁⁹_𝑖+𝑂⁹_𝑖+𝑊¹_𝑖^𝑇+𝐴^𝑇_𝑖𝑊⁹_𝑗+𝐵¹_𝑖^𝑇𝑊⁹_𝑗,Σ^1,10_𝑖,𝑗=𝑃¹_𝑖+𝑁_𝑖¹⁰+𝑂_𝑖¹⁰+𝑊¹_𝑖^𝑇+𝐴^𝑇_𝑖𝑊_𝑗¹⁰+𝐵¹_𝑖^𝑇𝑊_𝑗¹⁰,Σ^2,2_𝑖,𝑗=−1−ℎ_𝑑𝑒^−2𝑘ℎ𝑃³_𝑖−ℎ𝑀²_𝑖^𝑇−ℎ𝑀²_𝑖+ℎ²𝑀⁵_𝑖+ℎ²𝑅¹_𝑖−2ℎ𝑅²_𝑖−𝑒^−2𝑘ℎ𝑃⁹_𝑖+𝜖₂𝜌²𝐼−𝑁²_𝑖^𝑇−𝑁²_𝑖+𝑊²_𝑖^𝑇𝐵²_𝑗+𝐵²_𝑖^𝑇𝑊²_𝑗,Σ^2,3_𝑖,𝑗=−𝑁³_𝑖−𝑂²_𝑖^𝑇+𝐵²_𝑖^𝑇𝑊³_𝑗,Σ^2,4_𝑖,𝑗=−𝑁²_𝑖^𝑇−𝑁⁴_𝑖−𝑊²_𝑖^𝑇𝐵¹_𝑗+𝐵²_𝑖^𝑇𝑊⁴_𝑗,Σ^2,5_𝑖,𝑗=−𝑁⁵_𝑖−𝑂²_𝑖^𝑇+𝐵²_𝑖^𝑇𝑊⁵_𝑗,Σ^2,6_𝑖,𝑗=−𝑁⁶_𝑖−𝑊²_𝑖^𝑇+𝐵²_𝑖^𝑇𝑊⁶_𝑗,Σ^2,7_𝑖,𝑗=−𝑁⁷_𝑖+𝑊²_𝑖^𝑇𝐶_𝑗+𝐵²_𝑖^𝑇𝑊⁷_𝑗,Σ^2,8_𝑖,𝑗=−𝑁⁸_𝑖+𝑊²_𝑖^𝑇+𝐵²_𝑖^𝑇𝑊⁸_𝑗,Σ^2,9_𝑖,𝑗=−𝑁⁹_𝑖^𝑇−𝑊²_𝑖^𝑇+𝐵²_𝑖^𝑇𝑊⁹_𝑗,Σ^2,10_𝑖,𝑗=−𝑁_𝑖¹⁰+𝑊²_𝑖^𝑇+𝐵²_𝑖^𝑇𝑊_𝑗¹⁰,Σ_𝑖^3,3=−1−𝛽ℎ_𝑑𝑒^{−2𝛽𝑘ℎ}𝑃⁴_𝑖−𝑒^{−2𝛽𝑘ℎ}𝑃_𝑖¹⁰+𝛽²ℎ²𝑅⁴_𝑖−2𝛽ℎ𝑅⁵_𝑖−𝛽ℎ𝑀⁷_𝑖^𝑇−𝛽ℎ𝑀⁷_𝑖+𝛽²ℎ²𝑀_𝑖¹⁰−𝑂³_𝑖^𝑇−𝑂³_𝑖,Σ^3,4_𝑖,𝑗=−𝑁³_𝑖^𝑇−𝑂⁴_𝑖−𝑊³_𝑖^𝑇𝐵¹_𝑗,Σ_𝑖^3,5=−𝑂³_𝑖^𝑇−𝑂⁵_𝑖,Σ_𝑖^3,6=−𝑂⁹_𝑖+𝑊³_𝑖^𝑇,Σ_𝑖^3,7=−𝑂⁷_𝑖+𝑊³_𝑖^𝑇𝐶_𝑗,Σ_𝑖^3,8=−𝑂⁸_𝑖+𝑊³_𝑖^𝑇,Σ_𝑖^3,9=−𝑂⁹_𝑖+𝑊³_𝑖^𝑇,Σ_𝑖^3,10=−𝑂_𝑖¹⁰+𝑊³_𝑖^𝑇,Σ^4,4_𝑖,𝑗=−𝑒^2𝑘ℎ𝑃⁷_𝑖−𝑁⁴_𝑖^𝑇−𝑁⁴_𝑖+𝑊⁴_𝑖^𝑇𝐵¹_𝑗−𝐵¹_𝑖^𝑇𝑊⁴_𝑗,Σ^4,5_𝑖,𝑗=−𝑁⁵_𝑖−𝑂⁴_𝑖^𝑇−𝐵¹_𝑖^𝑇𝑊⁵_𝑗,Σ^4,6_𝑖,𝑗=−𝑁⁶_𝑖−𝑊⁴_𝑖^𝑇−𝐵¹_𝑖^𝑇𝑊⁶_𝑗,Σ^4,7_𝑖,𝑗=−𝑁⁷_𝑖+𝑊⁴_𝑖^𝑇𝐶_𝑗−𝐵¹_𝑖^𝑇𝑊⁷_𝑗,Σ_𝑖^4,8=−𝑂⁸_𝑖+𝑊³_𝑖^𝑇,Σ_𝑖^4,9=−𝑂⁹_𝑖+𝑊³_𝑖^𝑇,Σ_𝑖^4,10=−𝑂_𝑖¹⁰+𝑊³_𝑖^𝑇,Σ_𝑖^5,5=−𝑒^2𝛽𝑘ℎ𝑃⁸_𝑖−𝑂⁵_𝑖^𝑇−𝑂⁵_𝑖,Σ_𝑖^5,6=−𝑂⁶_𝑖−𝑊⁵_𝑖^𝑇,Σ^5,7_𝑖,𝑗=−𝑂⁷_𝑖+𝑊⁵_𝑖^𝑇𝐶_𝑗,Σ_𝑖^5,8=−𝑂⁸_𝑖+𝑊⁵_𝑖^𝑇,Σ_𝑖^5,9=−𝑂⁹_𝑖+𝑊⁵_𝑖^𝑇,Σ_𝑖^5,10=−𝑂_𝑖¹⁰+𝑊⁵_𝑖^𝑇,Σ_𝑖^6,6=𝑃²_𝑖+ℎ²𝑃⁵_𝑖+𝛽²ℎ²𝑃⁶_𝑖+ℎ²𝑃⁷_𝑖+𝛽²ℎ²𝑃⁸_𝑖+ℎ²𝑃⁹_𝑖+𝛽²ℎ²𝑃_𝑖¹⁰−𝑊⁶_𝑖^𝑇−𝑊⁶_𝑖,Σ^6,7_𝑖,𝑗=𝑊⁶_𝑖^𝑇𝐶_𝑗−𝑊⁷_𝑖,Σ_𝑖^6,8=𝑊⁶_𝑖^𝑇−𝑊⁸_𝑖,Σ_𝑖^6,9=𝑊⁶_𝑖^𝑇−𝑊⁹_𝑖,Σ_𝑖^6,10=𝑊⁶_𝑖^𝑇−𝑊_𝑖¹⁰,Σ^7,7_𝑖,𝑗=−1−𝑟_𝑑𝑒^−2𝑘𝑟𝑃²_𝑖+𝜖₃𝛾²𝐼+𝑊⁷_𝑖^𝑇𝐶_𝑗+𝐶^𝑇_𝑖𝑊⁷_𝑗,Σ^7,8_𝑖,𝑗=𝑊⁷_𝑖^𝑇−𝐶^𝑇_𝑖𝑊⁸_𝑗,Σ^7,9_𝑖,𝑗=𝑊⁷_𝑖^𝑇−𝐶^𝑇_𝑖𝑊⁹_𝑗,Σ^7,10_𝑖,𝑗=𝑊⁷_𝑖^𝑇−𝐶^𝑇_𝑖𝑊_𝑗¹⁰,Σ_𝑖^8,8=𝑊⁸_𝑖^𝑇+𝑊⁸_𝑖−𝜖₁Σ𝐼,_𝑖8,9=𝑊⁸_𝑖^𝑇+𝑊⁹_𝑖,Σ_𝑖^8,10=𝑊⁸_𝑖^𝑇+𝑊_𝑖¹⁰,Σ_𝑖^9,9=𝑊⁹_𝑖^𝑇+𝑊⁹_𝑖−𝜖₂𝐼,Σ_𝑖^9,10=𝑊⁹_𝑖^𝑇+𝑊_𝑖¹⁰,Σ_𝑖^10,10=𝑊_𝑖^𝑇10+𝑊_𝑖¹⁰−𝜖₃Σ𝐼,_𝑖7,7=−(1−𝑟𝑑)𝑒^2𝑘𝑟𝑃²_𝑖+𝑊⁷_𝑖^𝑇𝐶_𝑗+𝐶^𝑇_𝑖𝑊⁷_𝑗,𝑍¹_𝑖=𝑃¹_𝑖𝐺. (24)

Theorem 7. For ‖𝐶_𝑖‖+𝛾<1,  𝑖=1,2,…,𝑁 and given positive real constants ℎ, ℎ_𝑑, 𝑟, 𝑟_𝑑, 𝜂, 𝜌, 𝛾, and 𝛽, system (1) is robustly exponentially stable with a decay rate 𝑘, if there exist symmetric positive definite matrices 𝑃^𝑗_𝑖, any appropriate dimensional matrices 𝑅^𝑝_𝑖, 𝑁^𝑗_𝑖, 𝑂^𝑗_𝑖, 𝑊^𝑗_𝑖, 𝑀^𝑗_𝑖, 𝐺, 𝑝=1,2,…6, 𝑗=1,2,…,10, 𝑖=1,2,…,𝑁, and positive real constants 𝜖₁, 𝜖₂, and 𝜖₃ such that the following symmetric linear matrix inequalities hold: 𝑅¹_𝑖𝑅²_𝑖∗𝑅³_𝑖𝑅>0,𝑖=1,2,…,𝑁,_4𝑖𝑅⁵_𝑖∗𝑅⁶_𝑖𝑒>0,𝑖=1,2,…,𝑁,_−2𝑘ℎ𝑃³_𝑖−𝑅³_𝑖𝑀¹_𝑖𝑀²_𝑖∗𝑀³_𝑖𝑀⁴_𝑖∗∗𝑀⁵_𝑖𝑒≥0,𝑖=1,2,…,𝑁,_{−2𝛽𝑘ℎ}𝑃⁶_𝑖−𝑅⁶_𝑖𝑀⁶_𝑖𝑀⁷_𝑖∗𝑀⁸_𝑖𝑀⁹_𝑖∗∗𝑀_𝑖¹⁰≥0,𝑖=1,2,…,𝑁,_𝑖,𝑖<−𝐼,𝑖=1,2,…,𝑁,_𝑖,𝑗+_𝑗,𝑖<2(𝑁−1)𝐼,𝑖=1,2,…,𝑁−1,𝑗=𝑖+1,𝑖+2,…,𝑁. (25)
Moreover, the solution 𝑥(𝑡,𝜙,𝜓) satisfies the inequality 𝑥𝑡,𝜙,𝜓≤𝐿𝜆_min𝑃₁(𝛼)max𝜙,𝜓𝑒^−𝑘𝑡,∀𝑡∈𝑅⁺, (26) where 𝐿=𝜆_max(𝑃₁(𝛼))+𝑟𝜆_max(𝑃₂(𝛼))+ℎ𝜆_max(𝑃₃(𝛼)) + 𝛽ℎ𝜆_max(𝑃₄(𝛼))+ℎ³𝜆_max(𝑃₅(𝛼) + 𝑃₇(𝛼) + 𝑃₉(𝛼)) + (𝛽ℎ)³𝜆_max(𝑃₆(𝛼) + 𝑃₈(𝛼)+𝑃₁₀(𝛼))+ℎ³𝜆_max(^{𝑅1(𝛼)𝑅2(𝛼)}_{∗𝑅3(𝛼)}) + (𝛽ℎ)³𝜆_max(^{𝑅4(𝛼)𝑅5(𝛼)}_{∗𝑅6(𝛼)}).

Proof. Choose a parameter-dependent Lyapunov-Krasovskii functional candidate for system (9) as 𝑉(𝑡)=⁷_𝑖=1𝑉_𝑖(𝑡), (27) where 𝑉₁(𝑡)=𝑥^𝑇(𝑡)𝑃₁(𝛼)𝑥(𝑡),𝑉₂(𝑡)=^𝑡_{(𝑡)𝑡−𝑟}𝑒^{2𝑘(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑃₂(𝛼)𝑥̇(𝑠)𝑉𝑑𝑠,₃(𝑡)=^𝑡_{(𝑡)𝑡−ℎ}𝑒^{2𝑘(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑃₃(𝛼)𝑥(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝛽ℎ}𝑒^{2𝑘(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑃₄(𝛼)𝑥(𝑠)𝑉𝑑𝑠,₄(𝑡)=ℎ⁰_−ℎ^𝑡_𝑡+𝜃𝑒^{2𝑘(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑃₅(𝛼)𝑥̇(𝑠)𝑑𝑠𝑑𝜃+𝛽ℎ⁰_−𝛽ℎ^𝑡_𝑡+𝜃𝑒^{2𝑘(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑃₆(𝛼)𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₅(𝑡)=ℎ⁰_−ℎ^𝑡_𝑡+𝜃𝑒^{2𝑘(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑃₇(𝛼)𝑥̇(𝑠)𝑑𝑠𝑑𝜃+𝛽ℎ⁰_−𝛽ℎ^𝑡_𝑡+𝜃𝑒^{2𝑘(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑃₈(𝛼)𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₆(𝑡)=ℎ⁰_−ℎ^𝑡_𝑡+𝜃𝑒^{2𝑘(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑃₉(𝛼)𝑥̇(𝑠)𝑑𝑠𝑑𝜃+𝛽ℎ⁰_−𝛽ℎ^𝑡_𝑡+𝜃𝑒^{2𝑘(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑃₁₀(𝛼)𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₇(𝑡)=ℎ^𝑡_−ℎ^𝜃_{(𝜃)𝜃−ℎ}𝑒^{2𝑘(𝜃−𝑡)}𝑥((𝜃𝑥̇(𝑠)𝜃−ℎ))^𝑇×𝑅₁(𝛼)𝑅₂(𝛼)∗𝑅₃(𝛼)×𝑥((𝜃𝑥̇(𝑠)𝜃−ℎ))𝑑𝑠𝑑𝜃+𝛽ℎ^𝑡_−𝛽ℎ^𝜃_{(𝜃)𝜃−𝛽ℎ}𝑒^{2𝑘(𝜃−𝑡)}𝑥(𝜃)𝑥̇(𝑠)𝜃−𝛽ℎ^𝑇×𝑅₄(𝛼)𝑅₅(𝛼)∗𝑅₆(𝛼)×𝑥(𝜃)𝑥̇(𝑠)𝜃−𝛽ℎ𝑑𝑠𝑑𝜃. (28)
Calculating the time derivatives of 𝑉_𝑖(𝑡), 𝑖=1,2,3,…,7, along the trajectory of (9), yields 𝑉̇₁(𝑡)=2𝑥^𝑇(𝑡)𝑃₁(𝛼)𝑥̇(𝑡)=2𝑥^𝑇𝑃₁(𝛼)𝐴(𝛼)+𝐵₁(𝛼)+𝐺𝑥(𝑡)+𝐵₂(𝛼)𝑥((𝑡(𝑡((𝑡(((𝑡(𝑥̇((𝑡(𝛼)𝑥̇((𝑡𝑡−ℎ))−𝐺𝑥𝑡−𝛽ℎ)+𝑓𝑡,𝑥))+𝑔𝑡,𝑥𝑡−ℎ)))+𝑤𝑡,𝑡−𝑟)))+𝐶𝑡−𝑟))−𝐵₁(𝛼)^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠−𝐺^𝑡_{(𝑡)𝑡−𝛽ℎ}𝑥̇(𝑠)𝑑𝑠+2𝑘𝑥^𝑇(𝑡)𝑃₁(𝛼)𝑥(𝑡)−2𝑘𝑉₁(𝑡),𝑉̇₂(𝑡)=𝑥̇^𝑇(𝑡)𝑃₂(𝛼)𝑥̇(𝑡)−(1−𝑟̇(𝑡))𝑒^{−2𝑘𝑟(𝑡)}𝑥̇^𝑇(𝑡−𝑟(𝑡))×𝑃₂(𝛼)𝑥̇(𝑡−𝑟(𝑡))−2𝑘𝑉₂(𝑡)≤𝑥̇^𝑇(𝑡)𝑃₂(𝛼)𝑥̇(𝑡)−𝑒^−2𝑘𝑟𝑥̇^𝑇(𝑡−𝑟(𝑡))×𝑃₂(𝛼)𝑥̇(𝑡−𝑟(𝑡))+𝑟_𝑑𝑥̇^𝑇(𝑡−𝑟(𝑡))𝑃₂(𝛼)𝑥̇(𝑡−𝑟(𝑡))−2𝑘𝑉₂(𝑡). (29)
The time derivative of 𝑉₃(𝑡) is 𝑉̇₃(𝑡)=𝑥^𝑇(𝑡)𝑃₃(𝛼)𝑥(𝑡)−1−ℎ̇(𝑡)𝑒^{−2𝑘ℎ(𝑡)}𝑥^𝑇(𝑡−ℎ(𝑡))×𝑃₃(𝛼)𝑥(𝑡−ℎ(𝑡))+𝑥^𝑇(𝑡)𝑃₄(𝛼)𝑥(𝑡)−1−𝛽ℎ̇(𝑡)𝑒^{−2𝑘𝛽ℎ(𝑡)}𝑥^𝑇𝑡−𝛽ℎ(𝑡)×𝑃₄(𝛼)𝑥𝑡−𝛽ℎ(𝑡)−2𝑘𝑉₃(𝑡)≤𝑥^𝑇(𝑡)𝑃₃(𝛼)𝑥(𝑡)−𝑒^−2𝑘ℎ𝑥^𝑇(𝑡−ℎ(𝑡))×𝑃₃(𝛼)𝑥(𝑡−ℎ(𝑡))+ℎ_𝑑𝑒^−2𝑘ℎ𝑥^𝑇(𝑡−ℎ(𝑡))×𝑃₃(𝛼)𝑥(𝑡−ℎ(𝑡))+𝑥^𝑇(𝑡)𝑃₄(𝛼)𝑥(𝑡)−𝑒^{−2𝛼𝛽ℎ}𝑥^𝑇𝑡−𝛽ℎ(𝑡)𝑃₄(𝛼)𝑥𝑡−𝛽ℎ(𝑡)+𝛽ℎ_𝑑𝑒^{−2𝑘𝛽ℎ}𝑥^𝑇𝑡−𝛽ℎ(𝑡)𝑃₄(𝛼)𝑥𝑡−𝛽ℎ(𝑡)−2𝑘𝑉₃(𝑡). (30)
Obviously, for any scalar 𝑠∈[𝑡−ℎ,𝑡], we get 𝑒^−2𝑘ℎ≤𝑒^{−2𝑘(𝑠−𝑡)}≤1 and for any scalar 𝑠∈[𝑡−𝛽ℎ,𝑡], we obtain 𝑒^{−2𝛽𝑘ℎ}≤𝑒^{−2𝛽𝑘(𝑠−𝑡)}≤1. Together with Lemma 4, we obtain 𝑉̇₄(𝑡)=ℎ²𝑥̇^𝑇(𝑡)𝑃₅(𝛼)𝑥̇(𝑡)−ℎ⁰_−ℎ𝑒^2𝑘𝑠𝑥̇^𝑇(𝑡+𝑠)𝑃₅(𝛼)𝑥̇(𝑡+𝑠)𝑑𝑠+𝛽²ℎ²𝑥̇^𝑇(𝑡)𝑃₆(𝛼)𝑥̇(𝑡)−𝛽ℎ⁰_−𝛽ℎ𝑒^{2𝛽𝑘𝑠}𝑥̇^𝑇(𝑡+𝑠)𝑃₆(𝛼)𝑥̇(𝑡+𝑠)𝑑𝑠−2𝑘𝑉₄(𝑡)≤ℎ²𝑥̇^𝑇(𝑡)𝑃₅(𝛼)𝑥̇(𝑡)−ℎ^𝑡_𝑡−ℎ𝑒^{2𝑘(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑃₅(𝛼)𝑥̇(𝑠)𝑑𝑠+𝛽²ℎ²𝑥̇^𝑇(𝑡)𝑃₆(𝛼)𝑥̇(𝑡)−𝛽ℎ^𝑡_{𝑡−𝛽ℎ}𝑒^{2𝛽𝑘(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑃₆(𝛼)𝑥̇(𝑠)𝑑𝑠−2𝑘𝑉₄(𝑡)≤ℎ²𝑥̇^𝑇(𝑡)𝑃₅(𝛼)𝑥̇(𝑡)−ℎ𝑒^{−2𝑘ℎ𝑡}_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑃₅(𝛼)𝑥̇(𝑠)𝑑𝑠+𝛽²ℎ²𝑥̇^𝑇(𝑡)𝑃₆(𝛼)𝑥̇(𝑡)−𝛽ℎ𝑒^{−2𝛽𝑘ℎ𝑡}_{𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑃₆(𝛼)𝑥̇(𝑠)𝑑𝑠−2𝑘𝑉₄(𝑡),𝑉̇₅(𝑡)≤ℎ²𝑥̇^𝑇(𝑡)𝑃₇(𝛼)𝑥̇(𝑡)−ℎ𝑒^{−2𝑘ℎ𝑡}_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑃₇(𝛼)𝑥̇(𝑠)𝑑𝑠+𝛽²ℎ²𝑥̇^𝑇(𝑡)𝑃₈(𝛼)𝑥̇(𝑡)−𝛽ℎ𝑒^{−2𝛽𝑘ℎ𝑡}_{𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑃₈(𝛼)𝑥̇(𝑠)𝑑𝑠−2𝑘𝑉₅(𝑡)≤ℎ²𝑥̇^𝑇(𝑡)𝑃₇(𝛼)𝑥̇(𝑡)−𝑒^{−2𝑘ℎ𝑡}_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑑𝑠𝑃₇(𝛼)^𝑡_𝑡−ℎ𝑥̇(𝑠)𝑑𝑠+𝛽²ℎ²𝑥̇^𝑇(𝑡)𝑃₈(𝛼)𝑥̇(𝑡)−𝑒^{−2𝛽𝑘ℎ𝑡}_{𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑃₈(𝛼)^𝑡_{𝑡−𝛽ℎ}𝑥̇(𝑠)𝑑𝑠−2𝑘𝑉₅(𝑡)≤ℎ²𝑥̇^𝑇(𝑡)𝑃₇(𝛼)𝑥̇(𝑡)−𝑒^{−2𝑘ℎ𝑡}_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑃₇(𝛼)^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠+𝛽²ℎ²𝑥̇^𝑇(𝑡)𝑃₈(𝛼)𝑥̇(𝑡)−𝑒^{−2𝛽𝑘ℎ𝑡}_{(𝑡)𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑃₈(𝛼)^𝑡_{(𝑡)𝑡−𝛽ℎ}𝑥̇(𝑠)𝑑𝑠−2𝛼𝑉₅(𝑡),𝑉̇₆(𝑡)≤ℎ²𝑥̇^𝑇(𝑡)𝑃₉(𝛼)𝑥̇(𝑡)−𝑒^{−2𝑘ℎ𝑡}_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑃₉(𝛼)^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠+𝛽²ℎ²𝑥̇^𝑇(𝑡)𝑃₁₀(𝛼)𝑥̇(𝑡)−𝑒^{−2𝛽𝑘ℎ𝑡}_{(𝑡)𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑃₁₀(𝛼)^𝑡_{(𝑡)𝑡−𝛽ℎ}𝑥̇(𝑠)𝑑𝑠−2𝑘𝑉₆(𝑡)=ℎ²𝑥̇^𝑇(𝑡)𝑃₉(𝛼)𝑥̇(𝑡)−𝑒^−2𝑘ℎ𝑥^𝑇(𝑡)−𝑥^𝑇(𝑡−ℎ(𝑡))×𝑃₉(𝛼)[𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))]+𝛽ℎ²𝑥̇^𝑇(𝑡)𝑃₁₀(𝛼)𝑥̇(𝑡)−𝑒^{−2𝛽𝑘ℎ}𝑥^𝑇(𝑡)−𝑥^𝑇𝑡−𝛽ℎ(𝑡)×𝑃₁₀(𝛼)𝑥(𝑡)−𝑥𝑡−𝛽ℎ(𝑡)−2𝑘𝑉₆(𝑡). (31)
Taking the time derivative of 𝑉₇(𝑡), we obtain 𝑉̇₇(𝑡)=ℎℎ(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑅₁(𝛼)𝑥(𝑡−ℎ(𝑡))+2ℎ𝑥^𝑇(𝑡−ℎ(𝑡))𝑅₂(𝛼)𝑥(𝑡)−2ℎ𝑥^𝑇(𝑡−ℎ(𝑡))𝑅₂(𝛼)𝑥(𝑡−ℎ(𝑡))+ℎ^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑅₃(𝛼)𝑥̇(𝑠)𝑑𝑠+𝛽²ℎℎ(𝑡)𝑥^𝑇𝑡−𝛽ℎ(𝑡)𝑅₄(𝛼)𝑥𝑡−𝛽ℎ(𝑡)+2𝛽ℎ𝑥^𝑇(𝑡−ℎ(𝑡))𝑅₅(𝛼)𝑥(𝑡)−2𝛽ℎ𝑥^𝑇𝑡−𝛽ℎ(𝑡)𝑅₅(𝛼)𝑥𝑡−𝛽ℎ(𝑡)+𝛽ℎ^𝑡_{𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑅₆(𝛼)𝑥̇(𝑠)𝑑𝑠−2𝑘𝑉₇(𝑡)≤ℎ²𝑥^𝑇(𝑡−ℎ(𝑡))𝑅₁(𝛼)𝑥(𝑡−ℎ(𝑡))+2ℎ𝑥^𝑇(𝑡−ℎ(𝑡))𝑅₂(𝛼)𝑥(𝑡)−2ℎ𝑥^𝑇(𝑡−ℎ(𝑡))𝑅₂(𝛼)𝑥(𝑡−ℎ(𝑡))+ℎ^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑅₃(𝛼)𝑥̇(𝑠)𝑑𝑠+𝛽²ℎ²𝑥^𝑇𝑡−𝛽ℎ(𝑡)𝑅₄(𝛼)𝑥𝑡−𝛽ℎ(𝑡)+2𝛽ℎ𝑥^𝑇(𝑡−ℎ(𝑡))𝑅₅(𝛼)𝑥(𝑡)−2𝛽ℎ𝑥^𝑇𝑡−𝛽ℎ(𝑡)𝑅₅(𝛼)𝑥𝑡−𝛽ℎ(𝑡)+𝛽ℎ^𝑡_{𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑅₆(𝛼)𝑥̇(𝑠)𝑑𝑠−2𝑘𝑉₇(𝑡). (32)
By Lemma 5 and the integral term of the right hand side of 𝑉̇₄(𝑡) and 𝑉̇₇(𝑡), we obtain −ℎ^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑒^−2𝑘ℎ𝑃₅(𝛼)−𝑅₃(𝛼)𝑥̇(𝑠)𝑑𝑠−𝛽ℎ^𝑡_{𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑒^{−2𝛽𝑘ℎ}𝑃₆(𝛼)−𝑅₆(𝛼𝑥̇(𝑠)𝑥(𝑡)(𝑡))𝑑𝑠≤ℎ𝑥𝑡−ℎ^𝑇×𝑀^𝑇₁(𝛼)+𝑀₁(𝛼)−𝑀^𝑇₁(𝛼)+𝑀₂(𝛼)∗−𝑀^𝑇₂(𝛼)−𝑀₂(𝛼)×𝑥(𝑡)𝑥𝑡−ℎ(𝑡)+ℎ²𝑥(𝑡)𝑥𝑡−ℎ(𝑡)^𝑇𝑀₃(𝛼)𝑀₄(𝛼)∗𝑀₅(𝛼)×𝑥(𝑡)(𝑡)𝑥(𝑡)𝑥𝑡−ℎ(𝑡)+𝛽ℎ𝑥𝑡−𝛽ℎ^𝑇×𝑀^𝑇₆(𝛼)+𝑀₆(𝛼)−𝑀^𝑇₆(𝛼)+𝑀₇(𝛼)∗−𝑀^𝑇₇(𝛼)−𝑀₇(𝛼)×𝑥(𝑡)𝑥𝑡−𝛽ℎ(𝑡)+𝛽²ℎ²𝑥(𝑡)𝑥𝑡−𝛽ℎ(𝑡)^𝑇𝑀₈(𝛼)𝑀₉(𝛼)∗𝑀₁₀(𝛼)×𝑥(𝑡)(𝑡).𝑥𝑡−𝛽ℎ (33)
From the Leibniz-Newton formula, the following equations are true for any parameter-dependent real matrices 𝑁_𝑖(𝛼), 𝑂_𝑖(𝛼), 𝑖=1,2,…,10 with appropriate dimensions: 2𝑥^𝑇(𝑡)𝑁^𝑇₁(𝛼)+𝑥^𝑇(𝑡−ℎ(𝑡))𝑁^𝑇₂(𝛼)+𝑥^𝑇𝑡−𝛽ℎ(𝑡)𝑁^𝑇₃(𝛼)+^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑁^𝑇₄(𝛼)+^𝑡_{(𝑡)𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑁^𝑇₅(𝛼)+𝑥̇^𝑇(𝑡)𝑁^𝑇₆(𝛼)+𝑥̇^𝑇(𝑡−𝑟(𝑡))𝑁^𝑇₇(𝛼)+𝑓^𝑇(𝑡,𝑥(𝑡))𝑁^𝑇₈(𝛼)+𝑔^𝑇(𝑡,𝑥(𝑡−ℎ(𝑡)))𝑁^𝑇₉(𝛼)+𝑤^𝑇(𝑡,𝑥̇(𝑡−𝑟(𝑡)))𝑁^𝑇₁₀(𝛼×𝑥(𝑡)((𝑡−)−𝑥𝑡−ℎ))^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)2𝑥𝑑𝑠=0,_𝑇(𝑡)𝑂^𝑇₁(𝛼)+𝑥^𝑇(𝑡−ℎ(𝑡))𝑂^𝑇₂(𝛼)+𝑥^𝑇𝑡−𝛽ℎ(𝑡)𝑂^𝑇₃(𝛼)+^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑂^𝑇₄(𝛼)+^𝑡_{(𝑡)𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑂^𝑇₅(𝛼)+𝑥̇^𝑇(𝑡)𝑂^𝑇₆(𝛼)+𝑥̇^𝑇(𝑡−𝑟(𝑡))𝑂^𝑇₇(𝛼)+𝑓^𝑇(𝑡,𝑥(𝑡))𝑂^𝑇₈(𝛼)+𝑔^𝑇(𝑡,𝑥(𝑡−ℎ(𝑡)))𝑂^𝑇₉(𝛼)+𝑤^𝑇(𝑡,𝑥̇(𝑡−𝑟(𝑡)))𝑂^𝑇₁₀(𝛼×𝑥(𝑡)(𝑡−)−𝑥𝑡−𝛽ℎ)^𝑡_{(𝑡)𝑡−𝛽ℎ}𝑥̇(𝑠)𝑑𝑠=0. (34)
From the utilization of zero equation, the following equation is true for any parameter-dependent real matrices 𝑊_𝑖, 𝑖=1,2,…,10 with appropriate dimensions: 2𝑥^𝑇(𝑡)𝑊^𝑇₁(𝛼)+𝑥^𝑇(𝑡−ℎ(𝑡))𝑊^𝑇₂(𝛼)+𝑥^𝑇𝑡−𝛽ℎ(𝑡)𝑊^𝑇₃(𝛼)+^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑊^𝑇₄(𝛼)+^𝑡_{(𝑡)𝑡−𝛽ℎ}𝑥̇^𝑇(𝑠)𝑑𝑠𝑊^𝑇₅(𝛼)+𝑥̇^𝑇(𝑡)𝑊^𝑇₆(𝛼)+𝑥̇^𝑇(𝑡−𝑟(𝑡))𝑊^𝑇₇(𝛼)+𝑓^𝑇(𝑡,𝑥(𝑡))𝑊₈(𝛼)+𝑔^𝑇(𝑡,𝑥(𝑡−ℎ(𝑡)))𝑊^𝑇₉(𝛼)+𝑤^𝑇(𝑡,𝑥̇(𝑡−𝑟(𝑡)))𝑊^𝑇₁₀(𝛼×𝐴(𝛼))+𝐵₁(𝛼)𝑥(𝑡)+𝐵₂(𝛼)𝑥((𝑡(𝛼)𝑥̇((𝑡((𝑡(((𝑡(𝑥̇((𝑡𝑡−ℎ))+𝐶𝑡−𝑟))+𝑓𝑡,𝑥))+𝑔𝑡,𝑥𝑡−ℎ)))+𝑤𝑡,𝑡−𝑟)))−𝐵₁(𝛼)^𝑡_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠−𝑥̇(𝑡)=0. (35)
From (5), we obtain, for any positive real constants 𝜖₁, 𝜖₂, and 𝜖₃, 0≤𝜖₁𝜂²𝑥^𝑇(𝑡)𝑥(𝑡)−𝜖₁𝑓^𝑇(𝑡,𝑥(𝑡))𝑓(𝑡,𝑥(𝑡)),0≤𝜖₂𝜌²𝑥^𝑇(𝑡−ℎ(𝑡))𝑥(𝑡−ℎ(𝑡))−𝜖₂𝑔^𝑇(𝑡,𝑥(𝑡−ℎ(𝑡)))𝑔(𝑡,𝑥(𝑡−ℎ(𝑡))),0≤𝜖₃𝛾²𝑥̇^𝑇(𝑡−𝑟(𝑡))𝑥̇(𝑡−𝑟(𝑡))−𝜖₃𝑤^𝑇(𝑡,𝑥̇(𝑡−𝑟(𝑡)))𝑤(𝑡,𝑥̇(𝑡−𝑟(𝑡))). (36)
According to (29)–(36), it is straightforward to see that 𝑉̇(𝑡)≤𝜁^𝑇(𝑡)^𝑁𝑁_𝑖=1𝑗=1𝛼_𝑖𝛼_{𝑗𝑖,𝑗}𝜁(𝑡)−2𝑘𝑉(𝑡), (37) where 𝜁^𝑇(𝑡)=[𝑥^𝑇(𝑡), 𝑥^𝑇(𝑡−ℎ(𝑡)), 𝑥^𝑇(𝑡−𝛽ℎ(𝑡)), ∫^𝑡_{𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑑𝑠,∫^𝑡_{𝑡−𝛽ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑑𝑠,𝑥̇^𝑇(𝑡), 𝑥̇^𝑇(𝑡−𝑟(𝑡)), 𝑓^𝑇(𝑡,𝑥(𝑡)), 𝑔^𝑇(𝑡,𝑥(𝑡−ℎ(𝑡))), 𝑤^𝑇(𝑡,𝑥̇(𝑡−𝑟(𝑡))] and ∑_𝑖,𝑗 is defined in (23). From the fact that ∑^𝑁_𝑖=1𝛼_𝑖=1, ^𝑁_𝑖=1𝛼_𝑖𝐴_𝑖^𝑁_𝑖=1𝛼_𝑖𝐵_𝑖=^𝑁_𝑖=1𝛼²_𝑖𝐴_𝑖𝐵_𝑖+^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝛼_𝑖𝛼_𝑗𝐴_𝑖𝐵_𝑗+𝐴_𝑗𝐵_𝑖,(𝑁−1)^𝑁_𝑖=1𝛼²_𝑖−2^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝛼_𝑖𝛼_𝑗=^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝛼_𝑖−𝛼_𝑗²≥0. (38)
It is true that if conditions (25) hold, then 𝑉̇(𝑡)+2𝑘𝑉(𝑡)≤0,∀𝑡∈𝑅⁺, (39) which gives 𝑉(𝑡)≤𝑉(0)𝑒^−2𝑘𝑡,∀𝑡∈𝑅⁺. (40) From (40), it is easy to see that 𝜆_min𝑃₁(𝛼)‖𝑥(𝑡)‖²≤𝑉(𝑡)≤𝑉(0)𝑒^−2𝑘𝑡,𝑉(0)=⁷_𝑖=1𝑉_𝑖(0), (41)(42) where 𝑉₁(0)=𝑥^𝑇(0)𝑃₁(𝛼)𝑥(0),𝑉₂(0)=⁰_(0)−𝑟𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑃₂(𝛼)𝑥̇(𝑠)𝑉𝑑𝑠,₃(0)=⁰_(0)−ℎ𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑃₃(𝛼)𝑥(𝑠)+𝑑𝑠_{0(0)−𝛽ℎ}𝑒^2𝑘𝑠𝑥^𝑇(𝑠)𝑃₄(𝛼)𝑥(𝑠)𝑉𝑑𝑠,₄(0)=ℎ⁰_−ℎ⁰_𝜃𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑃₅(𝛼)𝑥̇(𝑠)𝑑𝑠𝑑𝜃+𝛽ℎ⁰_−𝛽ℎ⁰_𝜃𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑃₆(𝛼)𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₅(0)=ℎ⁰_−ℎ⁰_𝜃𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑃₇(𝛼)𝑥̇(𝑠)𝑑𝑠𝑑𝜃+𝛽ℎ⁰_−𝛽ℎ⁰_𝜃𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑃₈(𝛼)𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₆(0)=ℎ⁰_−ℎ^𝑡_𝜃𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑃₉(𝛼)𝑥̇(𝑠)𝑑𝑠𝑑𝜃+𝛽ℎ⁰_−𝛽ℎ⁰_𝜃𝑒^2𝑘𝑠𝑥̇^𝑇(𝑠)𝑃₁₀(𝛼)𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₇(0)=ℎ⁰_−ℎ^𝜃_{(𝜃)𝜃−ℎ}𝑒^2𝑘𝜃𝑥((𝜃𝑥̇(𝑠)𝜃−ℎ))^𝑇×𝑅₁(𝛼)𝑅₂(𝛼)∗𝑅₃(𝛼)×𝑥((𝜃𝑥̇(𝑠)𝜃−ℎ))𝑑𝑠𝑑𝜃+𝛽ℎ⁰_−𝛽ℎ^𝜃_{(𝜃)𝜃−𝛽ℎ}𝑒^2𝑘𝜃𝑥(𝜃)𝑥̇(𝑠)𝜃−𝛽ℎ^𝑇×𝑅₄(𝛼)𝑅₅(𝛼)∗𝑅₆(𝛼)×𝑥(𝜃)𝑥̇(𝑠)𝜃−𝛽ℎ𝑑𝑠𝑑𝜃. (43)
From (41), we conclude that 𝜆_min𝑃₁(𝛼)‖𝑥(𝑡)‖²≤𝑉(0)𝑒^−2𝑘𝑡≤𝐿max𝜙,𝜑²𝑒^−2𝑘𝑡, (44) where 𝐿=𝜆_max(𝑃₁(𝛼))+𝑟𝜆_max(𝑃₂(𝛼)) + ℎ𝜆_max(𝑃₃(𝛼))+𝛽ℎ𝜆_max(𝑃₄(𝛼)) + ℎ³𝜆_max(𝑃₅(𝛼)+𝑃₇(𝛼) + 𝑃₉(𝛼))+(𝛽ℎ)³𝜆_max(𝑃₆(𝛼)+𝑃₈(𝛼)+𝑃₁₀(𝛼)) +ℎ³𝜆_max(^{𝑅1(𝛼)𝑅2(𝛼)}_{∗𝑅3(𝛼)}) +(𝛽ℎ)³𝜆_max(^{𝑅4(𝛼)𝑅5(𝛼)}_{∗𝑅6(𝛼)}). From (44), this means that the system (1) is robustly exponentially stable. The proof of the theorem is complete.

Next, we consider the following system: 𝑥̇(𝑡)(𝛼)𝑥(𝑡)(𝛼)𝑥((𝑡(𝛼)𝑥̇((𝑡((𝑡(((𝑡𝑥(𝑡)(𝑡),𝑥̇(𝑡)(𝑡)[].=𝐴+𝐵𝑡−ℎ))+𝐶𝑡−𝑟))+𝑓𝑡,𝑥))+𝑔𝑡,𝑥𝑡−ℎ))),𝑡>0;=𝜙=𝜓,𝑡∈−𝑏,0 (45)

We introduce the following notations for later use: _𝑖,𝑗=Σ^1,1_𝑖,𝑗Σ^1,2_𝑖,𝑗Σ^1,3_𝑖,𝑗Σ^1,4_𝑖,𝑗Σ^1,5_𝑖,𝑗Σ^1,6_𝑖,𝑗Σ^1,7_𝑖,𝑗Σ^1,8_𝑖,𝑗Σ^1,9_𝑖,𝑗∗Σ^2,2_𝑖,𝑗Σ^2,3_𝑖,𝑗Σ^2,4_𝑖,𝑗Σ^2,5_𝑖,𝑗Σ^2,6_𝑖,𝑗Σ^2,7_𝑖,𝑗Σ^2,8_𝑖,𝑗Σ^2,9_𝑖,𝑗∗∗Σ_𝑖^3,3Σ^3,4_𝑖,𝑗Σ_𝑖^3,5Σ_𝑖^3,6Σ^3,7_𝑖,𝑗Σ_𝑖^3,8Σ_𝑖^3,9∗∗∗Σ^4,4_𝑖,𝑗Σ^4,5_𝑖,𝑗Σ^4,6_𝑖,𝑗Σ^4,7_𝑖,𝑗Σ_𝑖^4,8Σ_𝑖^4,9∗∗∗∗Σ_𝑖^5,5Σ_𝑖^5,6Σ^5,7_𝑖,𝑗Σ_𝑖^5,8Σ_𝑖^5,9∗∗∗∗∗Σ_𝑖^6,6Σ^6,7_𝑖,𝑗Σ_𝑖^6,8Σ_𝑖^6,9∗∗∗∗∗∗Σ^7,7_𝑖,𝑗Σ^7,8_𝑖,𝑗Σ^7,9_𝑖,𝑗∗∗∗∗∗∗∗Σ_𝑖^8,8Σ_𝑖^8,9∗∗∗∗∗∗∗∗Σ_𝑖^9,9. (46)

Corollary 8. For ‖𝐶_𝑖‖<1, 𝑖=1,2,…,𝑁 and given positive real constants ℎ, ℎ_𝑑, 𝑟, 𝑟_𝑑, 𝜂, 𝜌, and 𝛽, system (45) is robustly exponentially stable with a decay rate 𝑘, if there exist symmetric positive definite matrices 𝑃^𝑗_𝑖, any appropriate dimensional matrices 𝑅^𝑝_𝑖, 𝑀^𝑙_𝑖, 𝑁^𝑗_𝑖, 𝑂^𝑗_𝑖, 𝑊^𝑗_𝑖, 𝑝=1,2,…6, 𝑙=1,2,…,10, 𝑗=1,2,…,9, 𝑖=1,2,…,𝑁, and positive real constants 𝜖₁, 𝜖₂ such that the following symmetric linear matrix inequalities hold: 𝑅¹_𝑖𝑅²_𝑖∗𝑅³_𝑖𝑅>0,𝑖=1,2,…,𝑁,_4𝑖𝑅⁵_𝑖∗𝑅⁶_𝑖𝑒>0,𝑖=1,2,…,𝑁,_−2𝑘ℎ𝑃³_𝑖−𝑅³_𝑖𝑀¹_𝑖𝑀²_𝑖∗𝑀³_𝑖𝑀⁴_𝑖∗∗𝑀⁵_𝑖𝑒≥0,𝑖=1,2,…,𝑁,_{−2𝛽𝑘ℎ}𝑃⁶_𝑖−𝑅⁶_𝑖𝑀⁶_𝑖𝑀⁷_𝑖∗𝑀⁸_𝑖𝑀⁹_𝑖∗∗𝑀_𝑖¹⁰≥0,𝑖=1,2,…,𝑁,_𝑖,𝑖<−𝐼,𝑖=1,2,…,𝑁,_𝑖,𝑗+_𝑗,𝑖<2(𝑁−1)𝐼,𝑖=1,2,…,𝑁−1,𝑗=𝑖+1,𝑖+2,…,𝑁. (47)
Moreover, the solution 𝑥(𝑡,𝜙,𝜓) satisfies the inequality 𝑥𝑡,𝜙,𝜓≤𝐿𝜆_min𝑃₁(𝛼)max𝜙,𝜓𝑒^−𝑘𝑡,∀𝑡∈𝑅⁺, (48) where 𝐿=𝜆_max(𝑃₁(𝛼))+𝑟𝜆_max(𝑃₂(𝛼)) +ℎ𝜆_max(𝑃₃(𝛼))+𝛽ℎ𝜆_max(𝑃₄(𝛼)) +ℎ³𝜆_max(𝑃₅(𝛼)+𝑃₇(𝛼) +𝑃₉(𝛼)) + (𝛽ℎ)³𝜆_max(𝑃₆(𝛼) +𝑃₈(𝛼) +𝑃₁₀(𝛼)) + ℎ³𝜆_max(^{𝑅1(𝛼)𝑅2(𝛼)}_{∗𝑅3(𝛼)}) +(𝛽ℎ)³𝜆_max(^{𝑅4(𝛼)𝑅5(𝛼)}_{∗𝑅6(𝛼)}).

3.2. Exponential Stability Criteria

In this section, we study the exponential stability criteria for neutral systems with time-varying delays by using the combination of linear matrix inequality (LMI) technique and Lyapunov theory method. We introduce the following notations for later use: Σ=¹¹Σ₁₂Σ₁₃Σ₁₄Σ₁₅Σ₁₆Σ₁₇Σ₁₈Σ₁₉Σ_1,10∗Σ₂₂Σ₂₃Σ₂₄Σ₂₅Σ₂₆Σ₂₇Σ₂₈Σ₂₉Σ_2,10∗∗Σ₃₃Σ₃₄Σ₃₅Σ₃₆Σ₃₇Σ₃₈Σ₃₉Σ_3,10∗∗∗Σ₄₄Σ₄₅Σ₄₆Σ₄₇Σ₄₈Σ₄₉Σ_4,10∗∗∗∗Σ₅₅Σ₅₆Σ₅₇Σ₅₈Σ₅₉Σ_5,10∗∗∗∗∗Σ₆₆Σ₆₇Σ₆₈Σ₆₉Σ_6,10∗∗∗∗∗∗Σ₇₇Σ₇₈Σ₇₉Σ_7,10∗∗∗∗∗∗∗Σ₈₈Σ₈₉Σ_8,10∗∗∗∗∗∗∗∗Σ₉₉Σ_9,10∗∗∗∗∗∗∗∗∗Σ_10,10, (49) where Σ_1,1=𝐴^𝑇𝑃₁+𝐵^𝑇₁𝑃₁+𝑍^𝑇₁+𝑃₁𝐴+𝑃₁𝐵₁+𝑍₁+𝑃₃+𝑃₄+ℎ𝑀^𝑇₁+ℎ𝑀₁−𝑒^−2𝑘ℎ𝑃₉−𝑒^{−2𝑘𝛽ℎ}𝑃₁₀+ℎ²𝑀₃+𝛽ℎ𝑀₆+𝛽ℎ𝑀^𝑇₆+𝛽²ℎ²𝑀₈+𝑁^𝑇₁+𝑁₁+𝑂^𝑇₁+𝑂₁+𝐴^𝑇𝑊₁+𝑊^𝑇₁𝐴+𝐵^𝑇₁𝑊₁+𝑊^𝑇₁𝐵₁+𝜖₁𝜂²𝐼+2𝑘𝑃₁,Σ_1,2=𝑃₁𝐵₂−ℎ𝑀^𝑇₁+ℎ𝑀₂+ℎ²𝑀₄+𝑒^−2𝑘ℎ𝑃₉+ℎ𝑅₂−𝑁^𝑇₁+𝑁₂+𝑂₂+𝑊^𝑇₁𝐵₂+𝐴^𝑇𝑊₂+𝐵^𝑇₁𝑊₂,Σ_1,3=−𝑍₁−𝛽ℎ𝑀^𝑇₆+𝛽ℎ𝑀₇+𝛽ℎ²𝑀₉+𝛽ℎ𝑅₅+𝑒^{−2𝑘𝛽ℎ}𝑃₁₀+𝑁₃−𝑂^𝑇₁+𝑂₃+𝐴^𝑇𝑊₃+𝐵^𝑇₁𝑊₃,Σ_1,4=−𝑃₁𝐵₁−𝑁^𝑇₁+𝑁₄+𝑂₄−𝑊^𝑇₁𝐵₁+𝐴^𝑇𝑊₄+𝐵^𝑇₁𝑊₄,Σ_1,5=−𝑍₁+𝑁₅−𝑂^𝑇₁+𝑂₅+𝐴^𝑇𝑊₅+𝐵^𝑇₁𝑊₅,Σ_1,6=𝑁₆+𝑂₆−𝑊^𝑇₁+𝐴^𝑇𝑊₆+𝐵^𝑇₁𝑊₆,Σ_1,7=𝑃₁𝐶+𝑁₇+𝑂₇+𝑊^𝑇₁𝐶+𝐴^𝑇𝑊₇+𝐵^𝑇₁𝑊₇,Σ_1,8=𝑃₁+𝑁₈+𝑂₈+𝑊^𝑇₁+𝐴^𝑇𝑊₈+𝐵^𝑇₁𝑊₈,Σ_1,9=𝑃₁+𝑁₉+𝑂₉+𝑊^𝑇₁+𝐴^𝑇𝑊₉+𝐵^𝑇₁𝑊₉,Σ_1,10=𝑃₁+𝑁₁₀+𝑂₁₀+𝑊^𝑇₁+𝐴^𝑇𝑊₁₀+𝐵^𝑇₁𝑊₁₀,Σ_2,2=−1−ℎ_𝑑𝑒^−2𝑘ℎ𝑃₃−ℎ𝑀^𝑇₂−ℎ𝑀₂+ℎ²𝑀₅+ℎ²𝑅₁−2ℎ𝑅₂−𝑒^−2𝑘ℎ𝑃₉+𝜖₂𝜌²𝐼−𝑁^𝑇₂−𝑁₂+𝑊^𝑇₂𝐵₂+𝐵^𝑇₂𝑊₂,Σ_2,3=−𝑁₃−𝑂^𝑇₂+𝐵^𝑇₂𝑊₃,Σ_2,4=−𝑁^𝑇₂−𝑁₄−𝑊^𝑇₂𝐵₁+𝐵^𝑇₂𝑊₄,Σ_2,5=−𝑁₅−𝑂^𝑇₂+𝐵^𝑇₂𝑊₅,Σ_2,6=−𝑁₆−𝑊^𝑇₂+𝐵^𝑇₂𝑊₆,Σ_2,7=−𝑁₇+𝑊^𝑇₂𝐶+𝐵^𝑇₂𝑊₇,Σ_2,8=−𝑁₈+𝑊^𝑇₂+𝐵^𝑇₂𝑊₈,Σ_2,9=−𝑁^𝑇₉−𝑊^𝑇₂+𝐵^𝑇₂𝑊₉,Σ_2,10=−𝑁₁₀+𝑊^𝑇₂+𝐵^𝑇₂𝑊₁₀,Σ_3,3=−1−𝛽ℎ_𝑑𝑒^{−2𝛽𝑘ℎ}𝑃₄−𝑒^{−2𝛽𝑘ℎ}𝑃₁₀+𝛽²ℎ²𝑅₄−2𝛽ℎ𝑅₅−𝛽ℎ𝑀^𝑇₇−𝛽ℎ𝑀₇+𝛽²ℎ²𝑀₁₀−𝑂^𝑇₃−𝑂₃,Σ_3,4=−𝑁^𝑇₃−𝑂₄−𝑊^𝑇₃𝐵₁,Σ_3,5=−𝑂^𝑇₃−𝑂₅,Σ_3,6=−𝑂₉+𝑊^𝑇₃,Σ_3,7=−𝑂₇+𝑊^𝑇₃Σ𝐶,_3,8=−𝑂₈+𝑊^𝑇₃,Σ_3,9=−𝑂₉+𝑊^𝑇₃,Σ_3,10=−𝑂₁₀+𝑊^𝑇₃,Σ_4,4=−𝑒^2𝑘ℎ𝑃₇−𝑁^𝑇₄−𝑁₄+𝑊^𝑇₄𝐵₁−𝐵^𝑇₁𝑊₄,Σ_4,5=−𝑁₅−𝑂^𝑇₄−𝐵^𝑇₁𝑊₅,Σ_4,6=−𝑁₆−𝑊^𝑇₄−𝐵^𝑇₁𝑊₆,Σ_4,7=−𝑁₇+𝑊^𝑇₄𝐶−𝐵^𝑇₁𝑊₇,Σ_4,8=−𝑂₈+𝑊^𝑇₃,Σ_4,9=−𝑂₉+𝑊^𝑇₃,Σ_4,10=−𝑂₁₀+𝑊^𝑇₃,Σ_5,5=−𝑒^2𝛽𝑘ℎ𝑃₈−𝑂^𝑇₅−𝑂₅,Σ_5,6=−𝑂₆−𝑊^𝑇₅,Σ_5,7=−𝑂₇+𝑊^𝑇₅𝐶,Σ_5,8=−𝑂₈+𝑊^𝑇₅,Σ_5,9=−𝑂₉+𝑊^𝑇₅,Σ_5,10=−𝑂₁₀+𝑊^𝑇₅,Σ_6,6=𝑃₂+ℎ²𝑃₅+𝛽²ℎ²𝑃₆+ℎ²𝑃₇+𝛽²ℎ²𝑃₈+ℎ²𝑃₉+𝛽²ℎ²𝑃₁₀−𝑊^𝑇₆−𝑊₆,Σ_6,7=𝑊^𝑇₆𝐶−𝑊₇,Σ_6,8=𝑊^𝑇₆−𝑊₈,Σ_6,9=𝑊^𝑇₆−𝑊₉,Σ_6,10=𝑊^𝑇₆−𝑊₁₀,Σ_7,7=−1−𝑟_𝑑𝑒^−2𝑘𝑟𝑃₂+𝜖₃𝛾²𝐼+𝑊^𝑇₇𝐶+𝐶^𝑇𝑊₇,Σ_7,8=𝑊^𝑇₇−𝐶^𝑇𝑊₈,Σ_7,9=𝑊^𝑇₇−𝐶^𝑇𝑊₉,Σ_7,10=𝑊^𝑇₇−𝐶^𝑇𝑊₁₀Σ_8,8=𝑊^𝑇₈+𝑊₈−𝜖₁Σ𝐼,_8,9=𝑊^𝑇₈+𝑊₉,Σ_8,10=𝑊^𝑇₈+𝑊₁₀,Σ_9,9=𝑊^𝑇₉+𝑊₉−𝜖₂𝐼,Σ_9,10=𝑊^𝑇₉+𝑊₁₀,Σ_10,10=𝑊^𝑇₁₀+𝑊₁₀−𝜖₃Σ𝐼,_7,7=−1−𝑟_𝑑𝑒^−2𝑘𝑟𝑃₂+𝑊^𝑇₇𝐶+𝐶^𝑇𝑊₇,𝑍₁=𝑃₁𝐺. (50)

If 𝐴(𝛼)=𝐴, 𝐵(𝛼)=𝐵, and 𝐶(𝛼)=𝐶, where 𝐴,𝐵,𝐶∈𝑅^𝑛×𝑛 are real constant matrices, then system (1) reduces to the following system: 𝑥̇(𝑡)(𝑡)((𝑡𝑥̇((𝑡((𝑡(((𝑡(𝑥̇((𝑡𝑥(𝑡)(𝑡),𝑥̇(𝑡)(𝑡)[].=𝐴𝑥+𝐵𝑥𝑡−ℎ))+𝐶𝑡−𝑟))+𝑓𝑡,𝑥))+𝑔𝑡,𝑥𝑡−ℎ)))+𝑤𝑡,𝑡−𝑟))),𝑡>0;=𝜙=𝜓,𝑡∈−𝑏,0 (51)

Corollary 9. For ‖𝐶‖+𝛾<1 and given positive real constants ℎ, ℎ_𝑑, 𝑟, 𝑟_𝑑,𝜂, 𝜌, 𝛾, and 𝛽, system (51) is exponentially stable with a decay rate 𝑘, if there exist symmetric positive definite matrices 𝑃_𝑖, any appropriate dimensional matrices 𝑅_𝑠, 𝑁_𝑖, 𝑂_𝑖, 𝑊_𝑖, 𝑀_𝑖, 𝐺,   𝑠=1,2,…6, 𝑖=1,2,…,10, and positive real constants 𝜖₁, 𝜖₂, and 𝜖₃ such that the following symmetric linear matrix inequalities hold: 𝑅₁𝑅₂∗𝑅₃𝑅>0,₄𝑅₅∗𝑅₆𝑒>0,_−2𝑘ℎ𝑃₃−𝑅₃𝑀₁𝑀₂∗𝑀₃𝑀₄∗∗𝑀₅𝑒≥0,_{−2𝛽𝑘ℎ}𝑃₆−𝑅₆𝑀₆𝑀₇∗𝑀₈𝑀₉∗∗𝑀₁₀≥0,<0. (52)
Moreover, the solution 𝑥(𝑡,𝜙,𝜓) satisfies the inequality 𝑥𝑡,𝜙,𝜓≤𝐷𝜆_min𝑃₁max𝜙,𝜓𝑒^−𝑘𝑡,∀𝑡∈𝑅⁺, (53) where 𝐷=𝜆_max(𝑃₁) +𝑟𝜆_max(𝑃₂)+ ℎ𝜆_max(𝑃₃)+𝛽ℎ𝜆_max(𝑃₄) +ℎ³𝜆_max(𝑃₅+𝑃₇+𝑃₉) +(𝛽ℎ)³𝜆_max(𝑃₆+𝑃₈ +𝑃₁₀)+ℎ³𝜆_max(^𝑅1𝑅2_∗𝑅3) +(𝛽ℎ)³𝜆_max(^𝑅4𝑅5_∗𝑅6).

If 𝐴(𝛼)=𝐴, 𝐵(𝛼)=𝐵, 𝐶(𝛼)=𝐶, and 𝑤(𝑡,𝑥̇(𝑡−𝑟(𝑡)))=0, where 𝐴,𝐵,𝐶∈𝑅^𝑛×𝑛 are real constant matrices, then system (1) reduces to the following system: 𝑥̇(𝑡)(𝑡)((𝑡𝑥̇((𝑡((𝑡(((𝑡𝑥(𝑡)(𝑡),𝑥̇(𝑡)(𝑡)[].=𝐴𝑥+𝐵𝑥𝑡−ℎ))+𝐶𝑡−𝑟))+𝑓𝑡,𝑥))+𝑔𝑡,𝑥𝑡−ℎ))),𝑡>0;=𝜙=𝜓,𝑡∈−𝑏,0 (54)

Corollary 10. For ‖𝐶‖<1 and given positive real constants ℎ, ℎ_𝑑, 𝑟, 𝑟_𝑑, 𝜂, 𝜌, and 𝛽, system (54) is exponentially stable with a decay rate 𝑘, if there exist symmetric positive definite matrices 𝑃_𝑗, any appropriate dimensional matrices 𝑅_𝑠, 𝑁_𝑖, 𝑂_𝑖, 𝑊_𝑖, 𝑀_𝑗, and 𝐺, where 𝑠=1,2,…6, 𝑖=1,2,…,9, and 𝑗=1,2,…10, and positive real constants 𝜖₁, 𝜖₂ such that the following symmetric linear matrix inequalities hold: 𝑅₁𝑅₂∗𝑅₃𝑅>0,₄𝑅₅∗𝑅₆𝑒>0,_−2𝑘ℎ𝑃₃−𝑅₃𝑀₁𝑀₂∗𝑀₃𝑀₄∗∗𝑀₅𝑒≥0,_{−2𝛽𝑘ℎ}𝑃₆−𝑅₆𝑀₆𝑀₇∗𝑀₈𝑀₉∗∗𝑀₁₀≥0,<0, (55) where Σ=¹¹Σ₁₂Σ₁₃Σ₁₄Σ₁₅Σ₁₆Σ₁₇Σ₁₈Σ₁₉∗Σ₂₂Σ₂₃Σ₂₄Σ₂₅Σ₂₆Σ₂₇Σ₂₈Σ₂₉∗∗Σ₃₃Σ₃₄Σ₃₅Σ₃₆Σ₃₇Σ₃₈Σ₃₉∗∗∗Σ₄₄Σ₄₅Σ₄₆Σ₄₇Σ₄₈Σ₄₉∗∗∗∗Σ₅₅Σ₅₆Σ₅₇Σ₅₈Σ₅₉∗∗∗∗∗Σ₆₆Σ₆₇Σ₆₈Σ₆₉∗∗∗∗∗∗Σ₇₇Σ₇₈Σ₇₉∗∗∗∗∗∗∗Σ₈₈Σ₈₉∗∗∗∗∗∗∗∗Σ₉₉. (56)
Moreover, the solution 𝑥(𝑡,𝜙,𝜓) satisfies the inequality 𝑥𝑡,𝜙,𝜓≤𝐷𝜆_min𝑃₁max𝜙,𝜓𝑒^−𝑘𝑡,∀𝑡∈𝑅⁺, (57) where 𝐷=𝜆_max(𝑃₁) +𝑟𝜆_max(𝑃₂)+ℎ𝜆_max(𝑃₃) +𝛽ℎ𝜆_max(𝑃₄) +ℎ³𝜆_max(𝑃₅+ 𝑃₇+𝑃₉) +(𝛽ℎ)³ 𝜆_max(𝑃₆ +𝑃₈+ 𝑃₁₀)+ℎ³𝜆_max(^𝑅1𝑅2_∗𝑅3) +(𝛽ℎ)³𝜆_max(^𝑅4𝑅5_∗𝑅6).

4. Numerical Examples

In order to show the effectiveness of the approaches presented in Section 3, three numerical examples are provided.

Example 1. Consider the robust exponential stability of system (1) with 𝐴(𝛼)=𝛼₁−200−1+𝛼₂,𝐵(𝛼)−200−0.8=𝛼₁−10−1−0.8+𝛼₂,𝐶(𝛼)−10−1−1=𝛼₁0.1000.1+𝛼₂,𝑓((𝑡=(𝑡)𝑥0.2000.1𝑡,𝑥))0.05sin₁(𝑡)0.05cos(𝑡)𝑥₂(𝑡),𝑔(((𝑡=(𝑡)𝑡,𝑥𝑡−ℎ)))0.1cos²𝑥₁((𝑡(𝑡)𝑡−ℎ))0.1sin²𝑥₂((𝑡,ℎ(𝑡)𝑡−ℎ))=0.9sin²𝑡2,𝑤(𝑥̇((𝑡=(𝑡)𝑥̇𝑡,𝑡−𝑟)))0.1cos₁((𝑡(𝑡)𝑥̇𝑡−𝑟))0.1sin₂((𝑡,𝑟(𝑡)𝑡−𝑟))=cos²𝑡4, (58) where 𝑥(𝑡)∈𝑅². It is easy to see that ℎ=0.9, ℎ_𝑑=0.45, 𝑟=1, 𝑟_𝑑=0.25, 𝜂=0.05, 𝜌=0.1, 𝛾=0.1, and given rate of convergence 𝑘=0.1. Decompose matrix 𝐵(𝛼) as follows: 𝐵(𝛼)=𝐵₁(𝛼)+𝐵₂(𝛼), where 𝐵₁(𝛼)=𝛼₁−0.830−1−0.63+𝛼₂,𝐵−0.830−1−0.83₂(𝛼)=𝛼₁−0.1700−0.17+𝛼₂.−0.1700−0.17 (59)
The numerical solutions 𝑥₁(𝑡) and 𝑥₂(𝑡) of system (1) with (58)-(59) are plotted in Figure 1 where the states 𝑥₁(𝑡) and 𝑥₂(𝑡) are attracted to the stable origin.
Solution. By using the LMI Toolbox in MATLAB (with accuracy 0.01), we use conditions (25) in Theorem 7 for system (1) with (58)-(59). The solutions of LMIs verify as follows: 𝑃¹₁=10³×,𝑃3.3091−0.0455−0.04550.7941₁₂=10³×,𝑃3.3091−0.0455−0.04550.7941₂₁=10³×,𝑃4.3739−0.3929−0.39292.7610₂₂=10³×,𝑃4.7896−0.4618−0.46182.8307₃₁=10⁴×,𝑃1.59760.13840.13840.3189₃₂=10⁴×,𝑃1.59760.13840.13840.3189₄₁=10³×,𝑃9.8670−0.0635−0.06354.5726₄₂=10³×,𝑃8.8868−0.2421−0.24213.2473₅₁=10³×,𝑃1.3009−0.1372−0.13720.6505₅₂=10³×,𝑃1.0898−0.0980−0.09800.3986₆₁=10⁴×,𝑃2.8885−0.0338−0.03382.6935₆₂=10⁴×,𝑃2.8526−0.0449−0.04492.4922₇₁=10³×,𝑃2.6064−0.3045−0.30452.5259₇₂=10³×,𝑃2.3663−0.1722−0.17222.5747₈₁=10⁴×,𝑃1.4801−0.0277−0.02771.4063₈₂=10⁴×,𝑃1.4740−0.0862−0.08621.4207₉₁=10³×,𝑃2.6064−0.3045−0.30452.5259₉₂=10³×,𝑃2.3663−0.1722−0.17222.5747₁₁₀=10⁴×,𝑃1.4570−0.0000−0.00001.2074₂₁₀=10⁴×,𝑀1.43540.00190.00191.0978₁₁=10³×,𝑀−9.1839−1.2537−1.2537−2.0706₁₂=10⁴×,𝑀−1.0642−0.1844−0.1844−0.2293₂₁=10³×,𝑀8.84591.15031.15032.0066₂₂=10⁴×,𝑀1.02880.17630.17630.2238₃₁=10⁴×,𝑀1.04990.10900.10900.2928₃₂=10⁴×,𝑀1.18380.17660.17660.2881₄₁=10³×,𝑀−8.8058−0.9604−0.9604−2.6545₄₂=10⁴×,𝑀−1.0454−0.1677−0.1677−0.2680₅₁=10⁴×,𝑀1.03030.10180.10180.3190₅₂=10⁴×,𝑀1.15900.16220.16220.3078₆₁=,𝑀−34.9655−18.7513−18.751335.4129₆₂=,𝑀−39.8158−24.8476−24.847613.4547₇₁=2.09042.77172.7717−7.8109,𝑀⁷₂=,𝑀3.28035.29295.2929−4.0693₈₁=10⁴×,𝑀1.46790.00030.00031.2678₈₂=10⁴×,𝑀1.4438−0.0080−0.00801.1830₉₁=10³×,𝑀−0.89050.00580.0058−2.9028₉₂=10³×,𝑀−1.1307−0.0760−0.0760−3.7474₁₁₀=10⁴×,𝑀1.46840.00050.00051.2673₂₁₀=10⁴×,𝑅1.4444−0.0077−0.00771.1828₁₁=10³×,𝑅2.91310.17480.17480.7782₁₂=10³×,𝑅2.34630.03530.03530.5806₂₁=10³×,𝑅1.13930.05310.05310.1251₂₂=,𝑅876.562732.555032.555098.5599₃₁=10³×,𝑅2.6387−0.0759−0.07590.5866₃₂=10³×,𝑅2.0346−0.0762−0.07620.3589₄₁=10³×,𝑅5.6600−0.0415−0.04155.0541₄₂=10³×,𝑅5.5782−0.0710−0.07104.6031₅₁=10⁴×,𝑅−7.90710.05510.0551−3.8199₅₂=10⁴×,𝑅−7.39120.45760.4576−3.2145₆₁=10⁴×,𝑅1.4893−0.0162−0.01621.3778₆₂=10⁴×,𝑁1.4703−0.0223−0.02231.2713₁₁=10³×,𝑁8.74933.85343.8534−3.1296₁₂=10⁴×,𝑁0.66910.26860.2686−1.0299₂₁=10³×,𝑁−3.8698−0.3606−0.36061.4605₂₂=10³×,𝑁−4.5981−0.9753−0.97531.2772₃₁=,𝑁−114.8946−186.6948−186.6948−113.8034₃₂=,𝑁210.0606−115.9077−115.9077−174.4056₄₁=10⁴×,𝑁−1.1296−0.0266−0.02660.1446₄₂=10⁴×,𝑁−1.00620.10410.10410.8782₅₁=10³×,𝑁4.5589−2.3847−2.38470.8561₅₂=10³×,𝑁5.3887−2.4363−2.43631.1619₆₁=10³×,𝑁5.9564−1.9530−1.95301.5688₆₂=10³×,𝑁6.7696−1.9958−1.99582.0384₇₁=10³×,𝑁5.9602−0.2891−0.28912.9332₇₂=10³×,𝑁6.3929−0.0855−0.08553.1668₈₁=10³×,𝑁−9.6508−0.4780−0.47803.5901₈₂=10⁴×,𝑁−0.85050.05740.05741.1344₉₁=10³×,𝑁5.5789−2.0704−2.07041.3737₉₂=10³×,𝑁6.3923−2.1169−2.11691.8008₁₁₀=10³×,𝑁−3.65953.20323.2032−0.1648₂₁₀=10³×,𝑂−4.37813.21203.21200.1271₁₁=10⁴×,𝑂1.5923−0.7669−0.76691.6350₁₂=10⁴×,𝑂1.8134−0.6324−0.63242.3112₂₁=10⁴×,𝑂1.0371−0.0001−0.0001−0.3635₂₂=10⁴×,𝑂0.9152−0.1464−0.1464−1.0978₃₁=10³×,𝑂−1.03550.15150.1515−0.3626₃₂=10³×,𝑂−1.49900.31480.3148−0.4988₄₁=,𝑂−51.179771.716771.7167−390.4844₄₂=,𝑂−22.7818−177.9764−177.9764237.5644₅₁=10⁴×,𝑂−1.45690.23760.2376−0.4763₅₂=10⁴×,𝑂−1.50780.22770.2277−0.4312₆₁=10⁴×,𝑂−1.36670.24750.2475−0.4459₆₂=10⁴×,𝑂−1.41690.24410.2441−0.4015₇₁=10³×,𝑂5.50380.79000.7900−4.7294₇₂=10⁴×,𝑂0.4206−0.0170−0.0170−1.2249₈₁=,𝑂244.568488.459688.4596481.2263₈₂=,𝑂272.5078401.6259401.6259−0.2695₉₁=10⁴×,𝑂−1.39110.24470.2447−0.4543₉₂=10⁴×,𝑂−1.44150.23940.2394−0.4095₁₁₀=10⁴×,𝑂1.3674−0.2535−0.25350.3868₂₁₀=10⁴×,𝑊1.4277−0.2479−0.24790.3580₁₁=10⁴×,𝑊1.5106−0.0468−0.04680.7929₁₂=10⁴×,𝑊1.4523−0.0341−0.03410.7069₂₁=10³×,𝑊5.9653−2.4134−2.41341.7851₂₂=10³×,𝑊6.9091−2.5666−2.56661.9597₃₁=,𝑊−289.708524.420024.4200−461.3886₃₂=,𝑊−223.519393.656693.6566−234.3346₄₁=10⁴×,𝑊−1.27460.26530.2653−0.3694₄₂=10⁴×,𝑊−1.32420.26360.2636−0.3371₅₁=10³×,𝑊4.8778−0.0916−0.09163.8521₅₂=10³×,𝑊4.8245−0.0681−0.06813.9058₆₁=10³×,𝑊7.9418−0.4415−0.44156.9527₆₂=10³×,𝑊7.9864−0.4713−0.47137.2384₇₁=10³×,𝑊4.5999−2.3966−2.39660.1608₇₂=10³×,𝑊5.2537−2.3858−2.38580.0285₈₁=10⁴×,𝑊−1.38440.25770.2577−0.4094₈₂=10⁴×,𝑊−1.44260.24990.2499−0.3763₉₁=10³×,𝑊7.1086−0.3474−0.34746.1090₉₂=10³×,𝑊7.1257−0.3627−0.36276.3346₁₁₀=10³×,𝑊8.0468−0.7400−0.74006.5133₂₁₀=10³×,8.1365−0.8020−0.80206.8109𝐺=−2.15271.43896.5441−11.7139,𝜖₁=4.2164×10⁴,𝜖₂=5.7269×10⁴,𝜖₃=5.3163×10⁴. (60)

Figure 1 

The simulation solutions 𝑥1(𝑡) and 𝑥2(𝑡) are presented for system (1) with (58)-(59) in Example 1, 𝛼1=𝛼2=1/2, and initial conditions 𝑥1(𝑡)=2+3cos(𝑡), 𝑥2(𝑡)=1+2sin(𝑡), 𝑡∈[−1,0], by using the Runge-Kutta fourth order method with Matlab.

Example 2. Consider the following neutral system (54), which is considered in [3, 25, 26]: 𝑥̇(𝑡)(𝑡)((𝑡𝑥̇((𝑡((𝑡(((𝑡,=𝐴𝑥+𝐵𝑥𝑡−ℎ))+𝐶𝑡−𝑟))+𝑓𝑡,𝑥))+𝑔𝑡,𝑥𝑡−ℎ))) (61) with 𝐴=−200−0.9,𝐵=−10−1−1,𝐶=0.1000.1. (62)
Decompose matrix 𝐵 as follows: 𝐵=𝐵₁+𝐵₂, where 𝐵₁=−0.830−1−0.83,𝐵₂=−0.1700−0.17, (63)‖𝑓(𝑡,𝑥(𝑡))‖≤𝜂‖𝑥(𝑡)‖, and ‖𝑔(𝑡,𝑥(𝑡−ℎ(𝑡)))‖≤𝜌‖𝑥(𝑡−ℎ(𝑡))‖. The maximum value ℎ for exponential stability of system (61) with (62)-(63) is listed in the comparison in Table 1, for different values of ℎ_𝑑 and 𝑟_𝑑. In Table 1, we let 𝜂=0.05, 𝜌=0.1, 𝛽=0.1 and ℎ(𝑡)=𝑟(𝑡). We can see that our results in Corollary 8 are much less conservative than in [3, 25, 26].

Table 1 

The maximum allowed time delay ℎ for 𝑘=0.1, 𝜂=0.05, 𝜌=0.1, and 𝛽=0.1.

Example 3. Consider the following neutral system (51), which is considered in [14, 28]: 𝑥̇(𝑡)(𝑡)((𝑡𝑥̇((𝑡((𝑡(((𝑡(𝑥̇((𝑡,=𝐴𝑥+𝐵𝑥𝑡−ℎ))+𝐶𝑡−𝑟))+𝑓𝑡,𝑥))+𝑔𝑡,𝑥𝑡−ℎ)))+𝑤𝑡,𝑡−𝑟))) (64) with 𝐴=−200−2,𝐵=00.40.40,𝐶=0.1000.1. (65)
Decompose matrix 𝐵 as follows: 𝐵=𝐵₁+𝐵₂, where 𝐵₁=00.30.20,𝐵₂=00.10.20, (66)‖𝑓(𝑡,𝑥(𝑡))‖≤𝜂‖𝑥(𝑡)‖, ‖𝑔(𝑡,𝑥(𝑡−ℎ(𝑡)))‖≤𝜌‖𝑥(𝑡−ℎ(𝑡))‖, and ‖𝑤(𝑡,𝑥̇(𝑡−𝑟(𝑡)))‖≤𝛾‖𝑥̇(𝑡−𝑟(𝑡))‖. By Corollary 9 to system (64) with (65)-(66), one can obtain the maximum upper bounds of the time delay with different convergence rate 𝑘 as listed in Table 2. In Table 2, we let ℎ=0.1, 𝜂=0.1, 𝜌=0.05, 𝛾=0.05, and 𝛽=0.1. It is clear that the results in Corollary 9 give larger delay bounds than the recent results in [14, 28].

Table 2 

The maximum allowed time delay ℎ.

5. Conclusions

The problem of robust exponential stability for LPD neutral systems with mixed time-varying delays and nonlinear uncertainties has been presented. Based on combination of Leibniz-Newton formula, free-weighting matrices, linear matrix inequality, Cauchy’s inequality, modified version of Jensen's inequality, model transformation, and the use of suitable parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent robust exponential stability criteria are formulated in terms of LMIs. Numerical examples have shown significant improvements over some existing results.

Acknowledgments

This work was supported by the Higher Education Research Promotion and the National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education, Khon Kaen University, Khon Kaen, Thailand.