Open Access

Exponential synchronization of complex dynamical network with mixed time-varying and hybrid coupling delays via intermittent control

Advances in Difference Equations20142014:116

https://doi.org/10.1186/1687-1847-2014-116

Received: 8 December 2013

Accepted: 22 April 2014

Published: 6 May 2014

Abstract

In this paper, we shall investigate the problem of exponential synchronization for complex dynamical network with mixed time-varying and hybrid coupling delays, which is composed of state coupling, interval time-varying delay coupling and distributed time-varying delay coupling. The designed controller ensures that the synchronization of delayed complex dynamical network are proposed via either feedback control or intermittent feedback control. The constraint on the derivative of the time-varying delay is not required which allows the time-delay to be a fast time-varying function. We use common unitary matrices, and the problem of synchronization is transformed into the stability analysis of some linear time-varying delay systems. This is based on the construction of an improved Lyapunov-Krasovskii functional combined with the Leibniz-Newton formula and the technique of dealing with some integral terms. New synchronization criteria are derived in terms of LMIs which can be solved efficiently by standard convex optimization algorithms. Two numerical examples are included to show the effectiveness of the proposed feedback control and intermittent feedback control scheme.

Keywords

exponential synchronizationcomplex dynamical networkmixed time-varying delayshybrid couplingintermittent control

1 Introduction

Complex dynamical network, as an interesting subject, has been thoroughly investigated for decades. These networks show very complicated behavior and can be used to model and explain many complex systems in nature such as computer networks [1], the world wide web [2], food webs [3], cellular and metabolic networks [4], social networks [5], electrical power grids [6]etc. In general, a complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. As an implicit assumption, these networks are described by the mathematical term graph. In such graphs, each vertex represents an individual element in the system, while edges represent the relations between them. Two nodes are joined by an edge if and only if they interact.

In the last decade, the synchronization of complex dynamic networks has attracted much attention of researchers in this field [718]. Because the synchronization of complex dynamical networks can well explain many natural phenomena observed and is one of the important dynamical mechanisms for creating order in complex dynamical networks, the synchronization of coupled dynamical networks has come be a focal point in the study of nonlinear science. Wang and Chen introduced a uniform dynamical network model and also investigated its synchronization [1113]. They have shown that the synchronizability of a scale-free dynamical network is robust against random removal of nodes, and yet it is fragile to specific removal of the most highly connected nodes [12]. The authors in [14, 15] investigated synchronization of general complex dynamical network models with coupling delays. Li and Chen [8] considered the synchronization stability of complex dynamical network models with coupling delays for both continuous- and discrete-time, and they derived some synchronization conditions for both delay-independent and delay-dependent asymptotical stabilities. By utilizing Lyapunov functional method. Wang et al. [16] introduced several synchronization criteria for both delay-independent and delay-dependent asymptotical stability. Li and Yi [17] investigated synchronization of complex networks with time-varying couplings, the stability criteria were obtained by using Lyapunov-Krasovskii function method and subspace projection method. Yue and Li [18] studied the synchronization stability of continuous and discrete complex dynamical networks with interval time-varying delays in the dynamical nodes and the coupling term simultaneously, delay-dependent synchronization stability are derived in the form of linear matrix inequalities.

It is well known that the existence of time-delay in a system may cause instability and an example of oscillations can be found in systems such as chemical engineering systems, biological modeling, electrical networks, physical networks, and many others [1925]. The stability criteria for a system with time-delays can be classified into two categories: delay-independent and delay-dependent. Delay-independent criteria do not employ any information on the size of the delay; while delay-dependent criteria make use of such information at different levels. Delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the delay is small [25]. Recently, the delay-dependent stability for interval time-varying delay was investigated in [6, 18, 2022]. Interval time-varying delay is a time-delay that varies in an interval in which the lower bound is not restricted to be 0. Jiang and Han [22] considered the problem of robust H control for uncertain linear systems with interval time-varying delay based on Lyapunov functional approach in which restriction on the differentiability of the interval time-varying delay was removed. Shao [24] presented a new delay-dependent stability criterion for linear systems with interval time-varying delay, and stability criteria are derived in terms of linear matrix inequalities without introducing any free-weighting matrices. In order to reduce further the conservatism introduced by the descriptor model transformation and bounding techniques, a free-weighting matrix method is proposed in [20, 2629]. In [18], the synchronization problem has been investigated for continuous/discrete complex dynamical networks with interval time-varying delays. Based on a piecewise analysis method and the Lyapunov functional method, some new delay-dependent synchronization criteria are derived in the form of LMIs by introducing free-weighting matrices. It will be pointed out later that some existing results require more free-weighting matrix variables than our result.

Intermittent control is one of discontinuous control and has a nonzero control width. It is an engineering approach that has been widely used in engineering fields, such as manufacturing, air-quality control, transportation, and communication in practice. However, results using intermittent control to study exponential synchronization are few. In recent years, several synchronization criteria for complex dynamical networks with or without time-delays via feedback control or intermittent control have been presented; see [3041] and the references therein. Synchronization of a complex dynamical network with delayed nodes by pinning periodically intermittent control was also reported in [31]. A periodically intermittent control was applied to the complex dynamical networks with both time-varying delays dynamical nodes and time-varying delays coupling in [32, 33]. In [34], the authors investigated exponential synchronization of a complex network with nonidentical time-delayed dynamical nodes by applying open-loop control to all nodes and adding some intermittent controllers to partial nodes. The authors in [31] investigated synchronization of a general model of complex delayed dynamical networks. The periodically intermittent control scheme is introduced to drive the network to achieve synchronization. Based on the Lyapunov stability theory and pinning control method, some novel synchronization criteria for such dynamical network are derived. To the best of the authors’ knowledge, the problem of exponential synchronization for a complex dynamical network with mixed time-varying delays in the network hybrid coupling and time-varying delays in the dynamical nodes has not been fully investigated yet and remains open.

In this paper, inspired by the above discussions, we shall investigate the problem of exponential synchronization for a complex dynamical network with mixed time-varying and hybrid coupling delays, which is composed of constant coupling, interval time-varying delay coupling, and distributed time-varying delay coupling. The designed controller ensures that the synchronization of a delayed complex dynamical network is proposed via either feedback control or intermittent feedback control. The constraint on the derivative of the time-varying delay is not required, which allows the time-delay to be a fast time-varying function. We use common unitary matrices, and the problem of synchronization is transformed into the stability analysis of some linear time-varying delay systems. Based on the construction of an improved Lyapunov-Krasovskii functional is combined with the Leibniz-Newton formula and the technique of dealing with some integral terms. New synchronization criteria are derived in terms of LMIs which can be solved efficiently by standard convex optimization algorithms. Two numerical examples are included to show the effectiveness of the proposed feedback control and intermittent feedback control scheme.

The organization of the remaining part is as follows. In Section 2, a class of general complex dynamical network model with mixed time-varying and hybrid coupling delays and some useful lemmas are given. In Section 3, synchronization stability in complex dynamical network with mixed time-varying and hybrid coupling delays via feedback control and intermittent feedback control are investigated. Numerical examples illustrated the obtained results are given in Section 4. The paper ends with conclusions in Section 5.

2 Network model and mathematic preliminaries

Consider a complex dynamical network consisting of N identical coupled nodes, with each node being an n-dimensional dynamical system
x ˙ i ( t ) = f ( x i ( t ) , x i ( t h ( t ) ) , t k 1 ( t ) t x i ( s ) d s ) + c 1 j = 1 N a i j G 1 x j ( t ) + c 2 j = 1 N b i j G 2 x j ( t h ( t ) ) x ˙ i ( t ) = + c 3 j = 1 N c i j G 3 t k 1 ( t ) t x j ( s ) d s + U i ( t ) , t 0 , i = 1 , 2 , , N , x i ( t ) = ϕ i ( t ) , t [ τ max , 0 ] , τ max = max { h 2 , d , k 1 , k 2 } ,
(1)
where x i ( t ) = ( x i 1 ( t ) , x i 2 ( t ) , , x i n ( t ) ) T R n is the state vector of i th node; U i ( t ) R m are the control input of the node i; the constants c 1 , c 2 , c 3 > 0 are the coupling strength; G 1 = ( g 1 i j ) n × n , G 2 = ( g 2 i j ) n × n , G 3 = ( g 3 i j ) n × n R n × n are constant inner-coupling matrices, if some pairs ( i , j ) , 1 i , j n , with g 1 i j 0 , g 2 i j 0 , and g 3 i j 0 , which means two coupled nodes are linked through their i th and j th state variables, otherwise g 1 i j = 0 , g 2 i j = 0 , g 3 i j = 0 ; A = ( a i j ) N × N , B = ( b i j ) N × N , and C = ( c i j ) N × N R N × N are the outer-coupling matrices of the network, in which a i j , b i j are defined as follows: if there are a connection between node i and node j ( j i ), then a i j = a j i = 1 , b i j = b j i = 1 , c i j = c j i = 1 ; otherwise, a i j = a j i = 0 , b i j = b j i = 0 , c i j = c j i = 0 ( j i ), and the diagonal elements of matrices A, B, and C are defined by
a i i = j = 1 , i j N a i j = j = 1 , i j N a j i , b i i = j = 1 , i j N b i j = j = 1 , i j N b j i , c i i = j = 1 , i j N c i j = j = 1 , i j N c j i , i = 1 , 2 , , N .
(2)

It is assumed that network (1) is connected in the sense that there are no isolated clusters, that is, A, B, C are irreducible matrices.

Definition 2.1 [18]

The delayed dynamical network (1) is said to achieve asymptotical synchronization if
x 1 ( t ) = x 2 ( t ) = = s ( t ) as  t ,
(3)
where s ( t ) is a solution of an isolated node, satisfying
s ˙ ( t ) = f ( s ( t ) , s ( t h ( t ) ) , t k 1 ( t ) t s ( θ ) d θ ) .
In order to stabilize the origin of dynamical network (1) by means of the state feedback controller U i ( t ) satisfying either (H1) or (H2), for i = 1 , 2 , , n ,
( H 1 ) : U i ( t ) = D 1 i u i ( t ) + D 2 i u i ( t d ( t ) ) ( H 1 ) : U i ( t ) = + D 3 i t k 2 ( t ) t u i ( s ) d s , t t 0 , ( H 2 ) : U i ( t ) = { D 4 i u i ( t ) + D 5 i u i ( t d ( t ) ) + D 6 i t k 2 ( t ) t u i ( s ) d s , n ω t n ω + δ , 0 , n ω + δ < t ( n + 1 ) ω ,
where D j i , j = 1 , 2 , , 6 are given matrices of appropriate dimensions, u i ( t ) = K i ( x i ( t ) s ( t ) ) and K i is a constant matrix control gain, ω > 0 is the control period and δ > 0 is called the control width (control duration) and n is a non-negative integer. Then substituting it into dynamical network (1), it is easy to get the following:
x ˙ i ( t ) = f ( x i ( t ) , x i ( t h ( t ) ) , t k 1 ( t ) t x i ( s ) d s ) + c 1 j = 1 N a i j G 1 x j ( t ) + c 2 j = 1 N b i j G 2 x j ( t h ( t ) ) + c 3 j = 1 N c i j G 3 t k 1 ( t ) t x j ( s ) d s + D 1 i K i ( x i ( t ) s ( t ) ) + D 2 i u i ( t d ( t ) ) + D 3 i t k 2 ( t ) t u i ( s ) d s .
(4)
Namely, the dynamical network (1) is governed by the following system:
x ˙ i ( t ) = f ( x i ( t ) , x i ( t h ( t ) ) , t k 1 ( t ) t x i ( s ) d s ) + c 1 j = 1 N a i j G 1 x j ( t ) x ˙ i ( t ) = + c 2 j = 1 N b i j G 2 x j ( t h ( t ) ) + c 3 j = 1 N c i j G 3 t k 1 ( t ) t x j ( s ) d s x ˙ i ( t ) = + D 4 i K i ( x i ( t ) s ( t ) ) + D 5 i u i ( t d ( t ) ) + D 6 i t k 2 ( t ) t u i ( s ) d s , x ˙ i ( t ) = n ω t n ω + δ , x ˙ i ( t ) = f ( x i ( t ) , x i ( t h ( t ) ) , t k 1 ( t ) t x i ( s ) d s ) + c 1 j = 1 N a i j G 1 x j ( t ) x ˙ i ( t ) = + c 2 j = 1 N b i j G 2 x j ( t h ( t ) ) + c 3 j = 1 N c i j G 3 t k 1 ( t ) t x j ( s ) d s , x ˙ i ( t ) = n ω + δ < t ( n + 1 ) ω , i = 1 , 2 , , N .
(5)
It is clear that, if the zero solutions of the dynamical network (4) and (5) are globally exponentially stable, then exponential synchronization of the controlled dynamical network (1) is achieved. The time-varying delay functions h ( t ) , d ( t ) , k 1 ( t ) , and k 2 ( t ) satisfy the conditions
0 h 1 h ( t ) h 2 , 0 d ( t ) d , 0 k 1 ( t ) k 1 , 0 k 2 ( t ) k 2 .
(6)

The initial condition function ϕ i ( t ) denotes a continuous vector-valued initial function of t [ τ max , 0 ] .

In this paper, we assume that s ( t ) is an orbitally stable solution of the above system. Clearly, the stability of the synchronized states (3) of network (1) is determined by the dynamics of the isolate node, the coupling strength c 1 , c 2 , and c 3 , the inner-coupling matrices G 1 , G 2 , and G 3 , and the outer-coupling matrices A, B, and C.

The following lemmas are used in the proof of the main result.

Lemma 2.2 [42]

Let A, B be a family of diagonalizable matrices. Then A, B is a commuting family (under multiplication) if and only if it is a simultaneously diagonalizable family.

Lemma 2.3 [19]

For any constant symmetric matrix M R n × n , M = M T > 0 , 0 h 1 h ( t ) h 2 , t 0 , and any differentiable vector function x ( t ) R n , we have
( a ) [ t h 1 t x ˙ ( s ) d s ] T M [ t h 1 t x ˙ ( s ) d s ] h 1 t h 1 t x ˙ T ( s ) M x ˙ ( s ) d s , ( b ) [ t h ( t ) t h 1 x ˙ ( s ) d s ] T M [ t h ( t ) t h 1 x ˙ ( s ) d s ] ( h ( t ) h 1 ) t h ( t ) t h 1 x ˙ T ( s ) M x ˙ ( s ) d s [ t h ( t ) t h 1 x ˙ ( s ) d s ] T M [ t h ( t ) t h 1 x ˙ ( s ) d s ] ( h 2 h 1 ) t h ( t ) t h 1 x ˙ T ( s ) M x ˙ ( s ) d s .

Lemma 2.4 (Cauchy inequality [19])

For any symmetric positive definite matrix N M n × n and x , y R n we have
± 2 x T y x T N x + y T N 1 y .

3 Synchronization of delayed complex dynamical network via delayed feedback control and intermittent control

In this section, we shall obtain some delay-dependent exponential synchronization criteria for general complex dynamical network with discrete and distributed time-varying delays and hybrid coupling delays (1) by strict LMI approaches. Let us set
A ˜ i = J ( t ) + c 1 λ 1 i G 1 , B ˜ i = J h ( t ) + c 2 λ 2 i G 2 , C ˜ i = J k 1 ( t ) + c 3 λ 3 i G 3
and
  1. 1.

    J ( t ) = f ( s ( t ) , s ( t h ( t ) ) , t k 1 ( t ) t s ( ξ ) d ξ ) R n × n is the Jacobian of f ( x ( t ) , x ( t h ( t ) ) , t k 1 ( t ) t x ( s ) d s ) at s ( t ) with the derivative of f ( x ( t ) , x ( t h ( t ) ) , t k 1 ( t ) t x ( s ) d s ) respect to x ( t ) ,

     
  2. 2.

    J h ( t ) = f ( s ( t ) , s ( t h ( t ) ) , t k 1 ( t ) t s ( ξ ) d ξ ) R n × n is the Jacobian of f ( x ( t ) , x ( t h ( t ) ) , t k 1 ( t ) t x ( s ) d s ) at s ( t h ( t ) ) with the derivative of f ( x ( t ) , x ( t h ( t ) ) , t k 1 ( t ) t x ( s ) d s ) respect to x ( t h ( t ) ) ,

     
  3. 3.

    J k 1 ( t ) = f ( s ( t ) , s ( t h ( t ) ) , t k 1 ( t ) t s ( ξ ) d ξ ) R n × n is the Jacobian of f ( x ( t ) , x ( t h ( t ) ) , t k 1 ( t ) t x ( s ) d s ) at t k 1 ( t ) t s ( ξ ) d ξ with the derivative of f ( x ( t ) , x ( t h ( t ) ) , t k 1 ( t ) t x ( s ) d s ) respect to t k 1 ( t ) t x ( s ) d s .

     
Lemma 3.1 Consider the hybrid coupling delays dynamical network in (1). Let 0 = λ j 1 > λ j 2 λ j 3 λ j N , j = { 1 , 2 , 3 } , be the eigenvalues of the outer-coupling matrices A, B, and C, respectively. If the N 1 following n-dimensional linear time-varying delays differential equations are delay-dependent exponentially stable about their zero solutions:
z ˙ i ( t ) = ( A ˜ i + D 4 i K i ) z i ( t ) + B ˜ i z i ( t h ( t ) ) + C ˜ i t k 1 ( t ) t z i ( s ) d s z ˙ i ( t ) = + D 5 i K i z i ( t d ( t ) ) + D 6 i K i t k 2 ( t ) t z i ( s ) d s , n ω t n ω + δ , i = 2 , , N , z ˙ i ( t ) = A ˜ i z i ( t ) + B ˜ i z i ( t h ( t ) ) + C ˜ i t k 1 ( t ) t z i ( s ) d s , n ω + δ < t ( n + 1 ) ω , i = 2 , , N ,
(7)

then the dynamical networks (5) is exponentially stable, and then exponential synchronization of the controlled dynamical networks (1) is achieved.

Proof To investigate the stability of the synchronized states (3), set
e i ( t ) = x i ( t ) s ( t ) , i = 1 , 2 , , N .
(8)
Substituting (8) into (5), for 1 i N , we have
e ˙ i ( t ) = f ( x i ( t ) , x i ( t h ( t ) ) , t k 1 ( t ) t x i ( s ) d s ) f ( s ( t ) , s ( t h ( t ) ) , t k 1 ( t ) t s i ( ξ ) d ξ ) e ˙ i ( t ) = + c 1 j = 1 N a i j G 1 e j ( t ) + c 2 j = 1 N b i j G 2 e j ( t h ( t ) ) e ˙ i ( t ) = + c 3 j = 1 N c i j G 3 t k 1 ( t ) t e j ( s ) d s + D 4 i K i ( e i ( t ) ) + D 5 i K i ( e i ( t d ( t ) ) ) e ˙ i ( t ) = + D 6 i K i t k 2 ( t ) t e j ( s ) d s , n ω t n ω + δ , e ˙ i ( t ) = f ( x i ( t ) , x i ( t h ( t ) ) , t k 1 ( t ) t x i ( s ) d s ) f ( s ( t ) , s ( t h ( t ) ) , t k 1 ( t ) t s i ( ξ ) d ξ ) e ˙ i ( t ) = + c 1 j = 1 N a i j G 1 e j ( t ) + c 2 j = 1 N b i j G 2 e j ( t h ( t ) ) e ˙ i ( t ) = + c 3 j = 1 N c i j G 3 t k 1 ( t ) t e j ( s ) d s , n ω + δ < t ( n + 1 ) ω , i = 1 , 2 , , N .
(9)
Since f ( ) is continuous differentiable, it is easy to know that the origin of the nonlinear system (9) is an asymptotically stable equilibrium point if it is an asymptotically stable equilibrium point of the following linear time-varying delays systems:
e ˙ i ( t ) = J ( t ) e i ( t ) + J h ( t ) e i ( t h ( t ) ) + J k 1 ( t ) t k 1 ( t ) t e i ( s ) d s e ˙ i ( t ) = + c 1 G 1 ( e 1 ( t ) , e 2 ( t ) , , e N ( t ) ) ( a i 1 , , a i N ) T e ˙ i ( t ) = + c 2 G 2 ( e 1 ( t h ( t ) ) , , e N ( t h ( t ) ) ) ( b i 1 , , b i N ) T e ˙ i ( t ) = + c 3 G 3 t k 1 ( t ) t ( e 1 ( s ) , e 2 ( s ) , , e N ( s ) ) ( c i 1 , , c i N ) T d s e ˙ i ( t ) = + D 4 i K i e i ( t ) + D 5 i K i e i ( t d ( t ) ) + D 6 i K i t k 2 ( t ) t e j ( s ) d s , e ˙ i ( t ) = n ω t n ω + δ , e ˙ i ( t ) = J ( t ) e i ( t ) + J h ( t ) e i ( t h ( t ) ) + J k 1 ( t ) t k 1 ( t ) t e i ( s ) d s e ˙ i ( t ) = + c 1 G 1 ( e 1 ( t ) , e 2 ( t ) , , e N ( t ) ) ( a i 1 , , a i N ) T e ˙ i ( t ) = + c 2 G 2 ( e 1 ( t h ( t ) ) , , e N ( t h ( t ) ) ) ( b i 1 , , b i N ) T e ˙ i ( t ) = + c 3 G 3 t k 1 ( t ) t ( e 1 ( s ) , e 2 ( s ) , , e N ( s ) ) ( c i 1 , , c i N ) T d s , e ˙ i ( t ) = n ω + δ < t ( n + 1 ) ω .
Letting e ( t ) = ( e 1 ( t ) , , e N ( t ) ) R n × N , e ( t h ( t ) ) = ( e 1 ( t h ( t ) ) , , e N ( t h ( t ) ) ) R n × N , t k 1 ( t ) t e ( s ) d s = t k 1 ( t ) t ( e 1 ( s ) , e 2 ( s ) , , e N ( s ) ) d s R n × N , K = diag { K 1 , K 2 , , K N } , and D j = diag { D j 1 , D j 2 , , D j N } , j = { 4 , 5 , 6 } , we have
e ˙ ( t ) = ( J ( t ) + D K ) e ( t ) + J h ( t ) e ( t h ( t ) ) + J k 1 ( t ) t k 1 ( t ) t e ( s ) d s + c 1 G 1 e ( t ) A T e ˙ ( t ) = + c 2 G 2 e ( t h ( t ) ) B T + c 3 G 3 t k 1 ( t ) t e ( s ) C T d s + D 5 K e ( t d ( t ) ) e ˙ ( t ) = + D 6 K t k 2 ( t ) t e ( s ) d s , n ω t n ω + δ , e ˙ ( t ) = J ( t ) e ( t ) + J h ( t ) e ( t h ( t ) ) + J k 1 ( t ) t k 1 ( t ) t e ( s ) d s + c 1 G 1 e ( t ) A T e ˙ ( t ) = + c 2 G 2 e ( t h ( t ) ) B T + c 3 G 3 t k 1 ( t ) t e ( s ) C T d s , n ω + δ < t ( n + 1 ) ω .
(10)
Obviously, A, B, C are diagonalizable. If A, B, and C commute pairwise, i.e., A B = B A , then based on Lemma 2.2, one can get a common unitary matrix U ˆ R N × N with u ˆ i R n such that
U ˆ T A U ˆ = Γ 1 , U ˆ T B U ˆ = Γ 2 , U ˆ T C U ˆ = Γ 3 ,

where U ˆ T U ˆ = I , Γ j = diag { λ 1 j , , λ N j } , j = { 1 , 2 , 3 } . In addition, with (2) and the irreducible feature of A, B, and C we can select with u ˆ 1 = 1 N ( 1 , 1 , , 1 ) T such that λ 1 j = 0 , j = { 1 , 2 , 3 } .

Using the nonsingular transform e ( t ) U ˆ = z ( t ) = ( z 1 ( t ) , , z N ( t ) ) R N × N , from (10), we have the following matrix equation:
z ˙ ( t ) = ( J ( t ) + D K ) z ( t ) + J h ( t ) z ( t h ( t ) ) + J k 1 ( t ) t k 1 ( t ) t z ( s ) d s + c 1 G 1 z ( t ) Γ 1 z ˙ ( t ) = + c 2 G 2 z ( t h ( t ) ) Γ 2 + c 3 G 3 t k ( t ) t z ( s ) Γ 3 d s + D 5 K z ( t d ( t ) ) z ˙ ( t ) = + D 6 K t k 2 ( t ) t z ( s ) d s , n ω t n ω + δ , z ˙ ( t ) = J ( t ) z ( t ) + J h ( t ) z ( t h ( t ) ) + J k 1 ( t ) t k 1 ( t ) t z ( s ) d s + c 1 G 1 z ( t ) Γ 1 z ˙ ( t ) = + c 2 G 2 z ( t h ( t ) ) Γ 2 + c 3 G 3 t k 1 ( t ) t z ( s ) Γ 3 d s , n ω + δ < t ( n + 1 ) ω ,
that is,
z ˙ i ( t ) = ( A ˜ i + D 4 i K i ) z i ( t ) + B ˜ i z i ( t h ( t ) ) + C ˜ i t k 1 ( t ) t z i ( s ) d s z ˙ i ( t ) = + D 5 i K i z i ( t d ( t ) ) + D 6 i K i t k 2 ( t ) t z i ( s ) d s , n ω t n ω + δ , z ˙ i ( t ) = A ˜ 1 i z i ( t ) + B ˜ i z i ( t h ( t ) ) + C ˜ i t k 1 ( t ) t z i ( s ) d s , n ω + δ < t ( n + 1 ) ω , i = 1 , , N .
Thus, we have transformed the stability problem of the dynamical networks (5) to the stability problem of the N pieces of n-dimensional linear time-varying delays differential equations. Note that λ 1 k = 0 corresponding to the synchronization of the dynamical networks (5), where the state s ( t ) is an orbitally stable solution of the isolate node as assumed above in (3). If the following N 1 pieces of n-dimensional linear switched time-varying delays systems:
z ˙ i ( t ) = ( A ˜ i + D 4 i K i ) z i ( t ) + B ˜ i z i ( t h ( t ) ) + C ˜ i t k 1 ( t ) t z i ( s ) d s z ˙ i ( t ) = + D 5 i K i z i ( t d ( t ) ) + D 6 i K i t k 2 ( t ) t z i ( s ) d s , n ω t n ω + δ , z ˙ i ( t ) = A ˜ i z i ( t ) + B ˜ i z i ( t h ( t ) ) + C ˜ i t k 1 ( t ) t z i ( s ) d s , n ω + δ < t ( n + 1 ) ω , i = 2 , , N ,

are exponentially stable, then e ( t ) will tend to the origin exponentially, which is equivalent to the synchronization of the dynamical networks (5) being exponentially stable. This completes the proof. □

Lemma 3.2 Consider the hybrid coupling delays dynamical network in (1). Let 0 = λ j 1 > λ j 2 λ j 3 λ j N , j = { 1 , 2 , 3 } , be the eigenvalues of the outer-coupling matrices A, B, and C, respectively. If the N 1 following n-dimensional linear time-varying delays differential equations are delay-dependent exponentially stable about their zero solutions:
z ˙ i ( t ) = ( A ˜ i + D 1 i K i ) z i ( t ) + B ˜ i z i ( t h ( t ) ) + C ˜ i t k 1 ( t ) t z i ( s ) d s + D 2 i K i z i ( t d ( t ) ) + D 3 i K i t k 2 ( t ) t z i ( s ) d s , i = 2 , , N ,
(11)

then the dynamical networks (4) is exponentially stable, then exponential synchronization of the controlled dynamical networks (1) is achieved.

3.1 Linear delayed feedback control

Let us denote
ϕ i = z i ( 0 ) , φ i = sup τ max s 0 z i ( s ) , K i = L i P i 1 , γ i = λ min ( P i 1 ) , i = λ max ( P i 1 ) + [ 2 h 2 λ max ( P i 1 R i P i 1 ) + h 2 λ max ( P i 1 U i P i 1 ) ] 1 e 2 α h 2 2 α i = + d λ max ( P i 1 L i T T i 1 L i P i 1 ) 1 e 2 α d 2 α , ξ i = [ 2 λ max ( P i 1 Q i P i 1 ) + h 2 λ max ( P i 1 R i P i 1 ) + h 2 λ max ( P i 1 U i P i 1 ) ] ξ i = × 1 e 2 α h 2 2 α + k 1 λ max ( P i 1 S i P i 1 ) 1 e 2 α k 1 2 α ξ i = + d λ max ( P i 1 L i T T i 1 L i P i 1 ) 1 e 2 α h 2 2 α ξ i = + k 2 λ max ( P i 1 L i T W i 1 L i P i 1 ) 1 e 2 α d 2 α , N i = i ϕ i 2 + ξ i φ i 2 , γ = min { γ i , i = 2 , 3 , , N } , N = max { N i , i = 2 , 3 , , N } .
Theorem 3.3 For some given scalars 0 < α , the dynamical networks (11) with time-varying delay satisfying (6) are exponentially stable if there exist symmetric positive definite matrices P i > 0 , Q i > 0 , R i > 0 , S i > 0 , U i > 0 , T i > 0 , W i > 0 , and a matrix L i appropriately dimensioned such that the following symmetric linear matrix inequality holds:
Σ i 1 = Σ i [ 0 0 I I 0 ] T e 2 α h 2 U i [ 0 0 I I 0 ] < 0 ,
(12)
Σ i 2 = Σ i [ 0 0 0 I I ] T e 2 α h 2 U i [ 0 0 0 I I ] < 0 ,
(13)
Σ i 3 = [ 0.5 ( e 2 α h 1 + e 2 α h 2 ) R i 2 k 1 C ˜ i P i k 2 L i T 2 L i T 2 k 1 e 2 α k 1 S i 0 0 k 2 W i 0 2 e 2 α d T i ] < 0 ,
(14)
Σ i 4 = [ 0.5 P i 2 k 1 C ˜ i P i d 2 L i T 3 D 2 i T 2 k 2 D 3 i T 2 k 1 e 2 α k 1 S i 0 0 0 d 2 T i 0 0 3 e 2 α d T i 0 2 k 2 e 2 α k 2 W i ] < 0 ,
(15)
i = 2 , , N , where
Σ i = [ Σ i 11 Σ i 12 Σ i 13 Σ i 14 Σ i 15 Σ i 22 0 Σ i 24 0 Σ i 33 Σ i 34 0 Σ i 44 Σ i 45 Σ i 55 ] , Σ i 11 = P i T ( A ˜ i + α I ) + ( A ˜ i + α I ) T P i D 1 i L i L i T D 1 i T + 3 e 2 α d D 2 i T T i D 2 i Σ i 11 = + 2 k 2 e 2 α k 2 D 3 i T W i D 3 i + 2 Q i + k 1 S i 0.5 e 2 α h 1 R i 0.5 e 2 α h 2 R i , Σ i 12 = P i A ˜ i T , Σ i 13 = e 2 α h 1 R i , Σ i 14 = B ˜ i P i , Σ i 15 = e 2 α h 2 R i , Σ i 22 = h 1 2 R i + h 2 2 R i + η 2 U i 1.5 P i , Σ i 24 = B ˜ i P i , Σ i 33 = e 2 α h 1 Q i e 2 α h 1 R i e 2 α h 2 U i , Σ i 34 = e 2 α h 2 U i , Σ i 44 = 2 e 2 α h 2 U i , Σ i 45 = e 2 α h 2 U i , Σ i 55 = 2 e 2 α h 2 U i 2 e 2 α h 2 Q i 2 e 2 α h 2 R i ,
then the dynamical networks (11) have exponential synchronization. Moreover, the feedback control is
u i ( t ) = L i P i 1 z i ( t ) .
(16)
Proof Let Y i = P i 1 , y i ( t ) = Y i z i ( t ) . Using the feedback control (16) we consider the following Lyapunov-Krasovskii functional:
V i ( z i ( t ) ) = V i 1 ( t ) + V i 2 ( t ) + V i 3 ( t ) + V i 4 ( t ) + V i 5 ( t ) + V i 6 ( t ) + V i 7 ( t ) + V i 8 ( t ) + V i 9 ( t ) ,
(17)
where
V i 1 ( t ) = z i T ( t ) Y i z i ( t ) , V i 2 ( t ) = t h 1 t e 2 α ( s t ) z i T ( s ) Y i Q i Y i z i ( s ) d s , V i 3 ( t ) = t h 2 t e 2 α ( s t ) z i T ( s ) Y i Q i Y i z i ( s ) d s , V i 4 ( t ) = h 1 h 1 0 t + s t e 2 α ( τ t ) z ˙ i T ( τ ) Y i R i Y i z ˙ i ( τ ) d τ d s , V i 5 ( t ) = h 2 h 2 0 t + s t e 2 α ( τ t ) z ˙ i T ( τ ) Y i R i Y i z ˙ i ( τ ) d τ d s , V i 6 ( t ) = ( h 2 h 1 ) t h 2 t h 1 t + s t e 2 α ( τ t ) z ˙ i T ( τ ) Y i U i Y i z ˙ i ( τ ) d τ d s , V i 7 ( t ) = k 1 0 t + s t e 2 α ( τ t ) z i T ( τ ) Y i S i Y z i ( τ ) d τ d s , V i 8 ( t ) = d d 0 t + s t e 2 α ( τ t ) z ˙ i T ( τ ) K i T T i 1 K i z ˙ i ( τ ) d τ d s , V i 9 ( t ) = k 2 0 t + s t e 2 α ( τ t ) z i T ( τ ) K i T W i 1 K i z i ( τ ) d τ d s .
It easy to check that
γ z i ( t ) 2 V i ( z i ( t ) ) , t 0 .
(18)
By taking the derivative of V i 1 ( t ) along the trajectories of system (11), we have the following:
V ˙ i 1 ( t ) = 2 z i T ( t ) Y i z ˙ i ( t ) = 2 y i T ( t ) [ ( A ˜ i + D 1 i K i ) z i ( t ) + B ˜ i z i ( t h ( t ) ) + C ˜ i t k 1 ( t ) t z i ( s ) d s + D 2 i K i z i ( t d ( t ) ) + D 3 i K i t k 2 ( t ) t z i ( s ) d s ] = y i T ( t ) [ P i A ˜ i + A ˜ i T P i ] y i ( t ) + 2 y i T ( t ) B ˜ i P i y i ( t h ( t ) ) + 2 y i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s 2 y i T ( t ) D i L i T y i ( t ) + 2 y i T ( t ) D 2 i u i ( t d ( t ) ) + 2 y i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s + 2 y i T ( t ) α P i y i ( t ) 2 y i T ( t ) α P i y i ( t ) .
Applying Lemma 2.4 and Lemma 2.3 gives
2 y i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s 2 k 1 e 2 α k 1 y i T ( t ) C ˜ i P i S i 1 P i C ˜ i T y i ( t ) 2 y i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s + e 2 α k 1 2 k 1 ( t k 1 ( t ) t y i ( s ) d s ) T S i ( t k 1 ( t ) t y i ( s ) d s ) 2 y i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s 2 k 1 e 2 α k 1 y i T ( t ) C ˜ i P i S i 1 P i C ˜ i T y i ( t ) 2 y i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s + 1 2 e 2 α k 1 t k 1 ( t ) t y i T ( s ) S i y i ( s ) d s , 2 y i T ( t ) D 2 i u i ( t d ( t ) ) 3 e 2 α d y i T ( t ) D 2 i T i D 2 i T y i ( t ) 2 y i T ( t ) D 2 i u i ( t d ( t ) ) + e 2 α d 3 u i T ( t d ( t ) ) T i 1 u i ( t d ( t ) ) , 2 y i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s 2 k 2 e 2 α k 2 y i T ( t ) D 3 i W i D 3 i T y i ( t ) 2 y i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s + e 2 α k 2 2 k 2 ( t k 2 ( t ) t u i ( s ) d s ) T W i 1 ( t k 2 ( t ) t u i ( s ) d s ) 2 y i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s 2 k 2 e 2 α k 2 y i T ( t ) D 3 i W i D 3 i T y i ( t ) 2 y i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s + e 2 α k 2 2 t k 2 ( t ) t u i T ( s ) W i 1 u i ( s ) d s .
Therefore
V ˙ i 1 ( t ) + 2 α V i 1 ( t ) y i T ( t ) [ P i A ˜ i + A ˜ i T P i ] y i ( t ) + 2 y i T ( t ) α P i y i ( t ) + 2 y i T ( t ) B ˜ i P i y i ( t h ( t ) ) 2 y i T ( t ) D i L i T y i ( t ) + 2 k 1 e 2 α k 1 y i T ( t ) C ˜ i P i S i 1 P i C ˜ i T y i ( t ) + 1 2 e 2 α k 1 t k 1 ( t ) t y i T ( s ) S i y i ( s ) d s + 3 e 2 α d y i T ( t ) D 2 i T i D 2 i T y i ( t ) + e 2 α d 3 u i T ( t d ( t ) ) T i 1 u i ( t d ( t ) ) + 2 k 2 e 2 α k 2 y i T ( t ) D 3 i W i D 3 i T y i ( t ) + e 2 α k 2 2 t k 2 ( t ) t u i T ( s ) W i 1 u i ( s ) d s .
(19)
Next, by taking the derivative of V i j ( t ) , j = 2 , 3 , , 9 along the trajectories of system (11), we have the following:
V ˙ i 2 ( t ) y i T ( t ) Q i y i ( t ) e 2 α h 1 y i T ( t h 1 ) Q i y i ( t h 1 ) 2 α V i 2 ( t ) , V ˙ i 3 ( t ) y i T ( t ) Q i y i ( t ) e 2 α h 2 y i T ( t h 2 ) Q i y i ( t h 2 ) 2 α V i 3 ( t ) , V ˙ i 4 ( t ) h 1 2 y ˙ i T ( t ) R i y ˙ i ( t ) h 1 e 2 α h 1 t h 1 t y ˙ i T ( s ) R i y ˙ i ( s ) d s 2 α V i 4 ( t ) , V ˙ i 5 ( t ) h 2 2 y ˙ i T ( t ) R i y ˙ i ( t ) h 2 e 2 α h 2 t h 2 t y ˙ i T ( s ) R i y ˙ i ( s ) d s 2 α V i 5 ( t ) , V ˙ i 6 ( t ) η 2 y ˙ i T ( t ) U i y ˙ i ( t ) η e 2 α h 2 t h 2 t h 1 y ˙ i T ( s ) U i y ˙ i ( s ) d s 2 α V i 6 ( t ) , V ˙ i 7 ( t ) k 1 y i T ( t ) S i y i ( t ) e 2 α k 1 t k 1 ( t ) t y i T ( s ) S i y i ( s ) d s 2 α V i 7 ( t ) , V ˙ i 8 ( t ) d 2 z ˙ i T ( t ) K i T T i 1 K i z ˙ i T ( t ) d e 2 α d t d t z ˙ i T ( s ) K i T T i 1 K i z ˙ i T ( s ) d s 2 α V i 8 ( t ) V ˙ i 8 ( t ) d 2 y ˙ i T ( t ) P i K i T T i 1 K i P i y ˙ i T ( t ) d ( t ) e 2 α d t d ( t ) t u ˙ i T ( s ) T i 1 u ˙ i ( s ) d s 2 α V i 8 ( t ) V ˙ i 8 ( t ) = d 2 y ˙ i T ( t ) L i T T i 1 L i y ˙ i T ( t ) d ( t ) e 2 α d t d ( t ) t u ˙ i T ( s ) T i 1 u ˙ i ( s ) d s 2 α V i 8 ( t ) , V ˙ i 9 ( t ) k 2 z i T ( t ) K i T W i 1 K i z i T ( t ) e 2 α k 2 t k 2 t z i T ( s ) K i T W i 1 K i z i T ( s ) d s 2 α V i 9 ( t ) V ˙ i 9 ( t ) k 2 y i T ( t ) P i K i T W i 1 K i P i y i T ( t ) e 2 α k 2 t k 2 ( t ) t u i T ( s ) W i 1 u i T ( s ) d s 2 α V i 9 ( t ) V ˙ i 9 ( t ) k 2 y i T ( t ) L i T W i 1 L i y i T ( t ) e 2 α k 2 t k 2 ( t ) t u i T ( s ) W i 1 u i T ( s ) d s 2 α V i 9 ( t ) .
(20)
Applying Lemma 2.3 and the Leibniz-Newton formula, we have
h 1 t h 1 t y ˙ i T ( s ) R i y ˙ i ( s ) d s [ t h 1 t y ˙ i ( s ) d s ] T R i [ t h 1 t y ˙ i ( s ) d s ] [ y i ( t ) y i ( t h 1 ) ] T R i [ y i ( t ) y i ( t h 1 ) ] = y i T ( t ) R i y i ( t ) + 2 y i T ( t ) R i y i ( t h 1 ) y i T ( t h 1 ) R i y i ( t h 1 )
(21)
and
h 2 t h 2 t y ˙ i T ( s ) R i y ˙ i ( s ) d s [ t h 2 t y ˙ i ( s ) d s ] T R i [ t h 2 t y ˙ i ( s ) d s ] [ y i ( t ) y i ( t h 2 ) ] T R i [ y i ( t ) y i ( t h 2 ) ] = y i T ( t ) R i y i ( t ) + 2 y i T ( t ) R i y i ( t h 2 ) y i T ( t h 2 ) R i y i ( t h 2 ) .
(22)
On the other hand,
( h 2 h 1 ) t h 2 t h 1 y ˙ i T ( s ) U i y ˙ i ( s ) d s = ( h 2 h 1 ) t h 2 t h ( t ) y ˙ i T ( s ) U i y ˙ i ( s ) d s ( h 2 h 1 ) t h ( t ) t h 1 y ˙ i T ( s ) U i y ˙ i ( s ) d s = ( h 2 h ( t ) ) t h 2 t h ( t ) y ˙ i T ( s ) U i y ˙ i ( s ) d s ( h ( t ) h 1 ) t h 2 t h ( t ) y ˙ i T ( s ) U i y ˙ i ( s ) d s ( h ( t ) h 1 ) t h ( t ) t h 1 y ˙ i T ( s ) U i y ˙ i ( s ) d s ( h 2 h ( t ) ) t h ( t ) t h 1 y ˙ i T ( s ) U i y ˙ i ( s ) d s .
Using Lemma 2.3 gives
( h 2 h ( t ) ) t h 2 t h ( t ) y ˙ i T ( s ) U i y ˙ i ( s ) d s [ t h 2 t h ( t ) y ˙ i ( s ) d s ] T U i [ t h 2 t h ( t ) y ˙ i ( s ) d s ] [ y i ( t h ( t ) ) y i ( t h 2 ) ] T U i × [ y i ( t h ( t ) ) y i ( t h 2 ) ] = y i T ( t h ( t ) ) U i y i ( t h ( t ) ) + 2 y i T ( t h ( t ) ) U i y i ( t h 2 ) y i T ( t h 2 ) U i y i ( t h 2 )
(23)
and
( h ( t ) h 1 ) t h ( t ) t h 1 y ˙ i T ( s ) U i y ˙ i ( s ) d s [ t h ( t ) t h 1 y ˙ i ( s ) d s ] T U i [ t h ( t ) t h 1 y ˙ i ( s ) d s ] [ y i ( t h 1 ) y i ( t h ( t ) ) ] T U i × [ y i ( t h 1 ) y i ( t h ( t ) ) ] = y i T ( t h 1 ) U i y i ( t h 1 ) + 2 y i T ( t h 1 ) U i y i ( t h ( t ) ) y i T ( t h ( t ) ) U i y i ( t h ( t ) ) .
(24)
Let β = h 2 h ( t ) h 2 h 1 1 . Then
( h 2 h ( t ) ) t h ( t ) t h 1 y ˙ i T ( s ) U i y ˙ i ( s ) d s = β t h ( t ) t h 1 ( h 2 h 1 ) y ˙ i T ( s ) U i y ˙ i ( s ) d s β t h ( t ) t h 1 ( h ( t ) h 1 ) y ˙ i T ( s ) U i y ˙ i ( s ) d s β [ y i ( t h 1 ) y i ( t h ( t ) ) ] T U i × [ y i ( t h 1 ) y i ( t h ( t ) ) ]
(25)
and
( h ( t ) h 1 ) t h 2 t h ( t ) y ˙ i T ( s ) U i y ˙ i ( s ) d s = ( 1 β ) t h 2 t h ( t ) ( h 2 h 1 ) y ˙ i T ( s ) U i y ˙ i ( s ) d s ( 1 β ) t h 2 t h ( t ) ( h 2 h ( t ) ) y ˙ i T ( s ) U i y ˙ i ( s ) d s ( 1 β ) [ y i ( t h ( t ) ) y i ( t h 2 ) ] T U i × [ y i ( t h ( t ) ) y i ( t h 2 ) ] .
(26)
Therefore from (23)-(26), we obtain
( h 2 h 1 ) t h 2 t h 1 y ˙ i T ( s ) U i y ˙ i ( s ) d s [ y i ( t h ( t ) ) y i ( t h 2 ) ] T U i × [ y i ( t h ( t ) ) y i ( t h 2 ) ] [ y i ( t h 1 ) y i ( t h ( t ) ) ] T U i × [ y i ( t h 1 ) y i ( t h ( t ) ) ] β [ y i ( t h 1 ) y i ( t h ( t ) ) ] T U i × [ y i ( t h 1 ) y i ( t h ( t ) ) ] ( 1 β ) [ y i ( t h ( t ) ) y i ( t h 2 ) ] T U i × [ y i ( t h ( t ) ) y i ( t h 2 ) ] .
(27)
From V ˙ i 8 ( t ) , applying Lemma 2.3 and the Leibniz-Newton formula gives
d ( t ) e 2 α d t d ( t ) t u ˙ i T ( s ) T i 1 u ˙ i ( s ) d s e 2 α d ( t d ( t ) t u ˙ i ( s ) d s ) T T i 1 ( t d ( t ) t u ˙ i ( s ) d s ) e 2 α d u i T ( t ) T i 1 u i ( t ) + 2 e 2 α d u i T ( t ) i T i 1 u i ( t d ( t ) ) e 2 α d u i T ( t d ( t ) ) T i 1 u i ( t d ( t ) ) e 2 α d u i T ( t ) T i 1 u i ( t ) + 3 e 2 α d u i T ( t ) T i 1 u i ( t ) + e 2 α d 3 u i T ( t d ( t ) ) T i 1 T i T i 1 u i ( t d ( t ) ) e 2 α d u i T ( t d ( t ) ) T i 1 u i ( t d ( t ) ) = 2 e 2 α d z i T ( t ) K i T T i 1 K i z i ( t ) + e 2 α d 3 u i T ( t d ( t ) ) T i 1 T i T i 1 u i ( t d ( t ) ) e 2 α d u i T ( t d ( t ) ) T i 1 u i ( t d ( t ) ) = 2 e 2 α d y i T ( t ) L i T T i 1 L i y i ( t ) + e 2 α d 3 u i T ( t d ( t ) ) T i 1 u i ( t d ( t ) ) e 2 α d u i T ( t d ( t ) ) T i 1 u i ( t d ( t ) ) .
(28)
By using the following identity relation:
z ˙ i ( t ) + ( A ˜ i + D 1 i K i ) z i ( t ) + B ˜ i z i ( t h ( t ) ) + C ˜ i t k 1 ( t ) t z i ( s ) d s + D 2 i K i z i ( t d ( t ) ) + D 3 i K i t k 2 ( t ) t z i ( s ) d s = 0 ,
we have
2 y ˙ i T ( t ) P i y ˙ i ( t ) + 2 y ˙ i T ( t ) A ˜ i P i y i ( t ) 2 y ˙ i T ( t ) D 1 i L i y i ( t ) + 2 y ˙ i T ( t ) B ˜ i P i y i ( t h ( t ) ) + 2 y ˙ i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s + 2 y ˙ i T ( t ) D 2 i u i ( t d ( t ) ) + 2 y ˙ i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s = 0 .
(29)
Applying Lemma 2.4 and Lemma 2.3 gives
2 y ˙ i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s 2 k 1 e 2 α k 1 y ˙ i T ( t ) C ˜ i P i S i 1 P i C ˜ i T y ˙ i ( t ) 2 y ˙ i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s + 1 2 k 1 e 2 α k 1 ( t k 1 ( t ) t y i ( s ) d s ) T S i 2 y ˙ i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s × ( t k 1 ( t ) t y i ( s ) d s ) 2 y ˙ i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s 2 k 1 e 2 α k 1 y ˙ i T ( t ) C ˜ i P i S i 1 P i C ˜ i T y ˙ i ( t ) 2 y ˙ i T ( t ) C ˜ i P i t k 1 ( t ) t y i ( s ) d s + 1 2 e 2 α k 1 t k 1 ( t ) t y i T ( s ) S i y i ( s ) d s ,
(30)
2 y ˙ i T ( t ) D 2 i u i ( t d ( t ) ) 3 e 2 α d y ˙ i T ( t ) D 2 i T T i 1 D 2 i y ˙ i ( t ) 2 y ˙ i T ( t ) D 2 i u i ( t d ( t ) ) + e 2 α d 3 u i T ( t d ( t ) ) T i 1 u i ( t d ( t ) ) ,
(31)
3 y ˙ i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s 2 k 2 e 2 α k 2 y i T ( t ) D 3 i T W i 1 D 3 i y i ( t ) 3 y ˙ i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s + 1 2 k 2 e 2 α k 2 ( t k 2 ( t ) t u i ( s ) d s ) T W i 3 y ˙ i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s × ( t k 2 ( t ) t u i ( s ) d s ) 3 y ˙ i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s 2 k 2 e 2 α k 2 y i T ( t ) D 3 i T W i 1 D 3 i y i ( t ) 3 y ˙ i T ( t ) D 3 i t k 2 ( t ) t u i ( s ) d s + e 2 α k 2 2 t k 2 ( t ) t u i T ( s ) W i 1 u i ( s ) d s .
(32)
Hence, according to (19)-(28), (30)-(32), and adding the zero items of (29) we have
V ˙ i ( z i ( t ) ) + 2 α V i ( z i ( t ) ) ξ i T ( t ) [ ( 1 β ) Σ 1 i + β Σ 2 i ] ξ i ( t ) + y i T ( t ) M 3 i y i ( t ) + y ˙ i T ( t ) M 4 i y ˙ i ( t ) ,
(33)
where Σ 1 i and Σ 2 i are defined as in (12) and (13), respectively, and
ξ i T ( t ) = [ y i T ( t ) y ˙ i T ( t ) y i T ( t h 1 ) y i T ( t h ( t ) ) y i T ( t h 2 ) ] , M 3 i = 0.5 ( e 2 α h 1 + e 2 α h 2 ) R i + 2 k 1 e 2 α k 1 C ˜ i P i S i 1 P i C ˜ i T + k 2 L i T W i 1 L i M 3 i = + 2 e 2 α d L i T T i 1 L i , M 4 i = 0.5 P i + 2 k 1 e 2 α k 1 C ˜ i P i S i 1 P i C ˜ i T + d 2 L i T T i 1 L i + 3 e 2 α d D 2 i T T i 1 D 2 i M 4 i = + 2 k 2 e 2 α k 2 D 3 i T W i 1 D 3 i .
By ( 1 β ) Σ 1 i + β Σ 2 i < 0 holds if and only if Σ 1 i < 0 and Σ 2 i < 0 . Applying the Schur complement lemma, the inequalities M 3 i < 0 and M 4 i < 0 are equivalent to Σ 3 i < 0 and Σ 4 i < 0 , respectively. Therefore, it follows from (12)-(15), and (33), we obtain
V ˙ i ( z i ( t ) ) + 2 α V i ( z i ( t ) ) 0 , t 0 .
(34)
Integrating both sides of (34) from 0 to t, we have
V i ( z i ( t ) ) V i ( z i ( 0 ) ) e 2 α t , t 0 .
On the other hand, using the condition (18), we have
z i ( t ) V i ( z i ( 0 ) ) γ e α t , t 0 .
Estimating V i ( z i ( 0 ) ) gives
V i 1 ( z i ( 0 ) ) = z i T ( 0 ) P i 1 z i ( 0 ) λ max ( P i 1 ) ϕ i 2 , V i 2 ( z i ( 0 ) ) = h 1 0 e 2 α s z i T ( s ) Y i Q i Y i z i ( s ) d s λ max ( P i 1 Q i P i 1 ) h 1 0 e 2 α s d s φ i 2 V i 2 ( z i ( 0 ) ) = λ max ( P i 1 Q i P i 1 ) 1 e 2 α h 1 2 α φ i 2 λ max ( P i 1 Q i P i 1 ) 1 e 2 α h 2 2 α φ i 2 , V i 3 ( z i ( 0 ) ) λ max ( P i 1 Q i P i 1 ) 1 e 2 α h 2 2 α φ i 2 , V i 4 ( z i ( 0 ) ) = h 1 h 1 0 s 0 e 2 α τ z ˙ i T ( τ ) Y i R i Y i z ˙ i ( τ ) d τ d s V i 4 ( z i ( 0 ) ) = h 1 h 1 0 e 2 α s [ z i T ( 0 ) Y i R i Y i z i ( 0 ) z i T ( s ) Y i R i Y i z i ( s ) ] d s V i 4 ( z i ( 0 ) ) h 2 λ max ( Y i R i Y i ) h 1 0 e 2 α s d s ϕ i 2 h 2 λ max ( Y i R i Y i ) h 1 0 e 2 α s d s φ i 2 V i 4 ( z i ( 0 ) ) = h 2 λ max ( Y i R i Y i ) 1 e 2 α h 1 2 α ϕ i 2 h 2 λ max ( Y i R i Y i ) 1 e 2 α h 1 2 α φ i 2 V i 4 ( z i ( 0 ) ) h 2 λ max ( P i 1 R i P i 1 ) 1 e 2 α h 2 2 α ϕ i 2 + h 2 λ max ( P i 1 R i P i 1 ) V i 4 ( z i ( 0 ) ) = × 1 e 2 α h 2 2 α φ i 2 , V i 5 ( z i ( 0 ) ) h 2 λ max ( P i 1 R i P i 1 ) 1 e 2 α h 2 2 α ϕ i 2 + h 2 λ max ( P i 1 R i P i 1 ) V i 5 ( z i ( 0 ) ) × 1 e 2 α h 2 2 α φ i 2 , V i 6 ( z i ( 0 ) ) h 2 λ max ( P i 1 U i P i 1 ) 1 e 2 α h 2 2 α ϕ i 2 + h 2 λ max ( P i 1 U i P i 1</