Impulsive and pinning control synchronization of Markovian jumping complex dynamical networks with hybrid coupling and additive interval time-varying delays

Impulsive and pinning control synchronization of Markovian jumping complex dynamical networks with hybrid coupling and additive interval time-varying delays

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Impulsive and pinning control synchronization of Markovian jumping complex dynamical networks with hybrid coupling and additive interval time-varying delays J. Yogambigai, M. Syed Ali, Hamed Alsulami, Mohammed S. Alhodaly PII: DOI: Reference:

S1007-5704(20)30049-6 https://doi.org/10.1016/j.cnsns.2020.105215 CNSNS 105215

To appear in:

Communications in Nonlinear Science and Numerical Simulation

Please cite this article as: J. Yogambigai, M. Syed Ali, Hamed Alsulami, Mohammed S. Alhodaly, Impulsive and pinning control synchronization of Markovian jumping complex dynamical networks with hybrid coupling and additive interval time-varying delays, Communications in Nonlinear Science and Numerical Simulation (2020), doi: https://doi.org/10.1016/j.cnsns.2020.105215

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Highlights • Markovian jumping complex dynamical networks with hybrid coupling and additive interval timevarying delay is considered. • We constructed the appropriate controllers to achieve impulsive and pinning control synchronization. • New delay-dependent impulsive and pinning control synchronization criteria are established. • Master-slave Markovian jumping complex dynamical networks are expressed as the error dynamical system. • LMI presented allow simultaneous computation of two bounds that characterize the synchronization rate.

2

Impulsive and pinning control synchronization of Markovian jumping complex dynamical networks with hybrid coupling and additive interval time-varying delays J. Yogambigaia M. Syed Alib a

1

, Hamed Alsulami

c

and Mohammed S. Alhodaly

c

Department of Mathematics, M.M.E.S Women’s Arts and Science College, Melvisharam, Vellore, Tamilnadu, India b

Department of Mathematics, Thiruvalluvar University, Serkkadu, Vellore, Tamilnadu, India, c Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Abstract: This study examines the problem of impulsive and pinning control synchronization of Markovian jumping master-slave complex dynamical networks with hybrid coupling and additive interval time-varying delays. The linear couplings include both the discrete time-varying delay and the distributed time-varying delays. Two kinds of control schemes are utilized to synchronize the considered dynamical network system. Impulsive and pinning control strategies are designed to synchronize the master-slave complex networks. By applying Lyapunov stability theory, Jensens inequality, Schur complement and linear matrix inequality technique, some new delay-dependent conditions are derived to guarantee the asymptotic stability of this system. We provided numerical examples to illustrate the feasibility and effectiveness of the results obtained. Key Words: Complex dynamical networks, Impulsive and pinning control, Synchronization, Lyapunov-Krasovski method, Markovian jump, Linear Matrix Inequality. 1. Introduction A complex network consists of a large number of highly interconnected fundamental units and therefore exhibits very complicated dynamics. In recent decades, synchronization and control problems in complex systems have been widely studied in different fields of engineering and technology (see [1]– [2] and references there in). Many natural and technological systems can be modeled as complex networks, such as biological neural networks, genetic regulatory networks, cellular neural networks, wireless communication networks, World Wide Web, epidemiological models, electrical power grid, internet, metabolic networks, etc. Thus, complex networks attract the attention of researchers from the various fields in recent decades (For example [3]–[5], and references therein). Time delay can be found in many systems, such as nuclear reactors, dynamic population models, aircraft stabilization, biological systems, chemical engineering systems, ship stabilization and so on [6]–[8]. The existence of time delay can make the system unstable and degrade its performance. Recently, considerable attention has been devoted to time delay systems due to their extensive application in practical systems including circuit theory, chemical processing, bioengineering, complex dynamical networks [9], automatic control and so on. Some results on time-varying delay systems were provided in [10] and [11]. Particularly, interval time-varying delays are a common form of time-varying delays [12]. In the interval time-varying delay, the lower bound of the time-varying delay is not limited to zero. A network control system is an interval time-varying delay system (see [13]–[15] and references there in). Recently, more attention has been paid to stability analysis for systems with additive time-varying delays which arises from the so-called networked control systems. In networked control systems, the control loops are closed via a communication network. Also, the system components, such as sensors, actuators, controllers, are usually physically distributed over the network. For example, the time delay in the dynamical model such as x(t) ˙ = Ax(t)+Bx(t−τ1 (t)−τ2 (t)) where τ1 (t) is the time delay induced 1Corresponding author.

E-mail address: [email protected] (M. Syed Ali)

3

from sensor to controller and τ2 (t) is the delay induced from controller to the actuator. It should be mentioned that delays induced through different communication channels are generally time-varying and expose different physical characteristics. In this sense, it is not reasonable to combine these two delays and treat them as one delay, which gives rise to the study on stability analysis for systems with additive time-varying delays. The stability analysis for such systems has been carried out in [16]-[19] by using two additive time-varying delay components. Compared to the single-delay systems, this model has a stronger background to its application. Synchronization, the most important collective behavior of complex dynamical networks (CDNs), has received much of the focus (see [20]– [23] and references therein). Synchronization of coupled oscillators can explain well many of the natural phenomena. Some complex networks can be synchronized by itself, but the general case is that the whole network cannot be synchronized by itself [24]. Designing controllers is an effective method in control theory and one natural idea is to assign controllers to all nodes of the complex dynamical network [25] and [26] . In the past few decades, synchronization and control problems in complex networks have been widely studied in various fields of science and engineering [27]. More synchronization phenomena saw in everyday life. For instance, fireflies flashing in unison, crickets chirping in synchrony and heart cells beating in rhythm [28] and [29]. Also, some synchronization phenomena, such as synchronous transfer of digital or analog signals in communication networks, are very useful for everyday human life and work. Recently many control schemes are adopted to design effective controllers, such as adaptive control [30], intermittent control [31], impulsive control [32], pinning control [33], sampling data control [34], sliding mode control [35] and so on. Normally complex networks have a large number of nodes. Therefore, it is usually difficult to control a complex network by adding the controllers to all nodes. A natural approach to reduce the number of controllers is to control a complex network by pinning part of nodes [36]. Impulsive control is a kind of discontinuous control, i.e., the controllers are applied onto systems only at some discrete instants. Therefore, impulsive controllers are easy to implement since they have a relatively simple structure. Especially for those systems that cannot endure continuous control, the impulsive control scheme has been widely applied to design controllers [37]. In real-time systems, complex networks may exhibit a special characteristic called mode switching. This is also a universal phenomenon in complex dynamical networks (CDNs), and some times the network has finite modes that switch from one to another at different times. Since Markov jump model has great application potential in many areas, much attention has been devoted to the study of Markov jump CDNs (see [38]– [45] and references therein). The methods for synchronization of master-slave systems have been widely studied in recent years, and many different methods have been applied theoretically and experimentally to synchronize these systems, which can be seen in [46] , [47]. Recently, synchronization methods for a master-slave system with Markovian switching are developed in [48], [49]. The problem of synchronization for complex networks via an impulsive-pinning controller is not studied yet due to its mathematical complexity. Motivated by this, we investigated pinning control synchronization of Markovian jumping masterslave complex dynamical networks with additive interval time-varying delays. Besides, the effect of Markovian switching is coped with the impulsive pinning control of the transition rates of modes. The main contributions of this paper are summarized as follows: (1) Master-slave Markovian jumping complex dynamical networks is equivalently expressed as error dynamical system. (2) We construct appropriate controllers to achieve impulsive and pinning control synchronization of Markovian jumping master-slave complex dynamical system. By constructing a set of LyapunovKrasovskii functional, new delay-dependent impulsive and pinning control synchronization criteria for Markovian jumping complex dynamical networks with hybrid coupling and additive interval time-varying delay is established in terms of LMIs. It allows simultaneous computation of two bounds that characterize the synchronization rate of the solution and can be easily determined by utilizing Matlab LMI Control Toolbox.

4

Notation: 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 real matrices. The superscript 0 T 0 denotes matrix transposition and the notation U ≥ W (respectively, U < W ) where U and W are symmetric matrices, means that U-W is positive semidefinite (respectively, positive definite). k . k denotes the Euclidean norm in Rn . If u is a square matrix, λmax (u)(respectively, λmin (u)) means the largest(respectively, smallest) eigenvalue of u. Moreover, let (Ω, F, {Ft }t≥0 , P) be a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions(i.e., the filtration contains all P-null sets and is right continuous). The asterisk * in a symmetric matrix is used to denote term that is induced by symmetry. 2. Problem Formulation and Preliminaries In this paper, r(t), t = 0 is a right-continuous Markovian chain on the probability space taking values in a finite state space S= {1, 2, ...., m} with generator Γ = {γij }m×m (i, j ∈ S) given by γij ∆ + 0(∆), if i 6= j, P r{r(t + ∆) = j | r(t) = i} = 1 + γij ∆ + 0(∆), if i = j. P Here ∆ > 0 and γij ≥ 0 is the transition rate from i to j if j 6= i, while γii = − j6=i γij .

Consider the following Master Markovian jumping complex dynamical networks with additive timevarying delays consisting of N identical nodes, in which each node is an n-dimensional dynamical subsystem: x˙ k (t) = −C(r(t))xk (t) + D1 (r(t))f (xk (t)) + D2 (r(t))f (xk (t − ρ1 (t) − ρ2 (t))) + b0

N X

(1)

lkj E1 (r(t))xj (t) + b1

j=1

+ b2

N X

N X j=1

(3)

lkj E3 (r(t))

j=1

Z

t

(2)

lkj E2 (r(t))xj (t − ρ1 (t) − ρ2 (t))

xj (s)ds + uk (t) + Ik (t),

t−d(t)

xk (tκ ) = Jκ (r(t))xk (t− κ ), xk (t) = ϕk (t),

t 6= tκ t = tκ

t ∈ [−ω, 0],

k = 1, 2, ...., N,

(1)

where xk (t) = (xk1 (t), xk2 (t), . . . , xkn (t))T ∈ Rn represent state vector of the k th node of the master system; uk (t) denotes the control input and Ik (t) is the constant external input vector. f : Rn → Rn are continuously nonlinear vector functions which are with respect to the current state xk (t) and the delayed state xk (t − ρ1 (t) − ρ2 (t)). f (xk (t)), f (xk (t − ρ1 (t) − ρ2 (t))), are time-varying vectorvalued nonlinear functions. ϕk (t) is continuously differential vector-valued initial functions on [−ω, 0], ω ∈ max{ρ2 , d} of the master system. Consider the following slave Markovian jumping complex dynamical networks with additive timevarying delays consisting of N identical nodes, in which each node is an n-dimensional dynamical subsystem: y˙ k (t) = −C(r(t))yk (t) + D1 (r(t))f (yk (t)) + D2 (r(t))f (yk (t − ρ1 (t) − ρ2 (t))) + b0

N X

(1)

lkj E1 (r(t))yj (t) + b1

j=1

+ b2

N X

N X j=1

(3)

lkj E3 (r(t))

j=1

yk (tκ ) = Jκ (r(t))yk (t− κ ),

Z

t

t−d(t)

(2)

lkj E2 (r(t))yj (t − ρ1 (t) − ρ2 (t))

yj (s)ds + Ik (t),

t 6= tκ t = tκ

5

yk (t) = ψk (t),

t ∈ [−ω, 0],

k = 1, 2, ...., N,

(2)

where yk (t) = (yk1 (t), yk2 (t), . . . , ykn (t))T ∈ Rn represent state vector of the k th node of the slave system; Ik (t) is the constant external input vector. f : Rn → Rn are continuously nonlinear vector functions which are with respect to the current state yk (t) and the delayed state yk (t − ρ1 (t) − ρ2 (t)). f (yk (t)) and f (yk (t − ρ1 (t) − ρ2 (t))) are time-varying vector-valued nonlinear functions. ψk (t) is continuously differential vector-valued initial functions on [−ω, 0], ω ∈ max{ρ2 , d} of the slave system. C(r(t)), D1 (r(t)) and D2 (r(t)) ∈ Rn×n are matrix functions of the random process r(t), t = 0. Ea (r(t)) = diag{ua1 (r(t)), ua2 (r(t)), ...., uan (r(t))} (a = 1, 2, 3) are constant diagonal inner-coupling matrices. The positive constants b0 , b1 and b2 are the corresponding coupling strengths. L(a) = (a) (lkj )N ×N (a = 1, 2, 3) is the outer-coupling matrix representing the topological structure of the complex networks, in which lkj is defined as follows: if there is a connection between node k and node j (k 6= j), then lkj = ljk = 1; otherwise, lkj = ljk = 0 (k 6= j). The row sums of L(a) (a = 1, 2, 3) PN are zero, i.e., j=1 lkj = −lkk , k = 1, 2, ...., N. ρ(t) and d(t) are the time-varying additive retarded delays and distributed delays, respectively. Jκ (r(t)) is the impulse gain matrix at the moment of time tκ . The discrete set {tκ } satisfies 0 = t0 < + + − t1 < ... < tκ < ..., limκ→∞ tκ = ∞. xk (t− κ ), yk (tκ ) and xk (tκ ), yk (tκ ) denote the left-hand and right-hand limits at tκ of master and slave system, respectively. We assume that xk (tκ ) is right continu(a) (r(t)), Jκ (r(t)), Ea (r(t)), (a = ous, i.e., xk (t+ κ ) = xk (tκ ). We denote C(r(t)), D1 (r(t)), D2 (r(t)), L (a) 1, 2, 3) by Ci , D1i , D2i , Li , Jκi , Eai f or r(t) = i ∈ S. Assumption 1: ρ1 (t), ρ2 (t) and d(t) are time-varying delays satisfying 0 ≤ ρ11 ≤ ρ1 (t) ≤ρ12 , ρ˙ 1 (t) ≤µ1 ,

0 ≤ ρ21 ≤ ρ2 (t) ≤ ρ22 , ρ˙ 2 (t) ≤ µ2 ,

0 ≤ d(t) ≤ d, ˙ ≤ µ3 , d(t)

(3)

where ρ12 ≥ ρ11 , ρ22 ≥ ρ21 , µ1 , µ2 and µ3 are real constant scalars. Here, let us denote ρ(t) = ρ1 (t) + ρ2 (t), ρ1 = ρ11 + ρ21 , ρ2 = ρ12 + ρ22 , µ = µ1 + µ2 , τ1 = ρ12 − ρ11 , τ2 = ρ22 − ρ21 Assumption 2: [23] The nonlinear functions f (xk (t)) are globally Lipschitz, k f (xk (t)) − f (yk (t)) k ≤ νk k xk (t) − yk (t) k

(4)

where νk is a nonnegative Lipschitz constants. Remark 1 : (1) Assumption 1 is given to define the upper bounds of additive time-varying delays which are useful in constructing the Lyapunov functions. (2) Assumption 1 is given to ensure the existence of solution of the system concerned. Remark 2 : The delay in coupled complex dynamical networks is additive time-varying and the configuration matrix Ei are symmetric or asymmetric. Many authors consider delayed couplings as those in [5, 8, 20, 21]. In this paper we considered time-delayed couplings it makes the model too complex. The reason why we choose Assumption 2 is to reduce the complexity of the calculation, especially when the scale of the network is very large. Let ek (t) = xk (t)−yk (t) be the synchronization error of master-slave system (1) and (2) from initial mode r0 . Then the error dynamics, namely, the synchronization error system can be expressed by e˙ k (t) = −Ci ek (t) + D1i f (ek (t)) + D2i f (ek (t − ρ(t))) + b0

N X j=1

(1)

lkj E1i ej (t)

6

+ b1

N X j=1

(2)

lkj E2i ej (t − ρ(t)) + b2

ek (tκ ) = Jκi ek (t− κ ), ek (t) = ϕk (t) − ψk (t),

t ∈ [−ω, 0]

N X

(3)

lkj E3i

j=1

Z

t

t 6= tκ

ej (s)ds + uk (t),

t−d(t)

t = tκ

r(0) = r0 ,

k = 1, 2, ...., N,

(5)

where ek (t − ρ(t)) = xk (t − ρ(t)) − yk (t − ρ(t)), f (ek (t)) = f (xk (t)) − f (yk (t)), f (ek (t − ρ(t))) = f (xk (t − ρ(t))) − f (yk (t − ρ(t))). Definition 1. [52] The master system (1) and the slave system (2) are said to be global (asymptotically) synchronized, if lim kxk (t) − yk (t)k = 0,

k = 1, 2, ...., N.

t→∞

(6)

The aim of this paper is to design a controller to make the master system (1) synchronize with the slave system (2) with the synchronization rate as big as possible Definition 2. [53] The function V : [t0 , ∞) × Rn × S → R+ belongs to class ψ0 if (a) the function V is continuous on each of the sets [tκ−1 , tκ ) × Rn × S and for all t ≥ t0 , V (t, 0, i) ≡ 0, i ∈ S; (b)V(t, x, i) is locally Lipschitzian in x ∈ Rn , i ∈ S; (c) for each κ=1,2,...and i, j ∈ S, there exist finite limits lim(t,w,j)→(t− V (t− V (t+ κ , x, i) and lim(t,w,j)→(t+ κ , x, j) κ ,x,i) κ ,x,j)

with V (t+ κ , x, j) = V (tκ , x, j) satisfied. PN Lemma 1. [50] The eigenvalues of an irreducible matrix H = (hkw ) ∈ RN ×N with w6=k hkw = −hkk , k = 1, 2, ...., N satisfy the following properties: (i) Real parts of all eigenvalues of H are less than or equal to 0 with multiplicity 1. (ii) H has an eigenvalue 0 with multiplicity 1 and the right eigenvector (1, 1, ...., 1)T . Lemma 2. [51] For any constant positive-definite matrix W ∈ Rm×m , W = W T > 0 and α2 ≥ α1 , the following inequalities hold: Z α1 T Z α1  Z α1 T ˙ ˙ ˙ ˙ −(α1 − α2 ) φ (s)W φ(s)ds ≤ φ(s)ds W φ(s)ds , (7) −

(α1 − α2 ) 2

2

Z

α1

α2

Z

s

α2

α1

α2

˙ φ˙ T (u)W φ(u)duds ≤

Z

α1

α2

Z

α2

α1

s

T Z ˙ φ(u)duds W

α1

α2

Z

s

α1

 ˙ φ(u)duds .

(8)

3. Main Results Some concept defined in Appendix A. Remark 3 : The complex network model (1) which consists of nonlinear function and impulsive effects . It should be pointed out that the pinning synchronization problem of complex dynamical networks with additive interval time-varying delay and impulsive effects has not been addressed in the literature. Hence our system considered is more general. Here, we will investigate the impulsive and pinning control synchronization for the error dynamical network system (5) via different control schemes: pinning (adaptive) control and impulsive control which will be stated below in different subsections. We also discuss the impact of additive time-varying delays on the stability of the system. 3.1. Pinning synchronization In this section, pinning synchronization of network (5) is investigated. The pinning controller is designed as follows which is composed both by the open-loop control and by the feedback control:  −b3 σk E4 (xk (t) − yk (t)), k = 1, 2, ....l uk (t) = (9) 0, k = l + 1, l + 2, ....N.

7

where b3 > 0 for k = 1, 2,....,l and b3 = 0 when k = l + 1, l + 2, ....N which means that only l nodes are pinned by feedback control. E4 is the constant diagonal inner-coupling matrices and σk > 0 is the feedback gain So the error dynamical system (5) can be derived as follows: e˙ k (t) = −Ci ek (t) + D1i f (ek (t)) + D2i f (ek (t − ρ(t))) + b0 + b1

N X j=1

(2)

lkj E2i ej (t − ρ(t)) + b2

N X

(3)

lkj E3i

j=1

Z

t

t−d(t)

N X

(1)

lkj E1i ej (t)

j=1

ej (s)ds − b3 σk E4 ek (t)).

(10)

We may write the error system in its compact form as, e(t) ˙ = −Ci e(t) + D1i f (e(t)) + D2i f (e(t − ρ(t))) + b0 L(1) E1i e(t) Z t (2) (3) + b1 L E2i e(t − ρ(t)) + b2 L E3i e(s)ds − b3 σE4 e(t)),

(11)

t−d(t)

where e(t) = (e1 (t), e2 (t), ....., eN (t)), f (e(t)) = (f1 (e1 (t)), f2 (e2 (t)), ......, fN (eN (t))), f (e(t − ρ(t)) = (f1 (e1 (t − ρ(t)), f2 (e2 (t − ρ(t)), ....., fN (eN (t − ρ(t))) and σ = diag{σ1 , σ2 , ...., σN }. By the properties of the outer-coupling matrix L(a) (a = 1, 2, 3), there exists a unitary matrix U = (a) (a) (a) [U1 , U2 , ...., UN ] ∈ RN ×N such that U T L(a) = Λ(a) U T with Λ(a) = diag{λ1 , λ2 , ...., λN }(a = 1, 2, 3) and U U T = I. Using the nonsingular transform e(t)U = w(t) = [w1 (t), w2 (t), ....., wN (t)] ∈ RN ×N , from equation (11), it follows the matrix equation w(t) ˙ = −Ci w(t) + D1i f (e(t))U + D2i f (e(t − ρ(t)))U + b0 E1i Λ(1) w(t) Z t + b1 E2i Λ(2) w(t − ρ(t)) + b2 E3i Λ(3) w(s)ds − b3 σE4 w(t)).

(12)

t−d(t)

In a similar way, model (12) can be written as (1)

w(t) ˙ = (−Ci + b0 E1i λk − b3 σk E4 )w(t) + D1i hk (t) + D2i hk (t − ρ(t)) Z t (2) (3) + b1 E2i λk w(t − ρ(t)) + b2 E3i λk w(s)ds, k = 1, 2, ...., N,

(13)

t−d(t)

where hk (t) = f (e(t))Uk , hk (t − ρ(t)) = f (e(t − ρ(t))Uk . So far, we transformed the synchronization problem of the master-slave complex dynamical networks (1) and (2) into the synchronization problem of the N pieces of the corresponding error dynam(a) ical network (13). From Lemma 1, λ1 = 0, (a = 1, 2, 3) and w1 (t) = e(t)U1 = 0. Therefore, if the following (N-1) pieces of the corresponding error dynamical network (1)

w˙k (t) = (−Ci + b0 E1i λk − b3 σk E4 )wk (t) + D1i hk (t) + D2i hk (t − ρ(t)) Z t (2) (3) + b1 E2i λk wk (t − ρ(t)) + b2 E3i λk wk (s)ds, k = 2, 3, ...., N,

(14)

t−d(t)

are asymptotically stable, which implies that the synchronized states (1) and (2) are asymptotically stable. Theorem 3.1. Under Assumptions 1 and 2, the master system (1) and the slave system (2) under the pinning control input (22) with time-varying delays h(t) and τ (t) are asymptotically stable. For given positive scalars ρij > 0 (i, j = 1, 2), d > 0, µj > 1 (j = 1, 2, 3), δk , if there exist positive definite matrices Qki , Tkj , Xkj , Ykj , Zkj (j = 1, 2, 3, 4), Tk5 , Tk6 , Tk7 , Yk5 , positive diagonal matrices

8 (a)

Gk (a = 1, 2) and any appropriately dimensioned matrices Nkj , Mkj , Hkj , Pkj (j = 1, 2, ...., 15) such that the following LMI holds:   Ψk Σk1 Ξ= < 0, k = 2, 3, ..., N, (15) ∗ Σk2 where

Ψk = Ψk(r×s) , (r, s = 1, 2...., 19), (1)T

Σk1 = ηk

h τ1 Xk1

Σk2 = diag{−Xk1

with i τ12 τ22 τ12 τ22 Zk1 Zk2 Zk3 Zk4 , 2 2 2 2 − Zk1 − Zk2 − Zk3 − Zk4 },

(ρ2 − ρ1 )Xk4

τ2 Xk2

ρ2 Xk3

− Xk2

− Xk3

− Xk4

(1)

(1)

Ψk(1,1) = Qki (−Ci + b0 E1i λk − b3 σk E4 ) + (−Ci + b0 E1i λk − b3 σk E4 )T Qki +

m X

γij Qkj

j=1

+ Tk1 + Tk2 + Tk3 + Tk4 + Tk5 + τ12 Yk1 + τ22 Yk2 + ρ22 Yk3 + (ρ2 − ρ1 )2 Yk4 + d2 Yk5 (1)

T + Nk1 + Nk1 + δk (N − 1)ν¯k Gk ,

T Ψk(1,4) = Nk4 − Nk1 − Mk1 , T Ψk(1,8) = Nk8 − Pk1 ,

(2)

T Ψk(1,17) = Nk13 − Pk1 ,

+δk (N −

Ψk(2,7) = Pk2 ,

Ψk(2,19) = −Mk2 , +Hk3 ,

T Ψk(1,5) = Nk5 + Hk1 ,

T Ψk(1,9) = Qki D1i + Nk9 ,

Ψk(2,3) = Mk2 ,

Ψk(2,8) = −Pk2 ,

Ψk(2,16) = −Hk2 ,

T Ψk(3,18) = Mk14 − Nk3 ,

T T Ψk(4,8) = −Nk8 − Mk8 − Pk4 ,

T T Ψk(4,15) = −Nk11 − Mk11 ,

+ 2Hk5 ,

T Ψk(5,9) = Hk9 ,

T Ψk(3,16) = Mk12 − Hk3 ,

− Mk3 ,

T Ψk(3,5) = Mk5 T Ψk(3,9) = Mk9 ,

T T Ψk(4,7) = −Nk7 − Mk7 + Pk4 ,

T T Ψk(4,10) = −Nk10 − Mk10 ,

T T Ψk(4,17) = −Nk13 − Mk13 − Pk4 ,

T T Ψk(4,19) = −Nk15 − Mk15 − Mk4 ,

T Ψk(5,10) = Hk10 ,

Ψk(2,18) = −Nk2 ,

T Ψk(3,17) = Mk13 − P k3,

T T Ψk(4,16) = −Nk12 − Mk12 − Hk4 , T Hk6 ,

Ψk(2,6) = −Hk2 ,

Ψk(4,4) = −Tk3 − 2Nk4 − 2Mk4 ,

T T Ψk(4,9) = −Nk9 − Mk9 ,

Ψk(5,6) = −Hk5 +

T +Hk12 ,

T Mk4

Ψk(2,2) = −(1 − µ)Tk1

T Ψk(3,8) = Mk8 − Pk3 ,

T T Ψk(4,6) = −Nk6 − Mk6 − Hk4 ,

T T Ψk(4,18) = −Nk14 − Mk14 − Nk4 ,

−τ12 Zk1

Ψk(2,17) = −Pk2 ,

Ψk(3,4) = −Nk3 +

T Ψk(3,19) = Mk15 − Mk3 ,

T T Ψk(4,5) = −Nk5 − Mk5 + Hk4 ,

Ψk(2,5) = Hk2 ,

T Ψk(3,7) = Mk7 + Pk3 ,

T Ψk(3,15) = Mk11 ,

T Ψk(1,16) = Nk12 − Hk1 ,

T Ψk(1,19) = Nk15 − Mk1 ,

Ψk(2,4) = −Nk2 ,

Ψk(3,3) = −Tk2 + 2Mk3 ,

T Ψk(3,6) = Mk6 − Hk3 ,

T Ψk(3,10) = Mk10 ,

T Ψk(1,10) = Qki D2i + Nk10 ,

(3)

T Ψk(1,18) = Nk14 − Nk1 ,

T Ψk(1,3) = Nk3 + Mk1 ,

T T Ψk(1,6) = Nk6 − Hk1 , Ψk(1,7) = Nk7 + Pk1 ,

T Ψk(1,15) = Qki b2 E3i λk + Nk11 ,

Ψk(1,12) = Qki b1 E2i λk , (2) 1)ν¯k Gk ,

T Ψk(1,2) = Nk2 ,

Ψk(5,7) = Pk5 +

T Ψk(5,15) = Hk11 ,

T Hk7 ,

Ψk(5,5) = −Tk4 + Tk6

T Ψk(5,8) = −Pk5 + Hk8 ,

Ψk(5,11) = τ1 Zk1 ,

Ψk(5,16) = −Hk5

T T Ψk(5,17) = −Pk5 + Ψk(5,18) = −Nk5 + Hk14 , Ψk(5,19) = −Mk5 + Hk15 , 2 T T Ψk(6,6) = −Tk5 − Tk6 − τ1 Zk3 − 2Hk6 , Ψk(6,7) = −Hk7 + Pk6 , Ψk(6,8) = −Hk8 − Pk6 , T T T Ψk(6,9) = −Hk9 , Ψk(6,10) = −Hk10 , Ψk(6,11) = τ1 Zk3 , Ψk(6,15) = −Hk11 Ψk(6,16) =

−Hk6 ,

T Hk13 ,

T Ψk(6,17) = −Hk13 − Pk6 ,

Ψk(7,7) = Tk7 − τ22 Zk2 + 2Pk7 ,

Ψk(7,12) = τ2 Zk2 ,

T Ψk(6,18) = −Hk14 − Nk6 ,

T Ψk(7,8) = Pk8 − Pk7 ,

T Ψk(7,15) = Pk11 ,

T Ψk(6,19) = −Hk15 − Mk6 ,

T Ψk(7,9) = Pk9 ,

T Ψk(7,16) = Pk12 − Hk7 ,

T −Hk12

T Ψk(7,10) = Pk10 ,

T Ψk(7,17) = Pk13 − Pk7 ,

T T T Ψk(7,18) = Pk14 − Nk7 , Ψk(7,19) = Pk15 − Mk7 , Ψk(8,8) = −Tk7 − τ22 Zk4 − 2Pk8 , Ψk(8,9) = −Pk9 , T Ψk(8,10) = −Pk10 ,

Ψk(8,12) = τ2 Zk4 ,

T Ψk(8,17) = −Pk13 − Pk8 ,

T Ψk(8,15) = −Pk11 ,

T Ψk(8,18) = −Pk14 − Nk8 ,

T Ψk(8,16) = −Pk12 − Hk8 ,

(1)

T Ψk(8,19) = −Pk15 − Mk8 , Ψk(9,9) = −δk Gk ,

9

Ψk(9,16) = −Hk9 ,

Ψk(9,17) = −Pk9 ,

Ψk(10,16) = −Hk10 ,

Ψk(10,17) = −Pk10 ,

Ψk(11,11) = −Yk1 − Zk1 − Zk3 ,

(2)

Ψk(9,18) = −Nk9 ,

Ψk(9,19) = −Mk9 , Ψk(10,10) = −δk Gk ,

Ψk(10,18) = −Nk10 ,

Ψk(10,19) = −Mk10 ,

Ψk(12,12) = −Yk2 − Zk2 − Zk4 ,

Ψk(13,13) = −Yk3 ,

Ψk(14,14) = −Yk4 , Ψk(15,15) = −Yk5 , Ψk(15,16) = −Hk11 , Ψk(15,17) = −Pk11 , Ψk(15,18) = −Nk11 , Ψk(15,19) = −Mk11 , −Nk12 ,

T Ψk(16,17) = −Hk13 − Pk12 ,

Ψk(16,16) = −Xk1 − 2Hk12 ,

T Ψk(16,19) = −Hk15 − Mk12 ,

T Ψk(17,19) = −Pk15 − Mk13 ,

T Ψk(17,18) = −Pk14 − Nk13 ,

Ψk(17,17) = −Xk2 − 2Pk13 ,

T Ψk(18,19) = −Nk15 − Mk14 ,

Ψk(18,18) = −Xk3 − 2Nk14 ,

Ψk(19,19) = −Xk4 − 2Mk15 .

T Ψk(16,18) = −Hk14

proof: Construct the Lyapunov-Krasovskii functional: 5 X

Vk (wk (t), i, t) =

Vkr (wk (t), i, t)

(16)

r=1

where Vk1 (wk (t), i, t) = wkT (t)Qki wk (t), Z t Z Vk2 (wk (t), i, t) = wkT (s)Tk1 wk (s)ds + t−ρ(t) Z t

+

+ Vk3 (wk (t), i, t) = τ1

t−ρ1 t

wkT (s)Tk3 wk (s)ds

t−ρ2 Z t−ρ11 t−ρ12 Z −ρ11 −ρ12 Z 0

+ ρ2 Vk4 (wk (t), i, t) = τ1

−ρ12 Z 0

+ ρ2 +d τ2 Vk5 (wk (t), i, t) = 1 2

Z

Z

τ2 + 2 2 + +

Z

t

t+θ Z t

t+θ Z t

−ρ2 t+θ 0 Z t

wkT (s)Tk4 wk (s)ds

t−ρ11 Z t−ρ21 t−ρ22

w˙ kT (s)Xk1 w˙ k (s)dsdθ + τ2

Z

Z

−ρ21

−ρ22

w˙ kT (s)Xk3 w˙ k (s)dsdθ + (ρ2 − ρ1 ) wkT (s)Yk1 wk (s)dsdθ + τ2

Z

−ρ21

Z

−ρ22

t

w˙ kT (s)Zk1 w˙ k (s)dsdθdβ

−ρ12 β t+θ Z −ρ21 Z −ρ21 Z t

2

−ρ12 2 Z −ρ21 τ2 −ρ22

β Z β

−ρ12 Z β −ρ22

Z

t+θ t

t+θ Z t t+θ

+

t

t−ρ12

wkT (s)Tk5 wk (s)ds

w˙ kT (s)Zk2 w˙ k (s)dsdθdβ

w˙ kT (s)Zk3 w˙ k (s)dsdθdβ w˙ kT (s)Zk4 w˙ k (s)dsdθdβ.

t

w˙ kT (s)Xk2 z˙k (s)dsdθ

t+θ Z −ρ1

Z

wkT (s)Yk3 wk (s)dsdθ + (ρ2 − ρ1 )

−d t+θ −ρ11 Z −ρ11

Z

wkT (s)Tk7 wk (s)ds,

wkT (s)Yk5 wk (s)dsdθ,

−ρ22 Z τ12 −ρ11

2

+

wkT (s)Tk2 wk (s)ds

Z

wkT (s)Tk6 wk (s)ds +

−ρ2 t+θ −ρ11 Z t

Z

t

−ρ2 t

Z

t

t+θ

w˙ kT (s)Xk4 w˙ k (s)dsdθ

wkT (s)Yk2 wk (s)dsdθ

t+θ Z −ρ1 −ρ2

Z

t

t+θ

wkT (s)Yk4 wk (s)dsdθ

10

The derivative of Vkr (wk (t), i, t) along the trajectory of (14) with respect to t is given by: h (1) LVk1 (wk (t), i, t) = 2wkT (t)Qki (−Ci + b0 E1i λk − b3 σk E4 )wk (t) + D1i g(wk (t)) + D2i g(wk (t − ρ(t))) Z t m i X (2) (3) + b1 E2i λk wk (t − ρ(t)) + b2 E3i λk wk (s)ds + γij [wkT (t)Qkj wk (t)], t−d(t)

j=1

(17)

LVk2 (wk (t), i, t) ≤ wkT (t)[Tk1 + Tk2 + Tk3 + Tk4 + Tk5 ]wk (t) − (1 − µ)wkT (t − ρ(t))Tk1 wk (t − ρ(t)) − wkT (t − ρ1 )Tk2 wk (t − ρ1 ) − wkT (t − ρ2 )Tk3 wk (t − ρ2 ) + wkT (t − ρ11 ) × (Tk6 − Tk4 )wk (t − ρ11 ) + wkT (t − ρ12 )(−Tk6 − Tk5 )wk (t − ρ12 ) + wkT (t − ρ21 )Tk7 wk (t − ρ21 ) − wkT (t − ρ22 )Tk7 wk (t − ρ22 ), LVk3 (wk (t), i, t) ≤ w˙ kT (t)[τ12 Xk1 + τ22 Xk2 + ρ22 Xk3 + (ρ2 − ρ1 )2 Xk4 ]w˙ k (t) − τ1 − τ2

Z

t−ρ21

t−ρ22

− (ρ2 − ρ1 )

w˙ kT (s)Xk2 w˙ kT (s)ds − ρ2

Z

t−ρ1

t−ρ2

Z

t

t−ρ2

Z

w˙ kT (s)Xk3 w˙ kT (s)ds

w˙ kT (s)Xk4 w˙ kT (s)ds,

× Yk1 wkT (s)ds − τ2 Z

t−ρ1

t−ρ2

w˙ kT (s)Xk1 w˙ kT (s)ds

t−ρ12

(19)

LVk4 (wk (t), i, t) ≤ wkT (t)[τ12 Yk1 + τ22 Yk2 + ρ22 Yk3 + (ρ2 − ρ1 )2 Yk4 + d2 Yk5 ]wk (t) − τ1

− (ρ2 − ρ1 )

(18)

t−ρ11

Z

t−ρ21

t−ρ22

Z

wkT (s)Yk2 wkT (s)ds − ρ2

wkT (s)Yk4 wkT (s)ds − d

Z

t−ρ2

t

t−d

t

Z

t−ρ12

t−ρ12 t−ρ21

−τ2

Z

t−ρ22 Z t

−ρ2 −(ρ2 − ρ1 )

Z

t−ρ2 t−ρ1

t−ρ2

t−ρ11

t−ρ12 t−ρ21

Z T wk (s)Yk2 wk (s)ds ≤ −

t−ρ22 t

Z T wk (s)Yk3 wk (s)ds ≤ −

wkT (s)Yk5 wkT (s)ds,

t−ρ2

(20)

(21)

 Z wkT (s)ds Yk1

t−ρ11

t−ρ12 t−ρ21

 Z T wk (s)ds Yk2

 Z T wk (s)ds Yk3

t−ρ2 t−ρ1

Z wkT (s)Yk4 wk (s)ds ≤ −

wkT (s)

wkT (s)Yk3 wkT (s)ds

τ4 τ4 τ4 τ4 LVk5 (wk (t), i, t) = w˙ kT (t)[ 1 Zk1 + 2 Zk2 + 1 Zk3 + 2 Zk4 ]w˙ k (t) 4 4 4 4 Z Z τ12 −ρ11 t−ρ11 T − w˙ k (s)Zk1 w˙ k (s)dsdβ 2 −ρ12 t+β Z Z τ22 −ρ21 t−ρ21 T − w˙ k (s)Zk2 w˙ k (s)dsdβ 2 −ρ22 t+β Z Z τ 2 −ρ11 t+β T − 1 w˙ (s)Zk3 w˙ k (s)dsdβ 2 −ρ12 t−ρ12 k Z Z τ 2 −ρ21 t+β T − 2 w˙ (s)Zk4 w˙ k (s)dsdβ. 2 −ρ22 t−ρ22 k By Lemma 2, we have Z t−ρ11 Z −τ1 wkT (s)Yk1 wk (s)ds ≤ −

t−ρ11

t−ρ22

t

 wk (s)ds ,

 wk (s)ds ,

t−ρ2 t−ρ1

 Z wkT (s)ds Yk4

 wk (s)ds ,

t−ρ2

 wk (s)ds ,

11

−d −τ1 −τ2

Z

−

τ22 2

−

2

τ2 − 2 2

wkT (s)Yk5 wk (s)ds

Z ≤−

t−ρ12 t−ρ21 t−ρ22 Z t

Z w˙ kT (s)Xk3 w˙ k (s)ds ≤ −

Z

t−ρ2 t−ρ1

t−ρ2 t−ρ11

t−ρ12 t−ρ21

Z w˙ kT (s)Xk2 w˙ k (s)ds ≤ −

t−ρ22 t

t−ρ2 t−ρ1

t−ρ2 −ρ11

t+β −ρ22 −ρ11 Z t+β

Z T w˙ k (s)Zk3 w˙ k (s)dsdβ ≤ −

Z

Z

−ρ12 −ρ21

Z

−ρ22

t−ρ12 t+β

Z

t−ρ22

wkT (s)ds

Yk5

Z

−ρ12 −ρ21

Z w˙ kT (s)Zk2 w˙ k (s)dsdβ ≤ −

−ρ22 −ρ11 −ρ12 −ρ21

Z w˙ kT (s)Zk4 w˙ k (s)dsdβ ≤ −

−ρ22

 wk (s)ds ,

t

t−d(t) t−ρ11

 Z w˙ kT (s)ds Xk1

t−ρ12 t−ρ21

 Z w˙ kT (s)ds Xk2

 Z wkT (s)ds Xk3

Z T w˙ k (s)Xk4 w˙ k (s)ds ≤ −

t+β t−ρ21

−ρ12 −ρ21



t

t−d(t) t−ρ11

Z w˙ kT (s)Zk1 w˙ k (s)dsdβ ≤ −

−ρ11

Z

τ12

Z

t−d t−ρ11

Z

−(ρ2 − ρ1 ) Z

t

Z w˙ kT (s)Xk1 w˙ k (s)ds ≤ −

−ρ2

τ2 − 1 2

Z

t−ρ22

t−ρ2 t−ρ1 t−ρ2

t−ρ11

t+β t−ρ21

Z

t+β t+β

Z

t−ρ12 t+β

Z

t−ρ22

 w˙ k (s)ds ,

 w˙ k (s)ds ,

t

 Z T w˙ k (s)ds Xk4 Z

 w˙ k (s)ds ,

 w˙ k (s)ds ,

 Z w˙ kT (s)dsdβ Zk1 

w˙ kT (s)dsdβ Zk2

−ρ11

−ρ12 −ρ21

Z

 Z T w˙ k (s)dsdβ Zk3

−ρ22 −ρ11

−ρ12 −ρ21

 Z w˙ kT (s)dsdβ Zk4

−ρ22

Z

t−ρ11

t+β t−ρ21

Z

Z

t+β t+β

t−ρ12 t+β

Z

t−ρ22

 w˙ k (s)dsdβ ,  w˙ k (s)dsdβ ,

 w˙ k (s)dsdβ ,  w˙ k (s)dsdβ . (22)

From equations (16)-(22) and (44)-(48), we get LVk (wk (t), i, t) ≤

5 X r=1

LVkr (wk (t), i, t) − δk ξkT (t)Φk ξk (t) (1)T

= ξkT (t)Ψk ξk (t) + ξkT (t)ηk + (1)

Where ξk (t) and ηk

[τ12 Xk1 + τ22 Xk2 + ρ22 Xk3 + (ρ2 − ρ1 )2 Xk4

τ4 τ4 τ4 τ14 (1) Zk1 + 2 Zk2 + 1 Zk3 + 2 Zk4 ]ηk ξk (t) 4 4 4 4

are in (9) and (10). By the Schur complement Lemma, we obtain

LVk (wk (t), i, t) ≤ ξkT (t)Ξξk (t).

(23)

The following (24) is held, if (15) satisfies, LVk (wk (t), i, t) < 0.

(24)

From Definition 1, it implies that the system (14) is asymptotically stable. Then the proof is completed. Remark 4: The pinning controllers (9) are added to a part of the node in response networks by the information of the first node in complex dynamical networks, which is easy to achieve. The complex dynamical networks (1) and response networks (2) reaches synchronization by pinning controllers (9) as long as error system (10) is globally and asymptotically stable. 3.2. Impulsive synchronization In this subsection, to synchronize the master-slave hybrid coupled dynamical systems (1) and (2) using the impulsive control strategy. Using Lemma 1, the error dynamical system (5) with uk (t) = 0 can be derived as follows: (1)

(2)

w˙k (t) = (−Ci + b0 E1i λk )wk (t) + D1i hk (t) + D2i hk (t − ρ(t)) + b1 E2i λk wk (t − ρ(t))

12

+

(3) b2 E3i λk

wk (tκ ) = Jκi wk (t− κ ),

Z

t

wk (s)ds,

t−d(t)

t 6= tκ t = tκ

k = 2, 3, ...., N.

(25)

The corresponding error dynamical system are asymptotically stable, which implies that the synchronized states (1) and (2) are asymptotically stable. Theorem 3.2. Under Assumptions 1 and 2, the master-slave complex dynamical network (25) under the impulsive control input with time-varying delays h(t) and τ (t) are asymptotically stable. For given positive scalars ρij > 0 (i, j = 1, 2), d > 0, µj > 1 (j = 1, 2, 3), δk , if there exist positive definite matrices Qki , Tkj , Xkj , Ykj , Zkj (j = 1, 2, 3, 4), Tk5 , Tk6 , Tk7 , Yk5 , positive diagonal matrices (a) Gk (a = 1, 2) and any appropriately dimensioned matrices Nkj , Mkj , Hkj , Pkj (j = 1, 2, ...., 15) such that the following LMI holds:   Θk Υk1 U= < 0, k = 2, 3, ..., N, (26) ∗ Υk2 T Jκi Qk` Jκi − Qki ≤ 0,

where Υk1

[here r(tk ) = `]

(27)

Θk = Θk(r×s) , (r, s = 1, 2...., 19), with h (2)T = ηk τ1 Xk1 τ2 Xk2 ρ2 Xk3 (ρ2 − ρ1 )Xk4

τ12 Zk1 2

τ22 Zk2 2

τ12 Zk3 2

Υk2 = Σk2 ,

(1)

(1)

Θk(1,1) = Qki (−Ci + b0 E1i λk ) + (−Ci + b0 E1i λk )T Qki +

m X

i τ22 Zk4 , 2

γij Qkj + Tk1 + Tk2 + Tk3 + Tk4

j=1

(1)

T + Tk5 + τ12 Yk1 + τ22 Yk2 + ρ22 Yk3 + (ρ2 − ρ1 )2 Yk4 + d2 Yk5 + Nk1 + Nk1 + δk (N − 1)ν¯k Gk ,

Θk(j,l) = Ψk(j,l) ,

Θk(1,1) 6= Ψk(1,1)

(j, l = 1, 2..., 19)

proof: Define the Lyapunov-Krasovskii functional as Vk (wk (t), i, t) =

5 X

Vkr (wk (t), i, t)

(28)

r=1

P5 where r=1 Vkr (wk (t), i, t) are same as in Theorem 3.1. For t 6= tκ , the derivative of Vk1 (wk (t), i, t) along the trajectory of (25) with respect to t is given by: h (1) LVk1 (wk (t), i, t) = 2wkT (t)Qki (−Ci + b0 E1i λk )wk (t) + D1i g(wk (t)) + D2i g(wk (t − ρ(t))) Z t m i X (2) (3) + b1 E2i λk wk (t − ρ(t)) + b2 E3i λk wk (s)ds + γij [wkT (t)Qkj wk (t)]. t−d(t)

j=1

(29)

From equations (29), (17)-(21) and (44), we get LVk (wk (t), i, t) ≤

5 X r=1

LVkr (wk (t), i, t) − δk ξkT (t)Φk ξk (t), (2)T

= ξkT (t)Θk ξk (t) + ξkT (t)ηk +

[τ12 Xk1 + τ22 Xk2 + h22 Xk3 + (h2 − h1 )2 Xk4

τ14 τ4 τ4 τ4 (2) Zk1 + 2 Zk2 + 1 Zk3 + 2 Zk4 ]ηk ξk (t) 4 4 4 4

13 (2)

Where ξk (t) and ηk

are in (37) and (39). By the Schur complement Lemma, we obtain LVk (wk (t), i, t) ≤ ξkT (t)Uξk (t).

(30)

The following (31) is held for t ∈ [tκ , tκ+1 ], if (25) satisfies, LVk (wk (t), i, t) < 0.

(31)

For t = tκ , we get − T T − − Vk (wk (tκ ), `, tκ ) − Vk (wk (t− κ ), i, tκ ) = wk (tκ )Qk` wk (tκ ) − wk (tκ )Qki wk (tκ ),

T − T − − = wkT (t− κ )Jκi Qk` Jκi wk (tκ ) − wk (tκ )Qki wk (tκ ), T − = wkT (t− κ )[Jκi Qk` Jκi − Qki ]wk (tκ ) ≤ 0.

(32)

From (17) and (18), we can obtain the asymptotical stability condition of system (24) with uk (t) = 0. Remark 5 : The Theorem 3.1 provides results for asymptotically stablity of complex dynamical networks (1) and response networks (2) under pinning control. The Theorem 3.2 provides results for asymptotically stablity of complex dynamical networks (1) and response networks (2) under impulsive control. Remark 6 : In [31, 33, 36], the authors studied the synchronization of complex dynamical networks with time-varying delays via pinning control. In [22, 25, 26, 27, 32], the authors discussed the synchronization of complex dynamical networks with time-varying delays via impulsive control. In [42, 43, 44, 45], the authors analyzed the synchronization of Markovian jumping complex dynamical networks with time-varying delays. However, in our paper, we studied the impulsive and pinning control synchronization problem on Markovian jumping complex dynamical network. Hence this work is more general than those studied in [29, 37, 26, 27, 33]. Remark 7 : Most of the papers [4, 5, 7, 9, 12] used the Kronecker product method to study the complex dynamical networks. In this paper, we employed the transformation method. 4. Illustrative Examples In this section, numerical examples are presented to demonstrate the effectiveness of the synchronization for pinning and impulse control. Example 1 : Consider the following additive time-varying delayed Markovian jumping complex dynamical networks via pinning control with 3-node and mode s = {1, 2}. (1)

w˙k (t) = (−Ci + b0 E1i λk − b3 σk E4 )wk (t) + D1i hk (t) + D2i hk (t − ρ(t)) Z t (2) (3) + b1 E2i λk wk (t − ρ(t)) + b2 E3i λk wk (s)ds, k = 1, 2, 3

(33)

t−d(t)

T T T where wk (t) = (wk1 , wk2 ) and the relevant parameters are given as follows:

C1 = D12 =





0.17 −0.17 −0.8 0.9

E21 = E22 =



0.17 0.18 −1.1 0.8 1 0

0 1







, D11 =





−0.7 −1.2

0.1 0.9 −0.6 0.7

,

D22 =

,

E31 = E32 =





−0.7 0.6 2 0 0 2

  0.2 0.6 4.6 , C2 = −0.8 0.1 0   0.8 0 = E12 = , 0 0.8

, D21 = 



, E11

, E4 =





2 0 0 2



,

0 2.3



,

14

Γ=



−3 3 4 −4



,

 −2 1 1 = 2 1

L(a)

1 −2 1

 1 1  (a = 1, 2, 3) −2

Let us consider b0 = 1, b1 = b2 = 2, b3 = 3, σ1 = 0.4, σ2 = 0.2, σ3 = 0.6, ρ11 = 0.02, ρ12 = 0.04, ρ21 = 0.03, ρ22 = 0.06 µ1 = 0.03, µ2 = 0.05, d = 0.02 δ1 = δ2 = 12 ,. and the eigenvalues of L(a) (a = 1, 2, 3) are found to be λ1 = 0, λ2 = −0.5 and λ3 = −1.5. By using Matlab LMI Toolbox, we solve the LMIs (15) in Theorem 3.1, we obtain the feasible solutions for N = 3, k = 1, i = 1, 2. as follows

Q11 = T12 = T15 = X11 = X14 = Y13 = Z11 = Z14 =



 











4.1067 −0.9926

−0.9926 3.5997

3.2146 −1.6235

−1.6235 3.1572

−10.5386 −2.0955 18.7725 −0.0885 19.3338 −0.4531 19.1802 −0.1524 14.2254 0.0004 13.9727 0.0006





,

Q12 =

,

T13 =





4.0145 0.0410 0.0410 3.7881 2.3526 −1.0401

  −2.0955 18.6325 , T16 = −10.6550 0.9506   −0.0885 28.7889 , X12 = 18.8106 3.2058   −0.4531 5.3786 , Y11 = 19.4987 −0.0011   −0.1524 19.4410 , Y14 = 19.1733 −0.0392   0.0004 14.0083 , Z12 = 14.2254 0.0028  0.0006 . 13.9730



,



T11 =

 −1.0401 , 2.3077  0.9506 , 18.6591  3.2058 , 29.8671  −0.0011 , 5.3786  −0.0392 , 19.4393  0.0028 , 14.0082



T14 = T17 =



X13 =



Y12 = Y15 = Z13 =

 



22.4147 −2.6959

−2.6959 22.1434

20.6367 −1.1576

−1.1576 20.6034

0.1553 −0.0098 32.1790 8.1842 10.0316 −0.0369 42.7685 0.4071 14.2254 0.0004





, ,

 −0.0098 , 0.1570  8.1842 , 33.4116  −0.0369 , 10.0269  0.4071 , 41.7405  0.0004 , 14.2254

Example 2 : Consider the following additive time-varying delayed Markovian jumping complex dynamical networks via impulse control with 3-node and mode s = {1, 2}. (1)

(2)

w˙k (t) = (−Ci + b0 E1i λk )wk (t) + D1i hk (t) + D2i hk (t − ρ(t)) + b1 E2i λk wk (t − ρ(t)) Z t (3) + b2 E3i λk wk (s)ds, t 6= tκ t−d(t)

wk (tκ ) =

Jκi wk (t− κ ),

t = tκ

k = 1, 2, 3,

(34)

T T T where the relevant parameters are same as in Example 1 and wk (t) = (wk1 , wk2 ) , J11 = J12 = diag(0.4, 0.4). By using Matlab LMI Toolbox, we solve the LMIs (26)-(27) in Theorem 3.2, we obtain the feasible solutions for N = 3, k = 1, i = 1, 2. as follows

Q11 = T12 =





4.1826 −1.7107

−1.7107 3.8320

4.2525 −0.4432

−0.4432 3.8793





,

Q12 =

,

T13 =





7.7720 0.5061 0.5061 5.9658 3.4926 −0.2982



,

−0.2982 3.2181

T11 = 

,

T14 =





51.4668 −2.9717

−2.9717 51.4805

50.7673 −0.4985

−0.4985 50.2876





, ,

15

T15 = X11 = X14 = Y13 = Z11 = Z14 =



 







−32.5154 −0.8738 48.6137 −0.0064 49.9686 −0.0536 49.3834 −0.1224 37.0477 0.0003 36.2408 −0.0001

   −0.8738 50.4923 0.5026 , T16 = , −33.2663 0.5026 50.9764    −0.0064 84.8960 5.1587 , X12 = , 48.5014 5.1587 89.4471    −0.0536 14.1844 −0.0010 , Y11 = , 49.5494 −0.0010 14.1836    −0.1224 50.3089 −0.0318 , Y14 = , 49.2714 −0.0318 50.2798    0.0003 36.3575 0.0027 , Z12 = , 37.0480 0.0027 36.3586  −0.0001 . 36.2431

T17 = X13 = Y12 =









0.3602 −0.0094

−0.0094 0.3648

106.7346 12.2881 12.2881 107.0111 30.1783 −0.0469

−0.0469 30.0891



,





,

,

 125.0571 −4.2863 Y15 = , −4.2863 130.5853   37.0477 0.0003 Z13 = , 0.0003 37.0480

Example 3 : Consider the following additive time-varying delayed Markovian jumping complex dynamical networks via pinning control with 5-node and mode s = {1, 2}. (1)

w˙k (t) = (−Ci + b0 E1i λk − b3 σk E4 )wk (t) + D1i hk (t) + D2i hk (t − ρ(t)) Z t (2) (3) + b1 E2i λk wk (t − ρ(t)) + b2 E3i λk wk (s)ds, k = 1, 2, 3, 4, 5

(35)

t−d(t)

T T T T , wk2 , wk3 ) and the relevant parameters are given as follows: where wk (t) = (wk1



  0.17 0.17 0.12    0  , D11 =  C1 =  −0.17 0.18 0.13 −0.16 0.1    4.6 0 0.2    C2 =  0 2.3 0.4  , D12 =  1.2 0.5 1.4 E11 = E12





−0.8 0 0.17

  0.5 0.2 0.6 1.2   0  , D21 =  0.5 −0.8 0.1 0.12 0.4 0.1 −1.3   −1.1 0 −0.6 −0.7 0   0.9 0.8  , D22 =  1.3 0.7 0.6 0.17 0.12 0 −1.5 0.6

−0.7 −1.2 0.17

   0 1 0 0 0  , E21 = E22 =  0 1 0  , E31 = E32 0.8 0 0 1  −4 1 1    −2 1 1  1 −1 0 0   0 , Γ =  0 −1 L(a) =   3 −4 1  ,  1  1 0 0 2 1 1 −2 1 0 0

0.8 = 0 0

2 0 E4 =  0 2 0 0

0.1 0.9 0.17

0 0.8 0





 , 

 ,

 2 0 0 =  0 2 0 , 0 0 2  1 1 0 0   0 0   (a = 1, 2, 3) −1 0  0 −1

Let us consider b0 = 0.1, b1 = b2 = 0.2, b3 = 0.3, σ1 = 0.4, σ2 = 0.2, σ3 = 0.6, ρ11 = 0.02, ρ12 = 0.04, ρ21 = 0.03, ρ22 = 0.06 µ1 = 0.03, µ2 = 0.05, d = 0.02 δ1 = δ2 = 21 ,. and the eigenvalues of L(a) (a = 1, 2, 3) are found to be λ1 = 0, λ2 = λ3 = λ4 = −1 and λ5 = −5. By using Matlab LMI Toolbox, we solve the LMIs (15) in Theorem 3.1, we obtain the feasible solutions for N = 5, k = 1, i = 1, 2. as follows

16

Q11

T11

T13

T15

T17

X12

X14

Y12

Y14

Z11

Z13



 1.3057 −0.6982 0.2868 =  −0.6982 1.3186 −0.2701  , 0.2868 −0.2701 1.3148   10.7661 −0.2953 0.0256 =  −0.2953 10.8241 −0.1344  , 0.0256 −0.1344 10.8800   0.5840 −0.0385 0.0180 =  −0.0385 0.5966 −0.0263  , 0.0180 −0.0263 0.6206   −6.2110 −0.1445 0.0446 =  −0.1445 −6.1626 −0.0745  , 0.0446 −0.0745 −6.1694   0.0616 −0.0001 −0.0006 0.0002  , =  −0.0001 0.0634 −0.0006 0.0002 0.0587   19.8122 1.5035 −0.3365 =  1.5035 19.7507 0.7117  , −0.3365 0.7117 19.2600   11.5171 0.0969 −0.0042 =  0.0969 11.5440 0.0364  , −0.0042 0.0364 11.7330   5.8459 −0.0081 0.0075 =  −0.0081 5.8415 0.0021  , 0.0075 0.0021 5.8229   10.2653 −0.0034 0.0009 =  −0.0034 10.2668 −0.0017  , 0.0009 −0.0017 10.2667   7.5647 0.0000 −0.0000 =  0.0000 7.5647 0.0000  , −0.0000 0.0000 7.5647   7.5647 0.0000 −0.0000 =  0.0000 7.5647 0.0000  , −0.0000 0.0000 7.5647



2.0163 Q12 =  0.2382 −0.0083  0.7090 T12 =  −0.0506 0.0320  10.7919 T14 =  −0.0701 0.0251  9.8816 T16 =  0.0573 −0.0199  10.1626 X11 =  0.0223 −0.0033  27.4999 X13 =  3.6408 −0.0727  2.0611 Y11 =  −0.0001 0.0000  10.1112 Y13 =  −0.0133 0.0037  11.4052 Y15 =  −0.5013 −0.0655  7.2868 Z12 =  0.0005 −0.0002  7.2693 Z14 =  0.0004 −0.0004

0.2382 1.8888 −0.0242 −0.0506 0.7186 −0.0418 −0.0701 10.8238 −0.0392

 −0.0083 −0.0242  , 1.8937  0.0320 −0.0418  , 0.7606  0.0251 −0.0392  , 10.8423

 0.0573 − 0.0199 9.8563 0.0318  , 0.0318 9.8389  0.0223 −0.0033 10.1637 0.0068  , 0.0068 10.1953  3.6408 −0.0727 24.8913 1.5012  , 1.5012 31.4042  −0.0001 0.0000 2.0611 −0.0000  , −0.0000 2.0611  −0.0133 0.0037 10.1168 −0.0064  , −0.0064 10.1164  −0.5013 −0.0655 11.4116 −0.1977  , −0.1977 11.5915  0.0005 −0.0002 7.2866 −0.0001  , −0.0001 7.2883  0.0004 −0.0004 7.2697 −0.0001  . −0.0001 7.2702

Example 4 : Consider the following additive time-varying delayed Markovian jumping complex dynamical networks via impulse control with 5-node and mode s = {1, 2}. (1)

(2)

w˙k (t) = (−Ci + b0 E1i λk )wk (t) + D1i hk (t) + D2i hk (t − ρ(t)) + b1 E2i λk wk (t − ρ(t)) Z t (3) + b2 E3i λk wk (s)ds, t 6= tκ t−d(t)

wk (tκ ) =

Jκi wk (t− κ ),

t = tκ

k = 1, 2, 3, 4, 5

(36)

17 T T T T where the relevant parameters are same as in Example 1 and wk (t) = (wk1 , wk2 , wk3 ) , J11 = J12 = diag(0.4, 0.4, 0.4). By using Matlab LMI Toolbox, we solve the LMIs (26)-(27) in Theorem 3.2, we obtain the feasible solutions for N =5, k = 1, i = 1, 2. as follows

Q11

T11

T13

T15

T17

X12

X14

Y12

Y14

Z11

Z13



   0.9869 −0.5194 0.2370 1.6727 0.2526 −0.0389 =  −0.5194 1.0283 −0.2202  , Q12 =  0.2526 1.5401 0.0064  , 0.2370 −0.2202 1.0117 −0.0389 0.0064 1.4955     8.6432 −0.1997 0.0172 0.5554 −0.0284 0.0237 =  −0.1997 8.6492 −0.0848  , T12 =  −0.0284 0.5551 −0.0219  , 0.0172 −0.0848 8.7092 0.0237 −0.0219 0.5800     0.4559 −0.0181 0.0146 8.6109 −0.0276 0.0228 =  −0.0181 0.4559 −0.0129  , T14 =  −0.0276 8.6095 −0.0147  , 0.0146 −0.0129 0.4763 0.0228 −0.0147 8.6305     −5.0020 −0.0713 0.0364 7.9207 0.0228 −0.0189 =  −0.0713 −5.0000 −0.0342  , T16 =  0.0228 7.9224 0.0121  , 0.0364 −0.0342 −5.0003 −0.0189 0.0121 7.9037     0.0505 −0.0001 −0.0004 8.1284 0.0204 0.0005 0.0001  , X11 =  0.0204 8.1232 0.0075  , =  −0.0001 0.0521 −0.0004 0.0001 0.0482 0.0005 0.0075 8.1523     16.1378 1.0520 −0.2666 22.5645 2.6555 0.0262 =  1.0520 16.1932 0.5283  , X13 =  2.6555 20.4395 1.1339  , −0.2666 0.5283 15.7329 0.0262 1.1339 25.8050     9.2246 0.0956 0.0097 1.6490 −0.0000 0.0000  , Y11 =  −0.0000 1.6490 −0.0000  , =  0.0956 9.21120.0394 0.0097 0.0394 9.3909 0.0000 −0.0000 1.6490     4.3582 −0.0039 0.0049 8.0836 −0.0063 0.0035 =  −0.0039 4.3532 0.0021  , Y13 =  −0.0063 8.0841 −0.0029  , 0.0049 0.0021 4.3379 0.0035 −0.0029 8.0847     8.2093 −0.0016 0.0009 9.2501 −0.4612 −0.0764 =  −0.0016 8.2094 −0.0007  , Y15 =  −0.4612 9.2759 −0.1840  , 0.0009 −0.0007 8.2096 −0.0764 −0.1840 9.4249     6.0506 0.0000 −0.0000 5.8397 0.0003 −0.0002 5.8397 −0.0001  , =  0.0000 6.0506 0.0000  , Z12 =  0.0003 −0.0000 0.0000 6.0506 −0.0002 −0.0001 5.8412     5.8268 0.0002 −0.0003 6.0506 0.0000 −0.0000 5.8273 −0.0001  . =  0.0000 6.0506 0.0000  , Z14 =  0.0002 −0.0000 0.0000 6.0506 −0.0003 −0.0001 5.8278

18

2

1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1

Figure 1. Error trajectories of system in Example 1

2

1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1

Figure 2. Error trajectories of system in Example 2

5. conclusion In this paper, iimpulsive and pinning control synchronization criteria are proposed for a class of Markovian jumping master-slave complex dynamical networks with hybrid coupling and additive interval time-varying delay, where the linear couplings include both the discrete time-varying case and the distributed time-varying delay . Two kinds of control schemes are utilized to synchronize the considered dynamical network system. By developing Lyapunov-Krasovskii functional which contains triple integral terms and applying bounding techniques, novel delay dependent synchronization condition is derived in terms of linear matrix inequalities. We established sufficienct conditions for impulsive and pinning control synchronization. the numerical results given demonstrates the effectiveness of the obtained result. The idea and approach developed in this paper can be further generalized to deal with some other problems on sliding mode control and master-slave synchronization of complex dynamical networks with hybrid coupling.

19

6. Acknowledgement This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (DF − 536 − 130 − 1441). The authors, therefore, gratefully acknowledge DSR technical and financial support . Appendix A. Let us define h ξkT (t) = wkT (t) wkT (t − h(t)) wkT (t − ρ1 ) wkT (t − ρ2 ) wkT (t − ρ11 ) wkT (t − ρ12 ) wkT (t − ρ21 ) wkT (t − ρ22 ) Z t−ρ11 Z t−ρ21 Z t Z t−ρ1 g T (w(t)) g T (w(t − ρ(t))) wkT (s)ds wkT (s)ds wkT (s)ds wkT (s)ds Z

t

t−d(t)

wkT (s)ds

Z

t−ρ12

t−ρ11

t−ρ12

Z

w˙ kT (s)ds

t−ρ22

t−ρ21

t−ρ22

w˙ kT (s)ds

h (1) ηk = (−Ci + b0 E1i λk − b3 σk E4 ) b1 E2i λk 0

0

0

0

0

0

b2 E3i λk

0

0

(1)

w˙k (t) = ηk ξk (t), when uk (t) 6= 0. h (2) ηk = (−Ci + b0 E1i λk ) b1 E2i λk 0

0

0

b2 E3i λk

0

0

(2)

w˙k (t) = ηk ξk (t), when uk (t) = 0.

0 0 i ,

Z

t−ρ2

t

t−hρ2

w˙ kT (s)ds

Z

t−ρ2

0 0 0 0 0 0 i ,

0 0 0 0

D1i

t−ρ2

t−ρ1

D1i

i w˙ kT (s)ds , D2i

(37)

0 0 0 0 0 0 (38)

D2i

0 0 0 0 0 0 0 0 0 (39)

The inequality (4) and the Lipschitz continuity of hk (t) can be used to make hk (t) to satisfy k hk (t) k =k ≤ ≤ ≤

N X j=1

N X j=1

N X j=1

N X j=1

[f (xj (t)) − f (yj (t))]ukj k

k [f (xj (t)) − f (yj (t))] k| ukj | νk k [xj (t) − yj (t)] k= ν¯k k wj (t) k=

N X j=2

N X j=1

νk k ej (t) k

ν¯k k wj (t) k,

(40)

where ukj is the j-th element of Uk and ν¯k = maxνk . Therefore the following inequality N  X

k=2

 X XX ¯ X (1) k wj (t) k = k hk (t) k −ν¯k k wj (t) k k hk (t) k −νk N

N

j=2

k=2

=

N  X

k=2

N

N

k=2 j=2

k hk (t) k −(N − 1)ν¯k k wk (t) k



≤ 0,

holds, if the inequality k hk (t) k −(N − 1)ν¯k k wk (t) k≤ 0,

k = 2, 3, ...., N

(41)

20

is satisfied. Similarly, the following inequality hold: N  X

k=2

k hk (t − ρ(t)) k −(N − 1)ν¯k k wk (t − ρ(t)) k

if the following inequalities are satisfied k hk (t − ρ(t)) k −(N − 1)ν¯k k wk (t − ρ(t)) k≤ 0,



≤ 0,

(42)

k = 2, 3, ...., N.

(43)

From equation (9) and inequality (13)–(15), there exist positive diagonal matrices such that (1)

ξkT (t) diag{−(N − 1)ν¯k Gk , 0,

0,

0,

0,

0,

0,

(2)

−(N − 1)ν¯k Gk ,

0} ξk (t) = (1)

where Φk = diag{−(N − 1)ν¯k Gk , 0,

0,

0,

0,

0,

ξkT (t)

0,

0,

0 0,

0,

(1)

Gk ,

(2)

Gk ,

Φk ξk (t) ≤ 0, (2)

0,

0,

(2)

and Gk 0,

0, (44)

−(N − 1)ν¯k Gk , 0,

0,

(1) Gk

0,

0,

0 0,

0,

0}

0,

(1)

Gk ,

(2)

Gk , .

By Newton Leibniz formula, we have the following zero equations   Z t T 2ζk (t)Nk wk (t) − wk (t − ρ2 ) − w˙ k (s)ds = 0, t−ρ2

 Z 2ζkT (t)Mk wk (t − ρ1 ) − wk (t − ρ2 ) −

t−ρ1

 w˙ k (s)ds = 0,

t−ρ2 Z t−ρ11

 2ζkT (t)Hk wk (t − ρ11 ) − wk (t − ρ12 ) −

 2ζkT (t)Pk wk (t − ρ21 ) − wk (t − ρ22 ) −

t−ρ12 Z t−ρ21 t−ρ22

 w˙ k (s)ds = 0,

 w˙ k (s)ds = 0,

(45) (46) (47) (48)

where Nk = Nkj , Mk = Mkj , Hk = Hkj and Pk = Pkj (j = 1, 2, ...., 15) are any matrices with appropriate dimensions, and h ζkT (t) = wkT (t) wkT (t − ρ(t)) wkT (t − ρ1 ) wkT (t − ρ2 ) wkT (t − ρ11 ) wkT (t − ρ12 ) wkT (t − ρ21 ) wkT (t − ρ22 ) Z t Z t−ρ11 Z t−ρ21 g T (w(t)) g T (w(t − ρ(t))) wkT (s)ds w˙ kT (s)ds w˙ kT (s)ds Z

t

t−ρ2

w˙ kT (s)ds

Z

t−d(t)

t−h1

t−ρ2

t−ρ12

i w˙ kT (s)ds .

t−ρ22

(49) References

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