Exponential Synchronization for Neutral Complex Dynamical ...

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 639580, 21 pages http://dx.doi.org/10.1155/2013/639580

Research Article Exponential Synchronization for Neutral Complex Dynamical Networks with Interval Mode-Dependent Delays and Sampled Data Xinghua Liu and Hongsheng Xi Department of Auto, School of Information Science and Technology, University of Science and Technology of China, Anhui 230027, China Correspondence should be addressed to Xinghua Liu; [email protected] Received 26 May 2013; Revised 7 August 2013; Accepted 8 August 2013 Academic Editor: Jun-Juh Yan Copyright © 2013 X. Liu and H. Xi. 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 exponential synchronization and sampled-data controller problem for a class of neutral complex dynamical networks (NCDNs) with Markovian jump parameters, partially unknown transition rates and delays, is investigated in this paper. Both the discrete and neutral delays are considered to be interval mode dependent and time varying, while the sampling period is assumed to be time varying and bounded. Based on a new augmented stochastic Lyapunov functional, the delay-range-dependent and rate-dependent exponential stability conditions for the closed-loop error system are obtained by the Lyapunov-Krasovskii stability theory and reciprocally convex lemma. Then according to the proposed exponential stability conditions, the sampled-data synchronization controllers are designed in terms of the solution to linear matrix inequalities that can be solved effectively by using Matlab. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the proposed methods.

1. Introduction It is well known that many practical systems can be described as complex networks such as Internet networks, biological networks, epidemic spreading networks, collaborative networks, social networks and neural networks [1–4]. Thus, during the past decades, the research on the dynamics of complex dynamical networks (CDNs) has attracted extensive attention of scientific and engineering researchers in all fields domestic and overseas since the pioneering work of Watts and Strogatz [5]. As one of the most significant and important collective behaviors in CDNs, synchronization has received more attention; please refer to [6–10] and references therein for more details. Since time delay inevitably exists and has become an important issue in studying the CDNs, synchronization problems for complex networks with time delays have gained increasing research attention, and considerable progress has been made; see, for example, [6–17] and references therein for more details. However, in some practical applications of communication networks, signal transmission channels often

involve network-induced delays, packet dropouts, bit errors, environment disturbances, and so on, which will cause the error of transmission from one system to another. Then past change rate of the state variables affects the dynamics of nodes in the networks. This kind of complex dynamical network is termed as neutral complex dynamical network (NCDN), which contains delays both in its states and the derivatives of its states. There are some results about the synchronization design problem for neutral systems [18–23]. In these works, [19, 20] had studied the synchronization control for a kind of master-response setup and were further extended to the case of neutral-type neural networks with stochastic perturbation. The authors of [18, 22] had researched the synchronization problem for a class of complex networks with neutral-type coupling delays. The authors in [21] had studied the global asymptotic stability of neural networks of neutral type with mixed delays, which include constant delay in the leakage term, time-varying delays, and continuously distributed delays. The authors in [23] had investigated the robust global exponential synchronization problem for an array of neutraltype neural networks. However, much fewer results have been

2 proposed for neutral complex dynamical networks (NCDNs) compared with the rich results for CDNs with only discrete delays. On the other hand, network mode switching is also a universal phenomenon in CDNs of the actual systems, and sometimes the network has finite modes that switch from one to another with certain transition rate; then such switching can be governed by a Markovian chain. The stability and synchronization problems of complex networks and neural networks with Markovian jump parameters and delays are investigated in [16, 24–30], and references therein. The authors in [24] have established sufficient global exponential stability conditions on Markovian jump neural networks with impulse control and time varying delays. The authors in [27] have studied synchronization in an array of coupled neural networks with Markovian jumping and random coupling strength. Particularly, Ma et al. [25] have considered the stability and synchronization problems for Markovian jump delayed neural networks with partly unknown transition probabilities, which have not been fully investigated and need to propose more good results. Besides, sampled-data systems have attracted great attention because the digital signal processing methods require better reliability, accuracy, and stable performance with the rapid development in digital measurement and intelligent instrument. There are some important and essential results which have been reported in the literature [31–35]. What is worth mentioning is that the sampled-data synchronization control problem has been investigated for a class of general complex networks with time-varying coupling delays in [34, 35], where conditions have been presented to ensure the exponential stability of the closed-loop error system, and the desired sampled data feedback controllers have been designed. However, few results are available for neutral complex dynamical networks (NCDNs) with sampled data. To the best of the authors’ knowledge, the NCDNs are difficult to treat and they are very challenging, especially in the presence of Markovian jump parameters, mode-dependent time-varying delays, and sampled data. Motivated by the above analysis, the exponential synchronization and sampled-data controller problem for a class of NCDNs with Markovian jump parameters and modedependent time-varying delay is investigated in this paper. The addressed NCDNs consist of 𝑀 modes, and the networks switch from one mode to another according to a Markovian chain with partially known transition rate. In this paper, the synchronization and sampled-data controller problem is studied for NCDNs with Markovian jump parameters and partially known transition rates. The sampling period considered here is assumed to be time varying and bounded, while the neutral and discrete delays are interval mode-dependent and time varying. Firstly, by constructing a new augmented stochastic Lyapunov functional, exponential stability conditions are derived based on the Lyapunov stability theory and reciprocally convex lemma. Then the design method of the desired sampled-data controllers is solved on the basis of the obtained conditions. Moreover, all the derived results are in terms of LMIs that can be solved numerically, which are proved to be less conservative than the existing results.

Mathematical Problems in Engineering The remainder of the paper is organized as follows. Section 2 presents the problem and preliminaries. Section 3 gives the main results, which are then verified by numerical examples in Section 4. Section 5 concludes the paper. Notations. The following notations are used throughout the paper. R𝑛 denotes the 𝑛 dimensional Euclidean space and R𝑚×𝑛 is the set of all 𝑚 × 𝑛 matrices. 𝑋 < 𝑌 (𝑋 > 𝑌), where 𝑋 and 𝑌 are both symmetric matrices, which means that 𝑋 − 𝑌 is negative (positive) definite. 𝐼 is the identity matrix with proper dimensions. For a symmetric block matrix, we use ∗ to denote the terms introduced by symmetry. E stands for the mathematical expectation; ‖V‖ is the Euclidean norm of vector V, ‖V‖ = (V𝑇 V)1/2 , while ‖𝐴‖ is spectral norm of matrix 𝐴, ‖𝐴‖ = [𝜆 max (𝐴𝑇 𝐴)]1/2 . 𝜆 max(min) (𝐴) is the eigenvalue of matrix 𝐴 with maximum (minimum) real part. The Kronecker product of matrices 𝑃 ∈ R𝑚×𝑛 and 𝑄 ∈ R𝑝×𝑞 is a matrix in R𝑚𝑝×𝑛𝑞 which is denoted as 𝑃 ⊗ 𝑄. Let 𝜍 > 0 and 𝐶([−𝜍, 0], R𝑛 ) denote the family of continuous function 𝜑, from [−𝜍, 0] to R𝑛 with the norm |𝜑| = sup−𝜍≤𝜃≤0 ‖𝜑(𝜃)‖. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Problem Statement and Preliminaries Given a complete probability space {Ω, F, {F𝑡 }𝑡≥0 , P} with a natural filtration {F𝑡 }𝑡≥0 satisfying the usual conditions, where Ω is the sample space, F is the algebra of events and P is the probability measure defined on F. Let {𝑟(𝑡), 𝑡 ≥ 0} be a right-continuous Markov chain taking values in a finite state space 𝑆 = {1, 2, 3, . . . , 𝑀} with a generator Υ = (𝛾𝑖𝑗 )𝑀×𝑀, 𝑖, 𝑗 ∈ 𝑆, which is given by 𝑃 ( 𝑟 (𝑡 + Δ𝑡) = 𝑗 | 𝑟 (𝑡) = 𝑖) = {

𝑖 ≠ 𝑗, 𝛾𝑖𝑗 Δ𝑡 + 𝑜 (Δ𝑡) 1 + 𝛾𝑖𝑖 Δ𝑡 + 𝑜 (Δ𝑡) 𝑖 = 𝑗, (1)

where Δ𝑡 > 0, limΔ𝑡 → 0 (𝑜(Δ𝑡)/Δ𝑡) = 0, and 𝛾𝑖𝑗 ≥ 0 (𝑖, 𝑗 ∈ 𝑆, 𝑖 ≠ 𝑗) is the transition rate from mode 𝑖 to 𝑗, and for any state or mode 𝑖 ∈ 𝑆, it satisfies 𝑁

𝛾𝑖𝑖 = − ∑ 𝛾𝑖𝑗 .

(2)

𝑗=1,𝑗 ≠ 𝑖

It is assumed that 𝑟(𝑡) is irreducible and available at time 𝑡, but the transition rates of the Markov chain are partially known in this paper, which means that some elements in matrix Υ = (𝛾𝑖𝑗 )𝑀×𝑀 are inaccessible. For instance, in the system with six operation modes, the jump rates matrix Υ may be viewed as 0 𝛾11 [𝛾 [ 21 [ 0 [𝛾31 [ [𝛾0 [ 41 [ [𝛾51 [𝛾61

𝛾12 0 𝛾22 0 𝛾32 𝛾42 0 𝛾52 0 𝛾62

0 𝛾13 0 𝛾23 𝛾33 0 𝛾43 0 𝛾53 0 𝛾63

0 𝛾14 𝛾24 𝛾34 0 𝛾44 0 𝛾54 𝛾64

𝛾15 0 𝛾25 0 𝛾35 0 𝛾45 𝛾55 𝛾65

0 𝛾16 𝛾26 ] ] 0 ] ] 𝛾36 ], 𝛾46 ] ] 0 ] ] 𝛾56 0 𝛾66 ]

(3)

Mathematical Problems in Engineering

3

where 𝛾𝑖𝑗0 , 𝑖, 𝑗 ∈ 𝑆 represents the unknown element. Furthermore, let 𝛾𝑖𝑚 , 𝛾𝑖𝑀 be lower and upper bound for the diagonal element 𝛾𝑖𝑖 or 𝛾𝑖𝑖0 , for all 𝑖 ∈ 𝑆. For notation clarity, we denote that S𝑖 = S𝑖𝑘 ⋃ S𝑖𝑢𝑘 , for all 𝑖 ∈ 𝑆, and S𝑖𝑘 S𝑖𝑢𝑘 If

S𝑖𝑘

≜ {𝑗 : 𝜋𝑖𝑗 is known for 𝑗 ∈ 𝑆} ,

≜ {𝑗 : 𝜋𝑖𝑗 is unknown for 𝑗 ∈ 𝑆} .

(4)

=

𝑖 {𝑘1𝑖 , 𝑘2𝑖 , . . . , 𝑘𝑚 },

1 ≤ 𝑚 ≤ 𝑀,

(5)

where 𝑘𝑗𝑖 , (𝑗 = 1, 2, . . . , 𝑚) represent the 𝑗th known element of the set S𝑖𝑘 in the 𝑖th row of the transition rate matrix Υ. It should be noted that if S𝑖𝑘 = 0, S𝑖 = S𝑖𝑢𝑘 which means that any information between the 𝑖th mode and the other 𝑀 − 1 modes is not accessible, then MJSs with 𝑀 modes can be regarded as ones with 𝑀 − 1 modes. The following neutral complex dynamical network (NCDN) consisting of 𝑁 identical nodes with Markovian jump parameters and interval time-varying delays over the space {Ω, F, {F𝑡 }𝑡≥0 , P} is investigated in this paper: 𝑁

(1) 𝑥𝑘̇ (𝑡) = ∑𝑔𝑘𝑙 (𝑟 (𝑡)) 𝐴 (𝑟 (𝑡)) 𝑥𝑙 (𝑡) 𝑙=1

𝑁

(2) + ∑𝑔𝑘𝑙 (𝑟 (𝑡)) 𝐵 (𝑟 (𝑡)) 𝑥𝑙 (𝑡 − 𝑑 (𝑡, 𝑟 (𝑡))) 𝑙=1

(6)

𝑁

(3) + ∑𝑔𝑘𝑙 (𝑟 (𝑡)) 𝐶 (𝑟 (𝑡)) 𝑥𝑙̇ (𝑡 − 𝜏 (𝑡, 𝑟 (𝑡)))

𝑁

𝑙=1,𝑙 ≠ 𝑘

𝑚 = 1, 2, 3; 𝑘 = 1, 2, . . . , 𝑁.

(8)

when 𝑟 (𝑡) = 𝑖,

where 𝜏𝑖 , 0 ≤ ]𝑖 < 1, 𝑑1𝑖 , and 𝑑2𝑖 , 0 ≤ 𝜇𝑖 , are real constant scalars. 𝐷(𝑟(𝑡)) ∈ R𝑛×𝑛 is a parametric matrix with real values in mode 𝑟(𝑡). 𝑓: R𝑛 → R𝑛 is a continuous vector-valued nonlinear function. It is assumed to satisfy the following sector-bounded condition [36]: [𝑓 (𝑥) − 𝑓 (𝑦) − 𝐹1 (𝑥 − 𝑦)]

𝑇

× [𝑓 (𝑥) − 𝑓 (𝑦) − 𝐹2 (𝑥 − 𝑦)] ≤ 0,

(9)

∀𝑥, 𝑦 ∈ R𝑛 ,

where 𝐹1 , 𝐹2 are two constant matrices. Such a description of nonlinear functions has been exploited in [37–39] and is more general than the commonly used Lipschitz conditions. The sector-bounded condition would be possible to reduce the conservatism of the main results caused by quantifying the nonlinear functions via a matrix inequality technique. Let 𝑠(𝑡) ∈ R𝑛 be the state trajectory of the unforced ̇ = 𝐷(𝑟(𝑡))𝑓(𝑠(𝑡)); then the synchronization error node 𝑠(𝑡) is defined to be 𝑒𝑘 (𝑡) = 𝑥𝑘 (𝑡) − 𝑠(𝑡). So the error dynamics of NCDN (6) can be derived as follows:

𝑁

(2) + ∑𝑔𝑘𝑙 (𝑟 (𝑡)) 𝐵 (𝑟 (𝑡)) 𝑒𝑙 (𝑡 − 𝑑 (𝑡, 𝑟 (𝑡))) 𝑙=1

(7)

(10)

𝑁

(3) + ∑𝑔𝑘𝑙 (𝑟 (𝑡)) 𝐶 (𝑟 (𝑡)) 𝑒𝑙̇ (𝑡 − 𝜏 (𝑡, 𝑟 (𝑡))) 𝑙=1

+ 𝐷 (𝑟 (𝑡)) 𝑓𝑒 (𝑒𝑘 (𝑡)) + 𝑢𝑘 (𝑡) , where 𝑓𝑒 (𝑒𝑘 (𝑡)) = 𝑓(𝑥𝑘 (𝑡)) − 𝑓(𝑠(𝑡)), 𝑘 = 1, 2, . . . , 𝑁. The control signal is assumed to be generated by using a zero-order-hold (ZOH) function with a sequence of hold times 0 = 𝑡0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑠 < ⋅ ⋅ ⋅ . Thus the state feedback controller takes the following form: 𝑢𝑘 (𝑡) = 𝐾𝑘 𝑒𝑘 (𝑡𝑠 ) ,

(𝑚) (𝑚) 𝑔𝑘𝑘 (𝑟 (𝑡)) = − ∑ 𝑔𝑘𝑙 (𝑟 (𝑡))

(𝑟 (𝑡)) ,

𝑗∈𝑆

𝑖∈𝑆

𝑙=1

where 𝑥𝑘 (𝑡) ∈ R𝑛 , 𝑢𝑘 (𝑡) ∈ R𝑛 are state variable and the control input of the node 𝑘 ∈ {1, 2, . . . , 𝑁}, respectively. 𝑟(𝑡) describes the evolution of the mode. 𝐴(𝑟(𝑡)) ∈ R𝑛×𝑛 , 𝐵(𝑟(𝑡)) ∈ R𝑛×𝑛 , and 𝐶(𝑟(𝑡)) ∈ R𝑛×𝑛 represent the innercoupling matrices linking between the subsystems in mode (1) (2) ]𝑁×𝑁, 𝐺(2) (𝑟(𝑡)) = [𝑔𝑘𝑙 ]𝑁×𝑁, and 𝑟(𝑡). 𝐺(1) (𝑟(𝑡)) = [𝑔𝑘𝑙 (3) (3) 𝐺 (𝑟(𝑡)) = [𝑔𝑘𝑙 ]𝑁×𝑁 are the coupling configuration matrices of the networks representing the coupling strength and the topological structure of the NCDNs in mode 𝑟(𝑡), in (𝑚) is defined as follows: if there exists a connecwhich 𝑔𝑘𝑙 (𝑚) tion between 𝑘th and 𝑙th (𝑘 ≠ 𝑙) nodes, then 𝑔𝑘𝑙 (𝑟(𝑡)) = (𝑚) (𝑚) (𝑚) 𝑔𝑙𝑘 (𝑟(𝑡)) = 1; otherwise 𝑔𝑘𝑙 (𝑟(𝑡)) = 𝑔𝑙𝑘 (𝑟(𝑡)) = 0, and

=

max {𝑑1𝑖 } ≤ min {𝑑2𝑗 } ,

(1) 𝑒𝑘̇ (𝑡) = ∑𝑔𝑘𝑙 (𝑟 (𝑡)) 𝐴 (𝑟 (𝑡)) 𝑒𝑙 (𝑡)

+ 𝐷 (𝑟 (𝑡)) 𝑓 (𝑥𝑘 (𝑡)) + 𝑢𝑘 (𝑡) ,

(𝑚) − ∑ 𝑔𝑙𝑘 𝑙=1,𝑙 ≠ 𝑘

0 ≤ 𝑑1𝑖 ≤ 𝑑𝑖 (𝑡) ≤ 𝑑2𝑖 ,

𝜏𝑖̇ (𝑡) ≤ ]𝑖 < 1,

𝑁

𝑙=1

𝑁

0 ≤ 𝜏𝑖 (𝑡) ≤ 𝜏𝑖 ,

𝑑𝑖̇ (𝑡) ≤ 𝜇𝑖 ,

≠ ⌀, than it is further described as S𝑖𝑘

𝜏(𝑡, 𝑟(𝑡)) and 𝑑(𝑡, 𝑟(𝑡)) denote the mode-dependent timevarying neutral delay and retarded delay, respectively. They are assumed to satisfy

𝑘 = 1, 2, . . . , 𝑁, 𝑡𝑠 ≤ 𝑡 < 𝑡𝑠+1 ,

(11)

where 𝐾𝑘 is sampled-data feedback controller gain matrix to be determined. 𝑒𝑘 (𝑡𝑠 ) is discrete measurement of 𝑒𝑘 (𝑡) at the sampling instant 𝑡𝑠 satisfying lim𝑠 → +∞ 𝑡𝑠 = +∞. It is assumed that 𝑡𝑠+1 − 𝑡𝑠 = ℎ𝑠 ≤ ℎ for any integer 𝑠 ≥ 0, where ℎ > 0 represents the largest sampling interval.

4


With (10) and (11), the error dynamics of NCDN (6) can be replaced as 𝑁

(1) 𝑒𝑘̇ (𝑡) = ∑𝑔𝑘𝑙 (𝑟 (𝑡)) 𝐴 (𝑟 (𝑡)) 𝑒𝑙 (𝑡) 𝑙=1

Definition 2 (see [41]). Define the stochastic LyapunovKrasovskii function of the error system (13) as 𝑉(𝑒(𝑡), 𝑟(𝑡) = 𝑖, 𝑡 > 0) = 𝑉(𝑒(𝑡), 𝑖, 𝑡) where its infinitesimal generator is defined as Γ𝑉 (𝑒 (𝑡) , 𝑖, 𝑡)

𝑁

(2) + ∑𝑔𝑘𝑙 (𝑟 (𝑡)) 𝐵 (𝑟 (𝑡)) 𝑒𝑙 (𝑡 − 𝑑 (𝑡, 𝑟 (𝑡))) 𝑙=1

+

𝑁

(3) ∑𝑔𝑘𝑙 𝑙=1

= lim (12)

=

+ 𝐷 (𝑟 (𝑡)) 𝑓𝑒 (𝑒𝑘 (𝑡)) + 𝐾𝑘 𝑒𝑘 (𝑡 − ℎ (𝑡)) ,

𝜕 𝜕 𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) + 𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) 𝑒 ̇ (𝑡) 𝜕𝑡 𝜕𝑒 𝑁

where ℎ(𝑡) = 𝑡 − 𝑡𝑠 and 0 ≤ ℎ(𝑡) ≤ ℎ. For simplicity of notations, we denote 𝐴(𝑟(𝑡)), 𝐵(𝑟(𝑡)), 𝐶(𝑟(𝑡)), 𝐷(𝑟(𝑡)), 𝐺(𝑚) (𝑟(𝑡)), (𝑚 = 1, 2, 3) by 𝐴 𝑖 , 𝐵𝑖 , 𝐶𝑖 , 𝐷𝑖 , 𝐺𝑖(𝑚) , (𝑚 = 1, 2, 3) for 𝑟𝑡 = 𝑖 ∈ 𝑆. By utilizing the Kronecker product of the matrices, (12) can be written in a more compact form as + C𝑖 𝑒 ̇ (𝑡 − 𝜏𝑖 (𝑡)) + D𝑖 𝐹𝑒 (𝑒 (𝑡)) + K𝑒 (𝑡 − ℎ (𝑡)) ,

[E {𝑉 (𝑒 (𝑡 + Δ𝑡) , 𝑟 (𝑡 + Δ𝑡) , 𝑡 + Δ𝑡) | 𝑒 (𝑡) = 𝑒, 𝑟 (𝑡) = 𝑖} − 𝑉 (𝑒 (𝑡) , 𝑖, 𝑡)]

(𝑟 (𝑡)) 𝐶 (𝑟 (𝑡)) 𝑒𝑙̇ (𝑡 − 𝜏 (𝑡, 𝑟 (𝑡)))

𝑒 ̇ (𝑡) = A𝑖 𝑒 (𝑡) + B𝑖 𝑒 (𝑡 − 𝑑𝑖 (𝑡))

1

Δ𝑡 → 0 Δ𝑡

+ ∑𝜋𝑖𝑗 𝑉 (𝑒 (𝑡) , 𝑗, 𝑡) . 𝑗=1

(16) Definition 3. The NCDN (6) is said to be exponentially synchronized if the error system (13) is exponentially stable; that is, [42] there exist two constants 𝜀 > 0 and 𝜅 ≥ 1 such that for all, 𝑒(𝑡), ‖𝑒 (𝑡)‖ ≤ 𝜅 exp {−𝜀 (𝑡 − 𝑡0 )} sup ‖𝑒 (𝜃)‖ , 𝜃∈[−𝜍,𝑡0 ]

(13)

where A𝑖 = 𝐺𝑖(1) ⊗ 𝐴 𝑖 , B𝑖 = 𝐺𝑖(2) ⊗ 𝐵𝑖 , C𝑖 = 𝐺𝑖(3) ⊗ 𝐶𝑖 , D𝑖 = 𝐼𝑁 ⊗ 𝐷𝑖 , and

where 𝜀 is the exponential decay rate; 𝜍 = max{ℎ, max𝑖∈𝑆 {𝜏𝑖 }, max𝑖∈𝑆 {𝑑2𝑖 }}. Lemma 4 (see [43]). Given matrices 𝐴, 𝐵, 𝐶, and 𝐷 with appropriate dimensions and scalar 𝛼, by the definition of the Kronecker product, the following properties hold:

𝑒 (𝑡) = col {𝑒1 (𝑡) , 𝑒2 (𝑡) , . . . , 𝑒𝑁 (𝑡)} , 𝑒 (𝑡 − 𝑑𝑖 (𝑡)) = col {𝑒1 (𝑡 − 𝑑𝑖 (𝑡)) , 𝑒2 (𝑡 − 𝑑𝑖 (𝑡)) ,

(𝛼𝐴) ⊗ 𝐵 = 𝐴 ⊗ (𝛼𝐵) ,

. . . , 𝑒𝑁 (𝑡 − 𝑑𝑖 (𝑡))} ,

(𝐴 + 𝐵) ⊗ 𝐶 = 𝐴 ⊗ 𝐶 + 𝐵 ⊗ 𝐶,

𝑒 ̇ (𝑡 − 𝜏𝑖 (𝑡)) = col {𝑒1̇ (𝑡 − 𝜏𝑖 (𝑡)) , 𝑒2̇ (𝑡 − 𝜏𝑖 (𝑡)) , ̇ (𝑡 − 𝜏𝑖 (𝑡))} , . . . , 𝑒𝑁 𝐹𝑒 (𝑒 (𝑡)) = col {𝑓𝑒 (𝑒1 (𝑡)) ,

(𝐴 ⊗ 𝐵) (𝐶 ⊗ 𝐷) = (𝐴𝐶) ⊗ (𝐵𝐷) , (14)

𝑒 (𝑡 − ℎ (𝑡)) = col {𝑒1 (𝑡 − ℎ (𝑡)) , 𝑒2 (𝑡 − ℎ (𝑡)) , . . . , 𝑒𝑁 (𝑡 − ℎ (𝑡))} ,

(𝐴 ⊗ 𝐵)𝑇 = 𝐴𝑇 ⊗ 𝐵𝑇 .

(a)

K = diag {𝐾1 , 𝐾2 , . . . , 𝐾𝑁} .

Definition 1 (see [40]). Define operator D: 𝐶([−𝜌, 0], R𝑛 ) → R𝑛 as D(𝑥𝑡 ) = 𝑥(𝑡) − 𝐶𝑥(𝑡 − 𝜏). D is said to be stable if the homogeneous difference equation 𝑥0 = 𝜓 ∈ {𝜙 ∈ 𝐶 ([−𝜌, 0] , R𝑛 ) : D𝜙 = 0}

(15)

is uniformly asymptotically stable. In this paper, that is, ‖𝐺𝑖(3) ⊗ 𝐶𝑖 ‖ < 1.

exp {2𝜀ℎ2 (𝑡)} − exp {2𝜀ℎ1 (𝑡)} 2𝜀 𝑡−ℎ1 (𝑡)

The purpose of this paper is to design a serial of sampleddata controllers (11) to ensure the exponential synchronization of NCDN (6). Before proceeding with the main results, we present the following definitions and lemmas.

𝑡 ≥ 0,

(18)

Lemma 5. For any constant matrix 𝑄 = 𝑄𝑇 > 0, continuous functions 0 ≤ ℎ1 (𝑡) ≤ ℎ2 (𝑡), constant scalars 0 ≤ 𝜏1 < 𝜏2 , and constant 𝜀 > 0 such that the following integrations are well defined:

𝑓𝑒 (𝑒2 (𝑡)) , . . . , 𝑓𝑒 (𝑒𝑁 (𝑡))} ,

D (𝑥𝑡 ) = 0,

(17)

×∫

𝑡−ℎ2 (𝑡)

≥ [∫

𝑡−ℎ1 (𝑡)

𝑡−ℎ2 (𝑡)

(b)

exp {2𝜀 (𝑠 − 𝑡)} 𝑥𝑇 (𝑠) 𝑄𝑥 (𝑠) 𝑑𝑠

𝑥𝑇 (𝑠) 𝑑𝑠] 𝑄 [∫

𝑡−ℎ1 (𝑡)

𝑡−ℎ2 (𝑡)

𝑥 (𝑠) 𝑑𝑠] ,

𝜏22 − 𝜏12 −𝜏1 𝑡 ∫ ∫ exp {2𝜀 (𝑠 − 𝑡)} 𝑥𝑇 (𝑠) 𝑄𝑥 (𝑠) 𝑑𝑠 𝑑𝜃 2 −𝜏2 𝑡+𝜃 ≥ exp {−2𝜀𝜏2 } [∫

−𝜏1

−𝜏2

× [∫

−𝜏1

−𝜏2

∫

𝑡

𝑡+𝜃

∫

𝑡

𝑡+𝜃

𝑥𝑇 (𝑠) 𝑑𝑠 𝑑𝜃] 𝑄

𝑥 (𝑠) 𝑑𝑠 𝑑𝜃] . (19)


5

Proof. (a) is directly obtained from [44]. In addition, from −𝜏2 ≤ 𝜃 ≤ −𝜏1 and 𝑡 + 𝜃 ≤ 𝑠 ≤ 𝑡, it is held that −𝜏2 ≤ 𝜃 ≤ 𝑠 − 𝑡 ≤ 0. Then, −𝜏1

∫

−𝜏2

∫

𝑡

𝑡+𝜃

When 𝑖 ∈ S𝑖𝑘 ; then ∑ 𝛾𝑖𝑗 (𝑄1𝑗 − 𝑌1𝑖 ) − 𝛾𝑖𝑖 𝑌1𝑖 ≤ 0,

𝑗∈S𝑖𝑘 ,𝑗 ≠ 𝑖

exp {2𝜀 (𝑠 − 𝑡)} 𝑥𝑇 (𝑠) 𝑄𝑥 (𝑠) 𝑑𝑠 𝑑𝜃

≥ exp {−2𝜀𝜏2 } ∫

−𝜏1

−𝜏2

∫

𝑡

𝑡+𝜃

∑ 𝛾𝑖𝑗 (𝑄2𝑗 − 𝑌2𝑖 ) − 𝛾𝑖𝑖 𝑌2𝑖 ≤ 0,

𝑥𝑇 (𝑠) 𝑄𝑥 (𝑠) 𝑑𝑠 𝑑𝜃;

∑ 𝛾𝑖𝑗 (𝑅1𝑗 − 𝑍𝑖 ) − 𝛾𝑖𝑖 𝑍𝑖 ≤ 0,


(b) is thus true by [45]. Lemma 6 (see [46]). For functions 𝜆 1 (𝑡), 𝜆 2 (𝑡) ∈ [0, 1], 𝜆 1 (𝑡) + 𝜆 2 (𝑡) = 1, and 𝜂1 = 0 with 𝜆 1 (𝑡) = 0 and 𝜂2 = 0 with 𝜆 2 (𝑡) = 0 and matrices 𝑃 > 0, 𝑄 > 0, then there exists matrix 𝑇 such that [


(20)

𝑃 𝑇 ] > 0, 𝑇𝑇 𝑄

𝑄1𝑗 − 𝑌1𝑖 ≤ 0,

𝑗 ∈ S𝑖𝑢𝑘 ,

𝑄2𝑗 − 𝑌2𝑖 ≤ 0,


𝑅1𝑗 − 𝑍𝑖 ≤ 0,


𝑃𝑗 − 𝑋𝑖 ≤ 0,

𝑗 ∈ S𝑖𝑢𝑘 . (23)

When 𝑖 ∈ S𝑖𝑢𝑘 , then ∑ 𝛾𝑖𝑗 (𝑄1𝑗 − 𝑌1𝑖 ) ≤ 0,

(21)

𝑗∈S𝑖𝑘

∑ 𝛾𝑖𝑗 (𝑄2𝑗 − 𝑌2𝑖 ) ≤ 0,

𝑗∈S𝑖𝑘

and the following inequality holds:

∑ 𝛾𝑖𝑗 (𝑅1𝑗 − 𝑍𝑖 ) ≤ 0,

𝑃 𝑇 𝜂𝑇 1 1 𝜂1 𝑃𝜂1𝑇 + 𝜂2 𝑄𝜂2𝑇 ≥ [𝜂1 𝜂2 ] [ 𝑇 ] [ 1𝑇 ] . 𝑇 𝑄 𝜂2 𝜆 1 (𝑡) 𝜆 2 (𝑡) (22)

𝑗∈S𝑖𝑘

According to Definition 3, the aim will be achieved if we obtain the gain matrices K such that the error system (13) is exponentially stable. So we give the main results as follows.

−

𝑃𝑗 − 𝑋𝑖 ≤ 0,

𝑗 ∈ S𝑖𝑢𝑘 , 𝑗 ≠ 𝑖,

𝑃𝑗 − 𝑋𝑖 ≥ 0,

𝑗 ∈ S𝑖𝑢𝑘 , 𝑗 = 𝑖,

∑ 𝑗∈S𝑖𝑢𝑘 ,𝑗 ≠ 𝑖

−


3. Main Results In this section, sufficient conditions are presented to ensure that the error system (13) is exponentially stable. Then, we propose a design method of the sampled-data controllers for NCDN (6). 3.1. Exponential Stability for the Error System Theorem 7. For given scalars 𝛾𝑖𝑚 , 𝛾𝑖𝑀, 𝜀, 𝜏𝑖 , ]𝑖 , ℎ, 𝑑1𝑖 , 𝑑2𝑖 , and 𝜇𝑖 and constant scalar 𝑑𝑚𝑖 satisfying 𝑑1𝑖 < 𝑑𝑚𝑖 < 𝑑2𝑖 , the error system (13) with partially known transition rates and sectorbounded condition (9) is exponentially stable with decay rate 𝜀 ‖𝐺𝑖(3)

⊗ 𝐶𝑖 ‖ < 1, and there exist symmetric and 𝜅 = √𝜆/𝜆 if positive matrices 𝑃𝑖 > 0, 𝑄1𝑖 > 0, 𝑄2𝑖 > 0, 𝑅1𝑖 > 0, (𝑖 ∈ 𝑆), 𝑄𝑗 > 0, (𝑗 = 3, 4, 5), 𝑅𝑘 > 0, (𝑘 = 2, 3, 4), 𝑇𝑙 > 0, (𝑙 = 1, 2, 3), 𝑈𝑚 > 0, 𝑉𝑛 > 0, 𝑊𝑠 > 0, (𝑚, 𝑛, 𝑠 = 1, 2, 3, 4, 5) and matrices 𝑀𝑘1 , 𝑁𝑘2 , (𝑘1 , 𝑘2 = 1, 2, 3, 4), for any scalar 𝛿 > 0, and symmetric matrices 𝑋𝑖 , 𝑌1𝑖 , 𝑌2𝑖 , 𝑍𝑖 , (𝑖 ∈ 𝑆) such that the following matrix inequalities hold.

(24)

−

(25)

𝛾𝑖𝑀 (𝑄1𝑗 − 𝑌1𝑖 ) − 𝛾𝑖𝑚 𝑌1𝑖 ≤ 0, 𝛾𝑖𝑀 (𝑄2𝑗 − 𝑌2𝑖 ) − 𝛾𝑖𝑚 𝑌2𝑖 ≤ 0,


𝛾𝑖𝑀 (𝑅1𝑗 − 𝑍𝑖 ) − 𝛾𝑖𝑚 𝑍𝑖 ≤ 0,

(26)

(27)

and [

𝑈1 𝑀1 ] > 0, 𝑀1𝑇 𝑈1

[

𝑈 𝑀3 [ 2𝑇 ] > 0, 𝑀3 𝑈2 (i)

[

(ii) (iii)

𝑉 𝑀4 [ 2𝑇 ] > 0, 𝑀4 𝑉2

𝑈4 𝑁1 ] > 0, 𝑁1𝑇 𝑈4

[

Ω𝜀𝑖0

+

Ω𝜀𝑖1

𝑈5 𝑁3 ] > 0, 𝑁3𝑇 𝑈5

(iv)

Ω𝜀𝑖0

+

𝑉1 𝑀2 ] > 0, 𝑀2𝑇 𝑉1

Ω𝜀𝑖2

𝑉 𝑁 [ 4𝑇 2 ] > 0, 𝑁2 𝑉4

(28)

(29)

𝑇

+ Λ 𝐽Λ < 0, [ 𝑇

𝑉5 𝑁4 ] > 0, 𝑁4𝑇 𝑉5

+ Λ 𝐽Λ < 0,

(30)

6


where

̂2 = − 𝐹 24

𝑇 Ω𝜀𝑖0 = ∑ 𝐸𝑚 Π𝑚 𝐸𝑚 +✠

−

𝑇

̂1 𝐹 ̂2 𝐸1𝑇 𝐹 ][ 𝑇 ] ∗ 𝐼 𝐸24

+ ∑ 𝛾𝑖𝑗 (𝑃𝑗 − 𝑋𝑖 ) + 2𝜀𝑃𝑖 𝑗∈S𝑖𝑘

2𝜀𝑑1𝑖 (𝐸 − 𝐸3 ) 𝑉3 (𝐸1𝑇 − 𝐸3𝑇 ) exp {2𝜀𝑑1𝑖 } − 1 1

+ 𝑄1𝑖 + 𝑄3 + 𝑄5 2

+ 𝑅2 + 𝜏𝑖 2 𝑈1 + ℎ 𝑈2

𝑇 ) − exp {−2𝜀𝜏𝑖 } (𝜏𝑖 𝐸1 − 𝐸19 ) 𝑊1 (𝜏𝑖 𝐸1𝑇 − 𝐸19

2 2 2 + 𝑑1𝑖 𝑈3 + 󰜚1𝑖 𝑈4 + 󰜚2𝑖 𝑈5 ,

𝑇 ) − exp {−2𝜀ℎ} (ℎ𝐸1 − 𝐸23 ) 𝑊2 (ℎ𝐸1𝑇 − 𝐸23

Π2 = 𝜛 (𝜀, 𝜇𝑖 ) 𝑅1𝑖 ,

− exp {−2𝜀𝑑1𝑖 } (𝑑1𝑖 𝐸1 − 𝐸9 ) 𝑊3 (𝑑1𝑖 𝐸1𝑇 − 𝐸9𝑇 )

Π3 = exp {−2𝜀𝑑1𝑖 } (𝑅1𝑖 + 𝑅3 − 𝑅2 ) ,

𝑇 ) − exp {−2𝜀𝑑𝑚𝑖 } (󰜚1𝑖 𝐸1 − 𝐸12 ) 𝑊4 (󰜚1𝑖 𝐸1𝑇 − 𝐸12

Π4 = exp {−2𝜀𝑑𝑚𝑖 } (𝑅4 − 𝑅3 ) ,

𝑇 ) − exp {−2𝜀𝑑2𝑖 } (󰜚2𝑖 𝐸1 − 𝐸13 ) 𝑊5 (󰜚2𝑖 𝐸1𝑇 − 𝐸13

2𝜀𝜏𝑖 − [𝐸18 𝐸19 − 𝐸18 ] exp {2𝜀𝜏𝑖 } − 1

(31)

Π8 = − exp {−2𝜀𝑑2𝑖 } 𝑇3 , Π9 = −

𝑇 𝑉1 𝑀2 𝐸1𝑇 − 𝐸16 ][ 𝑇 𝑇 𝑇] 𝑀2 𝑉1 𝐸16 − 𝑒14

2𝜀ℎ exp {2𝜀ℎ} − 1

×[ −

Π7 = exp {−2𝜀𝑑𝑚𝑖 } (𝑇3 − 𝑇2 ) ,

2𝜀𝜏𝑖 [𝐸1 − 𝐸16 𝐸16 − 𝐸14 ] exp {2𝜀𝜏𝑖 } − 1

×[ −

Π5 = − exp {−2𝜀𝑑2𝑖 } 𝑅4 , Π6 = exp {−2𝜀𝑑1𝑖 } (𝑇2 − 𝑇1 ) ,

𝑈 𝑀1 𝐸𝑇 × [ 1𝑇 ] [ 𝑇 18 𝑇 ] 𝑀1 𝑈1 𝐸19 − 𝐸18 −

exp {2𝜀ℎ} − 1

Π15 = − exp {−2𝜀𝜏𝑖 } 𝑄4 ,

[𝐸22 𝐸23 − 𝐸22 ]

Π16 = − (1 − ]𝑖 ) exp {−2𝜀𝜏𝑖 } 𝑄1𝑖 , Π17 = − (1 − ]𝑖 ) exp {−2𝜀𝜏𝑖 } 𝑄2𝑖 , Π20 = − exp {−2𝜀ℎ} 𝑄5 , Π𝑚 = 0,

[𝐸1 − 𝐸21 𝐸21 − 𝐸20 ]

Ω𝜀𝑖1 = −

𝑇 𝑉 𝑀4 𝐸1𝑇 − 𝐸21 × [ 2𝑇 ][ 𝑇 𝑇 ], 𝑀4 𝑉2 𝐸21 − 𝐸20

✠ = 𝐸1 𝑃𝑖 (𝐺𝑖(2) ⊗ 𝐵𝑖 ) 𝐸2𝑇 + 𝐸1 𝑃𝑖 (𝐺𝑖(3) ⊗ 𝐶𝑖 ) 𝐸17 𝑇 𝑇 + 𝐸1 𝑃𝑖 K𝐸21 + 𝐸1 𝑃𝑖 (𝐼𝑁 ⊗ 𝐷𝑖 ) 𝐸24 𝑇

𝑇

𝑇

𝐼𝑁 ⊗ (𝐹1𝑇 𝐹2 + 𝐹2𝑇 𝐹1 ) 2

,

2𝜀󰜚2𝑖 𝐸 𝑈 𝐸𝑇 exp {2𝜀𝑑2𝑖 } − exp {2𝜀𝑑𝑚𝑖 } 13 5 13 2𝜀󰜚2𝑖 (𝐸 − 𝐸5 ) 𝑉5 (𝐸4𝑇 − 𝐸5𝑇 ) exp {2𝜀𝑑2𝑖 } − exp {2𝜀𝑑𝑚𝑖 } 4

−

2𝜀󰜚1𝑖 [𝐸10 𝐸12 − 𝐸10 ] exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑1𝑖 }

×[

+ 𝐸2 (𝐺𝑖(2) ⊗ 𝐵𝑖 ) 𝑃𝑖 𝐸1𝑇 + 𝐸17 (𝐺𝑖(3) ⊗ 𝐶𝑖 ) 𝑃𝑖 𝐸1𝑇 + 𝐸21 K𝑇𝑃𝑖 𝐸1𝑇 + 𝐸24 (𝐼𝑁 ⊗ 𝐷𝑖 ) 𝑃𝑖 𝐸1𝑇

(𝑚 = 10, 11, 12, 13, 18, 19, 21, . . . , 24) ,

− where

̂1 = 𝐹

2𝜀𝑑1𝑖 𝑈, exp {2𝜀𝑑1𝑖 } − 1 3

Π14 = − exp {−2𝜀𝜏𝑖 } 𝑄3 ,

𝑈2 𝑀3 𝐸𝑇 ] [ 𝑇 22 𝑇 ] 𝑇 𝑀3 𝑈2 𝐸23 − 𝐸22 2𝜀ℎ

𝑇

Π1 = (𝐺𝑖(1) ⊗ 𝐴 𝑖 ) 𝑃𝑖 + 𝑃𝑖 (𝐺𝑖(1) ⊗ 𝐴 𝑖 )

𝑚=1

− 𝛿 [𝐸1 𝐸24 ] [

𝑇

(𝐼𝑁 ⊗ 𝐹1 ) + (𝐼𝑁 ⊗ 𝐹2 ) , 2

−

𝑈4 𝑁1 𝐸𝑇 ] [ 𝑇 10 𝑇 ] 𝑇 𝑁1 𝑈4 𝐸12 − 𝐸10

2𝜀󰜚1𝑖 [𝐸3 − 𝐸2 𝐸2 − 𝐸4 ] exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑1𝑖 }

×[

𝑉4 𝑁2 𝐸3𝑇 − 𝐸2𝑇 ][ ], 𝑁2𝑇 𝑉4 𝐸2𝑇 − 𝐸4𝑇

Mathematical Problems in Engineering Ω𝜀𝑖2 = −

7

2𝜀󰜚1𝑖 𝐸 𝑈 𝐸𝑇 exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑1𝑖 } 12 4 12

2𝜀󰜚1𝑖 − exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑1𝑖 }

+

× (𝐸3 − 𝐸4 ) 𝑉4 (𝐸3𝑇 − 𝐸4𝑇 ) −

𝑖∈𝑆

󰜚1𝑖 = 𝑑𝑚𝑖 − 𝑑1𝑖 ,

2

󰜚3𝑖 =

4

𝑉1 (𝑒 (𝑡) , 𝑖, 𝑡) = 𝑒𝑇 (𝑡) 𝑃𝑖 𝑒 (𝑡) ,

(35)

where

𝑉2 (𝑒 (𝑡) , 𝑖, 𝑡) = ∫

𝑇 𝐶𝑖 ) 𝐸17

(32) where 𝐸𝑖 {𝑖 = 1, 2, . . . , 24} are block entry matrices; that is, = [0 0 0 𝐼 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] , with 𝜇𝑖 ≥ 1, with 𝜇𝑖 < 1,

1 − exp {−2𝜀𝜍} 𝜆 = max {𝜆 max (𝑃𝑖 )} + 𝑖∈𝑆 2𝜀 4 3 {5 } × { ∑𝜆 max (𝑄𝑗 ) + ∑ 𝜆 max (𝑅𝑘 ) + ∑𝜆 max (𝑇𝑙 )} 𝑘=2 𝑙=1 {𝑗=3 }

1 − exp {−2𝜀𝜍} {max {𝜆 max (𝑄1𝑖 )}+max {𝜆 max (𝑄2𝑖 )} 𝑖∈𝑆 𝑖∈𝑆 2𝜀 +max {𝜆 max (𝑅1𝑖 )}} 𝑖∈𝑆

exp {−2𝜀𝜍} + 2𝜀𝜍 − 1 + 4𝜀2

𝑡

𝑡−𝜏𝑖 (𝑡)

𝑇 𝑇 + K𝐸21 + (𝐼𝑁 ⊗ 𝐷𝑖 ) 𝐸24 ,

+

(34)

𝑘=1

Λ = (𝐺𝑖(1) ⊗ 𝐴 𝑖 ) 𝐸1𝑇 + (𝐺𝑖(2) ⊗ 𝐵𝑖 ) 𝐸2𝑇

(𝜇𝑖 − 1) exp {−2𝜀𝑑1𝑖 } (𝜇𝑖 − 1) exp {−2𝜀𝑑2𝑖 }

1 2 2 ). (𝑑 − 𝑑𝑚𝑖 2 2𝑖

𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) = ∑ 𝑉𝑘 (𝑒 (𝑡) , 𝑖, 𝑡) ,

𝜏𝑖 ℎ 𝑊 + 𝑊 4 1 4 2

⊗

󰜚4𝑖 =

7

𝑑4 2 2 + 1𝑖 𝑊3 + 󰜚3𝑖 𝑊4 + 󰜚4𝑖 𝑊5 , 4

𝜛 (𝜀, 𝜇𝑖 ) = {

1 2 2 ), (𝑑 − 𝑑1𝑖 2 𝑚𝑖

(33)

2 2 2 + 𝑑1𝑖 𝑉3 + 󰜚1𝑖 𝑉4 + 󰜚2𝑖 𝑉5

(𝐺𝑖(3)

󰜚2𝑖 = 𝑑2𝑖 − 𝑑𝑚𝑖 ,

Proof. Construct the following stochastic Lyapunov functional:

2

𝐽 = 𝑄2𝑖 + 𝑄4 + 𝑇1 + 𝜏𝑖 𝑉1 + ℎ 𝑉2

𝐸4𝑇

1 − exp {−2𝜀𝜍} + 2𝜀2 𝜍2 − 2𝜀𝜍 8𝜀3

𝜆 = min {𝜆 min (𝑃𝑖 )} ,

𝐸𝑇 − 𝐸2𝑇 𝑉 𝑁 × [ 5𝑇 4 ] [ 4𝑇 ], 𝑁4 𝑉5 𝐸2 − 𝐸5𝑇

+

𝑛=1

𝑠=1

2𝜀󰜚2𝑖 [𝐸4 − 𝐸2 𝐸2 − 𝐸5 ] exp {2𝜀𝑑2𝑖 } − exp {2𝜀𝑑𝑚𝑖 }

+

𝑚=1

5

2𝜀󰜚2𝑖 [𝐸11 𝐸13 − 𝐸11 ] exp {2𝜀𝑑2𝑖 } − exp {2𝜀𝑑𝑚𝑖 }

4

5

× {∑𝜆 max (𝑊𝑠 )} ,

𝑈 𝑁 𝐸𝑇 × [ 5𝑇 3 ] [ 𝑇 11 𝑇 ] 𝑁3 𝑈5 𝐸13 − 𝐸11 −

5

× { ∑ 𝜆 max (𝑈𝑚 ) + ∑ 𝜆 max (𝑉𝑛 )}

+∫

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑄1𝑖 𝑒 (𝑠) 𝑑𝑠

𝑡


+∫

𝑡

𝑡−𝜏𝑖

+∫

𝑡

𝑡−𝜏𝑖

+∫

𝑡

𝑡−ℎ

𝑉3 (𝑒 (𝑡) , 𝑖, 𝑡) = ∫

𝑡−𝑑1𝑖

𝑡−𝑑𝑖 (𝑡)

+∫

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑄3 𝑒 (𝑠) 𝑑𝑠 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑄4 𝑒 ̇ (𝑠) 𝑑𝑠 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑄5 𝑒 (𝑠) 𝑑𝑠, (36) exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑅1𝑖 𝑒 (𝑠) 𝑑𝑠

𝑡

𝑡−𝑑1𝑖

+∫

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑅2 𝑒 (𝑠) 𝑑𝑠

𝑡−𝑑1𝑖

𝑡−𝑑𝑚𝑖

+∫

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑄2𝑖 𝑒 ̇ (𝑠) 𝑑𝑠

𝑡−𝑑𝑚𝑖

𝑡−𝑑2𝑖

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑅3 𝑒 (𝑠) 𝑑𝑠 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑅4 𝑒 (𝑠) 𝑑𝑠, (37)

8

Mathematical Problems in Engineering 𝑉4 (𝑒 (𝑡) , 𝑖, 𝑡) = ∫

𝑡

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑇1 𝑒 ̇ (𝑠) 𝑑𝑠

𝑡−𝑑1𝑖

+∫

𝑡−𝑑1𝑖

𝑡−𝑑𝑚𝑖

+∫

+∫

−𝑑1𝑖

−𝑑𝑚𝑖

0

𝑡

𝜃

𝑡+𝜆

∫ ∫

× (𝑠) 𝑊4 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜆 𝑑𝜃

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑇2 𝑒 ̇ (𝑠) 𝑑𝑠

𝑡−𝑑𝑚𝑖

𝑡−𝑑2𝑖

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑇3 𝑒 ̇ (𝑠) 𝑑𝑠,

󰜚3𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇

+∫

−𝑑𝑚𝑖

−𝑑2𝑖

0

𝑡

𝜃

𝑡+𝜆

∫ ∫

(41)

𝑇

󰜚4𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒 ̇

× (𝑠) 𝑊5 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜆 𝑑𝜃.

(38) 𝑉5 (𝑒 (𝑡) , 𝑖, 𝑡) 0

𝑡

−𝜏𝑖

𝑡+𝜃

=∫ ∫ 0

𝑡

ℎ exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈2 𝑒 (𝑠) 𝑑𝑠 𝑑𝜃

+∫ ∫

−ℎ 𝑡+𝜃 0

+∫

−𝑑1𝑖

∫

𝑡

𝑡+𝜃

−𝑑1𝑖

+∫

∫

−𝑑𝑚𝑖

+∫

𝑡

∫

󰜚1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒 (𝑠) 𝑈4 𝑒 (𝑠) 𝑑𝑠 𝑑𝜃

𝑡

Γ𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) = ∑ Γ𝑉𝑘 (𝑒 (𝑡) , 𝑖, 𝑡) ,

(42)

𝑘=1

Γ𝑉1 (𝑒 (𝑡) , 𝑖, 𝑡)

𝑇

𝑡+𝜃

−𝑑2𝑖

7

𝑑1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈3 𝑒 (𝑠) 𝑑𝑠 𝑑𝜃

𝑡+𝜃

−𝑑𝑚𝑖

Taking Γ as its infinitesimal generator along the trajectory of error system (13), we obtain the following from Definition 3 and (34)-(35):

𝜏𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈1 𝑒 (𝑠) 𝑑𝑠 𝑑𝜃

= 2 [𝑒𝑇 (𝑡) A𝑇𝑖 + 𝑒𝑇 (𝑡 − 𝑑𝑖 (𝑡)) B𝑇𝑖 + 𝑒𝑇̇ (𝑡 − 𝜏𝑖 (𝑡)) C𝑇𝑖 + 𝐹𝑒𝑇 (𝑒 (𝑡)) D𝑇𝑖

󰜚2𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈5 𝑒 (𝑠) 𝑑𝑠 𝑑𝜃,

+𝑒𝑇 (𝑡 − ℎ (𝑡)) K𝑇 ] 𝑃𝑖 𝑒 (𝑡)

(39)

(43)

𝑉6 (𝑒 (𝑡) , 𝑖, 𝑡) 0

𝑡

−𝜏𝑖

𝑡+𝜃

=∫ ∫ 0

+∫ ∫

𝑡

+∫

−𝑑1𝑖

+∫

∫

−𝑑1𝑖

−𝑑𝑚𝑖

+∫

−𝑑𝑚𝑖

−𝑑2𝑖

𝑡

Γ𝑉2 (𝑒 (𝑡) , 𝑖, 𝑡)

𝑇

𝑡+𝜃

∫

+ 2𝜀𝑒𝑇 (𝑡) 𝑃𝑖 𝑒 (𝑡) − 2𝜀𝑉1 (𝑒 (𝑡) , 𝑖, 𝑡) ,

ℎ exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉2 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃

−ℎ 𝑡+𝜃 0

+ 𝑒𝑇 (𝑡) [ ∑ 𝛾𝑖𝑗 𝑃𝑗 + ∑ 𝛾𝑖𝑗 𝑃𝑗 ] 𝑒 (𝑡) 𝑖 𝑗∈S𝑖𝑢𝑘 ] [𝑗∈S𝑘

𝜏𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉1 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃

𝑑1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒 ̇ (𝑠) 𝑉3 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃

𝑡

𝑡+𝜃 𝑡

∫

𝑡+𝜃

󰜚1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉4 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃 󰜚2𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒 ̇ (𝑠) 𝑉5 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃, (40) 0

𝑡

−𝜏𝑖

𝜃

𝑡+𝜆

0

+∫ ∫ ∫ −ℎ 𝜃

+∫

0

−𝑑1𝑖

× 𝑒𝑇 (𝑡 − 𝜏𝑖 (𝑡)) 𝑄1𝑖 𝑒 (𝑡 − 𝜏𝑖 (𝑡)) − (1 − 𝜏𝑖̇ (𝑡)) exp {−2𝜀𝜏𝑖 (𝑡)}

𝜏𝑖 2 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ 2

× 𝑒𝑇̇ (𝑡 − 𝜏𝑖 (𝑡)) 𝑄2𝑖 𝑒 ̇ (𝑡 − 𝜏𝑖 (𝑡))


− exp {−2𝜀𝜏𝑖 } 𝑒𝑇 (𝑡 − 𝜏𝑖 ) 𝑄3 𝑒 (𝑡 − 𝜏𝑖 )

𝑉7 (𝑒 (𝑡) , 𝑖, 𝑡) = ∫ ∫ ∫

0

+ 𝑒𝑇̇ (𝑡) [𝑄2𝑖 + 𝑄4 ] 𝑒 ̇ (𝑡) − 2𝜀𝑉2 (𝑒 (𝑡) , 𝑖, 𝑡) − (1 − 𝜏𝑖̇ (𝑡)) exp {−2𝜀𝜏𝑖 (𝑡)}

𝑇

0

= 𝑒𝑇 (𝑡) [𝑄1𝑖 + 𝑄3 + 𝑄5 ] 𝑒 (𝑡)

2

𝑡

𝑡+𝜆

ℎ exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ 2

− exp {−2𝜀𝜏𝑖 } 𝑒𝑇̇ (𝑡 − 𝜏𝑖 ) 𝑄4 𝑒 ̇ (𝑡 − 𝜏𝑖 )


+ ∑𝛾𝑖𝑗 ∫

0

𝑡

𝜃

𝑡+𝜆

∫ ∫

2 𝑑1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ 2


𝑗∈𝑆

𝑡

𝑡−𝜏𝑗 (𝑡)

exp {2𝜀 (𝑠 − 𝑡)} × [𝑒𝑇 (𝑠) 𝑄1𝑗 𝑒 (𝑠) + 𝑒𝑇̇ (𝑠) 𝑄2𝑗 𝑒 ̇ (𝑠)] 𝑑𝑠

− exp {−2𝜀ℎ} 𝑒𝑇 (𝑡 − ℎ) 𝑄5 𝑒 (𝑡 − ℎ)


9

≤ 𝑒𝑇 (𝑡) [𝑄1𝑖 + 𝑄3 + 𝑄5 ] 𝑒 (𝑡)

× 𝑒𝑇 (𝑡 − 𝑑𝑖 (𝑡)) 𝑅1𝑖 𝑒 (𝑡 − 𝑑𝑖 (𝑡)) 𝑡−𝑑1𝑖

+ 𝑒𝑇̇ (𝑡) [𝑄2𝑖 + 𝑄4 ] 𝑒 ̇ (𝑡) − 2𝜀𝑉2 (𝑒 (𝑡) , 𝑖, 𝑡)

+ ∑ 𝛾𝑖𝑗 ∫

𝑡−𝑑2𝑗

𝑗 ≠ 𝑖

− (1 − 𝜏𝑖̇ (𝑡)) exp {−2𝜀𝜏𝑖 (𝑡)}

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑅1𝑗 𝑒 (𝑠) 𝑑𝑠, (45)

𝑇

× 𝑒 (𝑡 − 𝜏𝑖 (𝑡)) 𝑄1𝑖 𝑒 (𝑡 − 𝜏𝑖 (𝑡))

Γ𝑉4 (𝑒 (𝑡) , 𝑖, 𝑡) = 𝑒𝑇̇ (𝑡) 𝑇1 𝑒 ̇ (𝑡)

− (1 − 𝜏𝑖̇ (𝑡)) exp {−2𝜀𝜏𝑖 (𝑡)} × 𝑒𝑇̇ (𝑡 − 𝜏𝑖 (𝑡)) 𝑄2𝑖 𝑒 ̇ (𝑡 − 𝜏𝑖 (𝑡))

+ exp {−2𝜀𝑑1𝑖 } 𝑒𝑇̇ (𝑡 − 𝑑1𝑖 ) [𝑇2 − 𝑇1 ]

− exp {−2𝜀𝜏𝑖 } 𝑒𝑇 (𝑡 − 𝜏𝑖 ) 𝑄3 𝑒 (𝑡 − 𝜏𝑖 )

× 𝑒 ̇ (𝑡 − 𝑑1𝑖 ) − 2𝜀𝑉4 (𝑒 (𝑡) , 𝑖, 𝑡) + exp {−2𝜀𝑑𝑚𝑖 } 𝑒 ̇ (𝑡 − 𝑑𝑚𝑖 )

− exp {−2𝜀𝜏𝑖 } 𝑒𝑇̇ (𝑡 − 𝜏𝑖 ) 𝑄4 𝑒 ̇ (𝑡 − 𝜏𝑖 ) + ∑ 𝛾𝑖𝑗 ∫ 𝑗 ≠ 𝑖

𝑡

𝑡−𝜏𝑗

× [𝑇3 − 𝑇2 ] 𝑒 ̇ (𝑡 − 𝑑𝑚𝑖 )

exp {2𝜀 (𝑠 − 𝑡)}

− exp {−2𝜀𝑑2𝑖 } 𝑒𝑇̇ (𝑡 − 𝑑2𝑖 ) 𝑇3 𝑒 ̇ (𝑡 − 𝑑2𝑖 ) ,

× [𝑒𝑇 (𝑠) 𝑄1𝑗 𝑒 (𝑠) + 𝑒𝑇̇ (𝑠) 𝑄2𝑗 𝑒 ̇ (𝑠)] 𝑑𝑠 − exp {−2𝜀ℎ} 𝑒𝑇 (𝑡 − ℎ) 𝑄5 𝑒 (𝑡 − ℎ) ,

Γ𝑉5 (𝑒 (𝑡) , 𝑖, 𝑡) 2

= 𝑒𝑇 (𝑡) [𝜏𝑖 2 𝑈1 + ℎ 𝑈2 (44)

2 2 2 +𝑑1𝑖 𝑈3 + 󰜚1𝑖 𝑈4 + 󰜚2𝑖 𝑈5 ] 𝑒 (𝑡)

Γ𝑉3 (𝑒 (𝑡) , 𝑖, 𝑡)

− 2𝜀𝑉5 (𝑒 (𝑡) , 𝑖, 𝑡)

= 𝑒𝑇 (𝑡) 𝑅2 𝑒 (𝑡) + exp {−2𝜀𝑑1𝑖 } 𝑒𝑇 (𝑡 − 𝑑1𝑖 )

−∫ −∫

+ exp {−2𝜀𝑑𝑚𝑖 } 𝑒𝑇 (𝑡 − 𝑑𝑚𝑖 )

−∫

− exp {−2𝜀𝑑2𝑖 } 𝑒 (𝑡 − 𝑑2𝑖 ) × 𝑅4 𝑒 (𝑡 − 𝑑2𝑖 ) − 2𝜀𝑉3 (𝑒 (𝑡) , 𝑖, 𝑡) − (1 − 𝑑𝑖̇ (𝑡)) exp {−2𝜀𝑑𝑖 (𝑡)} × 𝑒𝑇 (𝑡 − 𝑑𝑖 (𝑡)) 𝑅1𝑖 𝑒 (𝑡 − 𝑑𝑖 (𝑡)) 𝑡−𝑑1𝑖

𝑗∈𝑆

𝑡−𝑑𝑗 (𝑡)

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑅1𝑗 𝑒 (𝑠) 𝑑𝑠

𝑇

≤ 𝑒 (𝑡) 𝑅2 𝑒 (𝑡) + exp {−2𝜀𝑑1𝑖 } 𝑇

× 𝑒 (𝑡 − 𝑑1𝑖 ) [𝑅1𝑖 + 𝑅3 − 𝑅2 ] 𝑒 (𝑡 − 𝑑1𝑖 ) + exp {−2𝜀𝑑𝑚𝑖 } 𝑒𝑇 (𝑡 − 𝑑𝑚𝑖 ) × [𝑅4 − 𝑅3 ] 𝑒 (𝑡 − 𝑑𝑚𝑖 ) − exp {−2𝜀𝑑2𝑖 } 𝑒𝑇 (𝑡 − 𝑑2𝑖 ) × 𝑅4 𝑒 (𝑡 − 𝑑2𝑖 ) − 2𝜀𝑉3 (𝑒 (𝑡) , 𝑖, 𝑡) − (1 − 𝑑𝑖̇ (𝑡)) exp {−2𝜀𝑑𝑖 (𝑡)}

𝜏𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈1 𝑒 (𝑠) 𝑑𝑠

𝑡

ℎ exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈2 𝑒 (𝑠) 𝑑𝑠

𝑡−ℎ 𝑡

−∫

𝑡−𝑑1𝑖

󰜚1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈4 𝑒 (𝑠) 𝑑𝑠

𝑡−𝑑𝑚𝑖

−∫

(47)

𝑑1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈3 𝑒 (𝑠) 𝑑𝑠

𝑡−𝑑1𝑖

× [𝑅4 − 𝑅3 ] 𝑒 (𝑡 − 𝑑𝑚𝑖 ) 𝑇

𝑡

𝑡−𝜏𝑖

× [𝑅1𝑖 + 𝑅3 − 𝑅2 ] 𝑒 (𝑡 − 𝑑1𝑖 )

+ ∑𝛾𝑖𝑗 ∫

(46)

𝑇

𝑡−𝑑𝑚𝑖

𝑡−𝑑2𝑖

󰜚2𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈5 𝑒 (𝑠) 𝑑𝑠,

Γ𝑉6 (𝑒 (𝑡) , 𝑖, 𝑡) 2

2 2 2 𝑉3 + 󰜚1𝑖 𝑉4 + 󰜚2𝑖 𝑉5 ] 𝑒 ̇ (𝑡) = 𝑒𝑇̇ (𝑡) [𝜏𝑖 2 𝑉1 + ℎ 𝑉2 + 𝑑1𝑖

− 2𝜀𝑉6 (𝑒 (𝑡) , 𝑖, 𝑡) −∫

𝑡

𝑡−𝜏𝑖

−∫

𝑡

𝑡−ℎ

−∫

𝜏𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉1 𝑒 ̇ (𝑠) 𝑑𝑠 ℎ exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉2 𝑒 ̇ (𝑠) 𝑑𝑠

𝑡

𝑡−𝑑1𝑖

−∫

𝑑1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉3 𝑒 ̇ (𝑠) 𝑑𝑠

𝑡−𝑑1𝑖

𝑡−𝑑𝑚𝑖

−∫

𝑡−𝑑𝑚𝑖

𝑡−𝑑2𝑖

󰜚1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉4 𝑒 ̇ (𝑠) 𝑑𝑠 󰜚2𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉5 𝑒 ̇ (𝑠) 𝑑𝑠, (48)

10


Γ𝑉7 (𝑒 (𝑡) , 𝑖, 𝑡)

0

−∫

−𝑑1𝑖

4

𝑑4 𝜏4 ℎ 2 2 = 𝑒 ̇ (𝑡) [ 𝑖 𝑊1 + 𝑊2 + 1𝑖 𝑊3 + 󰜚3𝑖 𝑊4 + 󰜚4𝑖 𝑊5 ] 𝑒 ̇ (𝑡) 4 4 4 𝑇

−𝑑1𝑖

−∫

−𝑑𝑚𝑖

− 2𝜀𝑉7 (𝑒 (𝑡) , 𝑖, 𝑡) 0

𝑡

−𝜏𝑖

𝑡+𝜃

−∫ ∫ 0

−𝑑𝑚𝑖

−∫

𝜏𝑖 2 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊1 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃 2

−𝑑2𝑖

𝑡

𝑡+𝜃

∫

2 𝑑1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊3 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃 2

𝑡

𝑡+𝜃

∫

𝑡

𝑡+𝜃

󰜚3𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊4 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃 󰜚4𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊5 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃. (49)

2

𝑡

−∫ ∫

∫

−ℎ 𝑡+𝜃

ℎ exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊2 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃 2

Define

𝜉 (𝑡) = col {𝑒 (𝑡) 𝑒 (𝑡 − 𝑑𝑖 (𝑡)) 𝑒 (𝑡 − 𝑑1𝑖 ) 𝑒 (𝑡 − 𝑑𝑚𝑖 ) 𝑒 (𝑡 − 𝑑2𝑖 ) 𝑒 ̇ (𝑡 − 𝑑1𝑖 ) 𝑒 ̇ (𝑡 − 𝑑𝑚𝑖 ) 𝑒 ̇ (𝑡 − 𝑑2𝑖 ) 𝑡

∫

𝑡−𝑑1𝑖

𝑒 (𝑠) 𝑑𝑠 ∫

𝑡−𝑑1𝑖




𝑡−𝑑2𝑖


𝑡−𝑑1𝑖

𝑡−𝑑𝑚𝑖

𝑒 (𝑡 − 𝜏𝑖 ) 𝑒 ̇ (𝑡 − 𝜏𝑖 ) 𝑒 (𝑡 − 𝜏𝑖 (𝑡)) 𝑒 ̇ (𝑡 − 𝜏𝑖 (𝑡)) ∫

𝑡


𝑡

𝑒 (𝑡 − ℎ) 𝑒 (𝑡 − ℎ (𝑡))∫

𝑡−ℎ(𝑡)

𝑡


𝑡−ℎ

Due to ∑𝑁 𝑗=1 𝛾𝑖𝑗 = 0, the following zero equations hold for arbitrary matrices: 𝑋𝑖 = 𝑋𝑖𝑇 , 𝑌2𝑖 = 𝑌2𝑖𝑇 ,

𝑌1𝑖 = 𝑌1𝑖𝑇 , 𝑍𝑖 = 𝑍𝑖𝑇 ,

𝑡−𝑑𝑚𝑖


𝑡−𝑑2𝑖


𝑡

𝑡−𝜏𝑖

𝑒 (𝑠) 𝑑𝑠 (50)

𝑒 (𝑠) 𝑑𝑠

𝑒 (𝑠) 𝑑𝑠 𝐹𝑒 (𝑒 (𝑡))}.

𝑡−𝑑1𝑖

−∫

𝑡−𝑑2𝑗

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) × [ ∑ 𝛾𝑖𝑗 𝑍𝑖 + ∑ 𝛾𝑖𝑗 𝑍𝑖 ] 𝑒 (𝑠) 𝑑𝑠 = 0. 𝑖 𝑗∈S𝑖𝑢𝑘 ] [𝑗∈S𝑘

(51)

𝑖 ∈ 𝑆;

(52) Based on (9), we can obtain the following inequality for any 𝛿 > 0:

that is,

−𝑒𝑇 (𝑡) [ ∑ 𝛾𝑖𝑗 𝑋𝑖 + ∑ 𝛾𝑖𝑗 𝑋𝑖 ] 𝑒 (𝑡) = 0, 𝑖 𝑗∈S𝑖𝑢𝑘 ] [𝑗∈S𝑘 −∫

𝑡

𝑡−𝜏𝑗

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠)

𝛿 [𝑒𝑇 (𝑡) 𝐹𝑒𝑇 (𝑒 (𝑡))] [

̂1 𝐹 ̂2 𝑒 (𝑡) 𝐹 ] ][ ∗ 𝐼 𝐹𝑒 (𝑒 (𝑡))

̂ 𝐸𝑇 ̂ 𝐹 𝐹 = 𝛿𝜉 (𝑡) [𝐸1 𝐸24 ] [ 1 2 ] [ 𝑇1 ] 𝜉 (𝑡) ≤ 0. ∗ 𝐼 𝐸24

(53)

𝑇

From (42) and (53), we have 7

Γ𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) ≤ ∑ Γ𝑉𝑘 (𝑒 (𝑡) , 𝑖, 𝑡) × [ ∑ 𝛾𝑖𝑗 𝑌1𝑖 + ∑ 𝛾𝑖𝑗 𝑌1𝑖 ] 𝑒 (𝑠) 𝑑𝑠 = 0, 𝑖 𝑗∈S𝑖𝑢𝑘 [𝑗∈S𝑘 ] 𝑡

−∫

𝑡−𝜏𝑗

𝑘=1

− 𝛿𝜉𝑇 (𝑡) [𝐸1 𝐸24 ] [

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠)

̂1 𝐹 ̂2 𝐸1𝑇 𝐹 ] [ 𝑇 ] 𝜉 (𝑡) . ∗ 𝐼 𝐸24 (54)

From (8) we easily obtain that × [ ∑ 𝛾𝑖𝑗 𝑌2𝑖 + ∑ 𝛾𝑖𝑗 𝑌2𝑖 ] 𝑒 ̇ (𝑠) 𝑑𝑠 = 0, 𝑖 𝑗∈S𝑖𝑢𝑘 [𝑗∈S𝑘 ]

− (1 − 𝜏𝑖̇ (𝑡)) exp {−2𝜀𝜏𝑖 (𝑡)} ≤ − (1 − ]𝑖 ) exp {−2𝜀𝜏𝑖 } , − (1 − 𝑑𝑖̇ (𝑡)) exp {−2𝜀𝑑𝑖 (𝑡)} ≤ 𝜛 (𝜀, 𝜇𝑖 ) .

(55)


11 For 𝑑𝑖 (𝑡) ∈ [𝑑1𝑖 , 𝑑𝑚𝑖 ], the following is held from (a) of Lemma 5:

Notice (a) of Lemma 5; then,

−∫

𝑡

𝑡−𝑑1𝑖

𝑑1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈3 𝑒 (𝑠) 𝑑𝑠

−∫

𝑡−𝑑1𝑖

𝑡−𝑑𝑚𝑖

2𝜀𝑑1𝑖 ≤− 𝜉𝑇 (𝑡) 𝐸9 𝑈3 𝐸9𝑇 𝜉 (𝑡) , exp {2𝜀𝑑1𝑖 } − 1 −∫

𝑡

𝑡−𝑑1𝑖

󰜚1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉4 𝑒 ̇ (𝑠) 𝑑𝑠

= − {∫

≤−

−𝜏𝑖

𝑡+𝜃

× [∫

𝑒𝑇̇ (𝑠) 𝑑𝑠] 𝑉4 [∫

=−


𝑒𝑇̇ (𝑠) 𝑑𝑠] 𝑉4 [∫

ℎ exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊2 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃 2

−ℎ 𝑡+𝜃

+

−𝑑1𝑖

∫

𝑡

𝑡+𝜃

2 𝑑1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊3 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃 2

≤ − exp {−2𝜀𝑑1𝑖 } 𝜉𝑇 (𝑡) (𝑑1𝑖 𝐸1 − 𝐸9 ) 𝑊3 × (𝑑1𝑖 𝐸1𝑇 − 𝐸9𝑇 ) 𝜉 (𝑡) , −𝑑1𝑖

−∫

−𝑑𝑚𝑖

∫

𝑡

𝑡+𝜃

󰜚3𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊4 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃 𝑇

≤ − exp {−2𝜀𝑑𝑚𝑖 } 𝜉 (𝑡) (󰜚1𝑖 𝐸1 − 𝐸12 ) 𝑊4 𝑇 × (󰜚1𝑖 𝐸1𝑇 − 𝐸12 ) 𝜉 (𝑡) , −𝑑𝑚𝑖

−∫

−𝑑2𝑖

∫

𝑡

𝑡+𝜃

󰜚4𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊5 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃

≤ − exp {−2𝜀𝑑2𝑖 } 𝜉𝑇 (𝑡) (󰜚2𝑖 𝐸1 − 𝐸13 ) 𝑊5 𝑇 × (󰜚2𝑖 𝐸1𝑇 − 𝐸13 ) 𝜉 (𝑡) .

1 (𝐸 − 𝐸4 ) 𝑉4 (𝐸2𝑇 − 𝐸4𝑇 )] 𝜉 (𝑡) , 𝜆 2𝑖 (𝑡) 2 (58)

where

𝑇 × (ℎ𝑖 𝐸1𝑇 − 𝐸23 ) 𝜉 (𝑡) , 0

𝑒 ̇ (𝑠) 𝑑𝑠]

1 (𝐸 − 𝐸2 ) 𝑉4 (𝐸3𝑇 − 𝐸2𝑇 ) 𝜆 1𝑖 (𝑡) 3

≤ − exp {−2𝜀ℎ} 𝜉𝑇 (𝑡) (ℎ𝐸1 − 𝐸23 ) 𝑊2

−∫


𝑡−𝑑𝑚𝑖

2

𝑡

𝑒 ̇ (𝑠) 𝑑𝑠]

2𝜀󰜚1𝑖 exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑1𝑖 }

× 𝜉𝑇 (𝑡) [

𝑇 × (𝜏𝑖 𝐸1𝑇 − 𝐸19 ) 𝜉 (𝑡) ,

𝑡−𝑑1𝑖


𝑡−𝑑𝑚𝑖

≤ − exp {−2𝜀𝜏𝑖 } 𝜉 (𝑡) (𝜏𝑖 𝐸1 − 𝐸19 ) 𝑊1

0

} 󰜚1𝑖

2𝜀󰜚1𝑖 exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑 (𝑡)}

−

𝑇

−∫ ∫

𝑡−𝑑1𝑖


𝜏𝑖 2 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑊1 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃 2

−∫ ∫

𝑡−𝑑𝑚𝑖

2𝜀󰜚1𝑖 exp {2𝜀𝑑𝑖 (𝑡)} − exp {2𝜀𝑑1𝑖 }

× [∫

Notice (b) of Lemma 5; then, 𝑡

+∫

× exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉4 𝑒 ̇ (𝑠) 𝑑𝑠

2𝜀𝑑1𝑖 𝜉𝑇 (𝑡) (𝐸1 − 𝐸3 ) 𝑉3 (𝐸1𝑇 − 𝐸3𝑇 ) 𝜉 (𝑡) . exp {2𝜀𝑑1𝑖 } − 1 (56)

0



𝑑1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉3 𝑒 ̇ (𝑠) 𝑑𝑠

≤−

𝑡−𝑑1𝑖

𝜆 1𝑖 (𝑡) =

exp {2𝜀𝑑𝑖 (𝑡)} − exp {2𝜀𝑑1𝑖 } , exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑1𝑖 }

exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑𝑖 (𝑡)} 𝜆 2𝑖 (𝑡) = . exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑1𝑖 }

(57)

(59)

By Lemma 6, there exists matrix 𝑁2 with appropriate dimensions such that −∫

𝑡−𝑑1𝑖

𝑡−𝑑𝑚𝑖


≤−

2𝜀󰜚1𝑖 𝜉𝑇 (𝑡) exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑1𝑖 }

× [𝐸3 − 𝐸2 𝐸2 − 𝐸4 ] [

𝑉4 𝑁2 𝐸3𝑇 − 𝐸2𝑇 ][ ] 𝜉 (𝑡) , 𝑁2𝑇 𝑉4 𝐸2𝑇 − 𝐸4𝑇

𝑉4 𝑁2 ] > 0. [ 𝑇 𝑁2 𝑉4

(60)

12

Mathematical Problems in Engineering 𝑡−𝑑

Similarly, considering − ∫𝑡−𝑑 1𝑖 󰜚1𝑖 exp{2𝜀(𝑠 − 𝑡)}𝑥𝑇 (𝑠)𝑈4 𝑚𝑖 𝑥(𝑠)𝑑𝑠 and following the same procedure, there exists matrix 𝑁1 with appropriate dimensions such that 𝑡−𝑑1𝑖

−∫

𝑡−𝑑𝑚𝑖

[ −∫

𝑡−ℎ


≤−

𝑡

× [𝐸10 𝐸12 − 𝐸10 ] [

ℎ exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑉2 𝑒 ̇ (𝑠) 𝑑𝑠

≤−

2𝜀󰜚1𝑖 𝜉𝑇 (𝑡) exp {2𝜀𝑑𝑚𝑖 } − exp {2𝜀𝑑1𝑖 } 𝑈4 𝑁1 ][ 𝑇 𝑇 ] 𝜉 (𝑡) , 𝑁1𝑇 𝑈4 𝐸12 − 𝐸10

𝑈 𝑁 [ 4𝑇 1 ] > 0. 𝑁1 𝑈4


×[

𝑇 𝐸10

𝑡−𝜏𝑖

(63)

𝑡−𝑑

Consider − ∫𝑡−𝑑 𝑚𝑖 󰜚2𝑖 exp{2𝜀(𝑠 − 𝑡)}𝑒𝑇 (𝑠)𝑈5 𝑒(𝑠)𝑑𝑠 and 2𝑖

2𝜀𝜏𝑖 𝜉𝑇 (𝑡) [𝐸18 𝐸19 − 𝐸18 ] exp {2𝜀𝜏𝑖 } − 1

×[

−∫

−∫

𝑡−𝑑𝑚𝑖

𝑡−𝑑2𝑖

𝑈1 𝑀1 𝐸𝑇 ] [ 𝑇 18 𝑇 ] 𝜉 (𝑡) , 𝑇 𝑀1 𝑈1 𝐸19 − 𝐸18 𝑈 𝑀1 [ 1𝑇 ] > 0, 𝑀1 𝑈1

𝑡

̇ − ∫𝑡−𝑑 𝑚𝑖 󰜚2𝑖 exp{2𝜀(𝑠 − 𝑡)}𝑒𝑇̇ (𝑠)𝑉5 𝑒(𝑠)𝑑𝑠, which are directly 2𝑖 estimated by (a) of Lemma 5; that is,

𝜏𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈1 𝑒 (𝑠) 𝑑𝑠

≤−


≤−

(62)

−∫

𝑡−𝑑𝑚𝑖

𝑡−𝑑2𝑖

𝜏𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒 ̇ (𝑠) 𝑉1 𝑒 ̇ (𝑠) 𝑑𝑠

≤−

2𝜀𝜏𝑖 𝜉𝑇 (𝑡) [𝐸1 − 𝐸16 𝐸16 − 𝐸14 ] exp {2𝜀𝜏𝑖 } − 1

𝑇 𝑉1 𝑀2 𝐸1𝑇 − 𝐸16 ×[ 𝑇 ][ 𝑇 𝑇 ] 𝜉 (𝑡) , 𝑀2 𝑉1 𝐸16 − 𝐸14

𝑉 𝑀2 [ 1𝑇 ] > 0. 𝑀2 𝑉1 For ℎ(𝑡) ∈ [0, ℎ], we obtain the following: 𝑡

−∫

𝑡−ℎ

ℎ exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑈2 𝑒 (𝑠) 𝑑𝑠

≤−


2𝜀󰜚2𝑖 𝑇 𝜉 (𝑡) , 𝜉𝑇 (𝑡) 𝐸13 𝑈5 𝐸13 exp {2𝜀𝑑2𝑖 } − exp {2𝜀𝑑𝑚𝑖 }


𝑇

𝑡−𝜏𝑖

𝑉2 𝑀4 ] > 0. 𝑀4𝑇 𝑉2

(61)

𝑡−𝑑

𝑡

𝜉𝑇 (𝑡) [𝐸1 − 𝐸21 𝐸21 − 𝐸20 ]

𝑇 𝑉2 𝑀4 𝐸1𝑇 − 𝐸21 ][ 𝑇 𝑇 𝑇 ] 𝜉 (𝑡) , 𝑀4 𝑉2 𝐸21 − 𝐸20

[

For 𝜏𝑖 (𝑡) ∈ [0, 𝜏𝑖 ], with the same matrix inequalities technique, we obtain the following: −∫

𝑈2 𝑀3 ] > 0, 𝑀3𝑇 𝑈2

≤−

2𝜀󰜚2𝑖 exp {2𝜀𝑑2𝑖 } − exp {2𝜀𝑑𝑚𝑖 }

× 𝜉𝑇 (𝑡) (𝐸4 − 𝐸5 ) 𝑉5 (𝐸4𝑇 − 𝐸5𝑇 ) 𝜉 (𝑡) . (64)

̇ = In addition, with the error system (53), we obtain 𝑒(𝑡) Λ𝜉(𝑡) and 𝑒𝑇̇ (𝑡) 𝐽𝑒 ̇ (𝑡) = 𝜉𝑇 (𝑡) Λ𝑇 𝐽Λ𝜉 (𝑡) ,

(65)

where Λ and 𝐽 have been defined in Theorem 7. Substituting (43)–(3.1) and (55)–(65) into (54), we obtain

𝜉𝑇 (𝑡) [𝐸22 𝐸23 − 𝐸22 ] 𝑇 𝐸22

𝑈 𝑀3 × [ 2𝑇 ][ 𝑇 𝑇 ] 𝜉 (𝑡) , 𝑀3 𝑈2 𝐸23 − 𝐸22

Γ𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) + 2𝜀𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) ≤𝜉

𝑇

(𝑡) (Ω𝜀𝑖0

+

Ω𝜀𝑖1

𝑇

+ Λ 𝐽Λ) 𝜉 (𝑡) + ℧,

(66)


13

where

Moreover, we have

℧= ∫

𝑡

𝑡−𝜏𝑗

7

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠)

𝑉 (𝑒 (𝑡0 ) , 𝑟 (𝑡0 ) , 𝑡0 ) = ∑ 𝑉𝑘 (𝑒 (𝑡0 ) , 𝑟 (𝑡0 ) , 𝑡0 ) , 𝑘=1

} { × { ∑ 𝛾𝑖𝑗 𝑄1𝑗 − ∑ 𝛾𝑖𝑗 𝑌1𝑖 − ∑ 𝛾𝑖𝑗 𝑌1𝑖 } 𝑒 (𝑠) 𝑑𝑠 𝑗∈S𝑖𝑘 𝑗∈S𝑖𝑢𝑘 } {𝑗 = ̸ 𝑖 +∫

𝑡

𝑡−𝜏𝑗

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠)

+∫

𝑡−𝑑2𝑗

−∫

𝑡−𝑑2𝑖

−∫

𝑡−𝑑𝑚𝑖

−∫

𝑡−𝑑1𝑖

𝑡−𝑑𝑚𝑖

𝑡0

exp {2𝜀 (𝑠 − 𝑡0 )} 𝑒𝑇̇ (𝑠) × {𝑄2𝑖 + 𝑄4 + ∑𝑇𝑙 } 𝑒 ̇ (𝑠) 𝑑𝑠 𝑙=1

0

+∫ ∫

𝑡0

𝜍 exp {2𝜀 (𝑠 − 𝑡0 )} 𝑒𝑇

−𝜍 𝑡0 +𝜃

5

× (𝑠) ( ∑ 𝑈𝑚 ) 𝑒 (𝑠) 𝑑𝑠 𝑑𝜃

(67)

𝑚=1

0

+∫ ∫

𝑡0

𝜍 exp {2𝜀 (𝑠 − 𝑡0 )} 𝑒𝑇̇

−𝜍 𝑡0 +𝜃

𝑇

5

󰜚2𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒 (𝑠) 𝑈5 𝑒 (𝑠) 𝑑𝑠,

× (𝑠) ( ∑ 𝑉𝑛 ) 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜃

(68) 𝑇

󰜚2𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒 ̇ (𝑠) 𝑉5 𝑒 ̇ (𝑠) 𝑑𝑠

𝑛=1

0

0

𝑡0

+∫ ∫ ∫ −𝜍 𝜃

are disposed and estimated by Lemma 6. 𝑡−𝑑1𝑖

𝑘=2

3

On the other hand, for 𝑑𝑖 (𝑡) ∈ [𝑑𝑚𝑖 , 𝑑2𝑖 ], the integral terms

𝑡−𝑑𝑚𝑖

4

𝑡0 −𝜍

{ } + 𝑒𝑇 (𝑡) { ∑ 𝛾𝑖𝑗 (𝑃𝑗 − 𝑋𝑖 )} 𝑒 (𝑡) . 𝑖 {𝑗∈S𝑢𝑘 }

𝑡−𝑑2𝑖

exp {2𝜀 (𝑠 − 𝑡0 )} 𝑒𝑇 (𝑠) × {𝑄1𝑖 + 𝑄3 + 𝑄5 + 𝑅1𝑖 + ∑ 𝑅𝑘 } 𝑒 (𝑠) 𝑑𝑠

+∫

𝑇

} { × { ∑ 𝛾𝑖𝑗 𝑅1𝑗 − ∑ 𝛾𝑖𝑗 𝑍𝑖 − ∑ 𝛾𝑖𝑗 𝑍𝑖 } 𝑒 (𝑠) 𝑑𝑠 𝑗∈S𝑖𝑘 𝑗∈S𝑖𝑢𝑘 } {𝑗 = ̸ 𝑖

−∫

𝑡0

𝑡0 −𝜍

exp {2𝜀 (𝑠 − 𝑡)} 𝑒 (𝑠)

𝑡−𝑑𝑚𝑖

𝑘=1

≤∫

} { × { ∑ 𝛾𝑖𝑗 𝑄2𝑗 − ∑ 𝛾𝑖𝑗 𝑌2𝑖 − ∑ 𝛾𝑖𝑗 𝑌2𝑖 } 𝑒 ̇ (𝑠) 𝑑𝑠 𝑗∈S𝑖𝑘 𝑗∈S𝑖𝑢𝑘 } {𝑗 = ̸ 𝑖 𝑡−𝑑1𝑖

7

∑ 𝑉𝑘 (𝑒 (𝑡0 ) , 𝑟 (𝑡0 ) , 𝑡0 )

𝑡0 +𝜆

𝜍2 exp {2𝜀 (𝑠 − 𝑡0 )} 𝑒𝑇̇ 2 5

× (𝑠) (∑𝑊𝑠 ) 𝑒 ̇ (𝑠) 𝑑𝑠 𝑑𝜆 𝑑𝜃

𝑇

󰜚1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒 (𝑠) 𝑈4 𝑒 (𝑠) 𝑑𝑠,

𝑠=1

(69) 𝑇

󰜚1𝑖 exp {2𝜀 (𝑠 − 𝑡)} 𝑒 ̇ (𝑠) 𝑉4 𝑒 ̇ (𝑠) 𝑑𝑠

+ 𝑒𝑇 (𝑡0 ) 𝑃𝑖 𝑒 (𝑡0 ) 󵄩 󵄩2 ≤ 𝜆󵄩󵄩󵄩𝑒 (𝑡0 )󵄩󵄩󵄩𝜍 .

(73)

are directly estimated by (a) of Lemma 5. Therefore, Γ𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) + 2𝜀𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) ≤ 𝜉𝑇 (𝑡) (Ω𝜀𝑖0 + Ω𝜀𝑖2 + Λ𝑇 𝐽Λ) 𝜉 (𝑡) + ℧.

Then from (72) and (73), it is readily seen that (70)

With (66) and (70), the following inequality is held for 𝑑𝑖 (𝑡) ∈ [𝑑1𝑖 , 𝑑2𝑖 ] if (23)–(28) are satisfied: Γ𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) + 2𝜀𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) < 0.

(71)

From the stochastic Lyapunov functional (34) and (71), it is held that 𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) ≥ min {𝜆 min (𝑃𝑖 )} ‖𝑒(𝑡)‖2 = 𝜆‖𝑒(𝑡)‖2 , 𝑖∈𝑆

𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) < exp {−2𝜀 (𝑡 − 𝑡0 )} 𝑉 (𝑒 (𝑡0 ) , 𝑟 (𝑡0 ) , 𝑡0 ) .

(72)

‖𝑒 (𝑡)‖ ≤ √

𝜆 󵄩 󵄩 exp {−𝜀 (𝑡 − 𝑡0 )} 󵄩󵄩󵄩𝑒 (𝑡0 )󵄩󵄩󵄩𝜍 , 𝜆

(74)

where 𝜅 = √𝜆/𝜆 ≥ 1. Therefore, the error system (13) with partially known transition rates and sector-bounded condition (9) is exponentially stable with a decay rate 𝜀. This completes the proof. In some situations, due to the complexity of NCDN (6), the information on the transition rates may be completely

14


unknown, which can be viewed as switched complex dynamical networks with arbitrary switching. The following corollary is therefore given to guarantee the exponential stability for this case. 𝛾𝑖𝑚 ,

𝛾𝑖𝑀,

𝜀, 𝜏𝑖 , ]𝑖 , ℎ, 𝑑1𝑖 , 𝑑2𝑖 , Corollary 8. For given scalars and 𝜇𝑖 and constant scalar 𝑑𝑚𝑖 satisfying 𝑑1𝑖 < 𝑑𝑚𝑖 < 𝑑2𝑖 , the error system (13) with entirely unknown transition rates and sector-bounded condition (9) is exponentially stable with ‖𝐺𝑖(3)

⊗ 𝐶𝑖 ‖ < 1, and there exist decay rate 𝜀 and 𝜅 = √𝜆/𝜆 if symmetric positive matrices 𝑃 > 0, 𝑄𝑗 > 0, (𝑗 = 1, 2, 3, 4, 5), 𝑅𝑘 > 0, (𝑘 = 1, 2, 3, 4), 𝑇𝑙 > 0, (𝑙 = 1, 2, 3), 𝑈𝑚 > 0, 𝑉𝑛 > 0, 𝑊𝑠 > 0, (𝑚, 𝑛, 𝑠 = 1, 2, 3, 4, 5) and matrices 𝑀𝑘1 , 𝑁𝑘2 , (𝑘1 , 𝑘2 = 1, 2, 3, 4), for any scalar 𝛿 > 0, and symmetric matrices 𝑌1 , 𝑌2 , 𝑍 such that (28) and the following matrix inequalities hold: −

∑ ∑ ∑

𝑗∈S𝑖𝑢𝑘 ,𝑗 ≠ 𝑖

(ii)

(iv)

(76)

𝑉4 𝑁2 [ 𝑇 ] > 0, 𝑁2 𝑉4

(77)

̃ < 0, ̃𝜀 + Ω ̃𝜀 + Λ ̃𝑇 𝐽̃Λ Ω 𝑖0 𝑖1

𝑈5 𝑁3 [ 𝑇 ] > 0, 𝑁3 𝑈5

(iii)


−

2𝜀𝜏𝑖 [𝐸18 𝐸19 − 𝐸18 ] exp {2𝜀𝜏𝑖 } − 1

𝑈 𝑀1 𝐸𝑇 × [ 1𝑇 ] [ 𝑇 18 𝑇 ] 𝑀1 𝑈1 𝐸19 − 𝐸18 −

−

𝛾𝑖𝑀 (𝑅1 − 𝑍) − 𝛾𝑖𝑚 𝑍 ≤ 0,

𝑈4 𝑁1 [ 𝑇 ] > 0, 𝑁1 𝑈4

(i)


2𝜀𝜏𝑖 [𝐸1 − 𝐸16 𝐸16 − 𝐸14 ] exp {2𝜀𝜏𝑖 } − 1

(75)

𝛾𝑖𝑀 (𝑄2 − 𝑌2 ) − 𝛾𝑖𝑚 𝑌2 ≤ 0,


−


𝑇 𝑉 𝑀2 𝐸1𝑇 − 𝐸16 × [ 1𝑇 ][ 𝑇 𝑇] 𝑀2 𝑉1 𝐸16 − 𝑒14

𝛾𝑖𝑀 (𝑄1 − 𝑌1 ) − 𝛾𝑖𝑚 𝑌1 ≤ 0,


−

𝑇 ) − exp {−2𝜀ℎ} (ℎ𝐸1 − 𝐸23 ) 𝑊2 (ℎ𝐸1𝑇 − 𝐸23

𝑉5 𝑁4 [ 𝑇 ] > 0, 𝑁4 𝑉5

̃ < 0, ̃𝜀 + Ω ̃𝜀 + Λ ̃𝑇 𝐽̃Λ Ω 𝑖2 𝑖0


×[ −

(78)

𝑈2 𝑀3 𝐸𝑇 ] [ 𝑇 22 𝑇 ] 𝑇 𝑀3 𝑈2 𝐸23 − 𝐸22 2𝜀ℎ

exp {2𝜀ℎ} − 1

×[

[𝐸22 𝐸23 − 𝐸22 ]

[𝐸1 − 𝐸21 𝐸21 − 𝐸20 ]

𝑇 𝑉2 𝑀4 𝐸1𝑇 − 𝐸21 ][ 𝑇 𝑇 𝑇 ], 𝑀4 𝑉2 𝐸21 − 𝐸20 𝑇

̃ 1 = (𝐺(1) ⊗ 𝐴 𝑖 ) 𝑃 + 𝑃 (𝐺(1) ⊗ 𝐴 𝑖 ) Π 𝑖 𝑖 + 2𝜀𝑃 + 𝑄1 + 𝑄3 + 𝑄5 + 𝑅2

where

2

2 2 2 + 𝜏𝑖 2 𝑈1 + ℎ 𝑈2 + 𝑑1𝑖 𝑈3 + 󰜚1𝑖 𝑈4 + 󰜚2i 𝑈5 ,

2 𝐽̃ = 𝑄2 + 𝑄4 + 𝑇1 + 𝜏𝑖 2 𝑉1 + ℎ 𝑉2 2 2 2 + 𝑑1𝑖 𝑉3 + 󰜚1𝑖 𝑉4 + 󰜚2𝑖 𝑉5 +

̃ 2 = 𝜛 (𝜀, 𝜇𝑖 ) 𝑅1 , Π 4

𝜏𝑖 4 ℎ 𝑊1 + 𝑊2 4 4

̃ 16 = − (1 − ]𝑖 ) exp {−2𝜀𝜏𝑖 } 𝑄1 , Π ̃ 17 = − (1 − ]𝑖 ) exp {−2𝜀𝜏𝑖 } 𝑄2 , Π

𝑑4 2 2 + 1𝑖 𝑊3 + 󰜚3𝑖 𝑊4 + 󰜚4𝑖 𝑊5 , 4 ̃ = Λ, Λ

̃ 𝜀 = Ω𝜀 , Ω 𝑖1 𝑖1

̃ 3 = exp {−2𝜀𝑑1𝑖 } (𝑅1 + 𝑅3 − 𝑅2 ) , Π

̃ 4 = Π4 , Π

̃ 7 = Π7 , Π

̃ 𝜀 = Ω𝜀 , Ω 𝑖2 𝑖2

24

̃ 𝜀 = ∑ 𝐸𝑚 Π ̃ 𝑚 𝐸𝑇 + ✠ − 𝛿 Ω 𝑖0 𝑚 𝑚=1

̂ 𝐹 ̂ 𝐸𝑇 𝐹 × [𝐸1 𝐸24 ] [ 1 2 ] [ 𝑇1 ] ∗ 𝐼 𝐸24 2𝜀𝑑1𝑖 − (𝐸 − 𝐸3 ) 𝑉3 (𝐸1𝑇 − 𝐸3𝑇 ) exp {2𝜀𝑑1𝑖 } − 1 1 𝑇 ) − exp {−2𝜀𝜏𝑖 } (𝜏𝑖 𝐸1 − 𝐸19 ) 𝑊1 (𝜏𝑖 𝐸1𝑇 − 𝐸19

̃ 5 = Π5 , Π

̃ 9 = Π9 , Π ̃ 𝑚 = 0, Π

̃ 14 = Π14 , Π

̃ 6 = Π6 , Π

̃ 8 = Π8 , Π ̃ 15 = Π15 , Π

̃ 20 = Π20 , Π

(𝑚 = 10, 11, 12, 13, 18, 19, 21, . . . , 24) ,

𝜆 = 𝜆 max (𝑃) +

1 − exp {−2𝜀𝜍} 2𝜀

4 3 } {5 × { ∑𝜆 max (𝑄𝑗 ) + ∑ 𝜆 max (𝑅𝑘 ) + ∑𝜆 max (𝑇𝑙 )} 𝑘=1 𝑙=1 } {𝑗=1 exp {−2𝜀𝜍} + 2𝜀𝜍 − 1 + 4𝜀2

Mathematical Problems in Engineering 5

15

5

3.2. The Sampled-Data Controllers for NCDN. In this subsection, we give the design method of the desired sampleddata controllers to ensure the NCDN (6) exponentially synchronized on the basis of Theorem 7.

× { ∑ 𝜆 max (𝑈𝑚 ) + ∑ 𝜆 max (𝑉𝑛 )} 𝑚=1

+

𝑛=1 2 2

1 − exp {−2𝜀𝜍} + 2𝜀 𝜍 − 2𝜀𝜍 8𝜀3 5

× {∑𝜆 max (𝑊𝑠 )} . 𝑠=1

(79) Other notations are the same as those in Theorem 7. Proof. Since the elements of transition rates are entirely unknown, we choose the Lyapunov functional as follows: 7

𝑉 (𝑒 (𝑡) , 𝑖, 𝑡) = 𝑒𝑇 (𝑡) 𝑃𝑒 (𝑡) + ∑ 𝑉𝑘 (𝑒 (𝑡) , 𝑖, 𝑡) ,

(80)

Theorem 9. The NCDN (6) with partially known transition rates and sector-bounded condition (9) is exponentially synchronized by controllers of the form (11) if there exist symmetric positive definite matrices 𝑃𝑖 = diag{𝑃𝑖(1) , 𝑃𝑖(2) , . . . , 𝑃𝑖(𝑁) } > 0, 𝑄1𝑖 > 0, 𝑄2𝑖 > 0, 𝑅1𝑖 > 0, (𝑖 ∈ 𝑆 = {1, 2, . . . , 𝑀}), 𝑄𝑗 > 0, (𝑗 = 3, 4, 5), 𝑅𝑘 > 0, (𝑘 = 2, 3, 4), 𝑇𝑙 > 0, (𝑙 = 1, 2, 3), 𝑈𝑚 > 0, 𝑉𝑛 > 0, 𝑊𝑠 > 0, (𝑚, 𝑛, 𝑠 = 1, 2, 3, 4, 5), matrices Y𝑖 = diag{𝑌𝑖(1) , 𝑌𝑖(2) , . . . , 𝑌𝑖(𝑁) } and matrices 𝑀𝑘1 , 𝑁𝑘2 , (𝑘1 , 𝑘2 = 1, 2, 3, 4), for any scalar 𝛿 > 0, and symmetric matrices 𝑋𝑖 , 𝑌1𝑖 , 𝑌2𝑖 , 𝑍𝑖 , (𝑖 ∈ 𝑆) such that (23)–(28) and the following linear matrix inequalities hold:

𝑘=2

[

(i)

where 𝑉2 (𝑒 (𝑡) , 𝑖, 𝑡) = ∫

𝑡

𝑇

𝑡−𝜏𝑖 (𝑡) 𝑡

𝑡−𝜏𝑖 (𝑡) 𝑡

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑄2 𝑒 ̇ (𝑠) 𝑑𝑠

+∫

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑄3 𝑒 (𝑠) 𝑑𝑠

+∫

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑄4 𝑒 ̇ (𝑠) 𝑑𝑠

+∫

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑄5 𝑒 (𝑠) 𝑑𝑠,

𝑡−𝜏𝑖 𝑡 𝑡−𝜏𝑖 𝑡

𝑉3 (𝑒 (𝑡) , 𝑖, 𝑡) = ∫

𝑡−ℎ 𝑡−𝑑1𝑖

𝑡−𝑑𝑖 (𝑡) 𝑡

𝑡−𝑑1𝑖

+∫

𝑡−𝑑𝑚𝑖 𝑡−𝑑𝑚𝑖

+∫

𝑡−𝑑2𝑖

𝑡−𝑑1𝑖

+∫

𝑡−𝑑𝑚𝑖 𝑡−𝑑𝑚𝑖

+∫

𝑡−𝑑2𝑖

𝑉 𝑁 [ 4𝑇 2 ] > 0, 𝑁2 𝑉4

Θ𝜀 + Ω𝜀𝑖1 Λ𝑇 𝑃𝑖 [ 𝑖0 ] < 0, 𝑃𝑖 Λ −2𝑃𝑖 + 𝐽

𝑈5 𝑁3 ] > 0, 𝑁3𝑇 𝑈5

[

𝑉5 𝑁4 ] > 0, 𝑁4𝑇 𝑉5

Θ𝜀 + Ω𝜀𝑖2 Λ𝑇 𝑃𝑖 [ 𝑖0 ] < 0. 𝑃𝑖 Λ −2𝑃𝑖 + 𝐽

(82)

(83)

In addition, the desired sampled-data controllers gain matrices are given by −1

𝐾𝑘 = (𝑃𝑖(𝑘) ) 𝑌𝑖(𝑘) ,

𝑘 = 1, 2, . . . , 𝑁,

(84)

where 24

𝑇

exp {2𝜀 (𝑠 − 𝑡)} 𝑒 (𝑠) 𝑅3 𝑒 (𝑠) 𝑑𝑠 exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇 (𝑠) 𝑅4 𝑒 (𝑠) 𝑑𝑠,

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑇1 𝑒 ̇ (𝑠) 𝑑𝑠

𝑡−𝑑1𝑖

[

(iv)


𝑡−𝑑1𝑖

𝑡

(iii)


+∫

𝑉4 (𝑒 (𝑡) , 𝑖, 𝑡) = ∫

(ii)

exp {2𝜀 (𝑠 − 𝑡)} 𝑒 (𝑠) 𝑄1 𝑒 (𝑠) 𝑑𝑠

+∫

𝑈4 𝑁1 ] > 0, 𝑁1𝑇 𝑈4

𝑇 Θ𝜀𝑖0 = ∑ 𝐸𝑚 Π𝑚 𝐸𝑚 + ✠Θ − 𝛿 [𝐸1 𝐸24 ] 𝑚=1

̂ 𝐹 ̂ 𝐸𝑇 2𝜀𝑑1𝑖 𝐹 × [ 1 2 ] [ 𝑇1 ] − ∗ 𝐼 exp {2𝜀𝑑1𝑖 } − 1 𝐸24 × (𝐸1 − 𝐸3 ) 𝑉3 (𝐸1𝑇 − 𝐸3𝑇 ) 𝑇 ) − exp {−2𝜀𝜏𝑖 } (𝜏𝑖 𝐸1 − 𝐸19 ) 𝑊1 (𝜏𝑖 𝐸1𝑇 − 𝐸19

𝑇

exp {2𝜀 (𝑠 − 𝑡)} 𝑒 ̇ (𝑠) 𝑇2 𝑒 ̇ (𝑠) 𝑑𝑠

𝑇 − exp {−2𝜀ℎ} (ℎ𝐸1 − 𝐸23 ) 𝑊2 (ℎ𝐸1𝑇 − 𝐸23 )

exp {2𝜀 (𝑠 − 𝑡)} 𝑒𝑇̇ (𝑠) 𝑇3 𝑒 ̇ (𝑠) 𝑑𝑠, (81)

and 𝑉5 (𝑒(𝑡), 𝑖, 𝑡), 𝑉6 (𝑒(𝑡), 𝑖, 𝑡), 𝑉7 (𝑒(𝑡), 𝑖, 𝑡) are the same as (39), (40), (41). Then we follow a similar line as in proof of Theorem 7 and obtain the result.

− exp {−2𝜀𝑑1𝑖 } (𝑑1𝑖 𝐸1 − 𝐸9 ) 𝑊3 (𝑑1𝑖 𝐸1𝑇 − 𝐸9𝑇 ) 𝑇 ) − exp {−2𝜀𝑑𝑚𝑖 } (󰜚1𝑖 𝐸1 − 𝐸12 ) 𝑊4 (󰜚1𝑖 𝐸1𝑇 − 𝐸12 𝑇 ) − exp {−2𝜀𝑑2𝑖 } (󰜚2𝑖 𝐸1 − 𝐸13 ) 𝑊5 (󰜚2𝑖 𝐸1𝑇 − 𝐸13

−

2𝜀𝜏𝑖 [𝐸18 𝐸19 − 𝐸18 ] exp {2𝜀𝜏𝑖 } − 1

16

Mathematical Problems in Engineering ×[ −

𝑈1 𝑀1 𝐸𝑇 ] [ 𝑇 18 𝑇 ] 𝑇 𝑀1 𝑈1 𝐸19 − 𝐸18

of (86) and (iv) of (87), we define Y𝑖 = 𝑃𝑖 K and obtain the following:

2𝜀𝜏𝑖 [𝐸1 − 𝐸16 𝐸16 − 𝐸14 ] exp {2𝜀𝜏𝑖 } − 1

X[

Θ𝜀 + Ω𝜀𝑖2 Λ𝑇 𝑃𝑖 Ω𝜀 + Ω𝜀𝑖2 Λ𝑇 𝐽 𝑇 X [ 𝑖0 ] < 0. ] X = [ 𝑖0 𝐽Λ −𝐽 𝑃𝑖 Λ −𝑃𝑖 𝐽−1 𝑃𝑖

𝑇 𝑉 𝑀2 𝐸1𝑇 − 𝐸16 × [ 1𝑇 ][ 𝑇 𝑇] 𝑀2 𝑉1 𝐸16 − 𝑒14

−


×[ −

×

2𝜀ℎ

[𝐸1 − 𝐸21 𝐸21 − 𝐸20 ]

Remark 10. It should be mentioned that the sampled-data synchronization problem has been solved for NCDN (6) in Theorem 9, and the desired controllers can be obtained when LMIs (23)–(26), (82), and (83) are feasible. In this paper, we construct the novel stochastic Lyapunov functional (34) containing some triple-integral terms; which is very effective in the reduction of conservatism [45]. Besides, we apply Lemma 5 to the corresponding terms, the method by using reciprocally convex lemma [46] can achieve less conservative results. Thus the obtained delay-range-dependent and decay rate-dependent stability condition for NCDN (6) in Theorem 7 is less conservative than the previous ones, which can achieve larger maximum value of sampling period ℎ and will be verified in Section 4.

𝑇 𝑉 𝑀4 𝐸1𝑇 − 𝐸21 [ 2𝑇 ][ 𝑇 𝑇 ], 𝑀4 𝑉2 𝐸21 − 𝐸20

✠Θ = 𝐸1 𝑃𝑖 (𝐺𝑖(2) ⊗ 𝐵𝑖 ) 𝐸2𝑇 + 𝐸1 𝑃𝑖 (𝐺𝑖(3) ⊗ 𝐶𝑖 ) 𝐸17 𝑇 𝑇 + 𝐸1 Y𝑖 𝐸21 + 𝐸1 𝑃𝑖 (𝐼𝑁 ⊗ 𝐷𝑖 ) 𝐸24 𝑇

+ 𝐸2 (𝐺𝑖(2) ⊗ 𝐵𝑖 ) 𝑃𝑖 𝐸1𝑇 𝑇

+ 𝐸17 (𝐺𝑖(3) ⊗ 𝐶𝑖 ) 𝑃𝑖 𝐸1𝑇 + 𝐸21 Y𝑖𝑇 𝐸1𝑇 𝑇

+ 𝐸24 (𝐼𝑁 ⊗ 𝐷𝑖 )

𝑃𝑖 𝐸1𝑇 . (85)

Other notations are the same as those in Theorem 7. Proof. Considering (27) and (28), by applying the Schu Complement Lemma, we have (i)

[

𝑈4 𝑁1 ] > 0, 𝑁1𝑇 𝑈4

(ii) (iii)

[

𝑉4 𝑁2 ] > 0, 𝑁2𝑇 𝑉4

Ω𝜀 + Ω𝜀𝑖1 Λ𝑇 𝐽 [ 𝑖0 ] < 0, 𝐽Λ −𝐽

𝑈5 𝑁3 ] > 0, 𝑁3𝑇 𝑈5

(iv)

[

𝑉 𝑁 [ 5𝑇 4 ] > 0, 𝑁4 𝑉5

Ω𝜀 + Ω𝜀𝑖2 Λ𝑇 𝐽 ] < 0. [ 𝑖0 𝐽Λ −𝐽

(86)

(87)

0 𝑖 ≠ 𝑗 𝑖=𝑗

Remark 11. It should be pointed out that the given results can be extended to more general neutral delay in the form of NCDN (6). For instance, 𝜏1𝑖 ≤ 𝜏𝑖 (𝑡) ≤ 𝜏2𝑖 ; we can choose 𝜏𝑚𝑖 satisfying 𝜏1𝑖 ≤ 𝜏𝑚𝑖 ≤ 𝜏2𝑖 and construct the stochastic Lyapunov functional on [𝑡 − 𝜏𝑖 (𝑡), 𝑡 − 𝜏1𝑖 ], [𝑡 − 𝜏1𝑖 , 𝑡], [𝑡 − 𝜏2𝑖 , 𝑡 − 𝜏𝑚𝑖 ], and [𝑡 − 𝜏𝑚𝑖 , 𝑡 − 𝜏1𝑖 ]. Following the same thought, the time delay ℎ(𝑡) in the control input can also be extended to be more general. Corollary 12. The NCDN (6) with completely unknown transition rates and sector-bounded condition (9) is exponentially synchronized by controllers of the form (11) if there exist symmetric positive definite matrices 𝑃 = diag{𝑃(1) , 𝑃(2) , . . . , 𝑃(𝑁) } > 0, 𝑄𝑗 > 0, (𝑗 = 1, 2, 3, 4, 5), 𝑅𝑘 > 0, (𝑘 = 1, 2, 3, 4), 𝑇𝑙 > 0, (𝑙 = 1, 2, 3), 𝑈𝑚 > 0, 𝑉𝑛 > 0, 𝑊𝑠 > 0, (𝑚, 𝑛, 𝑠 = 1, 2, 3, 4, 5), matrices Y = diag{𝑌(1) , 𝑌(2) , . . . , 𝑌(𝑁) } and matrices 𝑀𝑘1 , 𝑁𝑘2 , (𝑘1 , 𝑘2 = 1, 2, 3, 4), for any scalar 𝛿 > 0, and symmetric matrices 𝑌1 , 𝑌2 , 𝑍 such that (28), (75), (76), and the following linear matrix inequalities hold: (i)

Define the block diagonal matrix X = [ 𝐻 𝑃𝑖 𝐽−1 ], where

𝑇 𝑇 𝐻 = ∑24 𝑚=1 𝐸𝑚 𝐸𝑚 . Noting that 𝐸𝑖 𝐸𝑗 = { 𝐼

(88)

Due to 𝐽 > 0 and (𝑃𝑖 − 𝐽)𝐽−1 (𝑃𝑖 − 𝐽) ≥ 0, we know that −𝑃𝑖 𝐽−1 𝑃𝑖 ≤ −2𝑃𝑖 + 𝐽. Thus (88) are held if (ii) of (82) and (iv) of (83) are satisfied, which implies that (ii) of (27) and (iv) of (28) are held. According to Definition 3, by the sampled-data controllers gain matrices (84), the NCDN (6) with partially known transition rates and sector-bounded condition (9) is exponentially synchronized. This completes the proof.

[𝐸22 𝐸23 − 𝐸22 ]

𝑈2 𝑀3 𝐸𝑇 ] [ 𝑇 22 𝑇 ] 𝑇 𝑀3 𝑈2 𝐸23 − 𝐸22

exp {2𝜀ℎ} − 1

Θ𝜀 + Ω𝜀𝑖1 Λ𝑇 𝑃𝑖 Ω𝜀𝑖0 + Ω𝜀𝑖1 Λ𝑇 𝐽 𝑇 ] < 0, ] X = [ 𝑖0 𝐽Λ −𝐽 𝑃𝑖 Λ −𝑃𝑖 𝐽−1 𝑃𝑖

, with (ii)

𝑈4 𝑁1 ] > 0, [ 𝑇 𝑁1 𝑈4 (ii)

𝑉4 𝑁2 [ 𝑇 ] > 0, 𝑁2 𝑉4

̃𝜀 + Ω ̃𝜀 Θ Λ𝑇 𝑃 𝑖1 [ 𝑖0 ] < 0, 𝑃Λ −2𝑃 + 𝐽

(89)

Mathematical Problems in Engineering (iii)

[

𝑈5 𝑁3 ] > 0, 𝑁3𝑇 𝑈5

(iv)

[

[

𝑉5 𝑁4 ] > 0, 𝑁4𝑇 𝑉5

̃𝜀 + Ω ̃𝜀 Θ Λ𝑇 𝑃 𝑖0 𝑖2 ] < 0. 𝑃Λ −2𝑃 + 𝐽

17 ̃ Θ = 𝐸1 𝑃 (𝐺𝑖(2) ⊗ 𝐵𝑖 ) 𝐸2𝑇 + 𝐸1 𝑃 (𝐺𝑖(3) ⊗ 𝐶𝑖 ) 𝐸17 ✠ (90)

𝑇

−1

𝑘 = 1, 2, . . . , 𝑁,

𝑇

+ 𝐸2 (𝐺𝑖(2) ⊗ 𝐵𝑖 ) 𝑃𝐸1𝑇 + 𝐸17 (𝐺𝑖(3) ⊗ 𝐶𝑖 ) 𝑃𝐸1𝑇

In addition, the desired sampled-data controllers gain matrices are given by 𝐾𝑘 = (𝑃(𝑘) ) 𝑌(𝑘) ,

𝑇 𝑇 + 𝐸1 Y𝑖 𝐸21 + 𝐸1 𝑃 (𝐼𝑁 ⊗ 𝐷𝑖 ) 𝐸24

𝑇

+ 𝐸21 Y𝑖𝑇 𝐸1𝑇 + 𝐸24 (𝐼𝑁 ⊗ 𝐷𝑖 ) 𝑃𝐸1𝑇. (92)

(91)

Other notations are the same as those in Corollary 8.

where 24

̃ 𝜀 = ∑ 𝐸𝑚 Π ̃ 𝑚 𝐸𝑇 + ✠ ̃Θ Θ 𝑖0 𝑚 𝑚=1

̂ 𝐹 ̂ 𝐸𝑇 𝐹 − 𝛿 [𝐸1 𝐸24 ] [ 1 2 ] [ 𝑇1 ] ∗ 𝐼 𝐸24 −

2𝜀𝑑1𝑖 (𝐸 − 𝐸3 ) 𝑉3 (𝐸1𝑇 − 𝐸3𝑇 ) exp {2𝜀𝑑1𝑖 } − 1 1

𝑇 ) − exp {−2𝜀𝜏𝑖 } (𝜏𝑖 𝐸1 − 𝐸19 ) 𝑊1 (𝜏𝑖 𝐸1𝑇 − 𝐸19

4. Numerical Examples In this section, numerical examples are used to illustrate the effectiveness of the results derived above. Example 1. As shown in the example, a three-node NCDN (6) with Markovian switching between two modes is taken into consideration; that is, 𝑁 = 3 and 𝑀 = 2. The parametric matrices of the NCDN are given as follows: −0.50 −0.35 ], 0.10 −0.50

𝐴2 = [


0.20 −0.15 𝐵1 = [ ], 0.50 −0.50

𝐵2 = [


0.28 0.02 ], 𝐶1 = [ −0.06 0.11

0.51 0.24 𝐶2 = [ ], 0.02 −0.44

0.20 0 ], 𝐷1 = [ 0 −0.15

𝐷2 = [

𝑇 − exp {−2𝜀ℎ} (ℎ𝐸1 − 𝐸23 ) 𝑊2 (ℎ𝐸1𝑇 − 𝐸23 )


2𝜀𝜏𝑖 − [𝐸18 𝐸19 − 𝐸18 ] exp {2𝜀𝜏𝑖 } − 1 𝑈1 𝑀1 𝐸𝑇 ×[ 𝑇 ] [ 𝑇 18 𝑇 ] 𝑀1 𝑈1 𝐸19 − 𝐸18 2𝜀𝜏𝑖 − [𝐸1 − 𝐸16 𝐸16 − 𝐸14 ] exp {2𝜀𝜏𝑖 } − 1 𝑇 𝑉1 𝑀2 𝐸1𝑇 − 𝐸16 ×[ 𝑇 ][ 𝑇 𝑇] 𝑀2 𝑉1 𝐸16 − 𝑒14

𝐴1 = [

−0.30 0.09 ], 0.20 −0.40

0.31 0.23 ], −0.12 0.17

0.30 0 ], −0.10 0.23

−2 1 1 𝐺1(1) = [ 1 −2 1 ] , [ 1 1 −2]

0 0 0 𝐺2(1) = [1 −1 0 ] , [1 1 −2]

−2 1 1 𝐺1(2) = [ 0 −1 1 ] , [ 1 0 −1]

−2 1 1 𝐺2(2) = [ 0 0 0 ] , [ 1 1 −2]

0 0 0 𝐺1(3) = [1 −2 1 ] , [1 0 −1]

−1 0 1 𝐺2(3) = [ 1 −2 1] . [ 0 0 0] (93)

−


×[

−

𝑈2 𝑀3 𝐸𝑇 ] [ 𝑇 22 𝑇 ] 𝑇 𝑀3 𝑈2 𝐸23 − 𝐸22 2𝜀ℎ

exp {2𝜀ℎ} − 1

×[

[𝐸22 𝐸23 − 𝐸22 ]

[𝐸1 − 𝐸21 𝐸21 − 𝐸20 ]

𝑇 𝑉2 𝑀4 𝐸1𝑇 − 𝐸21 ] [ 𝑇 𝑇 ], 𝑀4𝑇 𝑉2 𝐸21 − 𝐸20

The partially known transition rate matrix is considered as in the following two cases: −2 2 Υ1 = [ 0 0 ] , 𝛾21 𝛾22

Υ2 = [

0 0 𝛾12 𝛾11 0 0 ]. 𝛾21 𝛾22

(94)

Furthermore, the nonlinear function 𝑓(𝑥𝑘 (𝑡)) is given by 𝑇

𝑓 (𝑥𝑘 (𝑡)) = [0.4𝑥𝑘1 (𝑡) − tanh (0.1𝑥𝑘1 (𝑡)) 0.1𝑥𝑘2 (𝑡)] . (95)

18

Mathematical Problems in Engineering For the case of Υ1 , with the above parameters, by Theorem 9, we can obtain the sampled-data controllers as follows:

Then, it is easy to verify that 𝐹1 = [

0.4 0 ], 0 0.1

𝐹2 = [

0.3 0 ]. 0 0.1

(96)

−0.7246 −0.0345 𝐾1 = [ ], −0.0253 −0.6358

The interval mode-dependent time-varying neutral delays and discrete delays are, respectively, assumed to be

−0.5149 −0.0326 𝐾2 = [ ], −0.0117 −0.5048

𝜏1 (𝑡) = 0.5 (1 + sin 𝑡) ,

−0.3557 −0.0263 ]. 𝐾3 = [ −0.0046 −0.4356

𝜏2 (𝑡) = 0.6 (1 + cos 𝑡) ,

𝑑1 (𝑡) = 0.4 (1 + sin4 (𝑡)) ,

𝑑2 (𝑡) = 0.8 (1 + cos4 (𝑡)) . (97)

They are governed by the Markov process {𝑟(𝑡), 𝑡 ≥ 0} and shown in Figures 1 and 2. It can be readily obtained that 𝜏1 = 1,

𝜏2 = 1.2,

𝑑11 = 0.4, 𝑑22 = 1.6,

]1 = 0.5,

𝑑21 = 0.8, 𝜇1 =

3√3 , 5

]2 = 0.6,

𝑑12 = 0.8, 𝜇2 =

(98)

Furthermore, the state trajectories of the error system (13) 𝑇 are given in Figure 3, where 𝑥1 (0) = [0.2 0.1] , 𝑥2 (0) = 𝑇 𝑇 𝑇 [0.3 −0.2] , 𝑥3 (0) = [−0.1 0.1] , and 𝑠0 = [0.1 0.2] . For another case of Υ2 , the sampled-data controllers can be readily obtained by Corollary 12. Example 2. Consider NCDN (6) with three nodes but only one mode; that is, 𝑁 = 3 and 𝑀 = 1. So the subscript of the mode 𝑖 is omitted here. The outer-coupling matrices are assumed to be −1 0 1 𝐺(1) = 𝐺(2) = [ 0 −1 1 ] , [ 1 1 −2]

3√3 . 10

Given decay rate 𝜀 = 0.3 and the maximum value of sampling period ℎ = 0.2, we choose 𝑑𝑚1 = 0.5 and 𝑑𝑚2 = 1.0 and seek to solve the sampled-data controllers.

𝐺(3) = 0.

𝑇

(101) can be easily verified that −0.5 0.2 ], 0 0.95

𝐹2 = [

−0.3 0.2 ]. 0 0.2

(102)

The time-varying delay is chosen as 𝑑(𝑡) = 0.2 + 0.05 sin(10𝑡); then we know that 𝑑1 = 0.15, 𝑑2 = 0.25, and 𝜇 = 0.5. Besides, we choose 𝐷 = 𝐼2 , 𝑑𝑚 = 0.18 satisfying 𝑑1 < 𝑑𝑚 < 𝑑2 and consider the following two cases. Case 1. The inner-coupling matrices are given as 𝐴 = 0 and 0 𝐵 = [ 0.5 0 0.5 ]. According to the above parameters, the maximum value of sampling period ℎ, which satisfies LMIs (23)–(26), (82), and (83) in Theorem 9, can be calculated by solving a quasiconvex optimization problem. The degraded NCDN has also been considered in [34, 35]. The results on the maximum value of sampling period ℎ are compared in Table 1, where it can be seen that our method is less conservative than [34, 35]. Case 2. The inner-coupling matrices are given as 𝐴 = diag{0.3, 0.3} and 𝐵 = diag{0.4, 0.4}. The maximum value of sampling period ℎ also can be solved by Theorem 9. The

(100)

In fact, the NCDN in consideration here has become a common CDN. The nonlinear function 𝑓(𝑥𝑘 (𝑡)) is taken as

𝑓 (𝑥𝑘 (𝑡)) = [−0.5𝑥𝑘1 (𝑡) + tanh (0.2𝑥𝑘1 (𝑡)) + 0.2𝑥𝑘2 (𝑡) 0.95𝑥𝑘2 (𝑡) − tanh (0.75𝑥𝑘2 (𝑡)) ] ,

𝐹1 = [

(99)

(101)

results are listed in Table 2. In addition, the proposed method in [34] is not applicable here. From Table 2, it also can be seen that our method is less conservative than existing ones.

5. Conclusions In this paper, the sampled-data synchronization problem has been solved for a class of NCDNs with Markovian jump parameters and partially known transition rates. The discrete and neutral delays are considered to be interval modedependent and time varying, while the sampling period here is bounded and time-varying. With a novel stochastic Lyapunov functional, the delay-range-dependent and rate-dependent exponential synchronization conditions have been proposed by Lyapunov theory and better technique of matrix inequalities. Then the sampled-data controllers have been designed on the basis of the obtained conditions. These theoretical results are successfully verified through numerical examples, whose simulation results are less conservative than the previous results. Finally, the main contributions of this


19

Table 1: Maximum value of sampling period ℎ with 𝑑𝑚 = 0.18 and 𝜀 = 0. Methods Li et al. [34]

Wu et al. [35]

Theorem 9

ℎ = 0.5409

Sampled-data controllers −0.4856 −0.1435 −0.4857 −0.1435 −0.2435 −0.1036 [ ] [ ] [ ] 𝐾1 = [−0.0987 −1.5662], 𝐾2 = [−0.0987 −1.5664], 𝐾3 = [−0.0063 −1.0367]

ℎ = 0.5573

[ ] [ ] [ ] −0.4201 −0.1614 −0.4201 −0.1614 0.1221 −0.2073 [ ] [ ] [ ] 𝐾1 = [ 0.0001 −1.1698], 𝐾2 = [ 0.0001 −1.1698], 𝐾3 = [−0.0024 −1.0093]

ℎ = 0.5764

[ ] [ ] [ ] −0.4096 −0.1523 −0.4096 −0.1523 0.1176 −0.2135 [ ] [ ] [ ] 𝐾1 = [ 0.0012 −1.1577], 𝐾2 = [ 0.0012 −1.1577], 𝐾3 = [−0.0046 −0.9438] [

]

]

[

]

1.4

1 0.9

1.2

0.8

Time-varying neutral delay

Time-varying neutral delay

[

0.7 0.6 0.5 0.4 0.3

1 0.8 0.6 0.4

0.2 0.2

0.1 0

2

0

4

6

8

0

10

0

2

4

6

Time t

8

10

Time t

(a)

(b)

0.8

1.6

0.75

1.5 Time-varying discrete delay

Time-varying discrete delay

Figure 1: Time-varying neutral delay 𝜏𝑖 (𝑡) at mode 1 and mode 2.

0.7 0.65 0.6 0.55 0.5 0.45 0.4

1.4 1.3 1.2 1.1 1 0.9

0

1

2

3 Time t (a)

4

5

6

0.8

0

1

2

3 Time t (b)

Figure 2: Time-varying discrete delay 𝑑𝑖 (𝑡) at mode 1 and mode 2.

4

5

6

20

Mathematical Problems in Engineering Table 2: Maximum value of sampling period ℎ with 𝑑𝑚 = 0.18 and 𝜀 = 0.

Methods Li et al. [34]

—

Wu et al. [35]

ℎ = 0.0500

Sampled-data controllers — −0.4330 −0.2074 −0.4330 −0.2071 −0.1631 −0.1744 [ ] [ ] [ ] 𝐾1 = [−0.0631 −1.5627], 𝐾2 = [−0.0631 −1.5627], 𝐾3 = [−0.0355 −1.0803]

ℎ = 0.0617

[ ] [ ] [ ] −0.4252 −0.1836 −0.4252 −0.1845 −0.1466 −0.1746 [ ] [ ] [ ] 𝐾1 = [ 0.0094 −1.1847], 𝐾2 = [ 0.0092 −1.1847], 𝐾3 = [−0.0243 −1.0125]

Theorem 9

[

]

[

]

[

]

of China (20093402110019), the Anhui Provincial Natural Science Foundation (11040606M143), the Fundamental Research Funds for the Central Universities, and the Program for New Century Excellent Talents in University.

0.3 0.2 0.1

References

0 −0.1 −0.2 −0.3 −0.4

0

5

10

15

Time (s) e11 (t) e12 (t) e21 (t)

e22 (t) e31 (t) e32 (t)

Figure 3: State trajectories of the error system in Example 1.

paper can be summarized as follows. (i) To achieve exponential synchronization for NCDN (6), the desired sampleddata feedback controllers have been designed in terms of the solution to certain LMIs. The proposed results are expressed in a new representation, which are theoretically and numerically proved to be less conservative than some existing ones. (ii) The constructed stochastic Lyapunov functional contains some triple-integral terms, which are very effective in the reduction of conservativeness and have not appeared in the context of NCDNs. (iii) The bound of the delay is fully utilized in this paper; that is, improved bounding technique is used to reduce the conservativeness. (iv) The reciprocally convex lemma is used to derive the delay-range-dependent and ratedependent stability conditions, which can well reduce the conservativeness of the investigated systems.

Acknowledgments This work was supported in part by the National Key Scientific Research Project (61233003), the National Natural Science Foundation of China (60935001, 61174061, and 61074033), the Doctoral Fund of the Ministry of Education

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