# Adaptive Synchronization of Complex Dynamical Networks Governed by Local Lipschitz Nonlinearlity on Switching Topology

Authors: Bo Liu; Xiaoling Wang; Yanping Gao; Guangming Xie; Housheng Su
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper investigates the adaptive synchronization of complex dynamical networks satisfying the local Lipschitz condition with switching topology. Based on differential inclusion and nonsmooth analysis, it is proved that all nodes can converge to the synchronous state, even though only one node is informed by the synchronous state via introducing decentralized adaptive strategies to the coupling strengths and feedback gains. Finally, some numerical simulations are worked out to illustrate the analytical results.


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## Body

## 1. Introduction

In recent years, the research on the synchronization of complex dynamical networks has attracted great attention and emerged a large number of outstanding works (e.g., [1–8] and the references therein).For a network, the synchronization implies that all nodes will converge to the same state, which can be a homogeneous equilibrium point or a periodic orbit. To enhance the synchronization of such network, a lot of research methods are developed, and one of the most significant methods is to design effective adaptive strategies for the relevant parameters, such as the coupling strengths and the feedback gains [1–4, 8–10]. In references [4, 10, 11], there must have information channel by using a special indicator function; however, in reality, the information channel between any two nodes of a network may be lost or changed. Driven by it, we will investigate the synchronization of the complex dynamical networks with switching topology, which can lead to some information channels that occurred as well as another information channels that disappeared, by introducing adaptive strategies to the coupling strengths and feedback gains. Different from [8], the coupling strengths cijσ(t)(t) are dynamic and variable, which can change according to the switching signal, the neighbor rule, and the switched coupling configuration aijσ(t). Yet, “fast switching” [5] is hardly to be realized and bears new features and difficulties. Here, we attack the problem by invoking the theory of differential inclusion and nonsmooth analysis [12–18].So far, most attentions have been focused on nonlinear dynamics satisfying the globally Lipschitz condition. However, many known systems only satisfy the local Lipschitz condition, such as Lorenz system, Chen system, and FitzHugh-Rinzel system [9–11]. In this paper, we will also examine nonlinear dynamics of such complex network satisfying the local Lipschitz condition.The main contribution of the current work lies in characterizing synchronization of complex dynamical networks. First, the nonlinear dynamics of all nodes and the synchronous goal satisfies the local Lipschitz condition. Second, the adaptive strategies are introduced to the coupling strengths and the feedback gains. Third, the topology of complex dynamical networks can switch even though the information is discontinue, we can also solves the synchronization based on differential inclusion and nonsmooth analysis. In this paper, we will prove that all nodes of the complex dynamical network which is steered by the adaptive strategies can converge to the synchronous goal, even though only one node is informed by the synchronous goal if the neighboring graph remains connected.This paper is organized as follows. Section2 describes the model with the nonlinear dynamics satisfying the local Lipschitz condition, and some preliminaries about the nonsmooth analysis and the local Lipschitz condition are given. The main results are shown in Section 3, while Section 4 presents some simulations to illustrate our theoretical results. The conclusion is given in Section 5.


## 2. Preliminaries and Model Formulation

### 2.1. Model Formulation

Consider a dynamical network withN nodes, and each node updates its state by
(1)x˙i(t)=f(xi(t))+∑j∈Ni(t)‍aijσ(t)cijσ(t)(t)(xj(t)-xi(t))+ui(t),i=1,…,N,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn (i=1,2,…,N) is the state vector of node i, t∈[0,+∞), f:Rn→Rn is a continuous map, and Ni(t) is the neighboring set of node i at the instant t. Notice that the coupling between the nodes may vary in time. Therefore, the interconnection graph 𝒢σ(t) of the model is time varying, where σ(t):[0,+∞)→M={1,2,…,m} is the switching signal. For all k∈M, Ak=(aijk)∈RN×N is the weighted coupling configuration of network (1), which has aiik=-∑j∈Ni(t)‍aijk. If there exists communication channel between node i and node j at the switching segment k, then aijk=ajik>0; otherwise, aijk=0. cijσ(t)(t) denotes the coupling strengths between node i and node j. ui(t) is the feedback controller designed as
(2)ui(t)=-hici(t)(xi(t)-x0(t)),i=1,…,N,
where hi=1, if the node i is controlled; otherwise, hi=0; ci(t) is the feedback gain. x-(t) is a desired synchronous state for network (1) with
(3)x˙0(t)=f(x0(t)).The adaptive strategies on the coupling strengths and the feedback gains are designed as(4)c˙ijσ(t)(t)=kijaijσ(t)(xi(t)-xj(t))T(xi(t)-xj(t)),c˙i(t)=kihi(xi(t)-x0(t))T(xi(t)-x0(t)),
where cij(0)≥0, ci(0)≥0, kij>0, and ki>0 are the adaptive parameters of the coupling strength and the feedback gains, respectively.In thekth (for allk∈M) time period, the weighted coupling configuration matrix of network (1) is defined as
(5)Bk=[a11kc11ka12kc12k⋯a1Nkc1Nka21kc21ka22kc22k⋯a2Nkc2Nk⋮⋮⋱⋮aN1kcN1kaN2kcN2k⋯aNNkcNNk]∈RN×N,
with aiikciik=-∑j=1,j≠iN‍aijkcijk.


### 2.2. Mathematical Preliminaries

In this section, we introduce some useful concepts, assumptions, and lemmas.


#### 2.2.1. Concepts of Nonsmooth Analysis

Definition 1 (see [16], Filippov).
A vector functionx(·) is called a solution of equation x˙=f(x,t) on [t0,t1] if x(·) is absolutely continuous on [t0,t1], and for almost all t∈[t0,t1],
(6)x˙∈K[f](x,t)≡K[f](x)≜co¯{limf(xi)∣xi⟶x,xi∉Nf∪N},
where Nf⊂Rn,μ(Nf)=0;N⊂Rn,μ(N)=0.Definition 2 (see [16], generalized directional derivative).
The generalized directional derivative off at x in the direction v, denoted by f∘(x;v), is defined as
(7)f∘(x;v)≔limy→x,supt↓0f(y+tv)-f(y)t,
where y is a vector in the Banach space X and t is a positive scalar.Definition 3 (see [18]).
f
(
x
)
:
R
m
×
R
→
R is called regular if(i)
for allv, the usual one sided directed derivative f′(x;v) exists,(ii)
for allv, f′(x;v)=f∘(x;v).


#### 2.2.2. Assumptions

Assumption 1.
If eachfi of the nonlinear function f(xi(t))=(f1(xi(t)),f2(xi(t)),…,fn(xi(t)))T in network (1) satisfies the local Lipschitz condition, for any compact set 𝕊∈Rm, there exists a positive constant η(𝕊), such that
(8)(x-y)T[f(x)-f(y)]≤η(𝕊)∥x-y∥2,∀x,y∈𝕊.Assumption 2.
The synchronous statex0 is bounded; that is, there exists a compact set 𝕊=𝕊(x0(0))∈Rn such that the trajectory of (3) starting from x0(0) is always in the compact set 𝕊.Remark 4.
Note that the nonlinear dynamics of network (1) only satisfies the local Lipschitz condition. If the Jacobian matrix of f is continuous, then f is at least local Lipschitz. Many famous systems may not be governed by global Lipschitz nonlinearity but by local Lipschitz nonlinearity, such as Lorenz system and Chen system; therefore, it is worthy of discussing the local Lipschitz nonlinearity dynamics.Assumption 3 (see [13]).
The interconnection graph𝒢 remains connected.


#### 2.2.3. Lemmas

Lemma 5 (see [2]).
IfA=(aij)∈RN×N is a symmetric irreducible matrix with aii=-∑j=1,j≠iN‍aij and the matrix E=diag(e,0,…,0) with e>0, then all eigenvalues of the matrix A-E are negative.Lemma 6 (see [18], Chain Rule).
Letx(·) be a Filippov solution to x˙=f(x,t) on an interval containing t and let V:Rn×R→R be a Lipschitz and regular function. Then V(x(t),t) is absolutely continuous, (d/dt)V(x(t),t) exists almost everywhere, and
(9)ddtV(x(t),t)∈a.e.V~˙(x,t),
where V~˙(x,t)≔⋂ξ∈∂V(x(t),t)ξT[Kf(x(t),t)1].Remark 7 (see [12]).
In Lemma6, the (global) Lipschitz continuity required to V(x) can be relaxed to the local condition.Lemma 8 (see [17], Lyapunov's theorem generalized).
If(i)
V:Rn→R,V(0)=0 and V(x)>0, for allx≠0;(ii)
x:R→Rn and V(x(t)) is absolutely continuous on [0,+∞) with
(10)ddt[V(x(t))]<-ε<0a.e.on{t∣x(t)≠0},then x converges to 0 in finite time.It is worth noting that if the conditionV(0)=0 in (i) of Lemma 8 cannot hold; then we can have the following lemma.Lemma 9.
If(i)
V:Rn→R, V(x)>0, for allx≠0;(ii)
x:R→Rn and V(x(t)) is absolutely continuous on [0,+∞) with
(11)ddt[V(x(t))]<-ε<0a.e.on{t∣x(t)≠0},then x converges to 0 as t→∞.Proof.
If Lemma9 is not true, then we have x(t)≠0, for allt∈[t0,+∞) and (d/dt)[V(x(t))]<-ε<0 a.e. on {t∣x(t)≠0}. Therefore,
(12)limt→∞V(t)=V(x(0))+∫0∞‍V˙(t)dt=-∞,
since V(x(0)) is bounded, which contradicts with V(x)>0, for allx≠0.
This completes the proof.


## 2.1. Model Formulation

Consider a dynamical network withN nodes, and each node updates its state by
(1)x˙i(t)=f(xi(t))+∑j∈Ni(t)‍aijσ(t)cijσ(t)(t)(xj(t)-xi(t))+ui(t),i=1,…,N,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn (i=1,2,…,N) is the state vector of node i, t∈[0,+∞), f:Rn→Rn is a continuous map, and Ni(t) is the neighboring set of node i at the instant t. Notice that the coupling between the nodes may vary in time. Therefore, the interconnection graph 𝒢σ(t) of the model is time varying, where σ(t):[0,+∞)→M={1,2,…,m} is the switching signal. For all k∈M, Ak=(aijk)∈RN×N is the weighted coupling configuration of network (1), which has aiik=-∑j∈Ni(t)‍aijk. If there exists communication channel between node i and node j at the switching segment k, then aijk=ajik>0; otherwise, aijk=0. cijσ(t)(t) denotes the coupling strengths between node i and node j. ui(t) is the feedback controller designed as
(2)ui(t)=-hici(t)(xi(t)-x0(t)),i=1,…,N,
where hi=1, if the node i is controlled; otherwise, hi=0; ci(t) is the feedback gain. x-(t) is a desired synchronous state for network (1) with
(3)x˙0(t)=f(x0(t)).The adaptive strategies on the coupling strengths and the feedback gains are designed as(4)c˙ijσ(t)(t)=kijaijσ(t)(xi(t)-xj(t))T(xi(t)-xj(t)),c˙i(t)=kihi(xi(t)-x0(t))T(xi(t)-x0(t)),
where cij(0)≥0, ci(0)≥0, kij>0, and ki>0 are the adaptive parameters of the coupling strength and the feedback gains, respectively.In thekth (for allk∈M) time period, the weighted coupling configuration matrix of network (1) is defined as
(5)Bk=[a11kc11ka12kc12k⋯a1Nkc1Nka21kc21ka22kc22k⋯a2Nkc2Nk⋮⋮⋱⋮aN1kcN1kaN2kcN2k⋯aNNkcNNk]∈RN×N,
with aiikciik=-∑j=1,j≠iN‍aijkcijk.


## 2.2. Mathematical Preliminaries

In this section, we introduce some useful concepts, assumptions, and lemmas.


### 2.2.1. Concepts of Nonsmooth Analysis

Definition 1 (see [16], Filippov).
A vector functionx(·) is called a solution of equation x˙=f(x,t) on [t0,t1] if x(·) is absolutely continuous on [t0,t1], and for almost all t∈[t0,t1],
(6)x˙∈K[f](x,t)≡K[f](x)≜co¯{limf(xi)∣xi⟶x,xi∉Nf∪N},
where Nf⊂Rn,μ(Nf)=0;N⊂Rn,μ(N)=0.Definition 2 (see [16], generalized directional derivative).
The generalized directional derivative off at x in the direction v, denoted by f∘(x;v), is defined as
(7)f∘(x;v)≔limy→x,supt↓0f(y+tv)-f(y)t,
where y is a vector in the Banach space X and t is a positive scalar.Definition 3 (see [18]).
f
(
x
)
:
R
m
×
R
→
R is called regular if(i)
for allv, the usual one sided directed derivative f′(x;v) exists,(ii)
for allv, f′(x;v)=f∘(x;v).


### 2.2.2. Assumptions

Assumption 1.
If eachfi of the nonlinear function f(xi(t))=(f1(xi(t)),f2(xi(t)),…,fn(xi(t)))T in network (1) satisfies the local Lipschitz condition, for any compact set 𝕊∈Rm, there exists a positive constant η(𝕊), such that
(8)(x-y)T[f(x)-f(y)]≤η(𝕊)∥x-y∥2,∀x,y∈𝕊.Assumption 2.
The synchronous statex0 is bounded; that is, there exists a compact set 𝕊=𝕊(x0(0))∈Rn such that the trajectory of (3) starting from x0(0) is always in the compact set 𝕊.Remark 4.
Note that the nonlinear dynamics of network (1) only satisfies the local Lipschitz condition. If the Jacobian matrix of f is continuous, then f is at least local Lipschitz. Many famous systems may not be governed by global Lipschitz nonlinearity but by local Lipschitz nonlinearity, such as Lorenz system and Chen system; therefore, it is worthy of discussing the local Lipschitz nonlinearity dynamics.Assumption 3 (see [13]).
The interconnection graph𝒢 remains connected.


### 2.2.3. Lemmas

Lemma 5 (see [2]).
IfA=(aij)∈RN×N is a symmetric irreducible matrix with aii=-∑j=1,j≠iN‍aij and the matrix E=diag(e,0,…,0) with e>0, then all eigenvalues of the matrix A-E are negative.Lemma 6 (see [18], Chain Rule).
Letx(·) be a Filippov solution to x˙=f(x,t) on an interval containing t and let V:Rn×R→R be a Lipschitz and regular function. Then V(x(t),t) is absolutely continuous, (d/dt)V(x(t),t) exists almost everywhere, and
(9)ddtV(x(t),t)∈a.e.V~˙(x,t),
where V~˙(x,t)≔⋂ξ∈∂V(x(t),t)ξT[Kf(x(t),t)1].Remark 7 (see [12]).
In Lemma6, the (global) Lipschitz continuity required to V(x) can be relaxed to the local condition.Lemma 8 (see [17], Lyapunov's theorem generalized).
If(i)
V:Rn→R,V(0)=0 and V(x)>0, for allx≠0;(ii)
x:R→Rn and V(x(t)) is absolutely continuous on [0,+∞) with
(10)ddt[V(x(t))]<-ε<0a.e.on{t∣x(t)≠0},then x converges to 0 in finite time.It is worth noting that if the conditionV(0)=0 in (i) of Lemma 8 cannot hold; then we can have the following lemma.Lemma 9.
If(i)
V:Rn→R, V(x)>0, for allx≠0;(ii)
x:R→Rn and V(x(t)) is absolutely continuous on [0,+∞) with
(11)ddt[V(x(t))]<-ε<0a.e.on{t∣x(t)≠0},then x converges to 0 as t→∞.Proof.
If Lemma9 is not true, then we have x(t)≠0, for allt∈[t0,+∞) and (d/dt)[V(x(t))]<-ε<0 a.e. on {t∣x(t)≠0}. Therefore,
(12)limt→∞V(t)=V(x(0))+∫0∞‍V˙(t)dt=-∞,
since V(x(0)) is bounded, which contradicts with V(x)>0, for allx≠0.
This completes the proof.


## 2.2.1. Concepts of Nonsmooth Analysis

Definition 1 (see [16], Filippov).
A vector functionx(·) is called a solution of equation x˙=f(x,t) on [t0,t1] if x(·) is absolutely continuous on [t0,t1], and for almost all t∈[t0,t1],
(6)x˙∈K[f](x,t)≡K[f](x)≜co¯{limf(xi)∣xi⟶x,xi∉Nf∪N},
where Nf⊂Rn,μ(Nf)=0;N⊂Rn,μ(N)=0.Definition 2 (see [16], generalized directional derivative).
The generalized directional derivative off at x in the direction v, denoted by f∘(x;v), is defined as
(7)f∘(x;v)≔limy→x,supt↓0f(y+tv)-f(y)t,
where y is a vector in the Banach space X and t is a positive scalar.Definition 3 (see [18]).
f
(
x
)
:
R
m
×
R
→
R is called regular if(i)
for allv, the usual one sided directed derivative f′(x;v) exists,(ii)
for allv, f′(x;v)=f∘(x;v).


## 2.2.2. Assumptions

Assumption 1.
If eachfi of the nonlinear function f(xi(t))=(f1(xi(t)),f2(xi(t)),…,fn(xi(t)))T in network (1) satisfies the local Lipschitz condition, for any compact set 𝕊∈Rm, there exists a positive constant η(𝕊), such that
(8)(x-y)T[f(x)-f(y)]≤η(𝕊)∥x-y∥2,∀x,y∈𝕊.Assumption 2.
The synchronous statex0 is bounded; that is, there exists a compact set 𝕊=𝕊(x0(0))∈Rn such that the trajectory of (3) starting from x0(0) is always in the compact set 𝕊.Remark 4.
Note that the nonlinear dynamics of network (1) only satisfies the local Lipschitz condition. If the Jacobian matrix of f is continuous, then f is at least local Lipschitz. Many famous systems may not be governed by global Lipschitz nonlinearity but by local Lipschitz nonlinearity, such as Lorenz system and Chen system; therefore, it is worthy of discussing the local Lipschitz nonlinearity dynamics.Assumption 3 (see [13]).
The interconnection graph𝒢 remains connected.


## 2.2.3. Lemmas

Lemma 5 (see [2]).
IfA=(aij)∈RN×N is a symmetric irreducible matrix with aii=-∑j=1,j≠iN‍aij and the matrix E=diag(e,0,…,0) with e>0, then all eigenvalues of the matrix A-E are negative.Lemma 6 (see [18], Chain Rule).
Letx(·) be a Filippov solution to x˙=f(x,t) on an interval containing t and let V:Rn×R→R be a Lipschitz and regular function. Then V(x(t),t) is absolutely continuous, (d/dt)V(x(t),t) exists almost everywhere, and
(9)ddtV(x(t),t)∈a.e.V~˙(x,t),
where V~˙(x,t)≔⋂ξ∈∂V(x(t),t)ξT[Kf(x(t),t)1].Remark 7 (see [12]).
In Lemma6, the (global) Lipschitz continuity required to V(x) can be relaxed to the local condition.Lemma 8 (see [17], Lyapunov's theorem generalized).
If(i)
V:Rn→R,V(0)=0 and V(x)>0, for allx≠0;(ii)
x:R→Rn and V(x(t)) is absolutely continuous on [0,+∞) with
(10)ddt[V(x(t))]<-ε<0a.e.on{t∣x(t)≠0},then x converges to 0 in finite time.It is worth noting that if the conditionV(0)=0 in (i) of Lemma 8 cannot hold; then we can have the following lemma.Lemma 9.
If(i)
V:Rn→R, V(x)>0, for allx≠0;(ii)
x:R→Rn and V(x(t)) is absolutely continuous on [0,+∞) with
(11)ddt[V(x(t))]<-ε<0a.e.on{t∣x(t)≠0},then x converges to 0 as t→∞.Proof.
If Lemma9 is not true, then we have x(t)≠0, for allt∈[t0,+∞) and (d/dt)[V(x(t))]<-ε<0 a.e. on {t∣x(t)≠0}. Therefore,
(12)limt→∞V(t)=V(x(0))+∫0∞‍V˙(t)dt=-∞,
since V(x(0)) is bounded, which contradicts with V(x)>0, for allx≠0.
This completes the proof.


## 3. Main Results

Denote(13)d0≜12∑i=1N‍(xi(0)-x0)2,
and construct a closed space
(14)𝔹(σd0,x0)={x∈RNn∣12∑i=1N‍∥xi-x0∥2≤σd0},
where x=(x1T,x2T,…,xNT)T and σ>1 is a constant. Under Assumption 1, for all x∈𝔹(σd0,x0), there exists a constant η(σ,x0) such that
(15)∑i=1N‍(xi-x0)T(f(xi)-f(x0))≤η(σ,x0)∥xi-x0∥2.In order to get the main result, we construct the following potential function:(16)V(t)=12∑i=1N‍x~iT(t)x~i(t)+∑i=1N‍∑j∈Ni(t)‍(cij(t)-m)24kij+∑i=1N‍(ci(t)-m)22ki,
where m>0 is sufficiently large. For convenience, we denote c=(cij)N×N and c~=(c1T,c2T,…,cNT)T. Then, we can have the following results.Theorem 10.
Supposing that Assumptions1–3 hold, then the combined trajectories of all nodes and the parameters (x,c,c~) in (1) and (2) are geared to a compact hyper-ellipsoid
(17)Ω(σd0,m,x0)={(x,c,c~)∈RNn×RNn×RN∣V<σd0},
if the initial value is selected from
(18)Ω0(σd0,m,x0)={(x,c,c~)∈RNn×RNn×RN∣V(0)<σd0}.Proof.
Taking the derivative ofV(t) in [t0,t1), we can get
(19)V˙(t)=∑i=1N‍x~iT(t)[∑j∈Ni‍f(xi(t))-f(x0(t))+∑j∈Ni‍aijcij(t)(x~j(t)-x~i(t))]+12∑i=1N‍∑j∈Ni‍(cij(t)-m)aij(x~i(t)-x~j(t))T×(x~i(t)-x~j(t))+∑i=1N‍(ci(t)-m)hix~iT(t)x~i(t)=∑i=1N‍x~iT(t)(η(σ,x0)IN)x~i(t)-12m∑i=1N‍∑j∈Ni‍aij(x~i(t)-x~j(t))T(x~i(t)-x~j(t))-m∑i=1N‍hix~iT(t)x~i(t)=x~T(t)[((η(σ,x0)IN)+m(A-H))⊗In]x~(t).
Under Assumption 3, Lemma 5 implies that A-H<0 in [t0,t1). Since m is a sufficiently large positive constant, then
(20)V˙≤0,
which implies
(21)V(t)≤V(0),t∈[t0,t1).
Similarly,
(22)V(t)≤V(tk-1),t∈[tk-1,tk),k=1,2,…,m.
Thus, we have
(23)V(t)≤V(0)<σd0.
Therefore, we can conclude that the combined trajectories of all nodes with the parameters (x,c,c~) are geared to a compact hyper-ellipsoid Ω(σd0,m,x0)={(x,c,c~)∈RNn×RNn×RN∣V<σd0} if the initial value is selected from Ω0(σd0,m,x0)={(x,c,c~)∈RNn×RNn×RN∣V(0)<σd0}.
This completes the proof.Theorem 11.
If network (1) steered by adaptive laws (3) and the initial value of (x,c,c~) is defined as (18), under Assumptions 1–3, then all nodes will converge to the synchronous state even when only one node is controlled by (2).Proof.
Letx~i(t)=xi(t)-x-(t). Construct the Lyapunov function, one has
(24)V(t)=12∑i=1N‍x~iT(t)x~i(t)+∑i=1N‍∑j∈Ni(t)‍(cij(t)-m)24kij+∑i=1N(ci(t)-m)22ki,
where m>0 is a sufficiently large constant.
By Definition1, we can obtain
(25)x˙i(t)∈a.e.K[f(xi(t))+∑j∈Ni(t)‍aijcij(t)(xj(t)-xi(t))∑j∈Ni(t)‍-hici(t)(xi(t)-x-(t))],c˙ij(t)∈a.e.K[kijaij(xi(t)-xj(t))T(xi(t)-xj(t))].
Because the topology of network (1) is switching, we have
(26)x˙i(t)⊆K[f(xi(t))+∑j∈Ni(t)‍aijcij(t)(xj(t)-xi(t))]-hici(t)(xi(t)-x-(t)).
Also by Definition3, we know that V(t) is regular, then
(27)∂V⊆[x~1T(t),…,x~NT(t),∑j=2N‍(c1j(t)-m)2k1j,…,∑j=1N-1‍(cN-1j(t)-m)2kN-1j,c1(t)-mk1,⋯,cN(t)-mkN]T.
Hence,
(28)V~˙⊂K[∑i=1N‍x~iT(t)[∑j∈Ni‍f(xi(t))-f(x0(t))+∑j∈Ni‍aijcij(t)(x~j(t)-x~i(t))]]+K[12∑i=1N‍∑j∈Ni‍(cij(t)-m)aij(x~i(t)-x~j(t))T×(x~i(t)-x~j(t))+∑i=1N‍(ci(t)-m)hix~iT(t)x~i(t)]=K{∑i=1N‍x~iT(t)(η(σ,x0)IN)x~i(t)-12m∑i=1N‍∑j∈Ni‍aij(x~i(t)-x~j(t))T(x~i(t)-x~j(t))-m∑i=1N‍hix~iT(t)x~i(t)}.
Consequently,(29)V~˙⊂K[x~T(t)[((η(σ,x0)IN)+m(A-H))⊗In]x~(t)].
Lemma 5 indicates that all the eigenvalues of A-H are negative, and there exists ε>0 such that
(30)ddt[V(x(t))]<-ε<0a.e.on{t∣x(t)≠0},
since m is a sufficiently large positive constant.
From Lemma9, we can get that all the nodes of network (1) can converge to the synchronous state x-(t). This completes the proof.Remark 12.
Note also that Theorem11 is still established when Ak is asymmetric.Similar to the proof of Theorem11, we also have
(31)V~˙⊂K[A+AT2x~T(t)[A+AT2(A+AT2(η(σ,x0)IN)+m(A+AT2-H))⊗In]x~(t)],
since (A+AT)/2 is symmetric.


## 4. Simulations

In this section, number simulations are given to illustrate our theoretical results. All nodes of network (1) and the synchronous goal share the same nonlinear dynamics described as the Lorenz system as follows:
(32)f(x(t))=f(x1,x2,x3)={x˙1=10(x2-x1),x˙2=28x1-x1x3-x2,x˙3=x1x2-83x3,
as shown in Figure 1. Then the network can be shown as follows:
(33)[x˙i1x˙i2x˙i3]=[10(xi2-xi1)28xi1-xi1xi3-xi2xi1xi2-83xi3]+∑j∈Ni(t)‍aijσ(t)cijσ(t)(t)(xj(t)-xi(t))-hici(t)(xi(t)-x0(t)).
Figure 2 shows the topology of network (1) with subgraphs (a), (b), and (c) during the time intervals [0,4], [4,12], [12,50], respectively.Figure 1
Trajectories of the synchronous state.The topologies of network (1).
(a)
(b)
(c)The initial values of the 10 nodes are chosen randomly, and the initial value of the synchronous goal isx0(0)=[10,10,10]T. There is only one node informed by the synchronous goal, and the 10th node is given, that is, hi(i=1,2,…,9)=0 and h10=1. The initial values of the adaptive parameters are cij(0)=0 with the weight of kij(0)=0 for all i and j, while ci(0)=0 with the weight of ki(0)=0 for all i.Figure3 describes the convergence of the state errors on the x-axis, y-axis, and z-axis, respectively. From this figure, we can see that all nodes of network (1) can synchronize to the synchronous state when the neighboring graph 𝔾 remains connected with switching topology even though only one node is informed by the synchronous state. Figure 4 shows the change trends of the adaptive coupling strengths and the adaptive feedback gains, respectively, and all these parameters converge to the constants.Figure 3
Trajectories convergence of 10 nodes with only one node informed by controllers (2).Convergence of the adaptive coupling strengths and the feedback gain.
(a)
Coupling strengths
(b)
Feedback gain


## 5. Conclusion

In this paper, we have investigated the synchronization of complex dynamical networks with switching topology via differential inclusion method. Different from the most previous work, all nodes and the synchronous state in this paper share the same intrinsic nonlinear dynamics governed by the local Lispchitz condition. By adding decentralized adaptive strategies to the coupling strengths and the feedback gains, all nodes can converge to the synchronous state even when only one node is pinning controlled by the synchronous state if the neighboring graph𝒢 of the switching topology remains connected.



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*Source: 818242-2013-09-18.xml*