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

Abstract

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 combinatorial 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 characteristics of various activation functions and network parameters. When neural networks are created for problem solving, 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 activation functions and more general conditions [7–9]. On the other hand, the problem of chaos synchronization has attracted a wide range of research activity in recent years. A chaotic 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 synchronization making two chaotic systems oscillate in a synchronized 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 stabilize 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 guaranteed 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 convergence rate to the equilibrium point and any information about the decay rates of the delayed neural networks. Therefore it is particularly important when the exponential convergence rate is used to determine the speed of neural computations. The exponential stability property guarantees that, whatever transformation occurs, the network’s ability to store rapidly the activity pattern is left invariant by self-organization. Thus, it is important to determine the exponential stability and to estimate the exponential convergence rate for delayed neural networks. Recently, exponential synchronization 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-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. 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 stability 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 interval 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 derived to ensure the asymptotical and exponential synchronization 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 difficult 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 distributed 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 consider 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 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 formula and the technique of dealing with some integral terms, new delay-dependent sufficient conditions for the exponential 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 problem 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. In Section 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 exponential synchronization of cellular neural networks with various activation functions and interval and distributed time-varying delays with memoryless hybrid feedback controls are presented in Section 3. Numerical examples illustrating 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 semipositive 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 ∗.

In this paper, the master-slave cellular neural networks (MSCNNs) with mixed time-varying delays are described as follows: 𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝐶𝑓^∼(𝑥(𝑡))+𝐷𝑔^∼𝑥𝑡−ℎ₁(𝑡)+𝐸ℎ^𝑡_{𝑡−𝑘1(𝑡)}^∼(𝑥(𝑠(𝑡),𝑥(𝑡)))𝑑𝑠+𝐼=𝜙₁(𝑡)[],𝑦̇(𝑡)(𝑡),𝑡∈−𝑑,0=−𝐴𝑦+𝐶𝑓^∼𝑦(𝑡)+𝐷𝑔^∼𝑦𝑡−ℎ₁(𝑡)+𝐸ℎ^𝑡_{𝑡−𝑘1(𝑡)}^∼𝑦(𝑠)(𝑡)(𝑡),𝑦(𝑡)𝑑𝑠+𝐼+𝒰=𝜙₂(𝑡),𝑡∈[𝑑,0], (1)(2) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]∈ℝ^𝑛 and 𝑦(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. 𝑛 is the number of neural, 𝑓^∼(𝑥(𝑡))=𝑓^∼₁𝑥₁(𝑡),𝑓^∼₂𝑥₂(𝑡),…,𝑓^∼_𝑛𝑥_𝑛(𝑡)^𝑇,_∼𝑔(𝑥(𝑡))=𝑔^∼₁𝑥₁(𝑡),𝑔^∼₂𝑥₂(𝑡),…,𝑔^∼_𝑛𝑥_𝑛(𝑡)^𝑇,_∼ℎ(𝑥(𝑡))=ℎ^∼₁𝑥₁(𝑡),ℎ^∼₂𝑥₂(𝑡),…,ℎ^∼_𝑛𝑥_𝑛(𝑡)^𝑇, (3) are the activation functions, 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛), 𝑎_𝑖>0 represents the self-feedback term, and 𝐶,𝐷, and 𝐸 denote the connection weights, the discretely delayed connection weights, and the distributively delayed connection weight, respectively.

The synchronization error 𝑒(𝑡) is the form 𝑒(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 𝑒̇(𝑡)=−𝐴𝑒(𝑡)+𝐶𝑓(𝑒(𝑡))+𝐷𝑔𝑒𝑡−ℎ₁(𝑡)+𝐸^𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠(𝑡),𝑒(𝑡)))𝑑𝑠+𝒰=𝜙₂(𝑡)−𝜙₁(𝑡)=𝜙(𝑡),𝑡∈[−𝑑,0], (4) where 𝑓(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_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠=∫^𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠)+𝑥(𝑠))−ℎ(𝑥(𝑠))𝑑𝑠. The state hybrid feedback controller 𝒰(𝑡) satisfies (H1): (H1):𝒰(𝑡)=𝐵₁𝑢(𝑡)+𝐵₂𝑢𝑡−ℎ₂(𝑡)+𝐵₃^𝑡_{𝑡−𝑘2(𝑡)}𝑢(𝑠)𝑑𝑠,∀𝑡≥0, (5) where 𝑢(𝑡)=𝐾𝑒(𝑡) and 𝐾 is a constant matrix control gain. In this paper, our goal is to design suitable 𝐾 such that system (2) synchronizes with system (1). Then, substituting it into (4), it is easy to get the following: 𝑒̇(𝑡)=−𝐴𝑒(𝑡)+𝐶𝑓(𝑒(𝑡))+𝐷𝑔𝑒𝑡−ℎ₁(𝑡)+𝐸^𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠+𝐵₁𝐾𝑒(𝑡)+𝐵₂𝐾𝑒𝑡−ℎ₂(𝑡)+𝐵₃𝐾^𝑡_{𝑡−𝑘2(𝑡)}𝑒(𝑠)𝑒(𝑡)𝑑𝑠,∀𝑡≥0,=𝜙₂(𝑡)−𝜙₁(𝑡)=𝜙(𝑡),𝑡∈[−𝑑,0]. (6) Throughout this paper, we consider various activation functions and the activation functions 𝑓^∼(⋅), 𝑔^∼(⋅), and ℎ^∼(⋅) satisfy the following assumption.

(A1) The activation functions 𝑓^∼(⋅), 𝑔^∼(⋅), and ℎ^∼(⋅) satisfy Lipschitzian with the Lipschitz constants 𝑓_𝑖,𝑔_𝑖>0, and ℎ_𝑖>0: 𝑓^∼_𝑖𝜉₁−𝑓^∼_𝑖𝜉₂≤𝑓_𝑖𝜉₁−𝜉₂,𝑖=1,2,…,𝑛,∀𝜉₁,𝜉₂∈ℝ,_∼𝑔_𝑖𝜉₁−𝑔^∼_𝑖𝜉₂≤𝑔_𝑖𝜉₁−𝜉₂,𝑖=1,2,…,𝑛,∀𝜉₁,𝜉₂∈ℝ,_∼ℎ_𝑖𝜉₁−ℎ^∼_𝑖𝜉₂≤ℎ_𝑖𝜉₁−𝜉₂,𝑖=1,2,…,𝑛,∀𝜉₁,𝜉₂∈ℝ, (7) and we denote that 𝐹=diag𝑓_𝑖,𝑔,𝑖=1,2,…,𝑛𝐺=diag_𝑖,ℎ,𝑖=1,2,…,𝑛𝐻=diag_𝑖,𝑖=1,2,…,𝑛. (8)

The time-varying delay functions ℎ_𝑖(𝑡) and 𝑘_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+𝜙̇(𝑡)². (10)

Define the following quadratic cost function of the associated system (4) as follows: 𝐽=^∞₀𝑒^𝑇(𝑡)𝑄₁𝑒(𝑡)+𝑢^𝑇(𝑡)𝑄₂𝑢(𝑡)𝑑𝑡, (11) where 𝑄₁∈ℝ^𝑛×𝑛 and 𝑄₂∈ℝ^𝑚×𝑚 are positive definite matrices.

Remark 1. If 𝐸=0, 𝐵₂=0, 𝐵₃=0, and 𝑓(⋅)=𝑔(⋅), the system model (6) turns into the cellular neural networks with activation functions and time-varying delays proposed by [21, 22] 𝑒̇(𝑡)=−𝐴𝑒(𝑡)+𝐶𝑓(𝑒(𝑡))+𝐷𝑓𝑒𝑡−ℎ₁(𝑡)+𝐵₁𝐾𝑒(𝑡),∀𝑡≥0. (12) Therefore, (6) is a general cellular neural networks model, with (12) as the special case.

Definition 2. Given 𝛼>0, the zero solution of system (6) with 𝑢(𝑡)=𝐾𝑒(𝑡) is 𝛼-stable if there exists a positive number 𝑁>0 such that every solution 𝑒(𝑡,𝜙) satisfies the following condition: 𝑒𝑡,𝜙≤𝑁𝑒^−𝛼𝑡𝜙,∀𝑡≥0. (13)

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

Proposition 3 (see [42] (Cauchy inequality)). For any symmetric positive definite matrix 𝑁∈𝑀^𝑛×𝑛 and 𝑥,𝑦∈ℝ^𝑛, we have ±2𝑥^𝑇𝑦≤𝑥^𝑇𝑁𝑥+𝑦^𝑇𝑁⁻¹𝑦. (14)

Proposition 4 (see [42]). 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. (15)

Proposition 5 (see [42] (Schur complement lemma)). Given constant symmetric matrices 𝑋, 𝑌, and 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. (16)

3. Main Results

Let us set 𝜆₁=𝜆_min𝑃⁻¹,𝜆₂=𝜆_max𝑃⁻¹+ℎ_1𝑚+ℎ_1𝑀𝜆_max𝑃⁻¹𝑄𝑃⁻¹+ℎ³_1𝑚+ℎ³_1𝑀𝜆_max𝑃⁻¹𝑅𝑃⁻¹+𝛿³𝜆_max𝑃⁻¹𝑈𝑃⁻¹+ℎ³₂𝜆_max𝑃⁻¹𝑌^𝑇𝑆₁⁻¹𝑌𝑃⁻¹+𝑘²₁𝜆_max𝐻𝑈₃⁻¹𝐻+𝑘²₂𝜆_max𝑃⁻¹𝑌^𝑇𝑆₂⁻¹𝑌𝑃⁻¹. (17)

Theorem 6. Given 𝛼>0, 𝑄₁>0 and 𝑄₂>0, 𝑢(𝑡)=𝐾𝑒(𝑡) is a guaranteed cost controller if there exist symmetric positive definite matrices 𝑃, 𝑄, 𝑅, 𝑈, 𝑆₁, and 𝑆₂, diagonal matrices 𝑈_𝑖,𝑖=1,2,3, and a matrix 𝑌 with appropriately dimensioned such that the following LMIs holds: Γ₁=Γ−00−𝐼𝐼0^𝑇×𝑒^{−2𝛼ℎ1𝑀}𝑈Γ00−𝐼𝐼0<0,₂=Γ−000𝐼−𝐼^𝑇×𝑒^{−2𝛼ℎ1𝑀}𝑈Γ000𝐼−𝐼<0,₃=−0.1𝑒^{−2𝛼ℎ1𝑚}+𝑒^{−2𝛼ℎ1𝑀}𝑅2𝑃𝐹^𝑇𝑃𝐻^𝑇2𝑌^𝑇𝑃𝑄₁𝑌^𝑇𝑄₂∗−2𝑈₁0000∗∗−𝑈₃000∗∗∗−2𝑒^−2𝛼ℎ2𝑆₁00∗∗∗∗−𝑄₁0∗∗∗∗∗−𝑄₂Γ<0,₄=−0.1𝑃ℎ²₂𝑌^𝑇∗−ℎ²₂𝑆₁Γ<0,₅=−0.1𝑒^{−2𝛼ℎ1𝑀}𝑈2𝑃𝐺^𝑇∗−2𝑈₂Γ<0,Γ=¹¹Γ₁₂Γ₁₃0Γ₁₅∗Γ₂₂000∗∗Γ₃₃Γ₃₄0∗∗∗Γ₄₄Γ₄₅∗∗∗∗Γ₅₅, (18)(19)(20)(21)(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}Γ𝑅,₁₂=−𝑃𝐴^𝑇−𝑌^𝑇𝐵^𝑇,Γ₁₃=𝑒^{−2𝛼ℎ1𝑚}Γ𝑅,₁₅=𝑒^{−2𝛼ℎ1𝑀}Γ𝑅,₂₂=ℎ²_1𝑚𝑅+ℎ²_1𝑀𝑅+𝛿²𝑈−1.9𝑃+𝐶^𝑇𝑈₁𝐶+𝐷^𝑇𝑈₂𝐷+2𝑘₁𝑒^2𝛼𝑘1𝐸^𝑇𝑈₃𝐸+3𝑒^2𝛼ℎ2𝐵^𝑇₂𝑆₁𝐵₂+2𝑘₂𝑒^2𝛼𝑘2𝐵^𝑇₃𝑆₂𝐵₃Γ₃₃=−𝑒^{−2𝛼ℎ1𝑚}𝑄−𝑒^{−2𝛼ℎ1𝑚}𝑅−𝑒^{−2𝛼ℎ1𝑀}Γ𝑈,₃₄=𝑒^{−2𝛼ℎ1𝑀}Γ𝑈,₄₄=−1.9𝑒^{−2𝛼ℎ1𝑀}Γ𝑈,₄₅=𝑒^{−2𝛼ℎ1𝑀}Γ𝑈,₅₅=−𝑒^{−2𝛼ℎ1𝑀}𝑄−𝑒^{−2𝛼ℎ1𝑀}𝑅−𝑒^{−2𝛼ℎ1𝑀}𝑈, (23) then the error system (6) is exponentially stabilizable. Moreover, 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^∗=𝜆₂𝜙². (25)

Proof. Let 𝑊=𝑃⁻¹ and let 𝑧(𝑡)=𝑊𝑒(𝑡). Using the feedback control (24), we consider the following Lyapunov-Krasovskii functional: 𝑉(𝑡,𝑒(𝑡))=⁹_𝑖=1𝑉_𝑖, (26) where 𝑉₁=𝑒^𝑇(𝑡)(𝑡),𝑉𝑊𝑒₂=^𝑡_{𝑡−ℎ1𝑚}𝑒^{2𝛼(𝑠−𝑡)}𝑒^𝑇(𝑠)(𝑠)𝑉𝑊𝑄𝑊𝑒𝑑𝑠,₃=^𝑡_{𝑡−ℎ1𝑀}𝑒^{2𝛼(𝑠−𝑡)}𝑒^𝑇(𝑠)(𝑠)𝑉𝑊𝑄𝑊𝑒𝑑𝑠,₄=ℎ_1𝑚⁰_−ℎ1𝑚^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡)}𝑒^𝑇̇(𝜃)𝑒̇(𝜃)𝑉𝑊𝑅𝑊𝑑𝜃𝑑𝑠,₅=ℎ_1𝑀⁰_−ℎ1𝑀^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡)}𝑒^𝑇̇(𝜃)𝑒̇(𝜃)𝑉𝑊𝑅𝑊𝑑𝜃𝑑𝑠,₆=𝛿^−ℎ1𝑚_−ℎ1𝑀^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡)}𝑒^𝑇̇(𝜃)𝑒̇(𝜃)𝑉𝑊𝑈𝑊𝑑𝜃𝑑𝑠,₇=ℎ₂⁰_ℎ2^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡)}𝑢̇^𝑇(𝜃)𝑆₁⁻¹𝑢̇(𝜃)𝑉𝑑𝜃𝑑𝑠,₈=⁰_−𝑘1^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡)}ℎ^𝑇(𝑒(𝜃))𝑈₃⁻¹ℎ(𝑒(𝜃𝑉))𝑑𝜃𝑑𝑠,₉=⁰_−𝑘2^𝑡_𝑡+𝑠𝑒^{2𝛼(𝜃−𝑡)}𝑢^𝑇(𝜃)𝑆₂⁻¹𝑢(𝜃)𝑑𝜃𝑑𝑠. (27) It is easy to check that 𝜆₁‖𝑒(𝑡)‖²≤𝑉(𝑡,𝑒(𝑡))≤𝜆₂𝑒_𝑡(𝑡)²,∀𝑡≥0. (28) Taking the derivative of 𝑉(𝑡,𝑒(𝑡)) along the solution of system (6), we have 𝑉̇₁=2𝑒^𝑇(𝑡)𝑊𝑒̇(𝑡)=2𝑧^𝑇(𝑡)−𝐴𝑒(𝑡)+𝐶𝑓(𝑒(𝑡))+𝐷𝑔𝑒𝑡−ℎ₁(𝑡)+𝐸^𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠−𝐵₁𝑌𝑃⁻¹𝑒(𝑡)+𝐵₂(𝑡)𝐾𝑒𝑡−ℎ₂(𝑡)+𝐵₃𝐾^𝑡_{𝑡−𝑘2(𝑡)}𝑒(𝑠)𝑑𝑠=𝑧^𝑇(𝑡)−𝐴𝑃−𝑃𝐴^𝑇−2𝐵₁𝑌𝑧(𝑡)+2𝑧^𝑇(𝑡)𝐶𝑓(𝑒(𝑡))+2𝑧^𝑇(𝑡)𝐷𝑔𝑒𝑡−ℎ₁(𝑡)+2𝑧^𝑇(𝑡)𝐸^𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠+2𝑧^𝑇(𝑡)𝐵₂𝑢𝑡−ℎ₂(𝑡)+2𝑧^𝑇(𝑡)𝐵₃^𝑡_{𝑡−𝑘2(𝑡)}𝑢(𝑠)𝑉̇𝑑𝑠,₂=𝑧^𝑇(𝑡)𝑄𝑧(𝑡)−𝑒^{−2𝛼ℎ1𝑚}𝑧^𝑇𝑡−ℎ_1𝑚𝑄𝑧𝑡−ℎ_1𝑚−2𝛼𝑉₂,𝑉̇₃=𝑧^𝑇(𝑡)𝑄𝑧(𝑡)−𝑒^{−2𝛼ℎ1𝑀}𝑧^𝑇𝑡−ℎ_1𝑀𝑄𝑧𝑡−ℎ_1𝑀−2𝛼𝑉₃,𝑉̇₄≤ℎ²_1𝑚𝑧̇^𝑇(𝑡)𝑅𝑧̇(𝑡)−ℎ_1𝑚𝑒^{−2𝛼ℎ1𝑚𝑡}_{𝑡−ℎ1𝑚}𝑧̇^𝑇(𝑠)𝑅𝑧̇(𝑠)𝑑𝑠−2𝛼𝑉₄,𝑉̇₅≤ℎ²_1𝑀𝑧̇^𝑇(𝑡)𝑅𝑧̇(𝑡)−ℎ_1𝑀𝑒^{−2𝛼ℎ1𝑀𝑡}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑅𝑧̇(𝑠)𝑑𝑠−2𝛼𝑉₅,𝑉̇₆≤𝛿²𝑧̇^𝑇(𝑡)𝑈𝑧̇(𝑡)−𝛿𝑒^{−2𝛼ℎ1𝑀𝑡−ℎ1𝑚}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑅𝑧̇(𝑠)𝑑𝑠−2𝛼𝑉₆,𝑉̇₇≤ℎ²₂𝑢̇^𝑇(𝑡)𝑆₁⁻¹𝑢̇(𝑡)−ℎ₂𝑒^{−2𝛼ℎ2𝑡}_𝑡−ℎ2𝑢̇^𝑇(𝑠)𝑆₁⁻¹𝑢̇(𝑠)𝑑𝑠−2𝛼𝑉₇,𝑉̇₈≤𝑘₁ℎ^𝑇(𝑒(𝑡))𝑈₃⁻¹ℎ(𝑒(𝑡))−𝑒^{−2𝛼𝑘1𝑡}_𝑡−𝑘1ℎ^𝑇(𝑒(𝑠))𝑈₃⁻¹ℎ(𝑒(𝑠))𝑑𝑠−2𝛼𝑉₈,𝑉̇₉≤𝑘²₂𝑢^𝑇(𝑡)𝑆₂⁻¹𝑢(𝑡)−𝑘₂𝑒^{−2𝛼𝑘2𝑡}_𝑡−𝑘2𝑢^𝑇(𝑠)𝑆₂⁻¹𝑢(𝑠)𝑑𝑠−2𝛼𝑉₉. (29) For assumption (A1), we can obtain the following three inequalities: 𝑓_𝑖𝑒_𝑖(𝑡)=𝑓^∼_𝑖𝑒_𝑖(𝑡)+𝑥_𝑖(𝑡)−𝑓^∼_𝑖𝑥_𝑖(𝑡)≤𝑓_𝑖𝑒_𝑖(𝑡)+𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡)=𝑓_𝑖𝑒_𝑖(𝑡,𝑔_𝑖)𝑒_𝑖(𝑡)=𝑔^∼_𝑖𝑒_𝑖(𝑡)+𝑥_𝑖(𝑡)−𝑔^∼_𝑖𝑥_𝑖(𝑡≤𝑔_𝑖)𝑒_𝑖(𝑡)+𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡)=𝑔_𝑖𝑒_𝑖(𝑡,ℎ_𝑖)𝑒_𝑖(𝑡)=ℎ^∼_𝑖𝑒_𝑖(𝑡)+𝑥_𝑖(𝑡)−ℎ^∼_𝑖𝑥_𝑖(𝑡)≤ℎ_𝑖𝑒_𝑖(𝑡)+𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡)=ℎ_𝑖𝑒_𝑖(𝑡). (30) Applying Propositions 3 and 4 and since the matrices 𝑈_𝑖, 𝑖=1,2,3 are 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)𝐷𝑔𝑒𝑡−ℎ₁(𝑡)≤𝑧^𝑇(𝑡)𝐷^𝑇𝑈₂𝐷𝑧(𝑡)+𝑔^𝑇𝑒𝑡−ℎ₁(𝑡)𝑈₂⁻¹×𝑔𝑒𝑡−ℎ₁(𝑡)≤𝑧^𝑇(𝑡)𝐷^𝑇𝑈₂𝐷𝑧(𝑡)+𝑒^𝑇𝑡−ℎ₁(𝑡)𝐺^𝑇𝑈₂⁻¹×𝐺𝑒𝑡−ℎ₁(𝑡)=𝑧^𝑇(𝑡)𝐷^𝑇𝑈₂𝐷𝑧(𝑡)+𝑧^𝑇𝑡−ℎ₁(𝑡)𝑃𝐺^𝑇𝑈₂⁻¹×𝐺𝑃𝑧𝑡−ℎ₁(𝑡),𝑘₁ℎ^𝑇(𝑒(𝑡))𝑈₃⁻¹ℎ(𝑒(𝑡))≤𝑘₁𝑒^𝑇(𝑡)𝐻^𝑇𝑈₃⁻¹𝐻𝑒(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^𝑇𝑈₃(𝑡)+1𝐸𝑧2𝑘₁𝑒^{−2𝛼𝑘1𝑡}_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠^𝑇×𝑈₃^−1𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠≤2𝑘₁𝑒^2𝛼𝑘1𝑧^𝑇(𝑡)𝐸^𝑇𝑈₃(𝑡)+𝑒𝐸𝑧_{−2𝛼𝑘1}2^𝑡_{𝑡−𝑘1(𝑡)}ℎ^𝑇(𝑒(𝑠))𝑈₃⁻¹ℎ(𝑒(𝑠))𝑑𝑠,2𝑧^𝑇(𝑡)𝐵₂𝑢𝑡−ℎ₂(𝑡)≤3𝑒^2𝛼ℎ2𝑧^𝑇(𝑡)𝐵^𝑇₂𝑆₁𝐵₂𝑧(𝑡)+𝑒^−2𝛼ℎ23𝑢^𝑇𝑡−ℎ₂(𝑡)𝑆₁⁻¹𝑢𝑡−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^−1𝑡_{𝑡−𝑘1(𝑡)}𝑢(𝑠)𝑑𝑠≤2𝑘₂𝑒^2𝛼𝑘2𝑧^𝑇(𝑡)𝐵^𝑇₃𝑆₂𝐵₃𝑧(𝑡)+𝑒^{−2𝛼𝑘2}2^𝑡_{𝑡−𝑘2(𝑡)}𝑢^𝑇(𝑠)𝑆₂⁻¹𝑢(𝑠)ℎ𝑑𝑠,₂₂𝑢̇^𝑇(𝑡)𝑆₁⁻¹𝑢̇(𝑡)=ℎ²₂𝑒^𝑇̇(𝑡)𝑃⁻¹𝑌^𝑇𝑆₁⁻¹𝑌𝑃⁻¹𝑒̇(𝑡)=ℎ²₂𝑧̇^𝑇(𝑡)𝑌^𝑇𝑆₁⁻¹𝑌𝑧̇(𝑡) (31) and the Leibniz-Newton formula gives −ℎ₂𝑒^{−2𝛼ℎ2𝑡}_𝑡−ℎ2𝑢̇^𝑇(𝑠)𝑆₁⁻¹𝑢̇(𝑠)𝑑𝑠≤−ℎ₂(𝑡)𝑒^{−2𝛼ℎ2𝑡}_{𝑡−ℎ2(𝑡)}𝑢̇^𝑇(𝑠)𝑆₁⁻¹𝑢̇(𝑠)𝑑𝑠≤−𝑒^{−2𝛼ℎ2𝑡}_{𝑡−ℎ2(𝑡)}𝑢̇(𝑠)𝑑𝑠^𝑇𝑆₁^−1𝑡_{𝑡−ℎ2(𝑡)}𝑢̇(𝑠)𝑑𝑠≤−𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡)𝑆₁⁻¹𝑢(𝑡)+2𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡)𝑆₁⁻¹𝑢𝑡−ℎ₂(𝑡)−𝑒^−2𝛼ℎ2𝑢^𝑇𝑡−ℎ₂(𝑡)𝑆₁⁻¹𝑢𝑡−ℎ₂(𝑡)≤−𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡)𝑆₁⁻¹𝑢(𝑡)+3𝑒^−2𝛼ℎ2𝑢^𝑇(𝑡)𝑆₁⁻¹𝑢(𝑡)+𝑒^−2𝛼ℎ23𝑢^𝑇𝑡−ℎ₂(𝑡)𝑆₁⁻¹𝑆₁𝑆₁⁻¹𝑢𝑡−ℎ₂(𝑡)−𝑒^−2𝛼ℎ2𝑢^𝑇𝑡−ℎ₂(𝑡)𝑆₁⁻¹𝑢𝑡−ℎ₂(𝑡)=2𝑒^−2𝛼ℎ2𝑧^𝑇(𝑡)𝑌^𝑇𝑆₁⁻¹(𝑡)+𝑒𝑌𝑧_−2𝛼ℎ23𝑢^𝑇𝑡−ℎ₂(𝑡)𝑆₁⁻¹𝑢𝑡−ℎ₂(𝑡)−𝑒^−2𝛼ℎ2𝑢^𝑇𝑡−ℎ₂(𝑡)𝑆₁⁻¹𝑢𝑡−ℎ₂(𝑡). (32) Applying Proposition 4 and the Leibniz-Newton formula, we have −ℎ_1𝑚𝑒^{−2𝛼ℎ1𝑚𝑡}_{𝑡−ℎ1𝑚}𝑧̇^𝑇(𝑠)𝑅𝑧̇(𝑠)𝑑𝑠≤−𝑒^{−2𝛼ℎ1𝑚𝑡}_{𝑡−ℎ1𝑚}𝑧̇(𝑠)^𝑇𝑅^𝑡_{𝑡−ℎ1𝑚}𝑧̇(𝑠)≤−𝑒^{−2𝛼ℎ1𝑚}𝑧(𝑡)−𝑧𝑡−ℎ_1𝑚^𝑇𝑅𝑧(𝑡)−𝑧𝑡−ℎ_1𝑚=−𝑒^{−2𝛼ℎ1𝑚}𝑧^𝑇(𝑡)𝑅𝑧(𝑡)−2𝑧^𝑇(𝑡)𝑅𝑧𝑡−ℎ_1𝑚+𝑧^𝑇𝑡−ℎ_1𝑚𝑅𝑧𝑡−ℎ_1𝑚,−ℎ_1𝑀𝑒^{−2𝛼ℎ1𝑀𝑡}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑅𝑧̇(𝑠)𝑑𝑠≤−𝑒^{−2𝛼ℎ1𝑀𝑡}_{𝑡−ℎ1𝑀}𝑧̇(𝑠)^𝑇𝑅^𝑡_{𝑡−ℎ1𝑀}𝑧̇(𝑠)≤−𝑒^{−2𝛼ℎ1𝑀}𝑧(𝑡)−𝑧𝑡−ℎ_1𝑀^𝑇𝑅𝑧(𝑡)−𝑧𝑡−ℎ_1𝑀=−𝑒^{−2𝛼ℎ1𝑀}𝑧^𝑇(𝑡)𝑅𝑧(𝑡)−2𝑧^𝑇(𝑡)𝑅𝑧𝑡−ℎ_1𝑀+𝑧^𝑇𝑡−ℎ_1𝑀𝑅𝑧𝑡−ℎ_1𝑀. (33) Note that −𝛿^{𝑡−ℎ1𝑚}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)ℎ𝑑𝑠=−_1𝑀−ℎ_1𝑚^{𝑡−ℎ(𝑡)}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)−ℎ𝑑𝑠_1𝑀−ℎ_1𝑚^{𝑡−ℎ1𝑚}_{(𝑡)𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)ℎ𝑑𝑠=−_1𝑀−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)−ℎ(𝑡)𝑑𝑠−ℎ_1𝑚^{𝑡−ℎ(𝑡)}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)−ℎ(𝑡)𝑑𝑠−ℎ_1𝑚^{𝑡−ℎ1𝑚}_{(𝑡)𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)−ℎ𝑑𝑠_1𝑀−ℎ(𝑡)^{𝑡−ℎ1𝑚}_{(𝑡)𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑑𝑠. (34) Using Proposition 4 gives −ℎ_1𝑀−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑑𝑠≤−^{𝑡−ℎ(𝑡)}_{𝑡−ℎ1𝑀}𝑧̇(𝑠)𝑑𝑠^𝑇𝑈^{𝑡−ℎ(𝑡)}_{𝑡−ℎ1𝑀}𝑧̇(𝑠)𝑧((𝑡𝑑𝑠≤−𝑡−ℎ))−𝑧𝑡−ℎ_1𝑀^𝑇𝑈×𝑧(𝑡−ℎ(𝑡))−𝑧𝑡−ℎ_1𝑀,−ℎ(𝑡)−ℎ_1𝑚^{𝑡−ℎ1𝑀}_{(𝑡)𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑑𝑠≤−^{𝑡−ℎ1𝑚}_{(𝑡)𝑡−ℎ}𝑧̇(𝑠)𝑑𝑠^𝑇𝑈^{𝑡−ℎ1𝑚}_{(𝑡)𝑡−ℎ}𝑧̇(𝑠)𝑧𝑑𝑠≤−𝑡−ℎ_1𝑚−𝑧(𝑡−ℎ(𝑡))^𝑇𝑈×𝑧𝑡−ℎ_1𝑚−𝑧(𝑡−ℎ(𝑡)). (35) Let 𝛽=(ℎ_1𝑀−ℎ(𝑡))/(ℎ_1𝑀−ℎ_1𝑚)≤1. Then −ℎ_1𝑀−ℎ(𝑡)^{𝑡−ℎ1𝑚}_{(𝑡)𝑡−ℎ}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑑𝑠=−𝛽^{𝑡−ℎ1𝑚}_{(𝑡)𝑡−ℎ}ℎ_1𝑀−1𝑚𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑑𝑠≤−𝛽^{𝑡−ℎ1𝑚}_{(𝑡)𝑡−ℎ}ℎ(𝑡)−ℎ_1𝑚𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑧𝑑𝑠≤−𝛽𝑡−ℎ_1𝑚−𝑧(𝑡−ℎ(𝑡))^𝑇𝑈×𝑧𝑡−ℎ_1𝑚((𝑡,−ℎ(𝑡)−𝑧𝑡−ℎ))−ℎ_1𝑚^{𝑡−ℎ(𝑡)}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑑𝑠=−1−𝛽^{𝑡−ℎ(𝑡)}_{𝑡−ℎ1𝑀}ℎ_1𝑀−ℎ_1𝑚𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑑𝑠≤−1−𝛽^{𝑡−ℎ(𝑡)}_{𝑡−ℎ1𝑀}ℎ₂−ℎ(𝑡)𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑧((𝑡𝑑𝑠≤−1−𝛽𝑡−ℎ))−𝑧𝑡−ℎ_1𝑀^𝑇×𝑈𝑧(𝑡−ℎ(𝑡))−𝑧𝑡−ℎ_1𝑀. (36) Therefore from (35)-(36), we obtain −𝛿^{𝑡−ℎ1𝑚}_{𝑡−ℎ1𝑀}𝑧̇^𝑇(𝑠)𝑈𝑧̇(𝑠)𝑧((𝑡𝑑𝑠≤−𝑡−ℎ))−𝑧𝑡−ℎ_1𝑀^𝑇×𝑈𝑧(𝑡−ℎ(𝑡))−𝑧𝑡−ℎ_1𝑀−𝑧𝑡−ℎ_1𝑚−𝑧(𝑡−ℎ(𝑡))^𝑇×𝑈𝑧𝑡−ℎ_1𝑚((𝑡𝑧−𝑧𝑡−ℎ))−𝛽𝑡−ℎ_1𝑚−𝑧(𝑡−ℎ(𝑡))^𝑇×𝑈𝑧𝑡−ℎ_1𝑚((𝑡−𝑧((𝑡−𝑧𝑡−ℎ))1−𝛽𝑡−ℎ))−𝑧𝑡−ℎ_1𝑀^𝑇×𝑈𝑧(𝑡−ℎ(𝑡))−𝑧𝑡−ℎ_1𝑀. (37) By using the following identity relation: 0=−𝑒̇(𝑡)−𝐴𝑒(𝑡)+𝐶𝑓(𝑒(𝑡))+𝐷𝑔𝑒𝑡−ℎ₁(𝑡)+𝐸^𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠+𝐵₁𝐾𝑒(𝑡)+𝐵₂𝐾𝑒𝑡−ℎ₂(𝑡)+𝐵₃𝐾^𝑡_{𝑡−𝑘2(𝑡)}𝑒(𝑠)𝑧̇(𝑡)(𝑡)(𝑒(𝑡𝑒𝑑𝑠,=−𝑃−𝐴𝑃𝑧+𝐶𝑓))+𝐷𝑔𝑡−ℎ₁(𝑡)+𝐸^𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠−𝐵₁𝑌𝑧(𝑡)+𝐵₂(𝑡)𝑢𝑡−ℎ₂(𝑡)+𝐵₃^𝑡_{𝑡−𝑘2(𝑡)}𝑢(𝑠)𝑑𝑠, (38) we have 0=−2𝑧̇^𝑇(𝑡)𝑃𝑧̇(𝑡)−2𝑧̇^𝑇(𝑡)𝐴𝑃𝑧(𝑡)+2𝑧̇^𝑇(𝑡)(𝑒(𝑡𝑧̇𝐶𝑓))+2^𝑇(𝑡)𝐷𝑔𝑒𝑡−ℎ₁(𝑡𝑧̇)+2^𝑇(𝑡)𝐸^𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠−2𝑧̇^𝑇(𝑡)𝐵₁(𝑡)𝑧̇𝑌𝑧+2^𝑇(𝑡)𝐵₂(𝑡)𝑢𝑡−ℎ₂(𝑡)+2𝑧̇^𝑇(𝑡)𝐵₃^𝑡_{𝑡−𝑘2(𝑡)}𝑢(𝑠)𝑑𝑠. (39) By using Propositions 3 and 4, we have 2𝑧̇^𝑇(𝑡)(𝑒(𝑡≤𝑧̇𝐶𝑓))_𝑇(𝑡)𝐶^𝑇𝑈₁𝐶𝑧̇(𝑡)+𝑓^𝑇(𝑒(𝑡))𝑈₁⁻¹𝑓(𝑒(𝑡≤𝑧̇))_𝑇(𝑡)𝐶^𝑇𝑈₁𝐶𝑧̇(𝑡)+𝑒^𝑇(𝑡)𝐹^𝑇𝑈₁⁻¹(𝑡)=𝑧̇𝐹𝑒_𝑇(𝑡)𝐶^𝑇𝑈₁𝐶𝑧̇(𝑡)+𝑧^𝑇(𝑡)𝑃𝐹^𝑇𝑈₁⁻¹(𝑡),2𝑧̇𝐹𝑃𝑧_𝑇(𝑡)𝐷𝑔𝑒𝑡−ℎ₁(𝑡)≤𝑧̇^𝑇(𝑡)𝐷^𝑇𝑈₂𝐷𝑧̇(𝑡)+𝑔^𝑇𝑒𝑡−ℎ₁(𝑡)×𝑈₂⁻¹𝑔𝑒𝑡−ℎ₁(𝑡)≤𝑧̇^𝑇(𝑡)𝐷^𝑇𝑈₂𝐷𝑧̇(𝑡)+𝑒^𝑇𝑡−ℎ₁(𝑡)𝐺^𝑇𝑈₂⁻¹×𝐺𝑒𝑡−ℎ₁(𝑡)=𝑧̇^𝑇(𝑡)𝐷^𝑇𝑈₂𝐷𝑧̇(𝑡)+𝑧^𝑇𝑡−ℎ₁(𝑡)𝑃𝐺^𝑇𝑈₂⁻¹×𝐺𝑃𝑧𝑡−ℎ₁(𝑡),2𝑧̇^𝑇(𝑡)𝐸^𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠≤2𝑘₁𝑒^2𝛼𝑘1𝑧̇^𝑇(𝑡)𝐸^𝑇𝑈₃𝐸𝑧̇(𝑡)+12𝑘₁𝑒^{−2𝛼𝑘1𝑡}_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠^𝑇×𝑈₃^−1𝑡_{𝑡−𝑘1(𝑡)}ℎ(𝑒(𝑠))𝑑𝑠≤2𝑘₁𝑒^2𝛼𝑘1𝑧̇^𝑇(𝑡)𝐸^𝑇𝑈₃𝐸𝑧̇(𝑡)+𝑒^{−2𝛼𝑘1}2^𝑡_{𝑡−𝑘1(𝑡)}ℎ^𝑇(𝑒(𝑠))𝑈₃⁻¹ℎ(𝑒(𝑠2𝑧̇))𝑑𝑠,_𝑇(𝑡)𝐵₂(𝑡)𝑢𝑡−ℎ₂(𝑡)≤3𝑒^2𝛼ℎ2𝑧̇^𝑇(𝑡)𝐵^𝑇₂𝑆₁𝐵₂𝑧̇(𝑡)+𝑒^−2𝛼ℎ23𝑢^𝑇𝑡−ℎ₂(𝑡)𝑆₁⁻¹𝑢𝑡−ℎ₂(𝑡),2𝑧̇^𝑇(𝑡)𝐵₃^𝑡_{𝑡−𝑘2(𝑡)}𝑢(𝑠)𝑑𝑠≤2𝑘₂𝑒^2𝛼𝑘2𝑧̇^𝑇(𝑡)𝐵^𝑇₃𝑆₂𝐵₃𝑧̇(𝑡)+𝑒^{−2𝛼𝑘2}2𝑘₂^𝑡_{𝑡−𝑘2(𝑡)}𝑢(𝑠)𝑑𝑠^𝑇𝑆₂^−1𝑡_{𝑡−𝑘1(𝑡)}𝑢(𝑠)𝑑𝑠≤2𝑘₂𝑒^2𝛼𝑘2𝑧̇^𝑇(𝑡)𝐵^𝑇₃𝑆₂𝐵₃𝑧̇(𝑡)+𝑒^{−2𝛼𝑘2}2^𝑡_{𝑡−𝑘2(𝑡)}𝑢^𝑇(𝑠)𝑆₂⁻¹𝑢(𝑠)𝑑𝑠. (40) From (29)–(40), we obtain 𝑉̇(𝑡,𝑒(𝑡))+2𝛼𝑉(𝑡,𝑒(𝑡))≤𝜉^𝑇(𝑡)1−𝛽ℳ₁+𝛽ℳ₂𝜉(𝑡)+𝑧^𝑇(𝑡)ℳ₃𝑧(𝑡)+𝑧̇^𝑇(𝑡)ℳ₄𝑧̇(𝑡)+𝑧^𝑇𝑡−ℎ₁(𝑡)ℳ₅𝑧𝑡−ℎ₁(𝑡)−𝑒^𝑇(𝑡)ℳ₆𝑒(𝑡), (41) where ℳ₃=−0.1𝑅𝑒^{−2𝛼ℎ1𝑚}+𝑒^{−2𝛼ℎ1𝑀}+2𝑃𝐹^𝑇𝑈₁⁻¹𝐹𝑃+𝑃𝐻^𝑇𝑈₃⁻¹𝐻𝑃+2𝑒^−2𝛼ℎ2𝑌^𝑇𝑆₁⁻¹𝑌+𝑃𝑄₁𝑃+𝑌^𝑇𝑄₂ℳ𝑌,₄=−0.1𝑃+ℎ²₂𝑌^𝑇𝑆₁⁻¹ℳ𝑌,₅=−0.1𝑒^{−2𝛼ℎ1𝑀}𝑈+2𝑃𝐺^𝑇𝑈₂⁻¹ℳ𝐺𝑃,₆=𝑄₁+𝑃⁻¹𝑌^𝑇𝑄₂𝑌𝑃⁻¹,𝜉(𝑡)=𝑧(𝑡),𝑧̇(𝑡),𝑧𝑡−ℎ_1𝑚,𝑧(𝑡−ℎ(𝑡)),𝑧𝑡−ℎ_1𝑀. (42) 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 Γ₁<0 and Γ₂<0. Applying Schur complement lemma, Proposition 5, the inequalities ℳ₃<0, ℳ₄<0, and ℳ₅<0 are equivalent to Γ₃<0, Γ₄<0, and Γ₅<0, respectively. Thus, from (18)–(21) and (41), we obtain 𝑉̇(𝑡,𝑒(𝑡))+2𝛼𝑉(𝑡,𝑒(𝑡))≤−𝑒^𝑇(𝑡)ℳ₆𝑒(𝑡),∀𝑡≥0. (43) Since ℳ₆>0, we have 𝑉̇(𝑡,𝑒(𝑡))≤−2𝛼𝑉(𝑡,𝑒(𝑡)),∀𝑡≥0. (44) Integrating both sides of (44) from 0 to 𝑡, we obtain 𝑉(𝑡,𝑒(𝑡))≤𝑉𝜙(𝑡)𝑒^−2𝛼𝑡,∀𝑡≥0. (45) Furthermore, taking condition (28) into account, we have 𝜆₁𝑒𝑡,𝜙²≤𝑉(𝑡,𝑒(𝑡))≤𝑉𝜙(𝑡)𝑒^−2𝛼𝑡≤𝜆₂𝑒^−2𝛼𝑡𝜙². (46) Because 𝑉(𝑡,𝑒(𝑡)) is radially unbounded, by the Lyapunov-Krasovskii theorem and the solution ‖𝑒(𝑡,𝜙)‖ of the error system (6) satisfy 𝑒𝑡,𝜙≤𝜆₂𝜆₁𝑒^−𝛼𝑡𝜙,∀𝑡≥0, (47) which implies the exponential stability of the error system (6) under the controller (H1). Consequentially, the controlled slave system (1) is synchronized with the master system (2).
Furthermore, from (43) and 𝑉(𝑡,𝑒(𝑡))>0, we have 𝑉̇(𝑡,𝑒(𝑡))≤−𝑒^𝑇(𝑡)ℳ₅𝑒(𝑡),∀𝑡≥0. (48) Integrating both sides of (48) from 0 to 𝑡, we obtain ^𝑡₀𝑒^𝑇(𝑡)ℳ₅𝑒(𝑡)𝑑𝑡≤𝑉0,𝜙−𝑉(𝑡,𝑒(𝑡))≤𝑉0,𝜙,∀𝑡≥0, (49) due to 𝑉(𝑡,𝑒(𝑡))≥0. Hence ^𝑡₀𝑒^𝑇(𝑡)ℳ₅𝑒(𝑡)𝑑𝑡≤𝑉0,𝜙=𝜆₂𝜙². (50) Given 𝑡→∞, we obtain 𝐽=^∞₀𝑒^𝑇(𝑡)ℳ₅𝑒(𝑡)𝑑𝑡≤𝜆₂𝜙²=𝐽^∗. (51) The proof is completed.

Remark 7. In our main results, guaranteed cost synchronization problem for cellular neural networks with interval nondifferentiable time-varying and distributed time-varying delays is considered. We first construct the improved Lyapunov-Krasovskii functionals 𝑉(𝑡,𝑒(𝑡)) as shown in (26). We give sufficient conditions for the exponential synchronization of the error systems are independent on the derivatives of the time-varying delays and without introducing any free-weighting matrices turn out to be less conservative with fewer matrix variables than see [7, 9, 20, 35, 38, 39, 41].

Remark 8. In most results on guaranteed cost synchronization problem for cellular neural networks, authors have considered only activation functions with time-varying delay [21, 22]. But in our works, we have considered a more complicated problem, namely, guaranteed cost synchronization of cellular neural networks with various activation functions and mixed time-varying delays in state and feedback control term simultaneously. To the best of our knowledge, our results are among the first results on guaranteed cost synchronization of cellular neural networks with various activation functions and mixed time-varying delays using hybrid feedback control. Therefore, our stability conditions are less conservative than other existing results [21, 22].

4. Numerical Examples

In this section, we now provide an example to show the effectiveness of the result in Theorem 6.

Example 9. Consider the cellular neural networks with various activation functions and mixed time-varying delays using hybrid feedback control with the following parameters: 𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝐶𝑓^∼(𝑥(𝑡))+𝐷𝑔^∼𝑥𝑡−ℎ₁(𝑡)+𝐸ℎ^𝑡_{𝑡−𝑘1(𝑡)}^∼(𝑥(𝑠(𝑡),𝑥(𝑡)))𝑑𝑠+𝐼=𝜙₁(𝑡)[],𝑦̇(𝑡)(𝑡),𝑡∈−𝑑,0=−𝐴𝑦+𝐶𝑓^∼𝑦(𝑡)+𝐷𝑔^∼𝑦𝑡−ℎ₁(𝑡)+𝐸ℎ^𝑡_{𝑡−𝑘1(𝑡)}^∼𝑦(𝑠)𝑑𝑠+𝐼(𝑡)+𝐵₁𝑢(𝑡)+𝐵₂𝑢𝑡−ℎ₂(𝑡)+𝐵₃^𝑡_{𝑡−𝑘2(𝑡)}𝑢(𝑠)𝑦(𝑡)𝑑𝑠,=𝜙₂(𝑡),𝑡∈[−𝑑,0], (52)(53) where ,,𝐴=1001,𝐶=0.2−0.20.1−0.3,𝐷=0.20.1−0.10.2𝐸=0.10.2−0.50.1,𝐹=0.5000.3,𝐺=0.5000.4𝐻=0.4000.2,𝐵₁=4004,𝐵₂=,𝐵1001₃=,𝜙1001₁(𝑡)=[−0.4cos𝑡,0.5cos𝑡],𝜙₂(𝑡)=[sin𝑡,sin𝑡]. (54)

Solution 1. From the conditions (18)–(22) of Theorem 6, we let 𝛼=0.02, ℎ_1𝑚=0.1, ℎ_1𝑀=0.3, ℎ₂=0.3, 𝑘₁=0.3, 𝑘₂=0.2, 𝑄₁=³⁰₀₂, and 𝑄₂=²⁰₀₁. By using the LMI Toolbox in MATLAB, we obtain ,,𝑆𝑃=00.07890.00660.00660.1024,𝑄=0.01080.00390.00390.0196𝑅=0.21390.03880.03880.2779,𝑈=0.13750.03360.03360.2546₁=0.00090.00020.00020.0012,𝑆₂=,𝑈0.00120.00090.00090.0027₁=0.1649000.0935,𝑈₂=,𝑈0.2482000.1464₃=0.2491000.0336,𝑌=10⁻³,,0.14910.12490.07260.3067𝐾=−0.0018−0.0011−0.0007−0.0030 (55) and accordingly the feedback control is 𝑢(𝑡)=^{−0.0018−0.0011}_{−0.0007−0.0030}(𝑦(𝑡)−𝑥(𝑡)). Thus, is 0.02 exponentially synchronization and the value 𝜆₂/𝜆₁=5.2860, so the solution of the closed-loop system satisfies 𝑦𝑡,𝜙₂−𝑥𝑡,𝜙₁≤5.2860𝑒^−0.02𝑡𝜙,∀𝑡∈ℝ⁺, (56) and the optimal guaranteed cost of the closed-loop system is as follows: 𝐽≤𝐽^∗=4.6365. (57) We let ℎ₁(𝑡)=0.1+0.2|sin𝑡|, ℎ₂(𝑡)=0.3𝑒^|sin𝑡|, 𝑘₁(𝑡)=0.3|cos𝑡|, 𝑘₂(𝑡)=0.2𝑒^|cos𝑡|, 𝜙₁(𝑡)=[−0.4cos𝑡,0.5cos𝑡], 𝜙₂(𝑡)=[sin𝑡,sin𝑡], for all 𝑡∈[−0.3,0], and the activation functions as follows: 𝑓₁𝑥₁(𝑡)=0.25𝑥₁(𝑡)+1−𝑥₁(𝑡),𝑓−1₂𝑥₂(𝑡)=0.15𝑥₁(𝑡)+1−𝑥₁(𝑡),𝑔−1₁𝑥₁(𝑡)=0.25𝑥₁(𝑡)+1−𝑥₁(𝑡),𝑔−1₂𝑥₂(𝑡)=0.2𝑥₂(𝑡)+1−𝑥₂(𝑡),ℎ−1₁𝑥₁(𝑠)=0.4tanh−6𝑥₁(𝑠),ℎ₂𝑥₂(𝑠)=0.2tanh7𝑥₂(𝑠). (58) Figure 1 shows the trajectories of solutions 𝑒₁(𝑡) and 𝑒₂(𝑡) of the cellular neural networks with various activation functions and mixed time-varying delays without hybrid feedback control (𝑢(𝑡)=0). Figure 2 shows the trajectories of solutions 𝑒₁(𝑡) and 𝑒₂(𝑡) of the cellular neural networks with various activation functions and mixed time-varying delays with hybrid feedback control 𝑢(𝑡)=^{−0.0018−0.0011}_{−0.0007−0.0030}(𝑦(𝑡)−𝑥(𝑡)).

Figure 1 

Synchronization error curves of the master system (52) and the slave system (53) without hybrid feedback control.

Figure 2 

Synchronization error curves of the master system (52) and the slave system (53) with hybrid feedback control input.

Remark 10. The advantage of Example 9 is the lower bound of the delay ℎ_𝑚≠0 and interval time-varying delay and distributed time-varying delay are nondifferentiable. Moreover, in these examples we still investigate various activation functions and mixed time-varying delays in state and feedback control term simultaneously; hence the synchronization conditions derived in [21, 22] cannot be applied to these examples.

5. Conclusions

In this paper, we have investigated the exponential synchronization of cellular neural networks with various activation functions and mixed time-varying delays via hybrid feedback control. The interval time-varying delay function is not necessary to be differentiable which allows time-delay function to be a fast time-varying function. A new class of Lyapunov-Krasovskii functional is constructed in new delay-dependent sufficient conditions for the exponential synchronization of the error systems, which have been derived by a set 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. Simulation results have been given to illustrate the effectiveness of the proposed method.

Acknowledgments

The authors thank anonymous reviewers for their valuable comments and suggestions. This work is supported by Coordinating Center for Thai Government Science and Technology Scholarship Students (CSTS) of the National Science and Technology Development Agency (NSTDA) (SCH-NR2011-841), the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand, and 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.