# Stability and Passivity of Spatially and Temporally Complex Dynamical Networks with Time-Varying Delays

Authors: Shun-Yan Ren; Yue-Hui Zhao
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper proposes a new complex dynamical network model, in which the state, input, and output variables are varied with the time and space variables. By utilizing the Lyapunov functional method combined with the inequality techniques, several criteria for passivity and global exponential stability are established. Finally, numerical simulations are given to illustrate the effectiveness of the obtained results.


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## Body

## 1. Introduction

Complex networks can be seen everywhere, which have become an important part of our daily life. Recently, the topology and dynamical behavior of complex dynamical networks have been extensively studied by the researchers. In particular, special attention has been focused on synchronization in complex dynamical networks, and many interesting results on synchronization were derived for various complex networks [1–10].To our knowledge, in most existing works on the complex networks, they always assume that the node state is only dependent on the time. However, such simplification does not match the peculiarities of some real networks. Food webs are among the most well-known examples of complex networks and hold a central place in ecology to study the dynamics of animal populations. A food web can be characterized by a model of complex network, in which a node represents a species. To our knowledge, species are usually inhomogeneously distributed in a bounded habitat and the different population densities of predators and preys may cause different population movements; thus it is important and interesting to investigate their spatial density in order to better protect and control their population. In such a case, the state variable of node will represent the spatial density of the species. Moreover, the input and output variables are also dependent on the time and space in many practical situations. Therefore, it is essential to study the complex networks, in which the state, input, and output variables are varied with the time and space variables.Recently, food web [11–24] has become a focal research topic and attracted increasing attention from many fields of scientific research. In [17], Pao discussed the asymptotic behavior of time-dependent solutions of a three-species reaction-diffusion system in a bounded domain under a Neumann boundary condition. Kim and Lin [21] considered the blowup properties of solutions for a parabolic system with homogeneous Dirichlet boundary conditions, which describes dynamic properties of three interacting species in a spatial habitat. As a natural extension of the existing network models, we propose a complex dynamical network with time-varying delays, which is described by a system of parabolic partial differential equations. In addition, we investigate the global exponential stability of the proposed network model.Stability problems are often linked to the theory of dissipative systems, which postulate that the energy dissipated inside a dynamic system is less than that supplied from external source. Passivity is part of a broader and a general theory of dissipativeness. The main point of passivity theory is that the passive properties of systems can keep the systems internally stable. The passivity theory has found successful applications in diverse areas such as complexity [25], signal processing [26], stability [27, 28], chaos control and synchronization [29, 30], and fuzzy control [31]. Although research on passivity has attracted so much attention, little of that had been devoted to the passivity properties of spatially and temporally complex dynamical networks until Wang et al. [32] obtained some sufficient conditions on passivity properties for a class of reaction-diffusion neural networks with Dirichlet boundary conditions. To the best of our knowledge, very few researchers have investigated the passivity of complex dynamical networks with spatially and temporally varying state variables. Therefore, we also study the passivity of the proposed network model, and some sufficient conditions ensuring input strict passivity and output strict passivity are obtained by constructing appropriate Lyapunov functionals and using inequality techniques.


## 2. Network Model and Preliminaries

In this paper, we consider a parabolic complex network consisting ofN nonidentical nodes with diffusive and delay coupling. The entire network is described by
(1)∂wi(x,t)∂t=diΔwi(x,t)+piwi(x,t)+ui(x,t)+∑j=1N‍Gijwj(x,t-τj(t)),yi(x,t)=giwi(x,t)+hiui(x,t),
where i=1,2,…,N, Ω={x=(x1,x2,…,xm)T∣|xk|<lk,k=1,2,…,m} is the bounded domain with smooth boundary ∂Ω, di∈R>0, pi,gi,hi∈R, (x,t)∈Ω×R+, τj(t) is the time-varying delay with 0⩽τj(t)⩽τ, wi(x,t)∈R is the state of node i, ui(x,t),yi(x,t) are the input and output of node i at time t and in space x, respectively, Δ=∑k=1m‍(∂2/∂xk2) is the Laplace diffusion operator on Ω, and G=(Gij)N×N represents the topological structure of network and coupling strength between nodes, where Gij is defined as follows: if there is a connection from node i to node j (i≠j), then Gij>0; otherwise, Gij=0(i≠j), and Gii=-∑j=1,j≠iNGijLetw(x,t)=(w1(x,t),w2(x,t),…,wN(x,t))T. The initial value and boundary value conditions associated with the network (1) are given in the form
(2)w(x,t)=Φ(x,t),(x,t)∈Ω×[-τ,0],Φ(x,t)=(ϕ1(x,t),ϕ2(x,t),…,ϕN(x,t))T,w(x,t)=0,(x,t)∈∂Ω×[-τ,+∞),
where ϕi(x,t)(i=1,2,…,N) is bounded and continuous on Ω×[-τ,0]. Let w(x,t,Φ) be the state trajectory of network (1) from the above initial value and boundary value conditions.Next, we introduce some notations and definitions.For anye(x,t)=(e1(x,t),e2(x,t),…,eN(x,t))T∈RN, ∥e(x,t)∥2 denotes
(3)∥e(x,t)∥2=(∫Ω‍∑i=1N‍ei2(x,t)dx)1/2;
for any Θ(x,t)=(θ1(x,t),θ2(x,t),…,θN(x,t))T∈RN, (x,t)∈Ω×[-τ,0], we define
(4)∥Θ∥τ=sup⁡-τ⩽t⩽0∥Θ(x,t)∥2.Definition 1.
The complex network (1) is said to be globally exponentially stable if there exist constants ϵ>0 and M⩾1 such that for any two solutions w(x,t,Φ),w(x,t,Ψ) of network (1) with initial functions Φ,Ψ, respectively, it holds that
(5)∥w(x,t,Ψ)-w(x,t,Φ)∥2⩽M∥Ψ-Φ∥τe-ϵt
for all t⩾0. If such w(x,t,Φ) is an equilibrium solution (or periodic solution), then this equilibrium solution (or periodic solution) is said to be globally exponentially stable.Definition 2 (see [32]).
A system with inputu(x,t) and output y(x,t) where u(x,t),y(x,t)∈Rp is said to be passive if there is a constant β∈R such that
(6)∫0tp‍∫Ω‍yT(x,s)u(x,s)dxds⩾-β2
for all tp⩾0, where Ω is a bounded compact set. If in addition, there are constants γ1⩾0 and γ2⩾0 such that
(7)∫0tp‍∫Ω‍yT(x,s)u(x,s)dxds⩾-β2+γ1∫0tp‍∫Ω‍uT(x,s)u(x,s)dxds+γ2∫0tp‍∫Ω‍yT(x,s)y(x,s)dxds
for all tp⩾0, then the system is input-strictly passive if γ1>0 and output-strictly passive if γ2>0.Remark 3.
In [23], Wang and Wu discussed the global exponential stability and passivity of a parabolic complex network. In network model (1), the coupling matrix G is diffusive. Namely, Gii=-∑j=1,j≠iNGij, i=1,2,…,N. Compared with the network model considered in [23], the network model considered in this paper may be more reasonable. On the other hand, we investigate the input strict passivity and output strict passivity of the complex network (1), which are totally different from the work in [23].Lemma 4 (see [33]).
LetΩ be a cube |xk|<lk(k=1,2,…,m) and let h(x) be a real-valued function belonging to C1(Ω) which vanishes on the boundary ∂Ω of Ω, that is, h(x)|∂Ω=0. Then,
(8)∫Ω‍h2(x)dx⩽lk2∫Ω‍(∂h∂xk)2dx,
where x=(x1,x2,…,xm)T.


## 3. Stability and Passivity of Parabolic Complex Network

### 3.1. Stability Analysis

Theorem 5.
Letτ˙j(t)⩽σ<1. The complex network (1) is globally exponentially stable if there exist constants λi>0 and ϵ>0 such that
(9)2λiϵ+2λipi-2λi∑k=1m‍dilk2+∑j=1N‍λj|Gji|e2ϵτ1-σ+λi∑j=1N‍|Gij|⩽0,
where i,j=1,2,…,N.Proof.
Firstly, we can get from (1) that
(10)∂wi(x,t)∂t=diΔwi(x,t)+piwi(x,t)+ui(x,t)+∑j=1N‍Gijwj(x,t-τj(t)),
where i=1,2,…,N.
Definez(x,t)=w(x,t,Ψ)-w(x,t,Φ), Ψz(x,t)=Ψ(x,t)-Φ(x,t), and then the dynamics of the difference vector z(x,t)=(z1(x,t),z2(x,t),…,zN(x,t))T is governed by the following equation:
(11)∂zi(x,t)∂t=diΔzi(x,t)+∑j=1N‍Gijzj(x,t-τj(t))+pizi(x,t),
where i=1,2,…,N. Define the following Lyapunov functional for the system (11):
(12)V(t)=∑i=1N‍λi{∫τjt‍e2ϵt∫Ω‍zi2(x,t)dx+e2ϵτ1-σ∑j=1N‍|Gij|nnnnnnn×∫t-τj(t)t‍∫Ω‍zj2(x,s)e2ϵsdxds}.
Calculating the time derivative of V(t) along the trajectory of system (11), we can get
(13)V˙(t)=∑i=1N‍λie2ϵt{2ϵ∫Ω‍zi2(x,t)dx+∫Ω‍2zi(x,t)∂zi(x,t)∂tdxnnnnnnnnnnnn+e2ϵτ1-σ∑j=1N‍|Gij|∫Ω‍zj2(x,t)dxnnnnnnnnnnnn-e2ϵτ1-σ∑j=1N‍(1-τ˙j(t))|Gij|e-2ϵτj(t)nnnnnnnnnnnn×∫Ω‍zj2(x,t-τj(t))dx∂zi(x,t)∂t}⩽∑i=1N‍λie2ϵt{∑j=1N‍(2ϵ+2pi)∫Ω‍zi2(x,t)dxnnnnnnnnnnnn+2di∫Ω‍zi(x,t)Δzi(x,t)dxnnnnnnnnnnnn+2∫Ω‍zi(x,t)∑j=1N‍Gijzj(x,t-τj(t))dxnnnnnnnnnnnn+e2ϵτ1-σ∑j=1N‍|Gij|∫Ω‍zj2(x,t)dxnnnnnnnnnnnn-∑j=1N‍|Gij|∫Ω‍zj2(x,t-τj(t))dx}.
From Green’s theorem and the boundary condition, we have
(14)∫Ω‍zi(x,t)Δzi(x,t)dx=-∑k=1m‍∫Ω‍(∂zi(x,t)∂xk)2dx.
According to Lemma 4, we can obtain
(15)∑k=1m‍∫Ω‍(∂zi(x,t)∂xk)2dx⩾∑k=1m‍∫Ω‍zi2(x,t)lk2dx.
Therefore,
(16)V˙(t)⩽∑i=1N‍{∑j=1N‍λi(2ϵ+2pi-2∑k=1m‍dilk2)∫Ω‍zi2(x,t)dxnnnnnnnn+λie2ϵτ1-σ∑j=1N‍|Gij|∫Ω‍zj2(x,t)dxnnnnnnnn+2λi∫Ω‍zi(x,t)∑j=1N‍Gijzj(x,t-τj(t))dxnnnnnnnn-λi∑j=1N‍|Gij|∫Ω‍zj2(x,t-τj(t))dx}e2ϵt⩽∑i=1N‍{∑j=1N‍λi(2ϵ+2pi-2∑k=1m‍dilk2)∫Ω‍zi2(x,t)dxnnnnnnnn+2λi∑j=1N‍|Gij|∫Ω‍zi2(x,t)dxnnnnnnnn×∫Ω‍zj2(x,t-τj(t))dxnnnnnnnn+λie2ϵτ1-σ∑j=1N‍|Gij|∫Ω‍zj2(x,t)dxnnnnnnnn-λi∑j=1N‍|Gij|∫Ω‍zj2(x,t-τj(t))dx}e2ϵt⩽e2ϵt∑i=1N‍{(2λiϵ+2λipi-2λi∑k=1m‍dilk2+λi∑j=1N‍|Gij|)nnnnnnnnnn×∫Ω‍zi2(x,t)dx+λie2ϵτ1-σ∑j=1N‍|Gij|nnnnnnnnnn×∫Ω‍zj2(x,t)dx∑j=1N‍|Gij|}=e2ϵt∑i=1N‍{(∑j=1N‍2λiϵ+2λipi-2λi∑k=1m‍dilk2+λi∑j=1N‍|Gij|nnnnnnnnnnnn+∑j=1N‍λj|Gji|e2ϵτ1-σ)×∫Ω‍zi2(x,t)dx∑j=1N‍}.
From (9), we can get
(17)V˙(t)⩽0.
So V(t)⩽V(0), t⩾0. Since
(18)V(0)=∑i=1N‍λi{∑j=1N‍∫Ω‍zi2(x,0)dxnnnnnnnnn.nn+e2ϵτ1-σ∑j=1N‍|Gij|∫-τj(0)0‍∫Ω‍zj2(x,s)e2ϵsdxds}⩽[max⁡i=1,2,…,N{λi}+max⁡i=1,2,…,N{τe2ϵτ1-σ∑j=1N‍λj|Gji|}]∥Ψz∥τ2,V(t)⩾min⁡i=1,2,…,N{λi}e2ϵt∥z(x,t)∥22.
Let λ-=min⁡i=1,2,…,N{λi}, λ+=max⁡i=1,2,…,N{λi}, and
(19)M=λ++max⁡i=1,2,…,N⁡{(τe2ϵτ/(1-σ))∑j=1N‍λj|Gji|}λ-.
Then M⩾1, and we can obtain
(20)∥z(x,t)∥2⩽Me-ϵt∥Ψz∥τ.
Namely,
(21)∥w(x,t,Ψ)-w(x,t,Φ)∥2⩽M∥Ψ-Φ∥τe-ϵt.
This completes the proof of Theorem 5.Practically, Theorem5 not only can judge the global exponential stability of complex network (1), but also can guarantee the existence and uniqueness of the periodic solution in some circumstances.Assume thatui(x,t) and τi(t)  (i=1,2,…,N) are periodic continuous functions with period ϖ. Then we can get from (1) that
(22)∂wi(x,t)∂t=diΔwi(x,t)+piwi(x,t)+ui(x,t)+∑j=1N‍Gijwj(x,t-τj(t)),
where ui(x,t+ϖ)=ui(x,t), τi(t+ϖ)=τi(t), i=1,2,…,N.Theorem 6.
Letτ˙j(t)⩽σ<1. The system (22) has a unique ϖ-periodic solution if there exist constants λi>0 and ϵ>0 such that
(23)2λiϵ+2λipi-2λi∑k=1m‍dilk2+∑j=1N‍λj|Gji|e2ϵτ1-σ+λi∑j=1N‍|Gij|⩽0,
where i,j=1,2,…,N.Proof.
Define(24)wt(Ψ)=w(x,t+s,Ψ),s∈[-τ,0],t⩾0,wt(Φ)=w(x,t+s,Φ),s∈[-τ,0],t⩾0.
Obviously, from Theorem 5, we can derive
(25)∥w(x,t,Ψ)-w(x,t,Φ)∥2⩽Me-ϵt∥Ψ-Φ∥τ
for t⩾0, where M⩾1. We can choose a positive integer N such that
(26)Me-ϵ(Nω-τ)⩽13.
Now define a Poincaré mapping P:C(Ω×[-τ,0],RN)→C(Ω×[-τ,0],RN) by
(27)Ψ⟶P(Ψ)=wϖ(Ψ).
Since the periodicity of system (22), one has PN(Ψ)=wNϖ(Ψ). From (25) and (26), one obtains
(28)∥PN(Ψ)-PN(Φ)∥τ⩽13∥Ψ-Φ∥τ.
This implies that PN is a contraction mapping, so there exists one unique fixed point Ψ*∈C(Ω×[-τ,0],RN) such that
(29)PN(Ψ*)=Ψ*.
Since PN(P(Ψ*))=P(PN(Ψ*))=P(Ψ*), so P(Ψ*) is also a fixed point of PN, then
(30)P(Ψ*)=Ψ*.
Let w(x,t,Ψ*) be the solution of system (22) with initial conditions Ψ*, then w(x,t+ω,Ψ*) is also a solution of system (22). Obviously,
(31)w(x,t+ω,Ψ*)=w(x,t,wω(x,Ψ*))=w(x,t,Ψ*)∀t⩾0.
This means that w(x,t,Ψ*) is exactly one ω-periodic solution of system (22). This completes the proof of Theorem 6.


### 3.2.  Passivity Analysis

Theorem 7.
Letτ˙j(t)⩽σ<1. The complex network (1) is input-strictly passive if there exist constants λi>0 and γ>0 such that
(32)(Miλi-giλi-giγ-2hi)⩽0,
where Mi=-∑k=1m‍(2λidi/lk2)+2piλi+λi∑j=1N‍|Gij|+∑j=1N‍(λj|Gji|/(1-σ)), i,j=1,2,…,N.Proof.
Firstly, construct a Lyapunov functional for the network (1) as follows:
(33)V(t)=∑i=1N‍λi{∑j=1N‍∫Ω‍wi2(x,t)dxnnnnnnnnn+11-σ∑j=1N‍|Gij|∫t-τj(t)t‍∫Ω‍wj2(x,s)dxds}.
Calculating the time derivative of V(t) along the trajectory of system (1), we can get
(34)V˙(t)=∑i=1N‍λi{2∫Ω‍wi(x,t)∂wi(x,t)∂tdxnnnnn.nnnnnn+11-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dxnnnn.nnnnnnn-11-σ∑j=1N‍|Gij|(1-τ˙j(t))nnnn.nnnnnnn×∫Ω‍wj2(x,t-τj(t))dx∂wi(x,t)∂t}⩽∑i=1N‍λi{∑j=1N‍2di∫Ω‍wi(x,t)Δwi(x,t)dxnnnnnnnnn+2pi∫Ω‍wi2(x,t)dx+2∫Ω‍wi(x,t)ui(x,t)dxnnnnnnnnn+2∫Ω‍wi(x,t)∑j=1N‍Gijwj(x,t-τj(t))dxnnnnnnnnn+11-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dxnnnnnnnnn-∑j=1N‍|Gij|∫Ω‍wj2(x,t-τj(t))dx}⩽∑i=1N‍λi{∑j=1N‍2di∫Ω‍wi(x,t)Δwi(x,t)dxnnnnnnnnn+2pi∫Ω‍wi2(x,t)dx+2∫Ω‍wi(x,t)ui(x,t)dxnnnnnnnnn+11-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dxnnnnnnnnn+2∑j=1N‍|Gij|∫Ω‍wi2(x,t)dxnnnnnnnnn×∫Ω‍wj2(x,t-τj(t))dxnnnnnnnnn-∑j=1N‍|Gij|∫Ω‍wj2(x,t-τj(t))dx}⩽∑i=1N‍λi{∑j=1N‍2di∫Ω‍wi(x,t)Δwi(x,t)dxnnnnnnnnnn+2pi∫Ω‍wi2(x,t)dx+2∫Ω‍wi(x,t)ui(x,t)dxnnnnnnnnnn+11-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dxnnnnnnnnnn+∑j=1N‍|Gij|∫Ω‍wi2(x,t)dx}.
From Green’s theorem and the boundary condition, we have
(35)∫Ω‍wi(x,t)Δwi(x,t)dx=-∑k=1m‍∫Ω‍(∂wi(x,t)∂xk)2dx.
According to Lemma 4, we can obtain
(36)∑k=1m‍∫Ω‍(∂wi(x,t)∂xk)2dx⩾∑k=1m‍∫Ω‍wi2(x,t)lk2dx.
Therefore,
(37)V˙(t)-2∑i=1N‍∫Ω‍yi(x,t)ui(x,t)dx+γ∑i=1N‍∫Ω‍ui2(x,t)dx⩽∑i=1N‍{(-2λi∑k=1m‍dilk2+λi∑j=1N‍|Gij|+2piλi)nnnnnnn×∫Ω‍wi2(x,t)dx+(γ-2hi)∫Ω‍ui2(x,t)dxnnnnnnn+2(λi-gi)∫Ω‍wi(x,t)ui(x,t)dxnnnnnnn+λi1-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dx}=∑i=1N‍{(-2λi∑k=1m‍dilk2+2piλi∑j=1N‍λj|Gji|1-σnnnnnnnn.nn+λi∑j=1N‍|Gij|+∑j=1N‍λj|Gji|1-σ)∫Ω‍wi2(x,t)dxnnnnnnnnn+2(λi-gi)∫Ω‍wi(x,t)ui(x,t)dxnnnnnnnnn+(γ-2hi)∫Ω‍ui2(x,t)dx∑j=1N‍λj|Gji|1-σ}=∑i=1N‍∫Ω‍(wi(x,t)ui(x,t))T(Miλi-giλi-giγ-2hi)(wi(x,t)ui(x,t))dx.
It follows from (32) that
(38)V˙(t)-2∑i=1N‍∫Ω‍yi(x,t)ui(x,t)dx+γ∑i=1N‍∫Ω‍ui2(x,t)dx⩽0.
By integrating (38) with respect to t over the time period 0 to tp, we can get
(39)2∑i=1N‍∫0tp‍∫Ω‍yi(x,t)ui(x,t)dxdt⩾V(tp)-V(0)+γ∑i=1N‍∫0tp‍∫Ω‍ui2(x,t)dxdt.
From the definition of V(t), we have V(tp)⩾0 and V(0)⩾0. Thus,
(40)∑i=1N‍∫0tp‍∫Ω‍yi(x,t)ui(x,t)dxdt⩾-β2+γ2∑i=1N‍∫0tp‍∫Ω‍ui2(x,t)dxdt
for all tp⩾0, where β=V(0)/2. The proof is completed.By a minor modification of the proof of Theorem7, we can easily get the following.Theorem 8.
Letτ˙j(t)⩽σ<1. The complex network (1) is output-strictly passive if there exist constants λi>0 and γ>0 such that
(41)(M^iλi-gi+γgihiλi-gi+γgihiγhi2-2hi)⩽0,
where i,j=1,2,…,N, M^i=-2λi∑k=1m‍(di/lk2)+2piλi+λi∑j=1N‍|Gij|+γgi2+∑j=1N‍(λj|Gji|/(1-σ)).Remark 9.
In [22], two kinds of impulsive parabolic complex networks were considered, in which the node states are dependent on the time and space variables. Several global exponential stability and robust global exponential stability criteria were derived for the impulsive parabolic complex networks. Both global exponential stability and passivity are taken into consideration in this paper, and some sufficient conditions are established.


## 3.1. Stability Analysis

Theorem 5.
Letτ˙j(t)⩽σ<1. The complex network (1) is globally exponentially stable if there exist constants λi>0 and ϵ>0 such that
(9)2λiϵ+2λipi-2λi∑k=1m‍dilk2+∑j=1N‍λj|Gji|e2ϵτ1-σ+λi∑j=1N‍|Gij|⩽0,
where i,j=1,2,…,N.Proof.
Firstly, we can get from (1) that
(10)∂wi(x,t)∂t=diΔwi(x,t)+piwi(x,t)+ui(x,t)+∑j=1N‍Gijwj(x,t-τj(t)),
where i=1,2,…,N.
Definez(x,t)=w(x,t,Ψ)-w(x,t,Φ), Ψz(x,t)=Ψ(x,t)-Φ(x,t), and then the dynamics of the difference vector z(x,t)=(z1(x,t),z2(x,t),…,zN(x,t))T is governed by the following equation:
(11)∂zi(x,t)∂t=diΔzi(x,t)+∑j=1N‍Gijzj(x,t-τj(t))+pizi(x,t),
where i=1,2,…,N. Define the following Lyapunov functional for the system (11):
(12)V(t)=∑i=1N‍λi{∫τjt‍e2ϵt∫Ω‍zi2(x,t)dx+e2ϵτ1-σ∑j=1N‍|Gij|nnnnnnn×∫t-τj(t)t‍∫Ω‍zj2(x,s)e2ϵsdxds}.
Calculating the time derivative of V(t) along the trajectory of system (11), we can get
(13)V˙(t)=∑i=1N‍λie2ϵt{2ϵ∫Ω‍zi2(x,t)dx+∫Ω‍2zi(x,t)∂zi(x,t)∂tdxnnnnnnnnnnnn+e2ϵτ1-σ∑j=1N‍|Gij|∫Ω‍zj2(x,t)dxnnnnnnnnnnnn-e2ϵτ1-σ∑j=1N‍(1-τ˙j(t))|Gij|e-2ϵτj(t)nnnnnnnnnnnn×∫Ω‍zj2(x,t-τj(t))dx∂zi(x,t)∂t}⩽∑i=1N‍λie2ϵt{∑j=1N‍(2ϵ+2pi)∫Ω‍zi2(x,t)dxnnnnnnnnnnnn+2di∫Ω‍zi(x,t)Δzi(x,t)dxnnnnnnnnnnnn+2∫Ω‍zi(x,t)∑j=1N‍Gijzj(x,t-τj(t))dxnnnnnnnnnnnn+e2ϵτ1-σ∑j=1N‍|Gij|∫Ω‍zj2(x,t)dxnnnnnnnnnnnn-∑j=1N‍|Gij|∫Ω‍zj2(x,t-τj(t))dx}.
From Green’s theorem and the boundary condition, we have
(14)∫Ω‍zi(x,t)Δzi(x,t)dx=-∑k=1m‍∫Ω‍(∂zi(x,t)∂xk)2dx.
According to Lemma 4, we can obtain
(15)∑k=1m‍∫Ω‍(∂zi(x,t)∂xk)2dx⩾∑k=1m‍∫Ω‍zi2(x,t)lk2dx.
Therefore,
(16)V˙(t)⩽∑i=1N‍{∑j=1N‍λi(2ϵ+2pi-2∑k=1m‍dilk2)∫Ω‍zi2(x,t)dxnnnnnnnn+λie2ϵτ1-σ∑j=1N‍|Gij|∫Ω‍zj2(x,t)dxnnnnnnnn+2λi∫Ω‍zi(x,t)∑j=1N‍Gijzj(x,t-τj(t))dxnnnnnnnn-λi∑j=1N‍|Gij|∫Ω‍zj2(x,t-τj(t))dx}e2ϵt⩽∑i=1N‍{∑j=1N‍λi(2ϵ+2pi-2∑k=1m‍dilk2)∫Ω‍zi2(x,t)dxnnnnnnnn+2λi∑j=1N‍|Gij|∫Ω‍zi2(x,t)dxnnnnnnnn×∫Ω‍zj2(x,t-τj(t))dxnnnnnnnn+λie2ϵτ1-σ∑j=1N‍|Gij|∫Ω‍zj2(x,t)dxnnnnnnnn-λi∑j=1N‍|Gij|∫Ω‍zj2(x,t-τj(t))dx}e2ϵt⩽e2ϵt∑i=1N‍{(2λiϵ+2λipi-2λi∑k=1m‍dilk2+λi∑j=1N‍|Gij|)nnnnnnnnnn×∫Ω‍zi2(x,t)dx+λie2ϵτ1-σ∑j=1N‍|Gij|nnnnnnnnnn×∫Ω‍zj2(x,t)dx∑j=1N‍|Gij|}=e2ϵt∑i=1N‍{(∑j=1N‍2λiϵ+2λipi-2λi∑k=1m‍dilk2+λi∑j=1N‍|Gij|nnnnnnnnnnnn+∑j=1N‍λj|Gji|e2ϵτ1-σ)×∫Ω‍zi2(x,t)dx∑j=1N‍}.
From (9), we can get
(17)V˙(t)⩽0.
So V(t)⩽V(0), t⩾0. Since
(18)V(0)=∑i=1N‍λi{∑j=1N‍∫Ω‍zi2(x,0)dxnnnnnnnnn.nn+e2ϵτ1-σ∑j=1N‍|Gij|∫-τj(0)0‍∫Ω‍zj2(x,s)e2ϵsdxds}⩽[max⁡i=1,2,…,N{λi}+max⁡i=1,2,…,N{τe2ϵτ1-σ∑j=1N‍λj|Gji|}]∥Ψz∥τ2,V(t)⩾min⁡i=1,2,…,N{λi}e2ϵt∥z(x,t)∥22.
Let λ-=min⁡i=1,2,…,N{λi}, λ+=max⁡i=1,2,…,N{λi}, and
(19)M=λ++max⁡i=1,2,…,N⁡{(τe2ϵτ/(1-σ))∑j=1N‍λj|Gji|}λ-.
Then M⩾1, and we can obtain
(20)∥z(x,t)∥2⩽Me-ϵt∥Ψz∥τ.
Namely,
(21)∥w(x,t,Ψ)-w(x,t,Φ)∥2⩽M∥Ψ-Φ∥τe-ϵt.
This completes the proof of Theorem 5.Practically, Theorem5 not only can judge the global exponential stability of complex network (1), but also can guarantee the existence and uniqueness of the periodic solution in some circumstances.Assume thatui(x,t) and τi(t)  (i=1,2,…,N) are periodic continuous functions with period ϖ. Then we can get from (1) that
(22)∂wi(x,t)∂t=diΔwi(x,t)+piwi(x,t)+ui(x,t)+∑j=1N‍Gijwj(x,t-τj(t)),
where ui(x,t+ϖ)=ui(x,t), τi(t+ϖ)=τi(t), i=1,2,…,N.Theorem 6.
Letτ˙j(t)⩽σ<1. The system (22) has a unique ϖ-periodic solution if there exist constants λi>0 and ϵ>0 such that
(23)2λiϵ+2λipi-2λi∑k=1m‍dilk2+∑j=1N‍λj|Gji|e2ϵτ1-σ+λi∑j=1N‍|Gij|⩽0,
where i,j=1,2,…,N.Proof.
Define(24)wt(Ψ)=w(x,t+s,Ψ),s∈[-τ,0],t⩾0,wt(Φ)=w(x,t+s,Φ),s∈[-τ,0],t⩾0.
Obviously, from Theorem 5, we can derive
(25)∥w(x,t,Ψ)-w(x,t,Φ)∥2⩽Me-ϵt∥Ψ-Φ∥τ
for t⩾0, where M⩾1. We can choose a positive integer N such that
(26)Me-ϵ(Nω-τ)⩽13.
Now define a Poincaré mapping P:C(Ω×[-τ,0],RN)→C(Ω×[-τ,0],RN) by
(27)Ψ⟶P(Ψ)=wϖ(Ψ).
Since the periodicity of system (22), one has PN(Ψ)=wNϖ(Ψ). From (25) and (26), one obtains
(28)∥PN(Ψ)-PN(Φ)∥τ⩽13∥Ψ-Φ∥τ.
This implies that PN is a contraction mapping, so there exists one unique fixed point Ψ*∈C(Ω×[-τ,0],RN) such that
(29)PN(Ψ*)=Ψ*.
Since PN(P(Ψ*))=P(PN(Ψ*))=P(Ψ*), so P(Ψ*) is also a fixed point of PN, then
(30)P(Ψ*)=Ψ*.
Let w(x,t,Ψ*) be the solution of system (22) with initial conditions Ψ*, then w(x,t+ω,Ψ*) is also a solution of system (22). Obviously,
(31)w(x,t+ω,Ψ*)=w(x,t,wω(x,Ψ*))=w(x,t,Ψ*)∀t⩾0.
This means that w(x,t,Ψ*) is exactly one ω-periodic solution of system (22). This completes the proof of Theorem 6.


## 3.2.  Passivity Analysis

Theorem 7.
Letτ˙j(t)⩽σ<1. The complex network (1) is input-strictly passive if there exist constants λi>0 and γ>0 such that
(32)(Miλi-giλi-giγ-2hi)⩽0,
where Mi=-∑k=1m‍(2λidi/lk2)+2piλi+λi∑j=1N‍|Gij|+∑j=1N‍(λj|Gji|/(1-σ)), i,j=1,2,…,N.Proof.
Firstly, construct a Lyapunov functional for the network (1) as follows:
(33)V(t)=∑i=1N‍λi{∑j=1N‍∫Ω‍wi2(x,t)dxnnnnnnnnn+11-σ∑j=1N‍|Gij|∫t-τj(t)t‍∫Ω‍wj2(x,s)dxds}.
Calculating the time derivative of V(t) along the trajectory of system (1), we can get
(34)V˙(t)=∑i=1N‍λi{2∫Ω‍wi(x,t)∂wi(x,t)∂tdxnnnnn.nnnnnn+11-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dxnnnn.nnnnnnn-11-σ∑j=1N‍|Gij|(1-τ˙j(t))nnnn.nnnnnnn×∫Ω‍wj2(x,t-τj(t))dx∂wi(x,t)∂t}⩽∑i=1N‍λi{∑j=1N‍2di∫Ω‍wi(x,t)Δwi(x,t)dxnnnnnnnnn+2pi∫Ω‍wi2(x,t)dx+2∫Ω‍wi(x,t)ui(x,t)dxnnnnnnnnn+2∫Ω‍wi(x,t)∑j=1N‍Gijwj(x,t-τj(t))dxnnnnnnnnn+11-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dxnnnnnnnnn-∑j=1N‍|Gij|∫Ω‍wj2(x,t-τj(t))dx}⩽∑i=1N‍λi{∑j=1N‍2di∫Ω‍wi(x,t)Δwi(x,t)dxnnnnnnnnn+2pi∫Ω‍wi2(x,t)dx+2∫Ω‍wi(x,t)ui(x,t)dxnnnnnnnnn+11-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dxnnnnnnnnn+2∑j=1N‍|Gij|∫Ω‍wi2(x,t)dxnnnnnnnnn×∫Ω‍wj2(x,t-τj(t))dxnnnnnnnnn-∑j=1N‍|Gij|∫Ω‍wj2(x,t-τj(t))dx}⩽∑i=1N‍λi{∑j=1N‍2di∫Ω‍wi(x,t)Δwi(x,t)dxnnnnnnnnnn+2pi∫Ω‍wi2(x,t)dx+2∫Ω‍wi(x,t)ui(x,t)dxnnnnnnnnnn+11-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dxnnnnnnnnnn+∑j=1N‍|Gij|∫Ω‍wi2(x,t)dx}.
From Green’s theorem and the boundary condition, we have
(35)∫Ω‍wi(x,t)Δwi(x,t)dx=-∑k=1m‍∫Ω‍(∂wi(x,t)∂xk)2dx.
According to Lemma 4, we can obtain
(36)∑k=1m‍∫Ω‍(∂wi(x,t)∂xk)2dx⩾∑k=1m‍∫Ω‍wi2(x,t)lk2dx.
Therefore,
(37)V˙(t)-2∑i=1N‍∫Ω‍yi(x,t)ui(x,t)dx+γ∑i=1N‍∫Ω‍ui2(x,t)dx⩽∑i=1N‍{(-2λi∑k=1m‍dilk2+λi∑j=1N‍|Gij|+2piλi)nnnnnnn×∫Ω‍wi2(x,t)dx+(γ-2hi)∫Ω‍ui2(x,t)dxnnnnnnn+2(λi-gi)∫Ω‍wi(x,t)ui(x,t)dxnnnnnnn+λi1-σ∑j=1N‍|Gij|∫Ω‍wj2(x,t)dx}=∑i=1N‍{(-2λi∑k=1m‍dilk2+2piλi∑j=1N‍λj|Gji|1-σnnnnnnnn.nn+λi∑j=1N‍|Gij|+∑j=1N‍λj|Gji|1-σ)∫Ω‍wi2(x,t)dxnnnnnnnnn+2(λi-gi)∫Ω‍wi(x,t)ui(x,t)dxnnnnnnnnn+(γ-2hi)∫Ω‍ui2(x,t)dx∑j=1N‍λj|Gji|1-σ}=∑i=1N‍∫Ω‍(wi(x,t)ui(x,t))T(Miλi-giλi-giγ-2hi)(wi(x,t)ui(x,t))dx.
It follows from (32) that
(38)V˙(t)-2∑i=1N‍∫Ω‍yi(x,t)ui(x,t)dx+γ∑i=1N‍∫Ω‍ui2(x,t)dx⩽0.
By integrating (38) with respect to t over the time period 0 to tp, we can get
(39)2∑i=1N‍∫0tp‍∫Ω‍yi(x,t)ui(x,t)dxdt⩾V(tp)-V(0)+γ∑i=1N‍∫0tp‍∫Ω‍ui2(x,t)dxdt.
From the definition of V(t), we have V(tp)⩾0 and V(0)⩾0. Thus,
(40)∑i=1N‍∫0tp‍∫Ω‍yi(x,t)ui(x,t)dxdt⩾-β2+γ2∑i=1N‍∫0tp‍∫Ω‍ui2(x,t)dxdt
for all tp⩾0, where β=V(0)/2. The proof is completed.By a minor modification of the proof of Theorem7, we can easily get the following.Theorem 8.
Letτ˙j(t)⩽σ<1. The complex network (1) is output-strictly passive if there exist constants λi>0 and γ>0 such that
(41)(M^iλi-gi+γgihiλi-gi+γgihiγhi2-2hi)⩽0,
where i,j=1,2,…,N, M^i=-2λi∑k=1m‍(di/lk2)+2piλi+λi∑j=1N‍|Gij|+γgi2+∑j=1N‍(λj|Gji|/(1-σ)).Remark 9.
In [22], two kinds of impulsive parabolic complex networks were considered, in which the node states are dependent on the time and space variables. Several global exponential stability and robust global exponential stability criteria were derived for the impulsive parabolic complex networks. Both global exponential stability and passivity are taken into consideration in this paper, and some sufficient conditions are established.


## 4. Example

In this section, an illustrative example is provided to verify the effectiveness of the proposed theoretical results.Consider a complex network model, in which each node is a 1-dimensional dynamical system described by(42)∂wi(x,t)∂t=di∂2wi(x,t)∂x2+piwi(x,t),i=1,2,3,4.
Take di=0.4,  pi=0.1i, gi=1, hi=2, τj(t)=0.2-(e-t/(4+j)), τ=0.2, σ=0.2, x∈(-0.5,0.5). The matrix G is chosen as
(43)(-0.30.10.150.050.2-0.30.100.050.05-0.30.20.100.1-0.2).
It is easy to verify that (9) is satisfied if λ1=λ2=λ3=λ4=1 and ϵ=0.25. From Theorem 5, complex network (1) with above given parameters is globally exponentially stable. Moreover, if we take λ1=λ2=λ3=λ4=1 and γ=1.5, then (32) is satisfied. According to Theorem 7, complex network (1) with above given parameters is also input-strictly passive. The simulation results are shown in Figures 1 and 2.Figure 1
The change processes of∥zi(x,t)∥2,i=1,2,3,4.Figure 2
The change process ofd(t)=∑i=14‍∫0t‍∫Ω‍yi(x,s)ui(x,s)dxds-0.75∑i=14‍∫0t‍∫Ω‍ui2(x,s)dxds, in which ui(x,t)=0.2isin(2πx),i=1,2,3,4.


## 5. Conclusion

A parabolic complex network model has been introduced, in which the state, input, and output variables are dependent on the time and space variables. The input strict passivity, output strict passivity, and global exponential stability of the proposed network model have been discussed in this paper, and several sufficient conditions have been established. Illustrative simulations have been provided to verify the correctness and effectiveness of the obtained results. In future work, we shall study the passivity and robust passivity of parabolic complex networks with impulsive effects.



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*Source: 412169-2014-09-03.xml*