Exponential Stabilization of Neutral-Type Neural Networks with Interval Nondifferentiable and Distributed Time-Varying Delays

Abstract

The problem of exponential stabilization of neutral-type neural networks with various activation functions and interval nondifferentiable and distributed time-varying delays is considered. The interval time-varying delay function is not required to be differentiable. By employing new and improved Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, the stabilizability criteria are formulated in terms of a linear matrix inequalities. Numerical examples are given to illustrate and show the effectiveness of the obtained results.

1. Introduction

In recent years, there have been great attentions on the stability analysis of neural networks due to its real world application to various systems such as signal processing, pattern recognition, content-addressable memory, and optimization [1–5]. In performing a periodicity or stability analysis of a neural network, the conditions to be imposed on the neural network are determined by the characteristics of various activation functions and network parameters. When neural networks are designed for problem solving, it is desirable that their activation functions are not too restrictive, [3, 6, 7]. It is known that time delays cannot be avoided in the hardware implementation of neural networks due to the finite switching speed of amplifies in electronic neural networks or the finite signal propagation time in biological networks. The existence of time delays may result in instability or oscillation of a neural network. Therefore, many researchers have focused on the study of stability analysis of delayed neural networks with various activation functions with more general conditions during the last decades [2, 8–11].

The stability criteria for system with time delays can be classified into two categories: delay independent and delay dependent. Delay-independent criteria do not employ any information on the size of the delay; while delay-dependent criteria make use of such information at different levels. Delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the delay is small. In many situations, time delays are time-varying continuous functions which vary from 0 to a given upper bound. In addition, the range of time delays may vary in a range for which the lower bound is not restricted to be 0; in which case time delays are called interval time-varying delay. A typical example with interval time delay is the networked control system, which has been widely studied in the recent literature (see, e.g., [2, 11–14]). Therefore, it is of great significance to investigate the stability of system with interval time-varying delay. Another important type of time delay is distributed delay where stability analysis of neural networks with distributed delayed has been studied extensively recently, see [2, 5, 8–11, 15–17].

It is know that exponential stability is more important than asymptotic stability since it provides information on convergence rate of solutions of systems to equilibrium points. It is particularly important for neural networks where the exponential convergence rate is used to determine the speed of neural computations. Therefore, it is important to determine the exponential stability and to estimate the exponential convergence rate for delayed neural networks. Consequently, many researchers have considered the exponential stability analysis problem for delayed neural networks and several results on this topic that have been reported in the literatures [3, 9, 13, 14, 17].

In practical control designs, due to systems uncertainty, failure modes, or systems with various modes of operation, the simultaneous stabilization problem has often to be taken into account. The problem is concerned with designing a single controller which can simultaneously stabilize a set of systems. Recently, the exponential stability and stabilization problems for time-delay systems have been studied by many researchers, see [8, 12, 18, 19]. Among the usual approach, there are many results on the stabilization problem of neural networks being reported in the literature (see [2, 9, 15, 16, 20, 21]). In [21], robust stabilization criterion are provided via designing a memoryless state feedback controller for the time-delay dynamical neural networks with nonlinear perturbation. However, time-delay is required to be constant. In [16], global robust stabilizing control are presented for neural network with time-varying delay with the lower bound restricted to be 0. For neural network with interval time-varying delay, global stability analysis is considered with control input in [2, 9, 12]. Nonetheless, in most studies, the time-varying delays are required to be differentiable [2, 9, 15, 16]. Therefore, their methods have a conservatism which can be improved upon.

It is noted that these stability conditions are either with testing difficulty or with conservatism to some extent. It is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions. Such systems are called neutral-type systems in which the systems contain both state delay as well as state derivative delay, so called neutral delay. The phenomena on neutral delay often appears in the study of heat exchanges, distributed networks containing lossless transmission lines, partial element equivalent circuits and population ecology are examples of neutral systems see [1, 10, 11, 22, 23] and references cited therein.

Based on the above discussions, we consider the problem of exponential stabilization of neutral-type neural networks with interval and distributed time-varying delays. There are various activation functions which are considered in the system and the restriction on differentiability of interval time-varying delays is removed, which means that a fast interval time-varying delay is allowed. Based on the construction of improved Lyapunov-Krasovskii functionals combined with Liebniz-Newton's formula and some appropriate estimation of integral terms, new delay-dependent sufficient conditions for the exponential stabilization of the system are derived in terms of LMIs without introducing any free-weighting matrices. The new stability conditions are much less conservative and more general than some existing results. Numerical examples are given to illustrate the effectiveness and less conservativeness of our theoretical results. To the best of our knowledge, our results are among the first on investigation of exponential stabilization for neutral-type neural networks with discrete, neutral, and distributed delays.

The rest of this paper is organized as follows. In Section 2, we give notations, definitions, propositions, and lemmas which will be used in the proof of the main results. Delay-dependent sufficient conditions for the exponential stabilization of neutral-type neural networks with various activation functions, interval and distributed time-varying delays, and designs of memoryless feedback controls are presented in Section 3. Numerical examples illustrated the obtained results are given in Section 4. The paper ends with conclusions in Section 5 and cited references.

2. Preliminaries

The following notation will be used in this paper: ℝ⁺ denotes the set of all real nonnegative numbers; ℝ^𝑛 denotes the 𝑛-dimensional space and the vector norm ‖⋅‖; 𝑀^𝑛×𝑟 denotes the space of all matrices of (𝑛×𝑟)-dimensions.

𝐴^𝑇 denotes the transpose of 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)}.

𝑥_𝑡∶={𝑥(𝑡+𝑠)∶𝑠∈[−ℎ,0]},‖𝑥_𝑡‖=sup_{𝑠∈[−ℎ,0]}‖𝑥(𝑡+𝑠)‖;𝐶([0,𝑡],ℝ^𝑛) denotes the set of all ℝ^𝑛-valued continuous functions on [0,𝑡]; 𝐿₂([0,𝑡],ℝ^𝑚) denotes the set of all the ℝ^𝑚-valued square integrable functions on [0,𝑡].

Matrix 𝐴 is called semi-positive definite (𝐴≥0) if ⟨𝐴𝑥,𝑥⟩≥0, for all 𝑥∈ℝ^𝑛;𝐴 is positive definite (𝐴>0) if ⟨𝐴𝑥,𝑥⟩>0 for all 𝑥≠0;𝐴>𝐵 means 𝐴−𝐵>0. The symmetric term in a matrix is denoted by ∗.

Consider the following neural networks with mixed time-varying delays and control input 𝑥̇(𝑡)=−(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+𝑊₀+Δ𝑊₀𝑓(𝑥(𝑡))+𝑊₁+Δ𝑊₁𝑔(𝑥((𝑡+𝑊𝑡−ℎ)))₂+Δ𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+𝐵₀(𝑡)(𝑡),𝑥(𝑡)(𝑡)[]ℎ𝑥̇𝑡−𝜂+𝐵𝑢=𝜙,𝑡∈−𝑑,0,𝑑=max₂,𝑘,𝜂, (2.1) where 𝑥(𝑡)∈ℝ^𝑛 is the state of the neural networks, 𝑢(⋅)∈𝐿₂([0,𝑡],ℝ^𝑚) is the control, 𝑛 is the number of neurals, and 𝑓(𝑥(𝑡))=𝑓₁𝑥₁(𝑡),𝑓₂𝑥₂(𝑡),…,𝑓_𝑛𝑥_𝑛(𝑡)^𝑇,𝑔(𝑥(𝑡))=𝑔₁𝑥₁(𝑡),𝑔₂𝑥₂(𝑡),…,𝑔_𝑛𝑥_𝑛(𝑡)^𝑇,ℎ(𝑥(𝑡))=ℎ₁𝑥₁(𝑡),ℎ₂𝑥₂(𝑡),…ℎ_𝑛𝑥_𝑛(𝑡)^𝑇 (2.2) are the activation functions, 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛),𝑎_𝑖>0 represents the self-feedback term and 𝑊₀,𝑊₁,𝑊₂ denote the connection weights, the discretely delayed connection weights, and the distributively delayed connection weight, respectively. In this paper, we consider various activation functions and assume that the activation functions 𝑓(⋅),𝑔(⋅),ℎ(⋅) are Lipschitzian with the Lipschitz constants 𝑎_𝑖,𝑏_𝑖,𝑐_𝑖>0𝑓_𝑖𝜉₁−𝑓_𝑖𝜉₂≤𝑎_𝑖𝜉₁−𝜉₂,𝑖=1,2,…,𝑛,∀𝜉₁,𝜉₂𝑔∈ℝ,_𝑖𝜉₁−𝑔_𝑖𝜉₂≤𝑏_𝑖𝜉₁−𝜉₂,𝑖=1,2,…,𝑛,∀𝜉₁,𝜉₂ℎ∈ℝ,_𝑖𝜉₁−ℎ_𝑖𝜉₂≤𝑐_𝑖𝜉₁−𝜉₂,𝑖=1,2,…,𝑛,∀𝜉₁,𝜉₂∈ℝ. (2.3) The time-varying delay functions ℎ(𝑡),𝑘(𝑡),𝜂(𝑡) satisfy the condition0≤ℎ₁≤ℎ(𝑡)≤ℎ₂,(𝑡)(𝑡)(𝑡)0≤𝑘≤𝑘,0≤𝜂≤𝜂,𝜂̇≤𝛿<1. (2.4) It is worth noting that the time delay is assumed to be a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is bounded but not restricted to being zero. The initial functions 𝜙(𝑡)∈𝐶¹([−𝑑,0],ℝ^𝑛), with the norm ‖𝜙‖=sup_{𝑡∈[−𝑑,0]}‖𝜙(𝑡)‖²+‖𝜙̇(𝑡)‖². (2.5) The uncertainties satisfy the following condition:Δ𝐴(𝑡)=𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎,Δ𝑊₀(𝑡)=𝐸₀𝐹₀(𝑡)𝐻₀,Δ𝑊₁(𝑡)=𝐸₁𝐹₁(𝑡)𝐻₁,Δ𝑊₂(𝑡)=𝐸₂𝐹₂(𝑡)𝐻₂, (2.6) where 𝐸_𝑖,𝐻_𝑖,𝑖=𝑎,0,1,2 are given constant matrices with appropriate dimensions, 𝐹_𝑖(𝑡),𝑖=𝑎,0,1,2 are unknown, real matrices with Lebesgue measurable elements satisfying𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼,𝑖=𝑎,0,1,2∀𝑡≥0. (2.7)

Definition 2.1. The zero solution of system (2.8) is exponentially stabilizable if there exists a feedback control 𝑢(𝑡)=𝐾𝑥(𝑡),𝐾∈ℝ^𝑚×𝑛 such that the resulting closed-loop system 𝑥̇(𝑡)=−[𝐴−𝐵𝐾]𝑥(𝑡)+𝑊₀𝑓(𝑥(𝑡))+𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))+𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+𝐵₀𝑥̇𝑡−𝜂(𝑡) (2.8) is 𝛼-stable.

Definition 2.2. Given 𝛼>0. The zero solution of system (2.1) with 𝑢(𝑡)=0 is 𝛼-stable if there exists a positive number 𝑁>0 such that every solution 𝑥(𝑡,𝜙) satisfies the following condition: 𝑥𝑡,𝜙≤𝑁𝑒^−𝛼𝑡𝜙,∀𝑡≥0. (2.9)

We introduce the following technical well-known propositions and lemma, which will be used in the proof of our results.

Proposition 2.3 (Cauchy inequality). For any symmetric positive definite matrix 𝑁∈𝑀^𝑛×𝑛 and 𝑥,𝑦∈ℝ^𝑛 one has ±2𝑥^𝑇𝑦≤𝑥^𝑇𝑁𝑥+𝑦^𝑇𝑁⁻¹𝑦. (2.10)

Proposition 2.4 (see [24]). For any symmetric positive definite matrix 𝑀>0, scalar 𝛾>0 and vector function 𝜔∶[0,𝛾]→ℝ^𝑛 such that the integrations concerned are well defined, the following inequality holds: ^𝛾₀𝜔(𝑠)𝑑𝑠^𝑇𝑀^𝛾₀𝜔(𝑠)𝑑𝑠≤𝛾^𝛾₀𝜔^𝑇(𝑠)𝑀𝜔(𝑠)𝑑𝑠. (2.11)

Proposition 2.5 ([24, Schur complement lemma]). Given constant symmetric matrices 𝑋,𝑌,𝑍 with appropriate dimensions satisfying 𝑋=𝑋^𝑇,𝑌=𝑌^𝑇>0. Then 𝑋+𝑍^𝑇𝑌⁻¹𝑍<0 if and only if 𝑋𝑍^𝑇𝑍−𝑌𝑍<0or−𝑌𝑍_𝑇𝑋<0. (2.12)

Lemma 2.6 (see [25]). Given matrices 𝑄=𝑄^𝑇,𝐻,𝐸, and 𝑅=𝑅^𝑇>0 with appropriate dimensions. Then 𝑄+𝐻𝐹𝐸+𝐸^𝑇𝐹^𝑇𝐻^𝑇<0, (2.13) for all 𝐹 satisfying 𝐹^𝑇𝐹≤𝑅, if and only if there exists an 𝜖>0 such that 𝑄+𝜖𝐻𝐻^𝑇+𝜖⁻¹𝐸^𝑇𝑅𝐸<0. (2.14)

3. Main Results

3.1. Exponential Stabilization for Nominal Interval Time-Varying Delay Systems

The nominal system is given by𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝑊₀𝑓(𝑥(𝑡))+𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))+𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+𝐵₀(𝑡)(𝑡)𝑥(𝑡)(𝑡)[].𝑥̇𝑡−𝜂+𝐵𝑢=𝜙,𝑡∈−𝑑,0 (3.1) First, we present a delay-dependent exponential stabilizability analysis conditions for the given nominal interval time-varying delay system (3.1) with Δ𝐴(𝑡)=Δ𝑊₀(𝑡)=Δ𝑊₁(𝑡)=Δ𝑊₂(𝑡)=0. Let us set 𝜆₁=𝜆_min𝑃⁻¹,𝜆₂=𝜆_max𝑃⁻¹+2ℎ₂𝜆_max𝑃⁻¹𝑄𝑃⁻¹+2ℎ²₂𝜆_max𝑃⁻¹𝑅𝑃⁻¹+𝜂𝜆_max𝑃⁻¹𝑄₁𝑃⁻¹+2𝜆_max𝐻𝐷₂⁻¹𝐻+ℎ₂−ℎ₁²𝜆_max𝑃⁻¹𝑈𝑃⁻¹. (3.2)

Assumption 3.1. All the eigenvalues of matrix 𝐵₀ are inside the unit circle.

Theorem 3.2. Given 𝛼>0. The system (3.1) is 𝛼-exponentially stabilizable if there exist symmetric positive definite matrices 𝑃,𝑄,𝑅,𝑈,𝑄1, three diagonal matrices 𝐷_𝑖,𝑖=0,1,2 such that the following LMI holds: ℳ₁=ℳ−000−𝐼0𝐼^𝑇×𝑒^−2𝛼ℎ2𝑈ℳ000−𝐼0𝐼<0,₂=ℳ−00𝐼00−𝐼^𝑇×𝑒^−2𝛼ℎ2𝑈ℳ00𝐼00−𝐼<0,₃=−0.1𝐵𝐵^𝑇−0.1𝑒^−2𝛼ℎ1+𝑒^−2𝛼ℎ2𝑅2𝑘𝑃𝐻2𝑃𝐹∗−2𝑘𝐷₂0∗∗−2𝐷₀ℳ<0,₄=−0.1𝑒^−2𝛼ℎ2𝑈2𝑃𝐺∗−2𝐷₁𝑀<0,ℳ=¹¹𝑀₁₂𝑀₁₃𝑀₁₄𝑀₁₅0∗𝑀₂₂00𝑀₂₅0∗∗𝑀₃₃00𝑀₃₆∗∗∗𝑀₄₄0𝑀₄₆∗∗∗∗𝑀₅₅0∗∗∗∗∗𝑀₆₆, (3.3)(3.4)(3.5)(3.6)(3.7) where 𝑀₁₁=[−𝐴+𝛼𝐼]𝑃+𝑃[−𝐴+𝛼𝐼]^𝑇−0.9𝐵𝐵^𝑇+2𝑄+𝑊₀𝐷₀𝑊^𝑇₀+𝑊₁𝐷₁𝑊^𝑇₁+𝑘𝑒^2𝛼𝑘𝑊₂𝐷₂𝑊^𝑇₂−0.9𝑒^−2𝛼ℎ1𝑅−0.9𝑒^−2𝛼ℎ2𝑅𝑀₁₂=𝐵₀𝑃,𝑀₁₃=𝑒^−2𝛼ℎ1𝑅,𝑀₁₄=𝑒^−2𝛼ℎ2𝑅,𝑀₁₅=−𝑃𝐴^𝑇−0.5𝐵𝐵^𝑇,𝑀₂₂=−(1−𝛿)𝑒^−2𝛼𝜂𝑄₁,𝑀₂₅=𝑃𝐵^𝑇₀,𝑀₃₃=−𝑒^−2𝛼ℎ1𝑄−𝑒^−2𝛼ℎ1𝑅−𝑒^−2𝛼ℎ2𝑀𝑈,₃₆=𝑒^−2𝛼ℎ2𝑈,𝑀₄₄=−𝑒^−2𝛼ℎ2𝑄−𝑒^−2𝛼ℎ2𝑅−𝑒^−2𝛼ℎ2𝑈,𝑀₄₆=𝑒^−2𝛼ℎ2𝑀𝑈,₅₅=ℎ²₁𝑅+ℎ²₂𝑅+ℎ₂−ℎ₁²𝑈+𝑄₁−2𝑃+𝑊₀𝐷₀𝑊^𝑇₀+𝑊₁𝐷₁𝑊^𝑇₁+𝑘𝑒^2𝛼𝑘𝑊₂𝐷₂𝑊^𝑇₂𝑀₆₆=−1.9𝑒^−2𝛼ℎ2𝑈. (3.8) Moreover, the memoryless feedback control is 𝑢(𝑡)=−0.5𝐵^𝑇𝑃⁻¹𝑥(𝑡),𝑡≥0, (3.9) and the solution 𝑥(𝑡,𝜙) of the system satisfies 𝑥𝑡,𝜙≤𝜆₂𝜆₁𝑒^−𝛼𝑡𝜙,∀𝑡≥0. (3.10)

Proof. Let 𝑌=𝑃⁻¹,𝑦(𝑡)=𝑌𝑥(𝑡). Using the feedback control (3.9) we consider the following Lyapunov-Krasovskii functional: 𝑉𝑡,𝑥_𝑡=⁸_𝑖=1𝑉_𝑖, (3.11) where 𝑉₁=𝑥^𝑇(𝑡)(𝑡),𝑉𝑌𝑥₂=^𝑡_𝑡−ℎ1𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)(𝑠)𝑉𝑌𝑄𝑌𝑥𝑑𝑠,₃=^𝑡_𝑡−ℎ2𝑒^{2𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)(𝑠)𝑉𝑌𝑄𝑌𝑥𝑑𝑠,₄=ℎ₁⁰_−ℎ1^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑡)}𝑥̇^𝑇(𝜏)(𝜏)𝑉𝑌𝑅𝑌𝑥̇𝑑𝜏𝑑𝑠,₅=ℎ₂⁰_−ℎ2^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑡)}𝑥̇^𝑇(𝜏)(𝜏)𝑉𝑌𝑅𝑌𝑥̇𝑑𝜏𝑑𝑠,₆=ℎ₂−ℎ₁^−ℎ1_−ℎ2^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑡)}𝑥̇^𝑇(𝜏)(𝜏)𝑉𝑌𝑈𝑌𝑥̇𝑑𝜏𝑑𝑠,₇=^𝑡_{𝑡−𝜂(𝑡)}𝑒^{2𝛼(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌𝑄₁(𝑠)𝑉𝑌𝑥̇𝑑𝑠,₈=2⁰_−𝑘^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑡)}ℎ^𝑇(𝑥(𝜏))𝐷₂⁻¹ℎ(𝑥(𝜏))𝑑𝜏𝑑𝑠. (3.12) It easy to check that 𝜆₁‖𝑥(𝑡)‖²≤𝑉𝑡,𝑥_𝑡≤𝜆₂‖𝑥_𝑡‖²,∀𝑡≥0. (3.13) Taking the derivative of 𝑉(𝑥_𝑡) along the solution of system (3.1) we have 𝑉̇₁=2𝑥^𝑇(𝑡)𝑌𝑥̇(𝑡),=2𝑦^𝑇(𝑡)−𝐴𝑥(𝑡)+𝑊₀𝑓(𝑥(𝑡))+𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))+𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+𝐵₀𝑥̇𝑡−𝜂(𝑡)−0.5𝐵𝐵^𝑇𝑃⁻¹𝑥(𝑡)=2𝑦^𝑇(𝑡)−𝐴𝑃𝑦(𝑡)+𝑊₀𝑓(𝑥(𝑡))+𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))+𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+𝐵₀𝑃𝑦̇𝑡−𝜂(𝑡)−0.5𝐵𝐵^𝑇𝑦(𝑡) (3.14)=𝑦^𝑇(𝑡)−𝐴𝑃−𝑃𝐴^𝑇𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑊₀𝑓(𝑥(𝑡))+2𝑦^𝑇(𝑡)𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))+2𝑦^𝑇(𝑡)𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+2𝑦^𝑇(𝑡)𝐵₀𝑃𝑦̇𝑡−𝜂(𝑡)−𝑦^𝑇(𝑡)𝐵𝐵^𝑇𝑦(𝑡)+2𝛼𝑦^𝑇(𝑡)𝑃𝑦(𝑡)−2𝛼𝑉₁,𝑉̇₂=𝑦^𝑇(𝑡)𝑄𝑦(𝑡)−𝑒^−2𝛼ℎ1𝑦^𝑇𝑡−ℎ₁𝑄𝑦𝑡−ℎ₁−2𝛼𝑉₂,𝑉̇₃=𝑦^𝑇(𝑡)𝑄𝑦(𝑡)−𝑒^−2𝛼ℎ2𝑦^𝑇𝑡−ℎ₂𝑄𝑦𝑡−ℎ₂−2𝛼𝑉₃,𝑉̇₄≤ℎ²₁𝑦^𝑇̇(𝑡)𝑅𝑦̇(𝑡)−ℎ₁𝑒^{−2𝛼ℎ1𝑡}_𝑡−ℎ1𝑦^𝑇̇(𝑠)𝑅𝑦̇(𝑠)𝑑𝑠−2𝛼𝑉₄,𝑉̇₅≤ℎ²₂𝑦^𝑇̇(𝑡)𝑅𝑦̇(𝑡)−ℎ₂𝑒^{−2𝛼ℎ2𝑡}_𝑡−ℎ2𝑦^𝑇̇(𝑠)𝑅𝑦̇(𝑠)𝑑𝑠−2𝛼𝑉₅,𝑉̇₆≤ℎ₂−ℎ₁²𝑦^𝑇̇(𝑡)𝑈𝑦̇(𝑡)−ℎ₂−ℎ₁𝑒^{−2𝛼ℎ2𝑡−ℎ1}_𝑡−ℎ2𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠−2𝛼𝑉₆,𝑉̇₇≤𝑦^𝑇̇(𝑡)𝑄₁𝑦̇(𝑡)−(1−𝛿)𝑒^−2𝛼𝜂𝑦^𝑇̇𝑡−𝜂(𝑡)𝑄₁𝑦̇𝑡−𝜂(𝑡)−2𝛼𝑉₇,𝑉̇₈≤2𝑘ℎ^𝑇(𝑥(𝑡))𝐷₂⁻¹ℎ(𝑥(𝑡))−2𝑒^{−2𝛼𝑘𝑡}_𝑡−𝑘ℎ^𝑇(𝑥(𝑠))𝐷₂⁻¹ℎ(𝑥(𝑠))𝑑𝑠−2𝛼𝑉₈. (3.15) Using the condition (2.3) and since the matrices 𝐷_𝑖⁻¹>0,𝑖=0,1,2 are diagonal, we have 𝑘ℎ^𝑇(𝑥(𝑡))𝐷₂⁻¹ℎ(𝑥(𝑡))≤𝑘𝑥^𝑇(𝑡)𝐻𝐷₂⁻¹𝐻𝑥(𝑡)=𝑘𝑦^𝑇(𝑡)𝑃𝐻𝐷₂⁻¹(𝑡),𝑓𝐻𝑃𝑦_𝑇(𝑥(𝑡))𝐷₀⁻¹𝑓(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝐹𝐷₀⁻¹𝐹𝑥(𝑡)=𝑦^𝑇(𝑡)𝑃𝐹𝐷₀⁻¹(𝑡),𝑔𝐹𝑃𝑦_𝑇(𝑥(𝑡−ℎ(𝑡)))𝐷₁⁻¹𝑔(𝑥(𝑡−ℎ(𝑡)))≤𝑥^𝑇(𝑡−ℎ(𝑡))𝐺𝐷₁⁻¹𝐺𝑥(𝑡−ℎ(𝑡)),=𝑦^𝑇(𝑡−ℎ(𝑡))𝑃𝐺𝐷₁⁻¹𝐺𝑃𝑦(𝑡−ℎ(𝑡)), (3.16) and using (2.3) and the Proposition (2.3) for the following estimations: 2𝑦^𝑇(𝑡)𝑊₀𝑓(𝑥(𝑡))≤𝑦^𝑇(𝑡)𝑊₀𝐷₀𝑊^𝑇₀𝑦(𝑡)+𝑓^𝑇(𝑥(𝑡))𝐷₀⁻¹𝑓(𝑥(𝑡))≤𝑦^𝑇(𝑡)𝑊₀𝐷₀𝑊^𝑇₀𝑦(𝑡)+𝑥^𝑇(𝑡)𝐹𝐷₀⁻¹𝐹𝑥(𝑡)≤𝑦^𝑇(𝑡)𝑊₀𝐷₀𝑊^𝑇₀𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝐹𝐷₀⁻¹𝐹𝑃𝑦(𝑡),2𝑦^𝑇(𝑡)𝑊₁g(𝑥(𝑡−ℎ(𝑡)))≤𝑦^𝑇(𝑡)𝑊₁𝐷₁𝑊^𝑇₁𝑦(𝑡)+𝑔^𝑇(𝑥(𝑡−ℎ(𝑡)))𝐷₁⁻¹𝑔(𝑥(𝑡−ℎ(𝑡)))≤𝑦^𝑇(𝑡)𝑊₁𝐷₁𝑊^𝑇₁𝑦(𝑡)+𝑥^𝑇(𝑡−ℎ(𝑡))𝐺𝐷₁⁻¹𝐺𝑥(𝑡−ℎ(𝑡))≤𝑦^𝑇(𝑡)𝑊₁𝐷₁𝑊^𝑇₁𝑦(𝑡)+𝑦^𝑇(𝑡−ℎ(𝑡))𝑃𝐺𝐷₁⁻¹𝐺𝑃𝑦(𝑡−ℎ(𝑡)),2𝑦^𝑇(𝑡)𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠≤𝑘𝑒^2𝛼𝑘𝑦^𝑇(𝑡)𝑊₂𝐷₂𝑊^𝑇₂𝑦(𝑡)+𝑘⁻¹𝑒^{−2𝛼𝑘𝑡}_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠)𝑑𝑠)^𝑇𝐷₂^−1𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠≤𝑘𝑒^2𝛼𝑘𝑦^𝑇(𝑡)𝑊₂𝐷₂𝑊^𝑇₂𝑦(𝑡)+𝑒^{−2𝛼𝑘𝑡}_𝑡−𝑘ℎ^𝑇(𝑥(𝑠))𝐷₂⁻¹ℎ(𝑥(𝑠))𝑑𝑠. (3.17) Applying Proposition 2.4 and the Leibniz-Newton formula, we have −ℎ₁^𝑡_𝑡−ℎ1𝑦^𝑇̇(𝑠)𝑅𝑦̇(𝑠)𝑑𝑠≤−^𝑡_𝑡−ℎ1𝑦̇(𝑠)^𝑇𝑅^𝑡_𝑡−ℎ1(𝑠𝑦(𝑡)𝑦̇)≤−−𝑦𝑡−ℎ₁^𝑇𝑅𝑦(𝑡)−𝑦𝑡−ℎ₁=−𝑦^𝑇(𝑡)𝑅𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑅𝑦𝑡−ℎ₁−𝑦^𝑇𝑡−ℎ₁𝑅𝑦𝑡−ℎ₁,−ℎ₂^𝑡_𝑡−ℎ2𝑦^𝑇̇(𝑠)𝑅𝑦̇(𝑠)𝑑𝑠≤−^𝑡_𝑡−ℎ2𝑦̇(𝑠)^𝑇𝑅^𝑡_𝑡−ℎ2(𝑠𝑦(𝑡)𝑦̇)≤−−𝑦𝑡−ℎ₂^𝑇𝑅𝑦(𝑡)−𝑦𝑡−ℎ₂=−𝑦^𝑇(𝑡)𝑅𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑅𝑦𝑡−ℎ₂−𝑦^𝑇𝑡−ℎ₂𝑅𝑦𝑡−ℎ₂. (3.18) Note that −ℎ₂−ℎ₁^𝑡−ℎ1_𝑡−ℎ2𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠=−ℎ₂−ℎ₁^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑦^𝑇̇(𝑠)(𝑠)−ℎ𝑈𝑦̇𝑑𝑠₂−ℎ₁^𝑡−ℎ1_{𝑡−ℎ(𝑡)}𝑦^𝑇̇(𝑠)(𝑠)ℎ𝑈𝑦̇𝑑𝑠=−₂−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑦^𝑇̇(𝑠)(𝑠)−ℎ(𝑡)𝑈𝑦̇𝑑𝑠−ℎ₁^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑦^𝑇̇(𝑠)(𝑠)−ℎ(𝑡)𝑈𝑦̇𝑑𝑠−ℎ₁^𝑡−ℎ1_{𝑡−ℎ(𝑡)}𝑦^𝑇̇(𝑠)(𝑠)−ℎ𝑈𝑦̇𝑑𝑠₂−ℎ(𝑡)^𝑡−ℎ1_{𝑡−ℎ(𝑡)}𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠. (3.19) Using Proposition 2.4 gives −ℎ₂−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠≤−^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑦̇(𝑠)𝑑𝑠^𝑇𝑈^{𝑡−ℎ(𝑡)}_𝑡−ℎ2(𝑠)𝑦((𝑡𝑦̇𝑑𝑠≤−𝑡−ℎ))−𝑦𝑡−ℎ₂^𝑇𝑈𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂,−ℎ(𝑡)−ℎ₁^𝑡−ℎ1_{𝑡−ℎ(𝑡)}𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠≤−^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑦̇(𝑠)𝑑𝑠^𝑇𝑈^𝑡−ℎ1_{(𝑡)𝑡−ℎ}(𝑠)𝑦𝑦̇𝑑𝑠≤−𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡)). (3.20)(3.21) Let 𝛽=(ℎ₂−ℎ(𝑡))/(ℎ₂−ℎ₁)≤1. Then −ℎ₂−ℎ(𝑡)^𝑡−ℎ1_{𝑡−ℎ(𝑡)}𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠=−𝛽^𝑡−ℎ1_{(𝑡)𝑡−ℎ}ℎ₂−ℎ₁𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠≤−𝛽^𝑡−ℎ1_{(𝑡)𝑡−ℎ}ℎ(𝑡)−ℎ₁𝑦^𝑇̇(𝑠)(𝑠)𝑦𝑈𝑦̇𝑑𝑠≤−𝛽𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈𝑦𝑡−ℎ₁((𝑡,−ℎ(𝑡)−𝑦𝑡−ℎ))−ℎ₁^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠=−1−𝛽^{𝑡−ℎ(𝑡)}_𝑡−ℎ2ℎ₂−ℎ₁𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠≤−1−𝛽^{𝑡−ℎ(𝑡)}_𝑡−ℎ2ℎ₂−ℎ(𝑡)𝑦^𝑇̇(𝑠)(𝑠)𝑦((𝑡𝑈𝑦̇𝑑𝑠≤−1−𝛽𝑡−ℎ))−𝑦𝑡−ℎ₂^𝑇×𝑈𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂. (3.22)(3.23) Therefore from (3.20)–(3.23), we obtain −ℎ₂−ℎ₁^𝑡−ℎ1_𝑡−ℎ2𝑦^𝑇̇(𝑠)𝑈𝑦̇(𝑠)𝑑𝑠≤−𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂^𝑇𝑈𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂−𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈𝑦𝑡−ℎ₁((𝑡𝑦−𝑦𝑡−ℎ))−𝛽𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈𝑦𝑡−ℎ₁((𝑡−𝑦((𝑡−𝑦𝑡−ℎ))1−𝛽𝑡−ℎ))−𝑦𝑡−ℎ₂^𝑇×𝑈𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂. (3.24) By using the following identity relation: −𝑃𝑦̇(𝑡)−𝐴𝑃𝑦(𝑡)+𝑊₀𝑓(𝑥(𝑡))+𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))+𝑊₀^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+𝐵₀𝑃𝑦̇𝑡−𝜂(𝑡)−0.5𝐵𝐵^𝑇𝑦(𝑡)=0, (3.25) we have −2𝑦^𝑇̇(𝑡)𝑃𝑦̇(𝑡)−2𝑦^𝑇̇(𝑡)𝐴𝑃𝑦(𝑡)+2𝑦^𝑇̇(𝑡)𝑊₀𝑓(𝑥(𝑡))+2𝑦^𝑇̇(𝑡)𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))+2𝑦^𝑇̇(𝑡)𝑊₀^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+2𝑦^𝑇̇(𝑡)𝐵₀𝑃𝑦̇𝑡−𝜂(𝑡)−𝑦^𝑇̇(𝑡)𝐵𝐵^𝑇𝑦(𝑡)=0. (3.26) By using Propositions 2.3 and 2.4, we have 2𝑦^𝑇̇(𝑡)𝑊₀𝑓(𝑥(𝑡))≤𝑦^𝑇̇(𝑡)𝑊₀𝐷₀𝑊^𝑇₀𝑦̇(𝑡)+𝑓^𝑇(𝑥(𝑡))𝐷₀⁻¹𝑓(𝑥(𝑡))≤𝑦^𝑇̇(𝑡)𝑊₀𝐷₀𝑊^𝑇₀𝑦̇(𝑡)+𝑦^𝑇(𝑡)𝑃𝐹𝐷₀⁻¹𝐹𝑃𝑦(𝑡),2𝑦^𝑇̇(𝑡)𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))≤𝑦^𝑇̇(𝑡)𝑊₁𝐷₁𝑊^𝑇₁𝑦̇(𝑡)+𝑔^𝑇(𝑥(𝑡−ℎ(𝑡)))𝐷₁⁻¹𝑔(𝑥(𝑡−ℎ(𝑡)))≤𝑦^𝑇̇(𝑡)𝑊₁𝐷₁𝑊^𝑇₁𝑦̇(𝑡)+𝑦^𝑇(𝑡−ℎ(𝑡))𝑃𝐺𝐷₁⁻¹𝐺𝑃𝑦(𝑡−ℎ(𝑡)),2𝑦^𝑇̇(𝑡)𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠≤𝑘𝑒^2𝛼𝑘𝑦^𝑇̇(𝑡)𝑊₂𝐷₂𝑊^𝑇₂𝑦̇(𝑡)+𝑒^{−2𝛼𝑘𝑡}_𝑡−𝑘ℎ^𝑇(𝑥(𝑠))𝐷₂⁻¹ℎ(𝑥(𝑠))𝑑𝑠. (3.27)(3.28)(3.29) From (3.15)–(3.29), we obtain 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡≤𝑦^𝑇(𝑡)−𝐴𝑃−𝑃𝐴^𝑇+2𝛼𝑃−𝐵𝐵^𝑇+2𝑄+2𝑘𝑃𝐻𝐷₂⁻¹𝐻𝑃+𝑊₀𝐷₀𝑊^𝑇₀+𝑊₁𝐷₁𝑊^𝑇₁+𝑘𝑒^2𝛼𝑘𝑊₂𝐷₂𝑊^𝑇₂−𝑒^−2𝛼ℎ1𝑅−𝑒^−2𝛼ℎ2𝑅+𝑃𝐹𝐷₁⁻¹𝐹𝑃𝑦(𝑡)+2𝑦^𝑇(𝑡)𝐵₀𝑃𝑦̇𝑡−𝜂(𝑡)+𝑦^𝑇𝑡−ℎ₁−𝑒^−2𝛼ℎ1𝑄−𝑒^−2𝛼ℎ1𝑅−𝑒^−2𝛼ℎ2𝑈𝑦𝑡−ℎ₁,+𝑡−ℎ₂−𝑒^−2𝛼ℎ2𝑄−𝑒^−2𝛼ℎ2𝑅−𝑒^−2𝛼ℎ2𝑈𝑦𝑡−ℎ₂,+𝑦^𝑇̇(𝑡)ℎ²₁𝑅+ℎ²₂𝑅+ℎ₂−ℎ₁^𝟐𝑈+𝑄₁−2𝑃+𝑊₀𝐷₀𝑊^𝑇₀+𝑊₁𝐷₁𝑊^𝑇₁+𝑘𝑒^2𝛼𝑘𝑊₂𝐷₂𝑊^𝑇₂(𝑡)−()𝑒𝑦̇1−𝛿^−2𝛼𝜂𝑦^𝑇̇𝑡−𝜂(𝑡)𝑄₁𝑦̇𝑡−𝜂(𝑡)+𝑦^𝑇(𝑡−ℎ(𝑡))2𝑃𝐺𝐷₁⁻¹𝐺𝑃−2𝑒^−2𝛼ℎ2𝑈𝑦(𝑡−ℎ(𝑡))+2𝑒^−2𝛼ℎ1𝑦^𝑇(𝑡)𝑅𝑦𝑡−ℎ₁+2𝑒^−2𝛼ℎ2𝑦^𝑇(𝑡)𝑅𝑦𝑡−ℎ₂+2𝑒^−2𝛼ℎ2𝑦^𝑇(𝑡−ℎ(𝑡))𝑈𝑦𝑡−ℎ₂+2𝑒^−2𝛼ℎ2𝑦^𝑇(𝑡−ℎ(𝑡))𝑈𝑦𝑡−ℎ₁−2𝑦^𝑇̇(𝑡)𝐴𝑃𝑦(𝑡)+2𝑦^𝑇̇(𝑡)𝐵₀𝑃𝑦̇𝑡−𝜂(𝑡)−𝑦^𝑇̇(𝑡)𝐵𝐵^𝑇𝑦(𝑡)−𝑒^−2𝛼ℎ2𝛽𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))−𝑒^−2𝛼ℎ21−𝛽𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂^𝑇𝑈𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂=𝜉^𝑇(𝑡)Ω𝜉(𝑡)−𝑒^−2𝛼ℎ2𝛽𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈×𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))−𝑒^−2𝛼ℎ21−𝛽𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂^𝑇×𝑈𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂=𝜉^𝑇(𝑡)1−𝛽ℳ₁+𝛽ℳ₂𝜉(𝑡)+𝑦^𝑇(𝑡)𝑀₃𝑦(𝑡)+𝑦^𝑇(𝑡−ℎ(𝑡))𝑀₄𝑦(𝑡−ℎ(𝑡)), (3.30) where 𝑀₃=−0.1𝐵𝐵^𝑇−0.1𝑒^−2𝛼ℎ1+𝑒^−2𝛼ℎ2𝑅+2𝑘𝑃𝐻𝐷₂⁻¹𝐻𝑃+2𝑃𝐹𝐷₀⁻¹𝑀𝐹𝑃,₄=−0.1𝑒^−2𝛼ℎ2𝑈+2𝑃𝐺𝐷₁⁻¹𝜁(𝑡)=𝑦(𝑡)(𝑡)𝐺𝑃,,𝑦̇𝑡−𝜂,𝑦𝑡−ℎ₁,𝑦𝑡−ℎ₂,𝑦̇(𝑡),𝑦(𝑡−ℎ(𝑡)). (3.31) Since 0≤𝛽≤1, (1−𝛽)ℳ₁+𝛽ℳ₂ is a convex combination of ℳ₁ and ℳ₂. Therefore, (1−𝛽)ℳ₁+𝛽ℳ₂<0 is equivalent to ℳ₁<0 and ℳ₂<0. Applying Schur complement lemma Proposition 2.5, the inequalities 𝑀₃<0 and 𝑀₄<0 are equivalent to ℳ₃<0 and ℳ₄<0, respectively. Thus, it follows from (3.3)–(3.6) and (3.30), we obtain 𝑉̇𝑡,𝑥_𝑡≤−2𝛼𝑉𝑡,𝑥_𝑡,∀𝑡≥0. (3.32) Integrating both sides of (3.32) from 0 to 𝑡, we obtain 𝑉𝑡,𝑥_𝑡≤𝑉𝜙𝑒^−2𝛼𝑡,∀𝑡≥0. (3.33) Furthermore, taking condition (3.13) into account, we have 𝜆₁𝑥𝑡,𝜙²≤𝑉𝑥_𝑡≤𝑉𝜙𝑒^−2𝛼𝑡≤𝜆₂𝑒^−2𝛼𝑡𝜙², (3.34) then 𝑥𝑡,𝜙≤𝜆₂𝜆₁𝑒^−𝛼𝑡𝜙,𝑡≥0. (3.35) Therefore, nominal system (3.1) is 𝛼-exponentially stabilizable. The proof is completed.

3.2. Exponential Stabilization for Interval Time-Varying Delay Systems

Based on Theorem 3.2, we derive robustly 𝛼-exponential stabilizability conditions of uncertain linear control systems with interval time-varying delay (2.1) in terms of LMIs.

Theorem 3.3. Given 𝛼>0. The system (2.1) is 𝛼-exponentially stabilizable if there exist symmetric positive definite matrices 𝑃,𝑄,𝑅,𝑈,𝑄1, three diagonal matrices 𝐷_𝑖,𝑖=0,1,2 and 𝜖_𝑖>0,𝑖=1,2,…,6 such that the following LMI holds: 𝒲₁=𝒲−000−𝐼0𝐼^𝑇×𝑒^−2𝛼ℎ2𝑈𝒲000−𝐼0𝐼<0,₂=𝒲−00𝐼00−𝐼^𝑇×𝑒^−2𝛼ℎ2𝑈𝒲00𝐼00−𝐼<0,₃=Δ4𝑘𝑃𝐻2𝑃𝐹𝑃𝐻^𝑇_𝑎𝑃𝐻^𝑇_𝑎𝑃𝐹𝐻^𝑇₀𝑃𝐹𝐻^𝑇₀∗−4𝑘𝐷₂00000∗∗−2𝐷₀0000∗∗∗−𝜖₁𝐼000∗∗∗∗−𝜖₄𝐼00∗∗∗∗∗−𝜖₂𝐼0∗∗∗∗∗∗−𝜖₅𝐼𝒲<0,₄=−0.1𝑒^−2𝛼ℎ2𝑈2𝑃𝐺𝑃𝐺𝐻^𝑇₁𝑃𝐺𝐻^𝑇₁∗−2𝐷₁00∗∗−𝜖₃𝐼0∗∗∗−𝜖₆𝐼𝑊<0,𝒲=¹¹𝑊₁₂𝑊₁₃𝑊₁₄𝑊₁₅0∗𝑊₂₂00𝑊₂₅0∗∗𝑊₃₃00𝑊₃₆∗∗∗𝑊₄₄0𝑊₄₆∗∗∗∗𝑊₅₅0∗∗∗∗∗𝑊₆₆, (3.36)(3.37)(3.38)(3.39) where Δ=−0.1𝐵𝐵^𝑇−0.1𝑒^−2𝛼ℎ1+𝑒^−2𝛼ℎ2𝑊𝑅,₁₁=[−𝐴+𝛼𝐼]𝑃+𝑃[−𝐴+𝛼𝐼]^𝑇−0.9𝐵𝐵^𝑇+2𝑄+𝑊₀𝐷₀𝑊^𝑇₀+𝑊₁𝐷₁𝑊^𝑇₁+𝑘𝑒^2𝛼𝑘𝑊₂𝐷₂𝑊^𝑇₂−0.9𝑒^−2𝛼ℎ1𝑅−0.9𝑒^−2𝛼ℎ2𝑅+𝜖₁𝐸^𝑇_𝑎𝐸_𝑎+𝜖₃𝐸^𝑇₁𝐸₁+𝜖₂𝐸^𝑇₀𝐸₀+𝑘𝑒^2𝛼𝑘𝐸^𝑇₂𝐻^𝑇₂𝐷₂𝐻₂𝐸₂,𝑊₁₂=𝐵₀𝑃,𝑊₁₃=𝑒^−2𝛼ℎ1𝑅,𝑊₁₄=𝑒^−2𝛼ℎ2𝑅,𝑊₁₅=−𝑃𝐴^𝑇−0.5𝐵𝐵^𝑇,𝑊₂₂=−(1−𝛿)𝑒^−2𝛼𝜂𝑄₁,𝑊₂₅=𝑃𝐵^𝑇₀,𝑊₃₃=−𝑒^−2𝛼ℎ1𝑄−𝑒^−2𝛼ℎ1𝑅−𝑒^−2𝛼ℎ2𝑊𝑈,₃₆=𝑒^−2𝛼ℎ2𝑈,𝑊₄₄=−𝑒^−2𝛼ℎ2𝑄−𝑒^−2𝛼ℎ2𝑅−𝑒^−2𝛼ℎ2𝑈,𝑊₄₆=𝑒^−2𝛼ℎ2𝑊𝑈,₅₅=ℎ²₁𝑅+ℎ²₂𝑅+ℎ₂−ℎ₁²𝑈+𝑄₁−2𝑃+𝑊₀𝐷₀𝑊^𝑇₀+𝑊₁𝐷₁𝑊^𝑇₁+𝑘𝑒^2𝛼𝑘𝑊₂𝐷₂𝑊^𝑇₂+𝜖₄𝐸^𝑇_𝑎𝐸_𝑎+𝜖₅𝐸^𝑇₀𝐸₀+𝜖₆𝐸^𝑇₁𝐸₁+𝑘𝑒^2𝛼𝑘𝐸^𝑇₂𝐻^𝑇₂𝐷₂𝐻₂𝐸₂,𝑊₆₆=−1.9𝑒^−2𝛼ℎ2𝑈. (3.40)

Proof. Choose Lyapunov-Krasovskii functional as in (3.11) but change 𝑉₈ to 𝑉₈=4∫⁰_−𝑘∫^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜏−𝑡)}ℎ^𝑇(𝑥(𝜏))𝐷₂⁻¹ℎ(𝑥(𝜏))𝑑𝜏𝑑𝑠, we may proof the Theorem by using a similar argument as in the proof of Theorem 3.2. By replacing 𝐴,𝑊₀,𝑊₁, and 𝑊₂ with 𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎,𝑊₀+𝐸₀𝐹₀(𝑡)𝐻₀,𝑊₁+𝐸₁𝐹₁(𝑡)𝐻₁, and 𝑊₂+𝐸₂𝐹₂(𝑡)𝐻₂, respectively. We have the following: 𝑉̇𝑡,𝑥_𝑡+2𝛼𝑉𝑡,𝑥_𝑡≤𝑦^𝑇(𝑡)−𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃+𝑃−𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎^𝑇−𝐵𝐵^𝑇+2𝛼𝑃+2𝑄−𝑒^−2𝛼ℎ1𝑅−𝑒^−2𝛼ℎ2𝑅𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑊₀+𝐸₀𝐹₀(𝑡)𝐻₀𝑓(𝑥(𝑡))+2𝑦^𝑇(𝑡)𝑊₁+𝐸₁𝐹₁(𝑡)𝐻₁𝑔(𝑥(𝑡−ℎ(𝑡)))+2𝑦^𝑇(𝑡)𝑊₂+𝐸₂𝐹₂(𝑡)𝐻₂×^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+2𝑦^𝑇(𝑡)𝐵₀𝑃𝑦̇𝑡−𝜂(𝑡)−𝑒^−2𝛼ℎ1𝑦^𝑇𝑡−ℎ₁𝑄𝑦𝑡−ℎ₁−𝑒^−2𝛼ℎ2𝑦^𝑇𝑡−ℎ₂𝑄𝑦𝑡−ℎ₂+ℎ²₁𝑦^𝑇̇(𝑡)𝑅𝑦̇(𝑡)+ℎ²₂𝑦^𝑇̇(𝑡)𝑅𝑦̇(𝑡)+ℎ₂−ℎ₁²𝑦^𝑇̇(𝑡)𝑈𝑦̇(𝑡)+𝑒^−2𝛼ℎ12𝑦^𝑇(𝑡)𝑅𝑦𝑡−ℎ₁−𝑒^−2𝛼ℎ1𝑦^𝑇𝑡−ℎ₁𝑅𝑦𝑡−ℎ₁+𝑒^−2𝛼ℎ22𝑦^𝑇(𝑡)𝑅𝑦𝑡−ℎ₂−𝑒^−2𝛼ℎ2𝑦^𝑇𝑡−ℎ₂𝑅𝑦𝑡−ℎ₂−𝑒^−2𝛼ℎ2𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂^𝑇𝑈𝑦𝑡−ℎ(𝑡)−𝑦𝑡−ℎ₂^𝑇−𝑒^−2𝛼ℎ2𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇−𝛽𝑒^−2𝛼ℎ2𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇𝑈𝑦𝑡−ℎ₁−𝑦(𝑡−ℎ(𝑡))^𝑇−𝑒^−2𝛼ℎ21−𝛽𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂^𝑇𝑈𝑦(𝑡−ℎ(𝑡))−𝑦𝑡−ℎ₂^𝑇+𝑦^𝑇̇(𝑡)𝑄₁𝑦̇(𝑡)−(1−𝛿)𝑒^−2𝛼𝜂𝑦^𝑇̇𝑡−𝜂(𝑡)𝑄₁𝑦̇𝑡−𝜂(𝑡)+4𝑘𝑦^𝑇(𝑡)𝑃𝐻𝐷₂⁻¹𝐻𝑃𝑦(𝑡)−4𝑒^{−2𝛼𝑘𝑡}_𝑡−𝑘ℎ^𝑇(𝑥(𝑠))𝐷₂⁻¹ℎ(𝑥(𝑠))𝑑𝑠−2𝑦^𝑇̇(𝑡)𝑃𝑦̇(𝑡)−2𝑦^𝑇̇(𝑡)𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)+2𝑦^𝑇̇(𝑡)𝑊₀+𝐸₀𝐹₀(𝑡)𝐻₀𝑓(𝑥(𝑡))+2𝑦^𝑇̇(𝑡)𝑊₁+𝐸₁𝐹₁(𝑡)𝐻₁𝑔(𝑥(𝑡−ℎ(𝑡)))+2𝑦^𝑇̇(𝑡)𝑊₂+𝐸₂𝐹₂(𝑡)𝐻₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+2𝑦^𝑇̇(𝑡)𝐵₀𝑃𝑦̇𝑡−𝜂(𝑡)−𝑦^𝑇̇(𝑡)𝐵𝐵^𝑇𝑦(𝑡)=𝜉^𝑇(𝑡)1−𝛽ℳ₁+𝛽ℳ₂𝜉(𝑡)+𝑦^𝑇(𝑡)𝑀₃𝑦(𝑡)+𝑦^𝑇(𝑡−ℎ(𝑡))𝑀₄𝑦(𝑡−ℎ(𝑡)). (3.41) Applying Proposition 2.3 and Lemma 2.6, the following estimations hold: 𝑦^𝑇(𝑡)−𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻)_𝑎𝑃+𝑃−𝐴^𝑇+𝐻^𝑇_𝑎𝐹^𝑇_𝑎(𝑡)𝐸^𝑇_𝑎𝑦(𝑡)≤𝑦^𝑇(𝑡)−𝑃𝐴^𝑇−𝐴𝑃𝑦(𝑡)+𝜖₁𝑦^𝑇(𝑡)𝐸^𝑇_𝑎𝐸_𝑎𝑦(𝑡)+𝜖₁⁻¹𝑦^𝑇(𝑡)𝑃𝐻^𝑇_𝑎𝐻_𝑎𝑃𝑦(𝑡),2𝑦^𝑇(𝑡)𝑊₀+𝐸₀𝐹₀(𝑡)𝐻₀𝑓(𝑥(𝑡))=2𝑦^𝑇(𝑡)𝑊₀𝑓(𝑥(𝑡))+2𝑦^𝑇(𝑡)𝐸₀𝐹₀(𝑡)𝐻₀𝑓(𝑥(𝑡))≤𝑦^𝑇(𝑡)𝑊₀𝐷₀𝑊^𝑇₀𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝐹𝐷₀⁻¹𝐹𝑃𝑦(𝑡)+𝜖₂𝑦^𝑇(𝑡)𝐸^𝑇₀𝐸₀𝑦(𝑡)+𝜖₂⁻¹𝑦^𝑇(𝑡)𝑃𝐹𝐻^𝑇₀𝐻₀𝐹𝑃𝑦(𝑡),2𝑦^𝑇(𝑡)𝑊₁+𝐸₁𝐹₁(𝑡)𝐻₁𝑔(𝑥(𝑡−ℎ(𝑡)))=2𝑦^𝑇(𝑡)𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))+2𝑦^𝑇(𝑡)𝐸₁𝐹₁(𝑡)𝐻₁𝑔(𝑥(𝑡−ℎ(𝑡)))≤𝑦^𝑇(𝑡)𝑊₁𝐷₁𝑊^𝑇₁𝑦(𝑡)+𝑦^𝑇(𝑡−ℎ(𝑡))𝑃𝐺𝐷₁⁻¹𝐺𝑃𝑦(𝑡−ℎ(𝑡))+𝜖₃𝑦^𝑇(𝑡)𝐸^𝑇₁𝐸₁𝑦(𝑡)+𝜖₃⁻¹𝑦^𝑇(𝑡−ℎ(𝑡))𝑃𝐺𝐻^𝑇₁𝐻₁𝐺𝑃𝑦^𝑇(𝑡−ℎ(𝑡)),2𝑦^𝑇(𝑡)𝑊₂+𝐸₂𝐹₂(𝑡)𝐻₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠=2𝑦^𝑇(𝑡)𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+2𝑦^𝑇(𝑡)𝐸₂𝐹₂(𝑡)𝐻₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠≤𝑘𝑒^2𝛼𝑘𝑦^𝑇(𝑡)𝑊₂𝐷₂𝑊^𝑇₂𝑦(𝑡)+𝑒^{−2𝛼𝑘𝑡}_𝑡−𝑘ℎ^𝑇(𝑥(𝑠))𝐷₂⁻¹ℎ(𝑥(𝑠))𝑑𝑠+𝑘𝑒^2𝛼𝑘𝑦^𝑇(𝑡)𝐸^𝑇₂𝐻₂𝐷₂𝐻^𝑇₂𝐸₂𝑦(𝑡)+𝑘⁻¹𝑒^{−2𝛼𝑘𝑡}_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠^𝑇𝐻^𝑇₂𝐻₂^−𝑇𝐷₂⁻¹𝐻₂⁻¹𝐻₂×^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠≤𝑘𝑒^2𝛼𝑘𝑦^T(𝑡)𝑊₂𝐷₂𝑊^𝑇₂𝑦(𝑡)+𝑒^{−2𝛼𝑘𝑡}_𝑡−𝑘ℎ^𝑇(𝑥(𝑠))𝐷₂⁻¹ℎ(𝑥(𝑠))𝑑𝑠+𝑘𝑒^2𝛼𝑘𝑦^𝑇(𝑡)𝐸^𝑇₂𝐻₂𝐷₂𝐻^𝑇₂𝐸₂𝑦(𝑡)+𝑒^{−2𝛼𝑘𝑡}_𝑡−𝑘ℎ^𝑇(𝑥(𝑠))𝐷₂⁻¹ℎ(𝑥(𝑠))𝑑𝑠−2𝑦^𝑇̇(𝑡)𝐴+𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)≤−2𝑦^𝑇̇(𝑡)𝐴𝑃𝑦(𝑡)−2𝑦^𝑇̇(𝑡)𝐸_𝑎𝐹_𝑎(𝑡)𝐻_𝑎𝑃𝑦(𝑡)≤−2𝑦^𝑇̇(𝑡)𝐴𝑃𝑦(𝑡)+𝜖₄𝑦^𝑇̇(𝑡)𝐸^𝑇_𝑎𝐸_𝑎𝑦̇(𝑡)+𝜖₄⁻¹𝑦^𝑇(𝑡)𝑃𝐻^𝑇_𝑎𝐻_𝑎𝑃𝑦(𝑡),2𝑦^𝑇̇(𝑡)𝑊₀+𝐸₀𝐹₀(𝑡)𝐻₀𝑓(𝑥(𝑡))=2𝑦^𝑇̇(𝑡)𝑊₀+𝐸₀𝐹₀(𝑡)𝐻₀𝑓(𝑥(𝑡))≤𝑦^𝑇̇(𝑡)𝑊₀𝐷₀𝑊^𝑇₀𝑦̇(𝑡)+𝑦^𝑇(𝑡)𝑃𝐹𝐷₀⁻¹𝐹𝑃𝑦(𝑡)+𝜖₅𝑦^𝑇̇(𝑡)𝐸^𝑇₀𝐸₀𝑦̇(𝑡)+𝜖₅⁻¹𝑦^𝑇(𝑡)𝑃𝐹𝐻^𝑇₀𝐻₀𝐹𝑃𝑦(𝑡),2𝑦^𝑇̇(𝑡)𝑊₁+𝐸₁𝐹₁(𝑡)𝐻₁𝑔(𝑥(𝑡−ℎ(𝑡)))≤𝑦^𝑇̇(𝑡)𝑊₁𝐷₁𝑊^𝑇₁𝑦̇(𝑡)+𝑦^𝑇(𝑡−ℎ(𝑡))𝑃𝐺𝐷₁⁻¹𝐺×𝑃𝑦(𝑡−ℎ(𝑡))+𝜖₆𝑦^𝑇̇(𝑡)𝐸^𝑇₁𝐸₁𝑦̇(𝑡)+𝜖₆⁻¹𝑦^𝑇(𝑡−ℎ(𝑡))𝑃𝐺𝐻^𝑇₁𝐻₁𝐺𝑃𝑦^𝑇(𝑡−ℎ(𝑡)),2𝑦^𝑇̇(𝑡)𝑊₂+𝐸₂𝐹₂(𝑡)𝐻₂≤𝑘𝑒^2𝛼𝑘𝑦^𝑇̇(𝑡)𝑊₂𝐷₂𝑊^𝑇₂𝑦̇(𝑡)+2𝑒^{−2𝛼𝑘𝑡}_𝑡−𝑘ℎ^𝑇(𝑥(𝑠))𝐷₂⁻¹ℎ(𝑥(𝑠))𝑑𝑠+𝑘𝑒^2𝛼𝑘𝑦^𝑇̇(𝑡)𝐸^𝑇₂𝐻^𝑇₂𝐷₂𝐻₂𝐸₂𝑦̇(𝑡). (3.42)

Remark 3.4. In [10, 13, 14], exponential stability of neutral-type neural networks with time-varying delays were investigated. However, the distributed delays have not been considered. Stability conditions in [13, 26–28] are not applicable to our work, since we consider more activation functions than them. Therefore, our stability conditions are less conservative than some other existing results.

Remark 3.5. In this paper, the restriction that the state delay is differentiable is not required which allows the state delay to be fast time-varying. Meanwhile, this restriction is required in some existing result, see [13, 14, 26–28].

4. Numerical Examples

In this section, we now provide an example to show the effectiveness of the result in Theorem 3.2.

Example 4.1. Consider the neural networks with interval time-varying delay and control input with the following parameters: 𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝑊₀𝑓(𝑥(𝑡))+𝑊₁𝑔(𝑥(𝑡−ℎ(𝑡)))+𝐵𝑢(𝑡), (4.1) where 𝐴=−0.2012,𝑊₀=0.40.10.1−0.2,𝑊₁=,.0.30.10.50.2,𝐹=0.3000.5𝐺=0.1000.4,𝐵=0.30.1 (4.2) It is worth noting that, the delay functions ℎ(𝑡)=0.1+0.1|sin𝑡|. Therefore, the methods used in [2, 9] are not applicable to this system. We have ℎ₁=0.1,ℎ₂=0.2. Given 𝛼=0.2 and any initial function 𝜙(𝑡)=𝐶¹([−0.2,0],ℝ²). Using the Matlab LMI toolbox, we obtain 𝑃=,0.03700.00100.00100.2938,𝑄=0.00080.00290.00290.0250,𝑈=0.01530.00800.0080.6201𝑅=0.03770.00550.00550.8173,𝐷₀=0.0353000.2833,𝐷₁=.0.0215000.5025 (4.3) Thus, the system (4.1) is 0.2-exponentially stabilizable and the value 𝜆₂/𝜆₁=1.6469, so the solution of the closed-loop system satisfies 𝑥𝑡,𝜙≤1.6469𝑒^−0.2𝑡𝜙,∀𝑡∈ℝ⁺. (4.4)

Example 4.2. Consider the neural networks with mixed interval time-varying delays and control input with the following parameters: 𝑥̇(𝑡)=−(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+𝑊₀+Δ𝑊₀𝑓(𝑥(𝑡))+𝑊₁+Δ𝑊₁𝑔(𝑥((𝑡+𝑊𝑡−ℎ)))₂+Δ𝑊₂^𝑡_{(𝑡)𝑡−𝑘}ℎ(𝑥(𝑠))𝑑𝑠+𝐵₀𝑥̇𝑡−𝜂(𝑡)+𝐵𝑢(𝑡), (4.5) where 𝐴=0.15001,𝑊₀=0.50.120.1−0.3,𝑊₁=0.20.10.10.2,𝑊₂=,𝐵0.10.20.50.1₀=,00.15000.15,𝐹=0.4000.5,𝐺=0.1000.2,𝐻=0.5000.3𝐵=0.1,𝐻_𝑎=𝐻₀=𝐻₁=𝐻₂=𝐸_𝑎=𝐸₀=𝐸₁=𝐸₂=.0.1000.1 (4.6) It is worth noting that, the delay functions ℎ(𝑡)=0.2+0.2|sin𝑡|,𝑘(𝑡)=|cos𝑡| are nondifferentiable and 𝜂(𝑡)=0.2sin²(𝑡). Therefore, the methods used in [13, 14] are not applicable to this system. We have ℎ₁=0.2,ℎ₂=0.4,𝑘=0.1,𝛿=0.1,𝜂=0.2. Given 𝛼=0.1 and any initial function 𝜙(𝑡)=𝐶¹([−0.4,0],ℝ²). Using the Matlab LMI toolbox, we obtain 𝑒₁=0.0173,𝑒₂=0.0128,𝑒₃=0.0111,𝑒₄=0.0263,𝑒₅=0.0209,𝑒₆=0.0192, 𝑃=0.00610.00020.00020.0228,𝑄=0.00030.00050.00010.0031,𝑄₁=,0.00050.00010.00010.0024𝑈=0.00280.00040.00040.00382,𝑅=0.00520.00080.00080.0543,𝐷₀=,𝐷0.0068000.0304₁=0.0038000.0145,𝐷₂=.0.0433000.0275 (4.7) Thus, the system (4.1) is 0.1-exponentially stabilizable and the value 𝜆₂/𝜆₁=2.2939, so the solution of the closed-loop system satisfies 𝑥𝑡,𝜙≤2.2939𝑒^−0.1𝑡𝜙,∀𝑡∈ℝ⁺. (4.8)

5. Conclusions

In this paper, we have investigated the exponential stabilization of neutral-type neural networks with various activation functions and interval nondifferentiable and distributed time-varying delays. The interval time-varying delay function is not necessary to be differentiable which allows time-delay function to be a fast time-varying function. By constructing a set of improved Lyapunov-Krasovskii functional combined with Leibniz-Newton's formula, the proposed stability criteria have been formulated in the form of a linear matrix inequalities. Numerical examples illustrate the effectiveness of the results.

Acknowledgments

Financial support from the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0355/2552) to W. Weera and P. Niamsup is acknowledged. The first author is also supported by the Graduate School, Chiang Mai University and the Center of Excellence in Mathematics. The second author is also supported by the the Center of Excellence in Mathematics, CHE, Thailand.