1.Introduction T.Botmart andW.Weera GuaranteedCostControlforExponentialSynchronizationofCellularNeuralNetworkswithMixedTime-VaryingDelaysviaHybridFeedbackControl ResearchArticle

Volume 2013, Article ID 175796,12pages http://dx.doi.org/10.1155/2013/175796

Research Article

Guaranteed Cost Control for Exponential Synchronization of Cellular Neural Networks with Mixed Time-Varying Delays via Hybrid Feedback Control

T. Botmart

and W. Weera

1Department of Mathematics, Srinakharinwirot University, Bangkok 10110, Thailand

2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

3Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand

Correspondence should be addressed to T. Botmart; [email protected] Received 30 November 2012; Revised 9 February 2013; Accepted 18 February 2013 Academic Editor: Yanni Xiao

Copyright © 2013 T. Botmart and W. Weera. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The problem of guaranteed cost control for exponential synchronization of cellular neural networks with interval nondifferentiable and distributed time-varying delays via hybrid feedback control is considered. The interval time-varying delay function is not necessary to be differentiable. Based on the construction of improved Lyapunov-Krasovskii functionals is combined with Leibniz- Newton’s formula and the technique of dealing with some integral terms. New delay-dependent sufficient conditions for the exponential synchronization of the error systems with memoryless hybrid feedback control are first established in terms of LMIs without introducing any free-weighting matrices. The optimal guaranteed cost control with linear error hybrid feedback is turned into the solvable problem of a set of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method.

1. Introduction

In the past decade, synchronization in neural networks (NNs), such as cellular NNs, Hopfield NNs, and bidirectional associative memory networks, has received a great deal of interest among scientists in a variety of areas, such as signal processing, pattern recognition, static image processing, associative memory, content-addressable memory and com- binatorial optimization [1–6]. In performing a periodicity or stability analysis of a neural network, the conditions to be imposed on the neural network are determined by the cha- racteristics of various activation functions and network para- meters. When neural networks are created for problem solv- ing, it is desirable for their activation functions not to be too restrictive. As a result, there has been considerable research work on the stability of neural networks with various activa- tion functions and more general conditions [7–9]. On the other hand, the problem of chaos synchronization has attract- ed a wide range of research activity in recent years. A chao- tic system has complex dynamical behaviors that possess

some special features, such as being extremely sensitive to tiny variations of initial conditions and having bounded trajectories in the phase space. The first concept of chaos syn- chronization making two chaotic systems oscillate in a syn- chronized manner was introduced by [2], and many different methods have been applied theoretically and experimentally to synchronize chaotic systems, for example, linear feedback method [10], active control [11], adaptive control [11, 12], impulsive control [13, 14], back-stepping design [15], time- delay feedback control [16] and intermittent control [17], sampled data control [18], and so forth.

The guaranteed cost control of uncertain systems was first put forward by Chang and Peng [19] and introduced by a lot of authors, which is to design a controller to robustly stabi- lize the uncertain system and guarantee an adequate level of performance. The guaranteed cost control approach has recently been extended to the neural networks with time delay (see [7,9,20–22] and references cited therein). In [7], author investigated the guaranteed cost control problem for a class of neural networks with various activation functions

and mixed time-varying delays in state and control. By using improved Lyapunov-Krasovskii functionals combined with LMIs technique. A delay-dependent criterion for existence of the guaranteed cost controller is derived in terms of LMIs.

Optimal guaranteed cost control for linear systems with mixed interval nondifferentiable time-varying delayed state and control has been studied in [20]. By constructing a set of augmented Lyapunov-Krasovskii functionals combined with Newton-Leibniz formula, the sufficient conditions for the existence of an optimal guaranteed cost state feedback for the system have been derived in terms of LMIs. Moreover, all this work has been developed for the guaranteed cost control synchronization of time-varying delay systems [21–24]. Based on the Lyapunov-Krasovskii analysis process and the zoned discussion and maximax synthesis (ZDMS) method, the quadratic matrix inequality (QMI) criterion for the guaran- teed cost synchronous controller is designed to synchronize the given neural networks with time-varying delay [21,22].

However, to the best of our knowledge, few published papers deal with the problem of guaranteed cost synchronization of cellular neural networks with time-varying delay by using feedback control. So, our paper presents cellular neural networks with various activation functions and mixed time- varying delays and we also approach to establish both delay and nondelay controllers to the system.

It is known that exponential stability is a more favorite property than asymptotic stability since it gives a faster con- vergence rate to the equilibrium point and any information about the decay rates of the delayed neural networks. There- fore it is particularly important when the exponential conver- gence rate is used to determine the speed of neural compu- tations. The exponential stability property guarantees that, whatever transformation occurs, the network’s ability to store rapidly the activity pattern is left invariant by self-organiza- tion. Thus, it is important to determine the exponential stability and to estimate the exponential convergence rate for delayed neural networks. Recently, exponential synchroniza- tion of neural networks has been widely investigated and many effective methods have been presented by [25–33].

A synchronization scheme for a class of delayed neural networks with time-varying delays based on the Lyapunov functional method and Hermitian matrices theory is derived in [25]. In [26,27], authors presented sufficient conditions for the exponential synchronization of neural networks with time-varying delays in terms of the feasible solution to the LMIs.

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-depend- ent criteria make use of such information at different levels.

Delay-dependent stability conditions are generally less con- servative than delay-independent ones especially when the delay is small. Recently, delay-dependent stability for interval time-varying delay was investigated in [8,9,34–37]. Interval time-varying delay is a time delay that varies in an interval in which the lower bound is not restricted to be 0. Tian and Zhou [36] considered the delay-dependent asymptotic stabi- lity criteria for neural networks (NNs) with time-varying

interval delay. By introducing a novel Lyapunov functional stability criteria of asymptotic stability is derived in terms of LMIs with adding the term free-weighting matrix. Delay- dependent robust exponential stabilization criteria for inter- val time-varying delay systems are proposed in [37], by using Lyapunov-Krasovskii functionals combined with the free- weighting matrices. It is noted that the former has more matrix variables than our result. Therefore, our result has less conservative and matrix variables than [36, 37]. Moreover, neural networks with distributed delays have been extensively discussed [29–33, 38–41]. In [38, 39, 41], a neural circuit has been designed with distributed delays, which solves the general problem of recognized patterns in a time-dependent signal. The master-slave synchronization problem has been investigated for neural networks with discrete and distributed time-varying delays in [29,30]; based on the drive-response concept, LMI approach, and the Lyapunov stability theorem, several delay-dependent feedback controllers were derived to achieve the exponential synchronization of the chaotic neural networks. In [33], by constructing proper Lyapunov- Krasovskii functional and employing a combination of the free-weighting matrix method, Leibniz-Newton, formulation and inequality technique, the feedback controllers were de- rived to ensure the asymptotical and exponential synchroni- zation of the addressed neural networks.

However, It is worth pointing out that the given criteria in [21,22] still have been based on the following conditions:

(1) the time-varying delays are continuously differentiable;

(2) the derivative of time-varying delay is bounded; (3) the time-varying delays with the lower bound are restricted to be 0. However, in most cases, these conditions are dif- ficult to satisfy. Therefore, in this paper we will employ some new techniques so that the above conditions can be removed. To the best of our knowledge, the guaranteed cost synchronization problem of the cellular neural networks with nondifferentiable interval time-varying discrete and dis- tributed delays and various activation functions is seldom discussed in terms of LMIs, which remains important and challenging.

In this paper, inspired by the above discussions, we con- sider the problem of guaranteed cost control for exponential synchronization of cellular neural networks with interval nondifferentiable and distributed time-varying delays via hybrid feedback control. There are various activation func- tions which are considered in the system, and the restric- tion 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 for- mula and the technique of dealing with some integral terms, new delay-dependent sufficient conditions for the expo- nential stabilization of the memoryless feedback controls are first established of LMIs without introducing any free- weighting matrices. The optimal guaranteed cost control with linear error hybrid feedback is turned into the solvable pro- blem of a set of LMIs. The new stability condition is much less conservative and more general than some existing results. A numerical example is also given to illustrate the effectiveness of the proposed method.

The rest of this paper is organized as follows. InSection 2, we give notations, definition, propositions, and lemma for using in the proof of the main results. Delay-dependent sufficient conditions of guaranteed cost control for exponen- tial synchronization of cellular neural networks with vari- ous activation functions and interval and distributed time- varying delays with memoryless hybrid feedback controls are presented inSection 3. Numerical examples illustrating the obtained results are given inSection 4. The paper ends with conclusions inSection 5and 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 the𝑛- dimensional space and the vector norm‖ ⋅ ‖;𝑀^𝑛×𝑟 denotes the space of all matrices of(𝑛 × 𝑟)-dimensions.

𝐴^𝑇denotes the transpose of matrix𝐴;𝐴is symmetric if 𝐴 = 𝐴^𝑇;𝐼denotes the identity matrix;𝜆(𝐴)denotes the set of all eigenvalues of𝐴;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 allR^𝑛-valued continuous functions on[0, 𝑡];𝐿₂([0, 𝑡],R^𝑚)denotes the set of all theR^𝑚- valued square integrable functions on[0, 𝑡].

Matrix 𝐴 is called semipositive definite (𝐴 ≥ 0) if

⟨𝐴𝑥, 𝑥⟩ ≥ 0for all𝑥 ∈ R^𝑛; 𝐴is positive definite(𝐴 > 0) if⟨𝐴𝑥, 𝑥⟩ > 0for all𝑥 ̸= 0;𝐴 > 𝐵means𝐴 − 𝐵 > 0. The sym- metric term in a matrix is denoted by∗.

In this paper, the master-slave cellular neural networks (MSCNNs) with mixed time-varying delays are described as follows:

̇𝑥 (𝑡) = − 𝐴𝑥 (𝑡) + 𝐶 ̃𝑓 (𝑥 (𝑡)) + 𝐷 ̃𝑔 (𝑥 (𝑡 − ℎ₁(𝑡))) + 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)̃ℎ (𝑥 (𝑠)) 𝑑 𝑠 + 𝐼 (𝑡) , 𝑥 (𝑡) = 𝜙₁(𝑡) , 𝑡 ∈ [−𝑑, 0] ,

(1)

̇𝑦 (𝑡) = − 𝐴𝑦 (𝑡) + 𝐶 ̃𝑓 (𝑦 (𝑡)) + 𝐷 ̃𝑔 (𝑦 (𝑡 − ℎ₁(𝑡))) + 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)̃ℎ (𝑦 (𝑠)) 𝑑 𝑠 + 𝐼 (𝑡) +U(𝑡) , 𝑦 (𝑡) = 𝜙₂(𝑡) , 𝑡 ∈ [𝑑 , 0] ,

(2)

where𝑥(𝑡) = [𝑥₁(𝑡), 𝑥₂(𝑡), . . . , 𝑥_𝑛(𝑡)] ∈R^𝑛and𝑦(𝑡) = [𝑦₁(𝑡), 𝑦₂(𝑡), . . . , 𝑦_𝑛(𝑡)] ∈ R^𝑛 are the master system’s state vector and the slave system’s state vector of the neural networks, respectively.𝑛is the number of neural,

𝑓 (𝑥 (𝑡)) = [ ̃̃ 𝑓₁(𝑥₁(𝑡)) , ̃𝑓₂(𝑥₂(𝑡)) , . . . , ̃𝑓_𝑛(𝑥_𝑛(𝑡))]^𝑇,

̃𝑔(𝑥 (𝑡)) = [ ̃𝑔₁(𝑥₁(𝑡)) , ̃𝑔₂(𝑥₂(𝑡)) , . . . , ̃𝑔_𝑛(𝑥_𝑛(𝑡))]^𝑇,

̃ℎ (𝑥 (𝑡)) = [̃ℎ₁(𝑥₁(𝑡)) , ̃ℎ₂(𝑥₂(𝑡)) , . . . , ̃ℎ_𝑛(𝑥_𝑛(𝑡))]^𝑇, (3)

are the activation functions,𝐴 =diag(𝑎₁, 𝑎₂, . . . , 𝑎_𝑛),𝑎_𝑖 > 0 represents the self-feedback term, and𝐶, 𝐷, and 𝐸denote the connection weights, the discretely delayed connection

weights, and the distributively delayed connection weight, respectively.

The synchronization error𝑒(𝑡)is the form𝑒(𝑡) = 𝑦(𝑡) − 𝑥(𝑡). Therefore, the cellular neural networks with mixed time- varying delays of synchronization error between the master- slave systems given in (1) and (2) can be described by

̇𝑒(𝑡) = − 𝐴𝑒 (𝑡) + 𝐶𝑓 (𝑒 (𝑡)) + 𝐷𝑔 (𝑒 (𝑡 − ℎ₁(𝑡))) + 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑 𝑠 +U(𝑡) , 𝑒 (𝑡) = 𝜙₂(𝑡) − 𝜙₁(𝑡) = 𝜙 (𝑡) , 𝑡 ∈ [−𝑑, 0] ,

(4)

where𝑓(𝑒(𝑡)) = ̃𝑓(𝑒(𝑡) + 𝑥(𝑡)) − ̃𝑓(𝑥(𝑡)),𝑔(𝑒(𝑡 − ℎ₁(𝑡))) =

̃𝑔(𝑒(𝑡 − ℎ₁(𝑡)) + 𝑥(𝑡 − ℎ₁(𝑡))) − ̃𝑔(𝑥(𝑡 − ℎ₁(𝑡))), and

∫_𝑡−𝑘^𝑡

1(𝑡)ℎ(𝑒(𝑠))𝑑𝑠 = ∫_𝑡−𝑘^𝑡

1(𝑡)ℎ(𝑒(𝑠) + 𝑥(𝑠)) − ℎ(𝑥(𝑠))𝑑𝑠. The state hybrid feedback controllerU(𝑡)satisfies(H1):

(H1):U(𝑡) = 𝐵₁𝑢 (𝑡) + 𝐵₂𝑢 (𝑡 − ℎ₂(𝑡)) + 𝐵₃∫^𝑡

𝑡−𝑘2(𝑡)𝑢 (𝑠) 𝑑𝑠, ∀𝑡 ≥ 0, (5) where𝑢(𝑡) = 𝐾𝑒(𝑡)and𝐾is a constant matrix control gain. In this paper, our goal is to design suitable𝐾such that system (2) synchronizes with system (1). Then, substituting it into (4), it is easy to get the following:

̇𝑒(𝑡) = − 𝐴𝑒 (𝑡) + 𝐶𝑓 (𝑒 (𝑡)) + 𝐷𝑔 (𝑒 (𝑡 − ℎ₁(𝑡))) + 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑 𝑠 + 𝐵₁𝐾𝑒 (𝑡) + 𝐵₂𝐾𝑒 (𝑡 − ℎ₂(𝑡)) + 𝐵₃𝐾 ∫^𝑡

𝑡−𝑘₂(𝑡)𝑒 (𝑠) 𝑑𝑠, ∀𝑡 ≥ 0,

𝑒 (𝑡) = 𝜙₂(𝑡) − 𝜙₁(𝑡) = 𝜙 (𝑡) , 𝑡 ∈ [−𝑑, 0] .

(6) Throughout this paper, we consider various activation func- tions and the activation functions𝑓(⋅),̃ ̃𝑔(⋅), and̃ℎ(⋅)satisfy the following assumption.

(A1) The activation functions𝑓(⋅),̃ ̃𝑔(⋅), and̃ℎ(⋅)satisfy Lipschitzian with the Lipschitz constants𝑓̂_𝑖, ̂𝑔_𝑖> 0, and̂ℎ_𝑖>

0:

󵄨󵄨󵄨󵄨󵄨𝑓̃_𝑖(𝜉₁) − ̃𝑓_𝑖(𝜉₂)󵄨󵄨󵄨󵄨󵄨 ≤ ̂𝑓^𝑖󵄨󵄨󵄨󵄨𝜉¹− 𝜉₂󵄨󵄨󵄨󵄨, 𝑖 = 1,2,...,𝑛,

∀𝜉₁, 𝜉₂∈R,

󵄨󵄨󵄨󵄨̃𝑔^𝑖(𝜉₁) − ̃𝑔_𝑖(𝜉₂)󵄨󵄨󵄨󵄨 ≤ ̂𝑔^𝑖󵄨󵄨󵄨󵄨𝜉¹− 𝜉₂󵄨󵄨󵄨󵄨, 𝑖 = 1,2,...,𝑛,

∀𝜉₁, 𝜉₂∈R,

󵄨󵄨󵄨󵄨󵄨̃ℎ_𝑖(𝜉₁) − ̃ℎ_𝑖(𝜉₂)󵄨󵄨󵄨󵄨󵄨 ≤ ̂ℎ^𝑖󵄨󵄨󵄨󵄨𝜉¹− 𝜉₂󵄨󵄨󵄨󵄨, 𝑖 = 1,2,...,𝑛,

∀𝜉₁, 𝜉₂∈R, (7)

and we denote that

𝐹 =diag{ ̂𝑓_𝑖, 𝑖 = 1, 2, . . . , 𝑛} , 𝐺 =diag{ ̂𝑔_𝑖, 𝑖 = 1, 2, . . . , 𝑛} , 𝐻 =diag{̂ℎ_𝑖, 𝑖 = 1, 2, . . . , 𝑛} .

(8)

The time-varying delay functionsℎ_𝑖(𝑡)and𝑘_𝑖(𝑡), 𝑖 = 1, 2, satisfy the condition

0 ≤ ℎ_1𝑚≤ ℎ₁(𝑡) ≤ ℎ_1𝑀, 0 ≤ ℎ₂(𝑡) ≤ ℎ₂,

0 ≤ 𝑘₁(𝑡) ≤ 𝑘₁, 0 ≤ 𝑘₂(𝑡) ≤ 𝑘₂. (9) 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),𝑑 =max{ℎ_1𝑀, ℎ₂, 𝑘₁, 𝑘₂}, with the norm

󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩 = sup

𝑡∈[−𝑑,0]√󵄩󵄩󵄩󵄩𝜙(𝑡)󵄩󵄩󵄩󵄩²+ 󵄩󵄩󵄩󵄩󵄩 ̇𝜙(𝑡)󵄩󵄩󵄩󵄩󵄩². (10) Define the following quadratic cost function of the asso- ciated system (4) as follows:

𝐽 = ∫^∞

0 [𝑒^𝑇(𝑡) 𝑄₁𝑒 (𝑡) + 𝑢^𝑇(𝑡) 𝑄₂𝑢 (𝑡)] 𝑑 𝑡, (11) where𝑄₁ ∈ R^𝑛×𝑛and𝑄₂ ∈ R^𝑚×𝑚are positive definite mat- rices.

Remark 1. If𝐸 = 0,𝐵₂ = 0,𝐵₃ = 0, and𝑓(⋅) = 𝑔(⋅), the system model (6) turns into the cellular neural networks with activation functions and time-varying delays proposed by [21, 22]

̇𝑒(𝑡) = −𝐴𝑒 (𝑡) + 𝐶𝑓 (𝑒 (𝑡)) + 𝐷𝑓 (𝑒 (𝑡 − ℎ₁(𝑡))) + 𝐵₁𝐾𝑒 (𝑡) ,

∀𝑡 ≥ 0.

(12) Therefore, (6) is a general cellular neural networks model, with (12) as the special case.

Definition 2. Given𝛼 > 0, the zero solution of system (6) with𝑢(𝑡) = 𝐾𝑒(𝑡)is𝛼-stable if there exists a positive number 𝑁 > 0such that every solution𝑒(𝑡, 𝜙)satisfies the following condition:

󵄩󵄩󵄩󵄩𝑒(𝑡,𝜙)󵄩󵄩󵄩󵄩 ≤ 𝑁𝑒^−𝛼𝑡󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩, ∀𝑡 ≥ 0. (13) We introduce the following technical well-known propo- sitions and lemma, which will be used in the proof of our results.

Proposition 3 (see [42] (Cauchy inequality)). For any sym- metric positive definite matrix𝑁 ∈ 𝑀^𝑛×𝑛and𝑥, 𝑦 ∈ R^𝑛, we have

±2𝑥^𝑇𝑦 ≤ 𝑥^𝑇𝑁𝑥 + 𝑦^𝑇𝑁⁻¹𝑦. (14)

Proposition 4 (see [42]). For any symmetric positive definite matrix𝑀 > 0, scalar𝛾 > 0, and vector function𝜔 : [0, 𝛾] → R^𝑛 such that the integrations concerned are well defined, the following inequality holds:

(∫^𝛾

0 𝜔 (𝑠) 𝑑𝑠)^𝑇𝑀(∫^𝛾

0 𝜔 (𝑠) 𝑑𝑠)≤𝛾 (∫^𝛾

0 𝜔^𝑇(𝑠) 𝑀𝜔 (𝑠) 𝑑𝑠) . (15) Proposition 5 (see [42] (Schur complement lemma)). Given constant symmetric matrices 𝑋,𝑌, and𝑍 with appropriate dimensions satisfying 𝑋 = 𝑋^𝑇, 𝑌 = 𝑌^𝑇 > 0, then 𝑋 + 𝑍^𝑇𝑌⁻¹𝑍 < 0if and only if

(𝑋 𝑍𝑍 −𝑌) < 0^𝑇 or (−𝑌 𝑍

𝑍^𝑇 𝑋) < 0. (16)

3. Main Results

Let us set

𝜆₁= 𝜆_min(𝑃⁻¹) ,

𝜆₂= 𝜆_max(𝑃⁻¹) + (ℎ_1𝑚+ ℎ_1𝑀) 𝜆_max(𝑃⁻¹𝑄𝑃⁻¹) + (ℎ³_1𝑚+ ℎ³_1𝑀) 𝜆_max(𝑃⁻¹𝑅𝑃⁻¹)

+ 𝛿³𝜆_max(𝑃⁻¹𝑈𝑃⁻¹) + ℎ³₂𝜆_max(𝑃⁻¹𝑌^𝑇𝑆⁻¹₁ 𝑌𝑃⁻¹) + 𝑘²₁𝜆_max(𝐻𝑈₃⁻¹𝐻) + 𝑘²₂𝜆_max(𝑃⁻¹𝑌^𝑇𝑆⁻¹₂ 𝑌𝑃⁻¹) .

(17)

Theorem 6. Given𝛼 > 0,𝑄₁ > 0and𝑄₂ > 0,𝑢(𝑡) = 𝐾𝑒(𝑡) is a guaranteed cost controller if there exist symmetric positive definite matrices 𝑃, 𝑄, 𝑅, 𝑈, 𝑆₁, and𝑆₂, diagonal matrices 𝑈_𝑖, 𝑖 = 1, 2, 3, and a matrix𝑌with appropriately dimensioned such that the following LMIs holds:

Γ₁= Γ − [0 0 −𝐼 𝐼 0]^𝑇× 𝑒^{−2𝛼ℎ1𝑀}𝑈 [0 0 −𝐼 𝐼 0] < 0, (18) Γ₂= Γ − [0 0 0 𝐼 −𝐼]^𝑇× 𝑒^{−2𝛼ℎ1𝑀}𝑈 [0 0 0 𝐼 −𝐼] < 0,

(19) Γ₃=

[[ [[ [

−0.1 (𝑒^{−2𝛼ℎ1𝑚}+ 𝑒^{−2𝛼ℎ1𝑀}) 𝑅 2𝑃𝐹^𝑇 𝑃𝐻^𝑇 2𝑌^𝑇 𝑃𝑄₁ 𝑌^𝑇𝑄₂

∗ −2𝑈₁ 0 0 0 0

∗ ∗ −𝑈₃ 0 0 0

∗ ∗ ∗ −2𝑒^−2𝛼ℎ2𝑆₁ 0 0

∗ ∗ ∗ ∗ −𝑄₁ 0

∗ ∗ ∗ ∗ ∗ −𝑄₂

]] ]] ]

< 0,

(20) Γ₄= [

[

−0.1𝑃 ℎ²₂𝑌^𝑇

∗ −ℎ²₂𝑆₁] ]

< 0, (21)

Γ₅= [−0.1𝑒^{−2𝛼ℎ1𝑀}𝑈 2𝑃𝐺^𝑇

∗ −2𝑈₂] < 0,

Γ = [[ [[ [ [

Γ₁₁ Γ₁₂ Γ₁₃ 0 Γ₁₅

∗ Γ₂₂ 0 0 0

∗ ∗ Γ₃₃ Γ₃₄ 0

∗ ∗ ∗ Γ₄₄ Γ₄₅

∗ ∗ ∗ ∗ Γ₅₅

]] ]] ] ] ,

(22)

where

Γ₁₁= [−𝐴 + 𝛼𝐼] 𝑃 + 𝑃[−𝐴 + 𝛼𝐼]^𝑇− 𝐵𝑌 − 𝑌^𝑇𝐵^𝑇 + 2𝑄 + 𝐶^𝑇𝑈₁𝐶 + 𝐷^𝑇𝑈₂𝐷 + 𝑘₁𝑒^2𝛼𝑘1𝐸^𝑇𝑈₃𝐸 + 3𝑒^2𝛼ℎ2𝐵^𝑇₂𝑆₁𝐵₂+ 2𝑘₂𝑒^2𝛼𝑘2𝐵^𝑇₃𝑆₂𝐵₃

− 0.9𝑒^{−2𝛼ℎ1𝑚}𝑅 − 0.9𝑒^{−2𝛼ℎ1𝑀}𝑅, Γ₁₂= −𝑃𝐴^𝑇− 𝑌^𝑇𝐵^𝑇,

Γ₁₃= 𝑒^{−2𝛼ℎ1𝑚}𝑅, Γ₁₅= 𝑒^{−2𝛼ℎ1𝑀}𝑅,

Γ₂₂= ℎ²_1𝑚𝑅 + ℎ²_1𝑀𝑅 + 𝛿²𝑈 − 1.9𝑃

+ 𝐶^𝑇𝑈₁𝐶 + 𝐷^𝑇𝑈₂𝐷 + 2𝑘₁𝑒^2𝛼𝑘1𝐸^𝑇𝑈₃𝐸 + 3𝑒^2𝛼ℎ2𝐵^𝑇₂𝑆₁𝐵₂+ 2𝑘₂𝑒^2𝛼𝑘2𝐵^𝑇₃𝑆₂𝐵₃ Γ₃₃= −𝑒^{−2𝛼ℎ1𝑚}𝑄 − 𝑒^{−2𝛼ℎ1𝑚}𝑅 − 𝑒^{−2𝛼ℎ1𝑀}𝑈, Γ₃₄= 𝑒^{−2𝛼ℎ1𝑀}𝑈,

Γ₄₄= −1.9𝑒^{−2𝛼ℎ1𝑀}𝑈, Γ₄₅= 𝑒^{−2𝛼ℎ1𝑀}𝑈,

Γ₅₅= −𝑒^{−2𝛼ℎ1𝑀}𝑄 − 𝑒^{−2𝛼ℎ1𝑀}𝑅 − 𝑒^{−2𝛼ℎ1𝑀}𝑈,

(23) then the error system(6)is exponentially stabilizable. More- over, the feedback control is

𝑢 (𝑡) = −𝑌𝑃⁻¹𝑒 (𝑡) , 𝑡 ≥ 0, (24) and the upper bound of the cost function(11)is as follows:

𝐽 ≤ 𝐽^∗= 𝜆₂󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩². (25) Proof. Let𝑊 = 𝑃⁻¹and let𝑧(𝑡) = 𝑊𝑒(𝑡). Using the feedback control (24), we consider the following Lyapunov-Krasovskii functional:

𝑉 (𝑡, 𝑒 (𝑡)) =∑⁹

𝑖=1

𝑉_𝑖, (26)

where

𝑉₁= 𝑒^𝑇(𝑡) 𝑊𝑒 (𝑡) , 𝑉₂= ∫^𝑡

𝑡−ℎ1𝑚

𝑒^{2𝛼(𝑠−𝑡)}𝑒^𝑇(𝑠) 𝑊𝑄𝑊𝑒 (𝑠) 𝑑𝑠,

𝑉₃= ∫^𝑡

𝑡−ℎ1𝑀

𝑒^{2𝛼(𝑠−𝑡)}𝑒^𝑇(𝑠) 𝑊𝑄𝑊𝑒 (𝑠) 𝑑𝑠,

𝑉₄= ℎ_1𝑚∫⁰

−ℎ1𝑚

∫^𝑡

𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡) 𝑇}̇𝑒 (𝜃) 𝑊𝑅𝑊 ̇𝑒(𝜃) 𝑑 𝜃 𝑑 𝑠, 𝑉₅= ℎ_1𝑀∫⁰

−ℎ_1𝑀∫^𝑡

𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡) 𝑇}̇𝑒 (𝜃) 𝑊𝑅𝑊 ̇𝑒(𝜃) 𝑑 𝜃 𝑑 𝑠, 𝑉₆= 𝛿 ∫^−ℎ1𝑚

−ℎ_1𝑀 ∫^𝑡

𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡) 𝑇}̇𝑒 (𝜃) 𝑊𝑈𝑊 ̇𝑒(𝜃) 𝑑 𝜃 𝑑 𝑠, 𝑉₇= ℎ₂∫⁰

ℎ₂∫^𝑡

𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡)} ̇𝑢^𝑇(𝜃) 𝑆₁⁻¹ ̇𝑢 (𝜃) 𝑑 𝜃 𝑑 𝑠, 𝑉₈= ∫⁰

−𝑘1

∫^𝑡

𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡)}ℎ^𝑇(𝑒 (𝜃)) 𝑈₃⁻¹ℎ (𝑒 (𝜃)) 𝑑 𝜃 𝑑 𝑠, 𝑉₉= ∫⁰

−𝑘2

∫^𝑡

𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡)}𝑢^𝑇(𝜃) 𝑆⁻¹₂ 𝑢 (𝜃) 𝑑 𝜃 𝑑 𝑠.

(27)

It is easy to check that

𝜆₁‖𝑒 (𝑡)‖²≤ 𝑉 (𝑡, 𝑒 (𝑡)) ≤ 𝜆₂󵄩󵄩󵄩󵄩𝑒^𝑡(𝑡)󵄩󵄩󵄩󵄩², ∀𝑡 ≥ 0. (28) Taking the derivative of𝑉(𝑡, 𝑒(𝑡))along the solution of system (6), we have

̇𝑉1= 2𝑒^𝑇(𝑡) 𝑊 ̇𝑒(𝑡)

= 2𝑧^𝑇(𝑡) [ − 𝐴𝑒 (𝑡) + 𝐶𝑓 (𝑒 (𝑡)) + 𝐷𝑔 (𝑒 (𝑡 − ℎ₁(𝑡)))

+ 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑 𝑠 − 𝐵₁𝑌𝑃⁻¹𝑒 (𝑡) +𝐵₂(𝑡) 𝐾𝑒 (𝑡 − ℎ₂(𝑡)) + 𝐵₃𝐾 ∫^𝑡

𝑡−𝑘2(𝑡)𝑒 (𝑠) 𝑑𝑠]

= 𝑧^𝑇(𝑡) [−𝐴𝑃 − 𝑃𝐴^𝑇− 2𝐵₁𝑌] 𝑧 (𝑡) + 2𝑧^𝑇(𝑡) 𝐶𝑓 (𝑒 (𝑡)) + 2𝑧^𝑇(𝑡) 𝐷𝑔 (𝑒 (𝑡 − ℎ₁(𝑡)))

+ 2𝑧^𝑇(𝑡) 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑𝑠 + 2𝑧^𝑇(𝑡) 𝐵₂𝑢 (𝑡 − ℎ₂(𝑡)) + 2𝑧^𝑇(𝑡) 𝐵₃∫^𝑡

𝑡−𝑘₂(𝑡)𝑢 (𝑠) 𝑑𝑠,

̇𝑉2= 𝑧^𝑇(𝑡) 𝑄𝑧 (𝑡) − 𝑒^{−2𝛼ℎ1𝑚}𝑧^𝑇(𝑡 − ℎ_1𝑚) 𝑄𝑧 (𝑡 − ℎ_1𝑚)

− 2𝛼𝑉₂,

̇𝑉3= 𝑧^𝑇(𝑡) 𝑄𝑧 (𝑡) − 𝑒^{−2𝛼ℎ1𝑀}𝑧^𝑇(𝑡 − ℎ_1𝑀) 𝑄𝑧 (𝑡 − ℎ_1𝑀)

− 2𝛼𝑉₃,

̇𝑉4≤ ℎ²_1𝑚 ̇𝑧^𝑇(𝑡) 𝑅 ̇𝑧 (𝑡) − ℎ_1𝑚𝑒^{−2𝛼ℎ1𝑚}∫^𝑡

𝑡−ℎ1𝑚

̇𝑧𝑇(𝑠) 𝑅 ̇𝑧 (𝑠) 𝑑𝑠

− 2𝛼𝑉₄,

̇𝑉5≤ ℎ²_1𝑀 ̇𝑧^𝑇(𝑡) 𝑅 ̇𝑧 (𝑡) − ℎ_1𝑀𝑒^{−2𝛼ℎ1𝑀}∫^𝑡

𝑡−ℎ_1𝑀 ̇𝑧^𝑇(𝑠) 𝑅 ̇𝑧 (𝑠) 𝑑𝑠

− 2𝛼𝑉₅,

̇𝑉6≤ 𝛿² ̇𝑧^𝑇(𝑡) 𝑈 ̇𝑧 (𝑡) − 𝛿𝑒^{−2𝛼ℎ1𝑀}∫^{𝑡−ℎ1𝑚}

𝑡−ℎ1𝑀

̇𝑧𝑇(𝑠) 𝑅 ̇𝑧 (𝑠) 𝑑𝑠

− 2𝛼𝑉₆,

̇𝑉7≤ ℎ²₂ ̇𝑢^𝑇(𝑡) 𝑆⁻¹₁ ̇𝑢 (𝑡) − ℎ₂𝑒^−2𝛼ℎ2∫^𝑡

𝑡−ℎ2

̇𝑢^𝑇(𝑠) 𝑆⁻¹₁ ̇𝑢 (𝑠) 𝑑𝑠

− 2𝛼𝑉₇,

̇𝑉8≤ 𝑘₁ℎ^𝑇(𝑒 (𝑡)) 𝑈₃⁻¹ℎ (𝑒 (𝑡))

− 𝑒^{−2𝛼𝑘1}∫^𝑡

𝑡−𝑘1

ℎ^𝑇(𝑒 (𝑠)) 𝑈₃⁻¹ℎ (𝑒 (𝑠)) 𝑑𝑠 − 2𝛼𝑉₈,

̇𝑉9≤ 𝑘²₂𝑢^𝑇(𝑡) 𝑆⁻¹₂ 𝑢 (𝑡) − 𝑘₂𝑒^{−2𝛼𝑘2}∫^𝑡

𝑡−𝑘2

𝑢^𝑇(𝑠) 𝑆⁻¹₂ 𝑢 (𝑠) 𝑑𝑠

− 2𝛼𝑉₉.

(29) For assumption (A1), we can obtain the following three inequalities:

󵄨󵄨󵄨󵄨𝑓^𝑖(𝑒_𝑖(𝑡))󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨󵄨𝑓̃_𝑖(𝑒_𝑖(𝑡) + 𝑥_𝑖(𝑡)) − ̃𝑓_𝑖(𝑥_𝑖(𝑡))󵄨󵄨󵄨󵄨󵄨

≤ ̂𝑓_𝑖󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡) + 𝑥_𝑖(𝑡) − 𝑥_𝑖(𝑡)󵄨󵄨󵄨󵄨 = ̂𝑓^𝑖󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨 ,

󵄨󵄨󵄨󵄨𝑔^𝑖(𝑒_𝑖(𝑡))󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨̃𝑔^𝑖(𝑒_𝑖(𝑡) + 𝑥_𝑖(𝑡)) − ̃𝑔_𝑖(𝑥_𝑖(𝑡))󵄨󵄨󵄨󵄨

≤ ̂𝑔_𝑖󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡) + 𝑥_𝑖(𝑡) − 𝑥_𝑖(𝑡)󵄨󵄨󵄨󵄨 = ̂𝑔^𝑖󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨 ,

󵄨󵄨󵄨󵄨ℎ^𝑖(𝑒_𝑖(𝑡))󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨󵄨̃ℎ_𝑖(𝑒_𝑖(𝑡) + 𝑥_𝑖(𝑡)) − ̃ℎ_𝑖(𝑥_𝑖(𝑡))󵄨󵄨󵄨󵄨󵄨

≤ ̂ℎ_𝑖󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡) + 𝑥_𝑖(𝑡) − 𝑥_𝑖(𝑡)󵄨󵄨󵄨󵄨 = ̂ℎ^𝑖󵄨󵄨󵄨󵄨𝑒^𝑖(𝑡)󵄨󵄨󵄨󵄨 . (30) Applying Propositions3and4and since the matrices𝑈_𝑖,𝑖 = 1, 2, 3are diagonal, we have

2𝑧^𝑇(𝑡) 𝐶𝑓 (𝑒 (𝑡))

≤ 𝑧^𝑇(𝑡) 𝐶^𝑇𝑈₁𝐶𝑧 (𝑡) + 𝑓^𝑇(𝑒 (𝑡)) 𝑈₁⁻¹𝑓 (𝑒 (𝑡))

≤ 𝑧^𝑇(𝑡) 𝐶^𝑇𝑈₁𝐶𝑧 (𝑡) + 𝑒^𝑇(𝑡) 𝐹^𝑇𝑈₁⁻¹𝐹𝑒 (𝑡)

= 𝑧^𝑇(𝑡) 𝐶^𝑇𝑈₁𝐶𝑧 (𝑡) + 𝑧^𝑇(𝑡) 𝑃𝐹^𝑇𝑈₁⁻¹𝐹𝑃𝑧 (𝑡) , 2𝑧^𝑇(𝑡) 𝐷𝑔 (𝑒 (𝑡 − ℎ₁(𝑡)))

≤ 𝑧^𝑇(𝑡) 𝐷^𝑇𝑈₂𝐷𝑧 (𝑡) + 𝑔^𝑇(𝑒 (𝑡 − ℎ₁(𝑡))) 𝑈₂⁻¹

× 𝑔 (𝑒 (𝑡 − ℎ₁(𝑡)))

≤ 𝑧^𝑇(𝑡) 𝐷^𝑇𝑈₂𝐷𝑧 (𝑡) + 𝑒^𝑇(𝑡 − ℎ₁(𝑡)) 𝐺^𝑇𝑈₂⁻¹

× 𝐺𝑒 (𝑡 − ℎ₁(𝑡))

= 𝑧^𝑇(𝑡) 𝐷^𝑇𝑈₂𝐷𝑧 (𝑡) + 𝑧^𝑇(𝑡 − ℎ₁(𝑡)) 𝑃𝐺^𝑇𝑈₂⁻¹

× 𝐺𝑃𝑧 (𝑡 − ℎ₁(𝑡)) , 𝑘₁ℎ^𝑇(𝑒 (𝑡)) 𝑈₃⁻¹ℎ (𝑒 (𝑡))

≤ 𝑘₁𝑒^𝑇(𝑡) 𝐻^𝑇𝑈⁻¹₃ 𝐻𝑒 (𝑡)

= 𝑘₁𝑧^𝑇(𝑡) 𝑃𝐻^𝑇𝑈₃⁻¹𝐻𝑃𝑧 (𝑡) , 2𝑧^𝑇(𝑡) 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑𝑠

≤ 2𝑘₁𝑒^2𝛼𝑘1𝑧^𝑇(𝑡) 𝐸^𝑇𝑈₃𝐸𝑧 (𝑡)

+ 1

2𝑘₁𝑒^{−2𝛼𝑘1}(∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑𝑠)^𝑇

× 𝑈₃⁻¹(∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑𝑠)

≤ 2𝑘₁𝑒^2𝛼𝑘1𝑧^𝑇(𝑡) 𝐸^𝑇𝑈₃𝐸𝑧 (𝑡) +𝑒^{−2𝛼𝑘1}

2 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ^𝑇(𝑒 (𝑠)) 𝑈₃⁻¹ℎ (𝑒 (𝑠)) 𝑑𝑠, 2𝑧^𝑇(𝑡) 𝐵₂𝑢 (𝑡 − ℎ₂(𝑡))

≤ 3𝑒^2𝛼ℎ2𝑧^𝑇(𝑡) 𝐵^𝑇₂𝑆₁𝐵₂𝑧 (𝑡) +𝑒^−2𝛼ℎ2

3 𝑢^𝑇(𝑡 − ℎ₂(𝑡)) 𝑆⁻¹₁ 𝑢 (𝑡 − ℎ₂(𝑡)) , 2𝑧^𝑇(𝑡) 𝐵₃∫^𝑡

𝑡−𝑘2(𝑡)𝑢 (𝑠) 𝑑𝑠

≤ 2𝑘₂𝑒^2𝛼𝑘2𝑧^𝑇(𝑡) 𝐵^𝑇₃𝑆₂𝐵₃𝑧 (𝑡) +𝑒^{−2𝛼𝑘2}

2𝑘₂ (∫^𝑡

𝑡−𝑘2(𝑡)𝑢 (𝑠) 𝑑𝑠)^𝑇𝑆₂⁻¹(∫^𝑡

𝑡−𝑘1(𝑡)𝑢 (𝑠) 𝑑𝑠)

≤ 2𝑘₂𝑒^2𝛼𝑘2𝑧^𝑇(𝑡) 𝐵^𝑇₃𝑆₂𝐵₃𝑧 (𝑡) +𝑒^{−2𝛼𝑘2}

2 ∫^𝑡

𝑡−𝑘2(𝑡)𝑢^𝑇(𝑠) 𝑆⁻¹₂ 𝑢 (𝑠) 𝑑𝑠, ℎ²₂ ̇𝑢^𝑇(𝑡) 𝑆⁻¹₁ ̇𝑢 (𝑡)

= ℎ²₂ ^𝑇̇𝑒 (𝑡) 𝑃⁻¹𝑌^𝑇𝑆₁⁻¹𝑌𝑃⁻¹ ̇𝑒(𝑡)

= ℎ²₂ ̇𝑧^𝑇(𝑡) 𝑌^𝑇𝑆⁻¹₁ 𝑌 ̇𝑧 (𝑡)

(31) and the Leibniz-Newton formula gives

− ℎ₂𝑒^−2𝛼ℎ2∫^𝑡

𝑡−ℎ2

̇𝑢^𝑇(𝑠) 𝑆⁻¹₁ ̇𝑢 (𝑠) 𝑑𝑠

≤ −ℎ₂(𝑡) 𝑒^−2𝛼ℎ2∫^𝑡

𝑡−ℎ₂(𝑡) ̇𝑢^𝑇(𝑠) 𝑆⁻¹₁ ̇𝑢 (𝑠) 𝑑𝑠

≤ −𝑒^−2𝛼ℎ2(∫^𝑡

𝑡−ℎ2(𝑡) ̇𝑢 (𝑠) 𝑑𝑠)^𝑇𝑆⁻¹₁ (∫^𝑡

𝑡−ℎ2(𝑡) ̇𝑢 (𝑠) 𝑑𝑠)

≤ −𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡) 𝑆⁻¹₁ 𝑢 (𝑡) + 2𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡) 𝑆⁻¹₁ 𝑢 (𝑡 − ℎ₂(𝑡))

− 𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡 − ℎ₂(𝑡)) 𝑆⁻¹₁ 𝑢 (𝑡 − ℎ₂(𝑡))

≤ −𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡) 𝑆⁻¹₁ 𝑢 (𝑡) + 3𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡) 𝑆⁻¹₁ 𝑢 (𝑡) +𝑒^−2𝛼ℎ2

3 𝑢^𝑇(𝑡 − ℎ₂(𝑡)) 𝑆⁻¹₁ 𝑆₁𝑆⁻¹₁ 𝑢 (𝑡 − ℎ₂(𝑡))

− 𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡 − ℎ₂(𝑡)) 𝑆⁻¹₁ 𝑢 (𝑡 − ℎ₂(𝑡))

= 2𝑒^−2𝛼ℎ2𝑧^𝑇(𝑡) 𝑌^𝑇𝑆⁻¹₁ 𝑌𝑧 (𝑡) +𝑒^−2𝛼ℎ2

3 𝑢^𝑇(𝑡 − ℎ₂(𝑡)) 𝑆⁻¹₁ 𝑢 (𝑡 − ℎ₂(𝑡))

− 𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡 − ℎ₂(𝑡)) 𝑆⁻¹₁ 𝑢 (𝑡 − ℎ₂(𝑡)) .

(32) ApplyingProposition 4and the Leibniz-Newton formula, we have

− ℎ_1𝑚𝑒^{−2𝛼ℎ1𝑚}∫^𝑡

𝑡−ℎ1𝑚

̇𝑧𝑇(𝑠) 𝑅 ̇𝑧 (𝑠) 𝑑𝑠

≤ −𝑒^{−2𝛼ℎ1𝑚}[∫^𝑡

𝑡−ℎ1𝑚

̇𝑧 (𝑠)]^𝑇𝑅 [∫^𝑡

𝑡−ℎ1𝑚

̇𝑧 (𝑠)]

≤ −𝑒^{−2𝛼ℎ1𝑚}[𝑧 (𝑡) − 𝑧 (𝑡 − ℎ_1𝑚)]^𝑇𝑅 [𝑧 (𝑡) − 𝑧 (𝑡 − ℎ_1𝑚)]

= −𝑒^{−2𝛼ℎ1𝑚}[𝑧^𝑇(𝑡) 𝑅𝑧 (𝑡) − 2𝑧^𝑇(𝑡) 𝑅𝑧 (𝑡 − ℎ_1𝑚) +𝑧^𝑇(𝑡 − ℎ_1𝑚) 𝑅𝑧 (𝑡 − ℎ_1𝑚)] ,

− ℎ_1𝑀𝑒^{−2𝛼ℎ1𝑀}∫^𝑡

𝑡−ℎ1𝑀

̇𝑧𝑇(𝑠) 𝑅 ̇𝑧 (𝑠) 𝑑𝑠

≤ −𝑒^{−2𝛼ℎ1𝑀}[∫^𝑡

𝑡−ℎ1𝑀

̇𝑧 (𝑠)]^𝑇𝑅 [∫^𝑡

𝑡−ℎ1𝑀

̇𝑧 (𝑠)]

≤ −𝑒^{−2𝛼ℎ1𝑀}[𝑧 (𝑡) − 𝑧 (𝑡 − ℎ1𝑀)]^𝑇𝑅 [𝑧 (𝑡) − 𝑧 (𝑡 − ℎ1𝑀)]

= −𝑒^{−2𝛼ℎ1𝑀}[𝑧^𝑇(𝑡) 𝑅𝑧 (𝑡) − 2𝑧^𝑇(𝑡) 𝑅𝑧 (𝑡 − ℎ_1𝑀) +𝑧^𝑇(𝑡 − ℎ_1𝑀) 𝑅𝑧 (𝑡 − ℎ_1𝑀)] .

(33) Note that

− 𝛿 ∫^{𝑡−ℎ1𝑚}

𝑡−ℎ1𝑀

̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

= − (ℎ_1𝑀− ℎ_1𝑚) ∫^{𝑡−ℎ(𝑡)}

𝑡−ℎ1𝑀

̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

− (ℎ_1𝑀− ℎ_1𝑚) ∫^{𝑡−ℎ1𝑚}

𝑡−ℎ(𝑡) ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

= − (ℎ_1𝑀− ℎ (𝑡)) ∫^{𝑡−ℎ(𝑡)}

𝑡−ℎ_1𝑀 ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

− (ℎ (𝑡) − ℎ_1𝑚) ∫^{𝑡−ℎ(𝑡)}

𝑡−ℎ1𝑀

̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

− (ℎ (𝑡) − ℎ_1𝑚) ∫^{𝑡−ℎ1𝑚}

𝑡−ℎ(𝑡) ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

− (ℎ_1𝑀− ℎ (𝑡)) ∫^{𝑡−ℎ1𝑚}

𝑡−ℎ(𝑡) ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠.

(34)

UsingProposition 4gives

− (ℎ_1𝑀− ℎ (𝑡)) ∫^{𝑡−ℎ(𝑡)}

𝑡−ℎ1𝑀

̇𝑧𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

≤ −[∫^{𝑡−ℎ(𝑡)}

𝑡−ℎ1𝑀

̇𝑧 (𝑠) 𝑑𝑠]^𝑇𝑈 [∫^{𝑡−ℎ(𝑡)}

𝑡−ℎ1𝑀

̇𝑧 (𝑠) 𝑑𝑠]

≤ −[𝑧 (𝑡 − ℎ (𝑡)) − 𝑧 (𝑡 − ℎ_1𝑀)]^𝑇𝑈

× [𝑧 (𝑡 − ℎ (𝑡)) − 𝑧 (𝑡 − ℎ1𝑀)] ,

− (ℎ (𝑡) − ℎ_1𝑚) ∫^{𝑡−ℎ1𝑀}

𝑡−ℎ(𝑡) ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

≤ −[∫^{𝑡−ℎ1𝑚}

𝑡−ℎ(𝑡) ̇𝑧 (𝑠) 𝑑𝑠]^𝑇𝑈 [∫^{𝑡−ℎ1𝑚}

𝑡−ℎ(𝑡) ̇𝑧 (𝑠) 𝑑𝑠]

≤ −[𝑧 (𝑡 − ℎ_1𝑚) − 𝑧 (𝑡 − ℎ (𝑡))]^𝑇𝑈

× [𝑧 (𝑡 − ℎ_1𝑚) − 𝑧 (𝑡 − ℎ (𝑡))] .

(35)

Let𝛽 = (ℎ_1𝑀− ℎ(𝑡))/(ℎ_1𝑀− ℎ_1𝑚) ≤ 1. Then

− (ℎ_1𝑀− ℎ (𝑡)) ∫^{𝑡−ℎ1𝑚}

𝑡−ℎ(𝑡) ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

= −𝛽 ∫^{𝑡−ℎ1𝑚}

𝑡−ℎ(𝑡)(ℎ_1𝑀− 1𝑚) ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

≤ −𝛽 ∫^{𝑡−ℎ1𝑚}

𝑡−ℎ(𝑡)(ℎ (𝑡) − ℎ1𝑚) ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

≤ −𝛽[𝑧 (𝑡 − ℎ_1𝑚) − 𝑧 (𝑡 − ℎ (𝑡))]^𝑇𝑈

× [𝑧 (𝑡 − ℎ_1𝑚) − 𝑧 (𝑡 − ℎ (𝑡))] ,

− (ℎ (𝑡) − ℎ_1𝑚) ∫^{𝑡−ℎ(𝑡)}

𝑡−ℎ1𝑀

̇𝑧𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

= − (1 − 𝛽) ∫^{𝑡−ℎ(𝑡)}

𝑡−ℎ1𝑀

(ℎ_1𝑀− ℎ_1𝑚) ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

≤ − (1 − 𝛽) ∫^{𝑡−ℎ(𝑡)}

𝑡−ℎ1𝑀

(ℎ₂− ℎ (𝑡)) ̇𝑧^𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

≤ − (1 − 𝛽) [𝑧 (𝑡 − ℎ (𝑡)) − 𝑧 (𝑡 − ℎ_1𝑀)]^𝑇

× 𝑈 [𝑧 (𝑡 − ℎ (𝑡)) − 𝑧 (𝑡 − ℎ_1𝑀)] .

(36) Therefore from (35)-(36), we obtain

− 𝛿 ∫^{𝑡−ℎ1𝑚}

𝑡−ℎ1𝑀

̇𝑧𝑇(𝑠) 𝑈 ̇𝑧 (𝑠) 𝑑𝑠

≤ −[𝑧 (𝑡 − ℎ (𝑡)) − 𝑧 (𝑡 − ℎ_1𝑀)]^𝑇

× 𝑈 [𝑧 (𝑡 − ℎ (𝑡)) − 𝑧 (𝑡 − ℎ1𝑀)]

− [𝑧 (𝑡 − ℎ_1𝑚) − 𝑧 (𝑡 − ℎ (𝑡))]^𝑇

× 𝑈 [𝑧 (𝑡 − ℎ_1𝑚) − 𝑧 (𝑡 − ℎ (𝑡))]

− 𝛽[𝑧 (𝑡 − ℎ_1𝑚) − 𝑧 (𝑡 − ℎ (𝑡))]^𝑇

× 𝑈 [𝑧 (𝑡 − ℎ_1𝑚) − 𝑧 (𝑡 − ℎ (𝑡))]

− (1 − 𝛽) [𝑧 (𝑡 − ℎ (𝑡)) − 𝑧 (𝑡 − ℎ1𝑀)]^𝑇

× 𝑈 [𝑧 (𝑡 − ℎ (𝑡)) − 𝑧 (𝑡 − ℎ_1𝑀)] .

(37) By using the following identity relation:

0 = − ̇𝑒(𝑡) − 𝐴𝑒 (𝑡) + 𝐶𝑓 (𝑒 (𝑡)) + 𝐷𝑔 (𝑒 (𝑡 − ℎ1(𝑡))) + 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑 𝑠 + 𝐵₁𝐾𝑒 (𝑡) + 𝐵₂𝐾𝑒 (𝑡 − ℎ₂(𝑡)) + 𝐵₃𝐾 ∫^𝑡

𝑡−𝑘2(𝑡)𝑒 (𝑠) 𝑑𝑠,

= − 𝑃 ̇𝑧 (𝑡) − 𝐴𝑃𝑧 (𝑡) + 𝐶𝑓 (𝑒 (𝑡)) + 𝐷𝑔 (𝑒 (𝑡 − ℎ1(𝑡))) + 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑 𝑠 − 𝐵₁𝑌𝑧 (𝑡) + 𝐵₂(𝑡) 𝑢 (𝑡 − ℎ₂(𝑡)) + 𝐵₃∫^𝑡

𝑡−𝑘2(𝑡)𝑢 (𝑠) 𝑑𝑠,

(38)

we have

0 = − 2 ̇𝑧^𝑇(𝑡) 𝑃 ̇𝑧 (𝑡) − 2 ̇𝑧^𝑇(𝑡) 𝐴𝑃𝑧 (𝑡) + 2 ̇𝑧^𝑇(𝑡) 𝐶𝑓 (𝑒 (𝑡)) + 2 ̇𝑧^𝑇(𝑡) 𝐷𝑔 (𝑒 (𝑡 − ℎ₁(𝑡)))

+ 2 ̇𝑧^𝑇(𝑡) 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑 𝑠 − 2 ̇𝑧^𝑇(𝑡) 𝐵₁𝑌𝑧 (𝑡) + 2 ̇𝑧^𝑇(𝑡) 𝐵₂(𝑡) 𝑢 (𝑡 − ℎ₂(𝑡)) + 2 ̇𝑧^𝑇(𝑡) 𝐵₃∫^𝑡

𝑡−𝑘2(𝑡)𝑢 (𝑠) 𝑑𝑠.

(39) By using Propositions3and4, we have

2 ̇𝑧^𝑇(𝑡) 𝐶𝑓 (𝑒 (𝑡))

≤ ̇𝑧^𝑇(𝑡) 𝐶^𝑇𝑈₁𝐶 ̇𝑧 (𝑡) + 𝑓^𝑇(𝑒 (𝑡)) 𝑈₁⁻¹𝑓 (𝑒 (𝑡))

≤ ̇𝑧^𝑇(𝑡) 𝐶^𝑇𝑈₁𝐶 ̇𝑧 (𝑡) + 𝑒^𝑇(𝑡) 𝐹^𝑇𝑈₁⁻¹𝐹𝑒 (𝑡)

= ̇𝑧^𝑇(𝑡) 𝐶^𝑇𝑈₁𝐶 ̇𝑧 (𝑡) + 𝑧^𝑇(𝑡) 𝑃𝐹^𝑇𝑈₁⁻¹𝐹𝑃𝑧 (𝑡) , 2 ̇𝑧^𝑇(𝑡) 𝐷𝑔 (𝑒 (𝑡 − ℎ₁(𝑡)))

≤ ̇𝑧^𝑇(𝑡) 𝐷^𝑇𝑈₂𝐷 ̇𝑧 (𝑡) + 𝑔^𝑇(𝑒 (𝑡 − ℎ₁(𝑡)))

× 𝑈₂⁻¹𝑔 (𝑒 (𝑡 − ℎ₁(𝑡)))

≤ ̇𝑧^𝑇(𝑡) 𝐷^𝑇𝑈₂𝐷 ̇𝑧 (𝑡) + 𝑒^𝑇(𝑡 − ℎ₁(𝑡)) 𝐺^𝑇𝑈₂⁻¹

× 𝐺𝑒 (𝑡 − ℎ₁(𝑡))

= ̇𝑧^𝑇(𝑡) 𝐷^𝑇𝑈₂𝐷 ̇𝑧 (𝑡) + 𝑧^𝑇(𝑡 − ℎ₁(𝑡)) 𝑃𝐺^𝑇𝑈₂⁻¹

× 𝐺𝑃𝑧 (𝑡 − ℎ₁(𝑡)) , 2 ̇𝑧^𝑇(𝑡) 𝐸 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑𝑠

≤ 2𝑘₁𝑒^2𝛼𝑘1 ̇𝑧^𝑇(𝑡) 𝐸^𝑇𝑈₃𝐸 ̇𝑧 (𝑡)

+ 1

2𝑘₁𝑒^{−2𝛼𝑘1}(∫^𝑡

𝑡−𝑘₁(𝑡)ℎ (𝑒 (𝑠)) 𝑑𝑠)^𝑇

× 𝑈₃⁻¹(∫^𝑡

𝑡−𝑘1(𝑡)ℎ (𝑒 (𝑠)) 𝑑𝑠)

≤ 2𝑘₁𝑒^2𝛼𝑘1 ̇𝑧^𝑇(𝑡) 𝐸^𝑇𝑈₃𝐸 ̇𝑧 (𝑡) +𝑒^{−2𝛼𝑘1}

2 ∫^𝑡

𝑡−𝑘1(𝑡)ℎ^𝑇(𝑒 (𝑠)) 𝑈₃⁻¹ℎ (𝑒 (𝑠)) 𝑑𝑠, 2 ̇𝑧^𝑇(𝑡) 𝐵₂(𝑡) 𝑢 (𝑡 − ℎ₂(𝑡))

≤ 3𝑒^2𝛼ℎ2 ̇𝑧^𝑇(𝑡) 𝐵^𝑇₂𝑆₁𝐵₂ ̇𝑧 (𝑡) +𝑒^−2𝛼ℎ2

3 𝑢^𝑇(𝑡 − ℎ₂(𝑡)) 𝑆⁻¹₁ 𝑢 (𝑡 − ℎ₂(𝑡)) , 2 ̇𝑧^𝑇(𝑡) 𝐵₃∫^𝑡

𝑡−𝑘₂(𝑡)𝑢 (𝑠) 𝑑𝑠