Intermittent control to stationary distribution and exponential stability for hybrid multi-stochastic-weight coupled networks based on aperiodicity

Available online at www.sciencedirect.com

Journal of the Franklin Institute 356 (2019) 7263–7289 www.elsevier.com/locate/jfranklin

Intermittent control to stationary distribution and exponential stability for hybrid multi-stochastic-weight coupled networks based on aperiodicity Yan Liu, Henglei Xu, Wenxue Li∗ Department of Mathematics, Harbin Institute of Technology, Weihai 264209, PR China Received 29 August 2018; received in revised form 2 April 2019; accepted 1 July 2019 Available online 12 July 2019

Abstract In this paper, the issue about the stationary distribution for hybrid multi-stochastic-weight coupled networks (HMSWCN) via aperiodically intermittent control is investigated. Specially, when stochastic disturbance gets to zero, the exponential stability in pth moment for hybrid multi-weight coupled networks (HMWCN) is considered. Under the framework of the Lyapunov method, M-matrix and Kirchhoff’s Matrix Tree Theorem in the graph theory, several sufficient conditions are derived to guarantee the existence of a stationary distribution and exponential stability. Different from previous work, the existing area of a stationary distribution is not only related to the topological structure of coupled networks, but also aperiodically intermittent control (the rate of control width and control duration). Subsequently, as an application to theoretical results, a class of hybrid multi-stochastic-weight coupled oscillators is studied. Ultimately, numerical examples are carried out to demonstrate the effectiveness of theoretical results and effects of the control schemes. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Coupled networks (so-called complex networks) can describe numerous phenomena in the real world, such as gene interaction networks in cancer cells [1], simultaneous localization ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (W. Li).

https://doi.org/10.1016/j.jfranklin.2019.07.001 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

7264

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

of software faults [2], global monsoon flows [3] and so forth. One of the focus topics is concerned with the stability of coupled networks. Nevertheless, the most of existing results are dependent on two core hypotheses: one is that the edge between two interconnected nodes is unique; the other is that the weights between nodes are absolutely fixed. Unfortunately, in our realistic life, such as signal processing, urban traffic, there usually exist more than one link between two nodes and every link possesses its own performances. Taking signal propagation in the brain as an example, neurons can transfer information through electrical impulse, neurotransmitter, etc. and different signal propagations result in different weights [4]. On this occasion, it seems unreasonable to consider single link into coupled networks merely. Therefore, it is important to consider multi-weight coupled networks and some results have been reported [5–7]. On the other hand, due to diverse reasons, the connection between nodes might be subject to certain switching behaviors and also exhibit stochastic perturbations, which results from the stochastic connectivity of nodes. Hence, it needs further exploration to reveal how noise influences coupled networks [5,8]. In addition, it is worth mentioning that stochastic disturbance can destroy the existence of the equilibrium, which may lead to the existence of a stationary distribution. Taking the predator-prey model [9] for example, when the equilibrium point of species number is broken due to the stochastic distribution, we can overcome the difficulty of fitting data perfectly and predict the fluctuation of species accurately through the existence of a stationary distribution. As a consequence, it is of significance to investigate the stationary distribution and many scholars have rationally considered the stationary distribution for systems in the real world. For instance, in [10], Yu and Liu indicated that there would be no equilibrium in a stochastic food-chain model with Lévy jumps while the system had a stationary distribution. The authors in [11], proved the existence of a stationary distribution for an SIS epidemic model with media coverage when the solution is fluctuating in a neighborhood of the equilibrium. Because the stationary distribution has been employed to solve multitude realistic problems, such as power-delay tradeoff [12], noisy chaotic systems [13], signal detection [14], substantial methods have been implemented to investigate the stationary distribution. By using the Lyapunov method and M-matrix, certain sufficient conditions were derived to prove the existence of an ergodic stationary distribution in SIR model in [15]. Zhu and Yin in [16], constructed appropriate Lyapunov functions to cope with the stationary distribution for stochastic systems with regime switching. Subsequently, motivated by Zhu and Yin [16], Zhang et al. in [17] established sufficient conditions to the existence of a unique ergodic stationary distribution for an SIS epidemic model with white and color noises by using the Lyapunov method. It is clear that the Lyapunov method has become a valid method to study stationary distribution, however, it is a tough work to directly construct an appropriate Lyapunov function. Fortunately, by using Kirchhoff’s Matrix Tree Theorem in the graph theory, a global Lyapunov function of coupled networks can be attained easily (the details can be seen in [18,19] and the references cited therein). In [20], we investigated the existence of a stationary distribution for stochastic coupled networks by combining the graph theory with the Lyapunov method based on the theory in [21]. Nowadays, control techniques have been gradually considered into the study about the dynamics of coupled networks, including feedback control [22], adaptive control [23], impulsive control [24,25], intermittent control [26–29]. Therein, intermittent control is a valuable technique since the transmitted signals are inevitably affected by external perturbations, which makes the signals interrupted intermittently. Moreover, intermittent control can also reduce the

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7265

control cost greatly in virtue of the fact that the controller merely needs to be activated during certain nonzero time interval and interdictory in the recumbent time. Hence, there are a wide range of applications in engineering fields. Nevertheless, it should be noticed that in periodically intermittent control strategy, each control period is deterministic and possesses the equal rate of control duration, which may be restrict and unreasonable in practical applications. For example, sea waves repeat irregularly, so any measurement of wave elevation time history requires treating as aperiodically intermittent. To remove this constraint, it seems necessary to consider aperiodical scheme. Aperiodically intermittent control (AIC) [6,8,30–32] has been presented recently, in which both the control time and the rate of control width are allowed to be changeable. The authors in [8] derived several novel synchronization conditions for complex-valued dynamical networks based on the Lyapunov method and the graph theory as well as AIC. In [33], Gan proposed an AIC strategy to ensure global exponential synchronization for the addressed generalized reaction-diffusion neural networks via the Lyapunov method. However, there have been scattered results about the stationary distribution for hybrid multi-stochastic-weight coupled networks (HMSWCN) via AIC. Under the framework of the above discussions, this paper aims to investigate the stationary distribution and exponential stability for HMSWCN via AIC by utilizing the Lyapunov method, Kirchhoff’s Matrix Tree Theorem, M-matrix and the theory of [16]. The chief contributions of our work can be summarized as: (1) there has been a rich body of literature on analyzing coupled networks with multi-groups [34,35], which are established in multiple groups and every group is independent. Consequently, every network needs to reach strongly connected by using the graph theory. While in this paper, coupled networks with multiple weights are built on one network, which is composed of many subnetworks. In this condition, when using the graph theory, it is unnecessary to require every subnetwork to be strongly connected. (2) In [20], the existence of a stationary distribution of stochastic coupled systems was investigated, which results reflected that the existing area of a stationary distribution was closely related to the topological structure of coupled networks and only the lower bound of the existing area of a stationary distribution was given. Except that, in this paper, AIC we adopt, including the control intensity, the control width and the rate of control duration, will affect the existing area of a stationary distribution and its lower as well as upper bounds are derived. (3) Zhang and Chen [5] provided new sufficient conditions to ascertain the exponential stability for stochastic complex networks with multiple weights without Markovian switching. In this paper, when we take Markovian switching into account, multi-weight coupled networks are not able to achieve stability while AIC helps to stabilize multi-weight coupled networks with Markovian switching (HMWCN). The remainder of this article is arranged as follows. In Section 2, the model of HMSWCN based on AIC is presented and several necessary lemmas, assumptions and definitions are given. Main results are formulated including the existence of a stationary distribution for HMSWCN and the exponential stability in pth moment for HMWCN in Section 3. Then, in Section 4, we are absorbed in applying our analytical results to multi-stochastic-weight oscillators networks. In Section 5, two numerical examples are given to demonstrate the effectiveness of the theoretical results and proposed controllers. Notations: Unless otherwise specified, let Rm stand for the m-dimensional Euclidean space and write | · | for the Euclidean norm in Rm . To simplify the description, M = {1, 2, . . . , m}, m 2 m 1 I = {1, 2, . . . , s} and Rm + = {xl ∈ R : xl > 0, l ∈ M} are used. Notation C (R × I, R+ ) m stands for the family of all nonnegative functions V(x, i) on R × I that are continuously twice differentiable in x. The transpose of a vector or matrix, M, is denoted by MT . Make

7266

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

(, F, {Ft }t≥0 , P ) be a complete probability space with a filtration {Ft }t≥0 satisfying usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets)  andE(·) is the mathematical expectation with respect to P. Let θ1 θ2  max{θ1 , θ2 } and θ1 θ2  min{θ1 , θ2 }. Additionally, λmin ( · ) and λmax ( · ) represent the minimum and maximum eigenvalues of a real symmetric matrix, respectively. 2. Model formulation Our model will be introduced in this section and before that, we briefly recall several essential concepts about the graph theory that will serve as a basis for the development of our model. Write (G, A ) for a digraph with m (m ≥ 2) vertices and weight matrix A = (aln )m×m , whose entry aln > 0 is equal to the weight of arc (n, l) if it exists, and 0 otherwise. A digraph G is strongly connected if there exists a directed path from one to the other for any pair of distinct  vertices. The Laplacian matrix of (G, A ) is defined as LA = (pln )m×m , where pll = n=l aln and pln = −aln for l = n. More concepts about the graph theory can be found in [36]. Moreover, let r(t) be a right-continuous Markov  chain on the probability space taking values in a finite state set I with generator  = ϒi j s×s given by  if i = j, ϒi j + o(δ), P{r(t + δ) = j|r(t ) = i} = 1 + ϒi j δ + o(δ), if i = j,  where δ > 0 and ϒ ij ≥ 0 is the transition rate from i to j, if i = j, whereas ϒii = − j=i ϒi j . In this paper, we make the assumption that ϒ ij > 0 for i = j. Given a network represented by digraph G with m (m ≥ 2) vertices, suppose the multiweight coupled network is split into k subnetworks. The coupled network with multiple weights can be described as z˙l = Pl (zl ) +

m  k 

α j Clnj N j zn

(1)

n=1 j=1

 T for l ∈ M, where zl = zl1 , zl2 , . . . , zlS ∈ RS is an S-dimensional state vector of the lth subnetwork, Pl : RS → RS is continuous, which is used to describe the dynamics of each uncoupled subnetwork. For the j-th subnetwork ( j = 1, 2, . . . , k), α j is the coupling strength, N j = diag(N11j , N22j , . . . , NSSj ) is a positive definite diagonal S × S matrix, which indicates the inner coupling and the outer coupling matrix (Clnj )S×S stands for the coupled network topology, e.g., if Clnj = 0, then vertex l receives direct information from vertex n and 0 otherwise. As we all know, coupled networks in real world are inevitably perturbed by the environmental noise, while taking valid control strategy can help to investigate the dynamical behaviors of coupled networks. As a result, it is necessary to study the following HMSWCN via AIC corresponding to system (1) as ⎡ ⎤ m  k  dzl = ⎣Pl (zl , r(t )) + α j (r (t ))Clnj (r (t ))N j (r (t ))zn + Gl (zl , r(t ))⎦dt + ql (r(t ))dwl n=1 j=1

(2)   for l ∈ M, ql (r(t )) = diag ql1 (r(t )), ql2 (r(t )), . . . , qlS (r(t )) is stochastic disturbance, wl is an S-dimensional Brownian motion defined on the complete probability space

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7267

(, F, {Ft }t≥0 , P ) and Gl is AIC described as  V (z , r(t )), t ∈ [tm˜ , xm˜ ), Gl (zl , r(t )) = l l 0, t ∈ [xm˜ , tm˜ +1 ), here, In the meanwhile, Pl + m k0 = t0
 l  dxk = fk (xk )dt + gk (xk ) + Hkh (xk , xh ) dBk h=1

established in [20] and other homologous models, the coupled factor is considered into the drift coefficient. Furthermore, the coupled networks are deemed as multiple weights rather than single weight, which have significance to handle realistic problems, such as human connection networks, transportation networks and so on. Additionally, on the basis of the model of stochastic coupled networks with multiple weights ⎡ ⎤ N N N    dxi (t ) = ⎣ fi (xi (t ), t ) + δ1 b1i j H1 x j (t ) + δ2 b2i j H2 x j (t ) + · · · + δl bli j Hl x j (t )⎦dt j=1

j=1

j=1

+ gi (xi (t ), t )dW (t ) proposed in [5], Markovian switching is taken into account, which is more practical in modeling real-world systems, where they may experience abrupt changes in their structure and parameters. Therefore, the model we consider is more general and meaningful. In order to derive our main results, the following lemma is introduced. Lemma 1. (Kirchhoff’s Matrix Tree Theorem) [18] Assume n ≥ 2. Then  ci = W (T ), i = 1, 2, . . . , n, T ∈T i

where Ti is the set of all spanning trees T of (G, A ) that are rooted at vertex i, and W (T ) is the weight of T . In particular, if (G, A ) is strongly connected, then ci ≥ 0 for i = 1, 2, . . . , n. We refer the reader to [18] to obtain more details about Lemma 1. Besides, the following basic definitions and an assumption are also needed. Definition 1 (nonsingular M-matrix). [37] A square matrix A = (ai j )n×n is called a nonsingular M-matrix if A can be expressed in the form A = sI − G with some G ≥ 0 and s > ρ(G), where I is the identity n × n matrix and ρ(G) the spectral radius of G. More properties about nonsingular M-matrix can be found in [37]. Definition 2. If πl (zl ) ∈ C 2 (RS × I, R1+ ), we define a differential operator L with respect to system (2) by

7268

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

   s 1 ∂ 2 πl (zl , i) T + Lπl (zl , i) = trace ql (i) q (i ) ϒih πl (zl , h) l 2 ∂zl2 h=1 ⎡ ⎤ m  k  ∂πl (zl , i) ⎣ + Pl (zl , i) + α j (i )Clnj (i )N j (i )zn + Gl (zl , i)⎦, ∂zl n=1 j=1 where ∂πl (zl ) = ∂zl



 ∂πl (zl ) ∂πl (zl ) ∂πl (zl ) , , ,··· , ∂zl1 ∂zl2 ∂zlS

∂ 2 πl (zl ) = ∂zl2



∂ 2 πl (zl ) ∂ zli¯ ∂ zl j¯

 . S×S

Definition 3. For AIC, we define δ = inf (tm˜ +1 − tm˜ ), m˜

ϕ = lim inf m˜ →∞

xm˜ − tm˜ , tm˜ +1 − tm˜

ϕ˜ = lim inf m˜ →∞

tm˜ +1 − xm˜ . tm˜ +1 − tm˜

Assumption 1. For AIC, there exist two positive scalars θ , ω, where 0 < θ < ω, such that θ = inf (xm˜ − tm˜ ) > 0, m˜

ω = sup(tm˜ +1 − tm˜ ) < +∞. m˜

Remark 2. In this article, AIC is considered into our model, in which Gl is an AIC we take. For any time span [tm˜ , tm˜ +1 ), [tm˜ , xm˜ ) is the control time, [xm˜ , tm˜ +1 ) is the rest time and xm˜ − tm˜ is called the m˜ th control duration, tm˜ +1 − xm˜ is called the m˜ th rest width. Apparently, the above requirement of tm˜ , xm˜ has a large scope and both control duration and rest width are variable. When xm˜ − tm˜ ≡ con1 , tm˜ +1 − xm˜ ≡ con2 , where con1 and con2 are constants, the intermittent control becomes the periodic one [6,26,27]. When tm˜ +1 − xm˜ → 0, AIC turns into impulsive control [24], while xm˜ − tm˜ ≡ 0, AIC becomes the usual continuous control [22]. The theoretical results of this type controller have a wider application scope. Different from the aforementioned remarkable work of AIC [8,30–32], the infimum of control duration rate ϕ and the infimum of rest width rate ϕ˜ are considered, which has a close relationship with the existing area of a stationary distribution. The more detailed effect of AIC will be introduced in Theorems 1 and 3. 3. Main results The purpose of this section is to investigate the existence of a stationary distribution and pth moment exponential stability based on the Lyapunov method, Kirchhoff’s Matrix Tree Theorem, M-matrix and the theory proposed in [16]. 3.1. Stationary distribution for HMSWCN When taking stochastic disturbance and Markovian switching into consideration, the existence of equilibrium point of system (2) may be destroyed, in this case, it seems significant to study the existence of a stationary distribution. Next, we introduce two kinds of theorems on whether a stationary distribution will exist in some specific conditions. The first theorem is given as follows. Before that, some basic hypotheses are needed.

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7269

1.1. There exist positive constants Cl (i), ξl (i ), ξ¯l (i ), aln ≥ 0, p > 0, functions Ml (zl , i) ∈ C 2 (RS × I, R1+ ) and Qln (zl , zn ) such that  m  ξl (i), t ∈ [tm˜ , xm˜ ), ξ`l (i) = LMl (zl , i) ≤ −ξ`l (i)|zl | p + aln Qln (zl , zn ) + Cl (i), ξ¯l (i), t ∈ [xm˜ , tm˜ +1 ). n=1

1.2. There are positive constants εl (i), ε¯l (i) satisfying εl (i)|zl | p ≤ Ml (zl , i) ≤ ε¯l (i)|zl | p . 1.3. Along each directed cycle CQ of digraph (G, (aln )m×m ), there is  Qln (zl , zn ) ≤ 0. (n,l )∈E (CQ )

Theorem 1. For any i ∈ I, l ∈ M, suppose digraph (G, (aln )m×m ) is strongly connected and Assumptions 1.1–1.3 hold. Then, it has a stationary distribution for system (2). Proof. According to the assumption of ϒ ij and the definition of q(i), we can observe that   ϒ ij > 0, i = j and q(i) is positive definite. Then, b|bˆ |2 ≤ q(i), bˆ ≤ b−1 |bˆ |2 , bˆ ∈ Rm , b ∈ (0, 1] is satisfied. Let mS

   D = (−ρ, ρ)c × (−ρ, ρ)c × · · · × (−ρ, ρ)c , ρ > 0, c

where D is a nonempty open set. Define Lyapunov function M(z, i) = m c μ M l l (zl , i), (z, i) ∈ D × I, in which μl > 0 is the cofactor of the lth diagonal l=1 element of the Laplacian matrix of (G, (aln )m×m ). When tm˜ ≤ t < xm˜ , by employing Assumption 1.1, it follows that   m m   p LM(z, i) ≤ μl −ξl (i)|zl | + aln Qln (zl , zn ) + Cl (i) n=1

l=1

and through Assumption 1.3, Lemma 1 as well as Theorem 2.2 in [18], it is not complicated to see that m  m    μl aln Qln (zl , zn ) = W (Q ) Qln (zl , zn ) ≤ 0, Q∈Q

l=1 n=1

(n,l )∈E (CQ )

where W (Q ) is the weight of Q. Consequently, we can immediately get m  LM(z, i) ≤ − μl (ξl (i)|zl | p − Cl (i) ).

(3)

l=1

For any zl j ∈ (−ρ, ρ)c , in which zl j is the jth component of zl , we have m    p LM(z, i) ≤ − μl ξl (i)S 2 ρ p − Cl (i) , l=1

as a result of which there exists ρ >



Cl (i) ξl (i)S

p 2

 1p

p

such that ξl (i)S 2 ρ p − Cl (i) > 0. Next, the

upper bound of ρ will be found. With the help of Assumption 1.2 and elementary inequality m  p  p  m (1− 2 ) 0 |z| p ≤ |zl | p ≤ m (1− 2 ) 0 |z| p , l=1

7270

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

it follows that m  p  M(z, i) ≤ μl ε¯l (i)|zl | p ≤ max {μl ε¯l (i)}m (1− 2 ) 0 |z| p  ε¯|z| p l∈M,i∈I

l=1

and we can compute M(z, i) ≥

m 

p

μl εl (i)|zl | p ≥ min {μl εl (i)}m (1− 2 ) l∈M,i∈I

l=1



0

|z| p  ε|z| p .

Therefore, Eq. (3) can turn into 1 LM(z, i) ≤ −ξ (i )|z| p − C(i ) ≤ −ξ M(z, i) − C(i), (4) ε¯  p  where ξ = minl∈M,i∈I {μl ξl (i)}m (1− 2 ) 0 , C(i) = m l=1 μl Cl (i). Integrating from t to t + ε and taking expectation on both sides of Eq. (4), the following derivation holds    1 EM(z(t + ε), r(t + ε)) − EM(z(t ), r(t )) 1 t+ε −ξ EM(z(s), r(s)) + C ds, ≤ ε ε t ε¯ where C = maxi∈I {C(i)}. Let ε be so sufficiently small that t + ε ∈ [tm˜ , xm˜ ). Making ε → 0+ gives that 1 D+ EM(z(t ), r(t )) ≤ −ξ EM(z(t ), r(t )) + C. ε¯ Subsequently, considering  + D EM(z(t ), r(t )) = −ξ 1ε¯ EM(z(t ), r(t )) + C, M(z(0), r(0)) = M(z0 ) = 0, and using variation of constants formula, it holds that     ξ ε¯ ε¯ exp − (t − tm˜ ) + C . EM(z(t ), r(t )) = M(z0 ) − C ξ ε¯ ξ According to comparison principle, it further shows that   ξ ε¯ ε¯ EM(z(t ), r(t )) < M(z0 ) exp − (t − tm˜ ) + C < M(z0 ) + C . ε¯ ξ ξ In addition, from

  1 1 ε¯ M(z0 ) + C , E|z(t )| < EM(z(t ), r(t )) < ε ε ξ p

p

in which E|z(t )| p > (mS) 2 ρ p , it is straightforward to produce that

  1p 1 ε¯ M(z ρ< ) + C . p 0 ξ ε(mS) 2 In summary, there exists ρ such that  1

  1p Cl (i) p 1 ε¯ <ρ< . M(z0 ) + C p p ξ ξl (i)S 2 ε(mS) 2

(5)

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289 c (1) Then, for  (z, × I, we  i) ∈ D  can obtain LM(z, i) ≤ −τ , p p 2 mini∈I,l∈M μl ξl (i)S ρ − Cl (i) > 0. When xm˜ ≤ t < tm˜ +1 , by the same analysis, we can get m    p LM(z, i) ≤ − μl ξ¯l (i)S 2 ρ p − Cl (i) .

in

which

7271

τ (1) =

(6)

l=1

Then, there exists ρ >



Cl (i) p ξ¯l (i)S 2

 1p

p such that ξ¯l (i)S 2 ρ p − Cl (i) > 0. On the other side,

Eq. (6) can be expressed as

1 LM(z, i) ≤ −ξ¯ M(z, i) − C(i), ε¯   p  in which ξ¯(i) = mini∈I,l∈M μl ξ¯l (i) m (1− 2 ) 0 . Similarly, we can derive

  1p 1 ε¯ ε¯(i ) M(z0 ) + C(i ) EM(z(t ), r(t )) < M(z0 ) + C(i) , ρ < , p ξ¯ ξ¯(i ) ε(mS) 2 in which ξ¯ = maxi∈I {ξ¯(i)}. Hence, there exists ρ satisfying  1

  1p Cl (i) p 1 ε¯ < ρ < ) + C . M(z p p 0 ξ¯ ξ¯l (i)S 2 ε(mS) 2 c Then for  (z, × I, we  i) ∈ D  can get p ¯ 2 mini∈I,l∈M μl ξl (i)S ρ p − Cl (i) > 0. For t ∈ [0, x0 ), we have   ξ ε¯ EM(z(t ), r(t )) < M(z0 ) exp − t + C . ε¯ ξ

For t ∈ [x0 , t1 ), we acquire

LM(z, i) ≤ −τ (2) ,

(7) in

which

τ (2) =

 ε¯ ξ ξ¯ EM(z(t ), r(t )) < M(z0 ) exp − x0 − (t − x0 ) + C . ε¯ ε¯ ξ¯ For t ∈ [t1 , x1 ), we get



 ε¯ ξ ξ¯ ξ EM(z(t ), r(t )) < M(z0 ) exp − x0 − (t1 − x0 ) − (t − t1 ) + C . ε¯ ε¯ ε¯ ξ For t ∈ [x1 , t2 ), we know





 ε¯ ξ ξ¯ ξ ξ¯ EM(z(t ), r(t )) < M(z0 ) exp − x0 − (t1 − x0 ) − (x1 − t1 ) − (t − x1 ) + C . ε¯(i) ε¯ ε¯ ε¯ ξ¯ Consequently, by induction, we can conclude that when t ∈ [tm˜ , xm˜ ), ⎛ ⎞ m˜ −1 m˜ −1   ε¯ EM(z(t ), r(t )) < M(z0 ) exp ⎝−β1 (x j˜ − t j˜ ) − β2 (t j˜+1 − x j˜ ) − β1 (t − tm˜ )⎠ + C ξ j˜=0

j˜=0

ε¯ < M(z0 ) exp (−(β1 ϕ + β2 ϕ˜ )δ) + C , ξ

7272

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289 ¯

and β2 = ξε¯ . When t ∈ [xm˜ , tm˜ +1 ), ⎛ ⎞ m˜ m˜ −1   ε¯ EM(z(t ), r(t )) < M(z0 ) exp ⎝−β1 (x j˜ − t j˜ ) − β2 (t j˜+1 − x j˜ ) − β2 (t − xm˜ )⎠ + C ξ¯ ˜ ˜ in which β1 =

ξ ε¯

j =0

j =0

ε¯ < M(z0 ) exp (−(β1 ϕ + β2 ϕ˜ )δ) + C . ξ¯ Therefore, for any t ≥ 0, it holds that EM(z(t ), r(t )) < M(z0 ) exp (−(β1 ϕ + β2 ϕ˜ )δ) + C¯ , " # where C¯ = max C ξε¯ , C ξε¯¯ . Then, it can be obtained that

ρ<

1

1   p ¯ ˜ )δ) + C  πˆ . p M(z0 ) exp (−(β1 ϕ + β2 ϕ

ε(mS) 2

(8)

Hence, taking Eqs. (5), (7) and (8) into account, it derives that $ 1  1 % Cl (i) p Cl (i) p min , < ρ < πˆ . p p ξ¯l (i)S 2 ξl (i)S 2   Ultimately, setting τ = min τ (1) , τ (2) , we can get LM(z, i) ≤ −τ, for (z, i) ∈ D c × I. In the light of Theorem 3.13 in [16], system (2) has a stationary distribution. This completes the proof.  Remark 3. Many results have been reported in literature regarding the coupled networks with multiple weights, see [5–7]. Note that most of the corresponding results are concerned with the stability and synchronization problems while few have explored the stationary distribution. The stationary distribution, as a kind of dynamic behaviors, can be applied not only on explaining natural phenomena [10,38], but also on power-delay tradeoff [12], signal detection [14] and so on. As a consequence, it seems more necessary to investigate the stationary distribution for multi-weight coupled networks. In this paper, Theorem 1 provides a theory to ensure that. Remark 4. There have been substantial methods to the study of the stationary distribution, such as the Lyapunov method, the Hasminskii’s method, the level set method [16,21,39]. Differing from the aforementioned methods, we combine the Lyapunov method with Kirchhoff’s Matrix Tree Theorem in the graph theory to research the existence of a stationary distribution for HSMWCN, which avoids some complex analysis. In addition, letting  M(z, i) = m μ l=1 l Ml (zl , i) provides a method to systematically construct a global Lyapunov function M(z, i) related to system (2) through making use of Lyapunov functions Ml (zl , i) of subnetworks and the topological structure of digraph (G, (aln )m×m ), which averts the difficulty of constructing a Lyapunov function directly. Remark 5. In Theorem 1, it can be observed that Assumption 1.2 is not unusual, which can be found in a lot of references [5,8,20]. We can conclude that the existing area of a stationary distribution closely depends on control strength through the proof of Theorem 1. In detail, the existing area of a stationary distribution is proportional to the range of ρ. When other parameters are fixed, ϕ and ϕ˜ get larger, then the existing area of a stationary distribution becomes smaller.

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7273

The theorem above is based on Lyapunov functions of subnetworks. For the next step, a theorem whose conditions are dependent on coefficients of the system is established as follows, which are easy to check and helpful in practice. Theorem 2. Let aln = max1≤ j≤k,i∈I {α j (i)Clnj (i)N j (i)} and assume that digraph (G, (aln )m×m ) is strongly connected. For any i ∈ I, l ∈ M, if the following conditions hold. 2.1. There are positive constants η(i), η(i) ¯ and a positive definite matrix B such that zlT B(Pl (zl , i) + Gl (zl , i)) ≤ −η(i) ˜ zlT Bzl , where η(i) ˜ =

 η(i), t ∈ η(i) ¯ ,t ∈

[tm˜ , xm˜ ), [xm˜ , tm˜ +1 ).

2.2. Let p ≥ 2 and suppose Al = −diag{Al (1), Al (2), . . . , Al (s)} −  is a nonsingular Mmatrix where p m p  2  λmin (B) Al (i) = −p max{η(i ), η(i ¯ )} p + kp aln + − 1 (p − 1). 2 2 λmax (B) n=1 Then, it has a stationary distribution for system (2). Proof. For any l ∈ M, i ∈ I, in virtue of that Al is a nonsingular M-matrix, we can find β l (i) > 0 and ψ l (i) > 0 satisfying ψl = Al βl , in which ψl = Definite Ml : RS × I → R1+ , (ψl (1), ψl (2), . . . , ψl (s) ), βl = (βl (1), βl (2), . . . , βl (s) ).  T  2p where Ml (zl , i) = βl (i) zl Bzl , (zl , i) ∈ Dlc × I and S

Dlc

   = (−ρ, ρ)c × (−ρ, ρ)c × · · · × (−ρ, ρ)c .

When tm˜ ≤ t < xm˜ , making use of Definition 2 and condition 2.1 yields that ⎡ ⎛ ⎞ m  k   T  2p −1 T ⎣zl B⎝Pl (zl , i) + LMl (zl , i) = βl (i) p zl Bzl α j (i )Clnj (i )N j (i )zn + Vl (zl , i)⎠ ⎤   1 + trace qlT (i)Bql (i) ⎦ 2

n=1 j=1

s p  &2   p −2 &  p + βl (i) p − 1 zlT Bzl 2 &zlT Bql (i)& + ϒih βl (h) zlT Bzl 2 2 h=1 p p   −1 p T 2 ≤ − η(i)βl (i) pλmax (B)|zl | + βl (i) p zl Bzl 2 ⎞⎤ ⎡ ⎛ m  k   T  1 j T × ⎣z l B ⎝ α j (i)Cln (i)N j (i)zn + trace ql (i)Bql (i) ⎠⎦ 2 n=1 j=1 s p  &2   2p −2 & T  p T & & zl Bql (i) + + βl (i) p − 1 zl Bzl ϒih βl (h) zlT Bzl 2 . 2 h=1

(9)

7274

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

Besides, we can deduce that ⎛ ⎞ m  k m  p  T  2p −1 T  j 2 ⎝ ⎠ βl (i) p zl Bzl zl B α j (i )Cln (i )N j (i )zn ≤ βl (i) pλmax (B)k aln |zl | p−1 |zn | n=1 j=1 p 2

≤ βl (i)kλmax (B)

m 

n=1

aln ( p|zl | p + |zn | p − |zl | p ),

(10)

n=1

  p  T  2p −1 1  T  1 2 βl (i) p zl Bzl trace ql (i)Bql (i) ≤ βl (i) pλmax (B)|zl | p−2 |ql (i)|2 2 2 p 1 2 ≤ βl (i)λmax (B)( (p − 2)|zl | p + 2|ql (i)| p ), 2

(11)

p  p  p &2  p −2 & 2 βl (i) p − 1 zlT Bzl 2 &zlT Bql (i)& ≤ βl (i) p − 1 λmax (B)|zl | p−2 |ql (i)|2 2 2 p  p 2 ≤ βl (i) p − 1 λmax (B)( (p − 2)|zl | p + 2|ql (i)| p ). 2 (12) Applying Eqs. (10), (11) and (12) into Eq. (9) implies that '   p m p  2  p λmin (B) 2 LMl (zl , i) ≤ βl (i)λmax (B) −pη(i) p + kp aln + − 1 (p − 1) 2 2 λmax (B) n=1 ( s  p 2 + λmax (B) ϒih βl (h) |zl | p h=1 p 2

+ βl (i)kλmax (B)

m 

p

2 aln (|zn | p − |zl | p ) + βl (i)λmax (B)(p − 1)|ql (i)| p

n=1

 −ψl (i)|zl | p +

m 

Aln Qln (zl , zn ) + Cl (i),

n=1 p

p

2 2 where Aln = βl kλmax (B)aln , βl = max{βl (i)}, Cl (i) = βl (i)λmax (B)(p − 1)|ql (i)| p , Qln (zl , p p zn ) = |zn | − |zl | and '   p m p  2  p λmin (B) 2 ψl (i) = − βl (i)λmax (B) −pη(i) p + kp aln + − 1 (p − 1) 2 2 λmax (B) n=1 ( s  p 2 + λmax (B) ϒih βl (h) .

h=1

According to condition 2.1, one can get ψ l (i) > 0. When xm˜ ≤ t < tm˜ +1 , by repeating the aforementioned procedure, it can be expressed as follows m  LMl (zl , i) ≤ −ψ¯ l (i)|zl | p + Aln Qln (zl , zn ) + Cl (i), n=1

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

in which

'



p

p 2

ψ¯ l (i) = − βl (i)λmax (B) −pη(i) ¯ p 2

+ λmax (B)

s 

(

2 λmin (B) p

2 λmax (B)

+ kp

m  n=1

aln +

p 2



7275



− 1 (p − 1)

ϒih βl (h) > 0.

h=1

In view of above calculations, we can easily get Assumption 1.1 holds. In addition, by p 2 the definition of Ml (zl , i), it is not difficult to know that βl (i)λmin (B)|zl | p ≤ Ml (zl , i) ≤ p  2 βl (i)λmax (B)|zl | p , which means Assumption 1.2 holds. And through (n,l )∈E (CQ ) Qln (zl , zn ) =  p p (n,l )∈E (CQ ) (|zn | − |zl | ) = 0, we can get Assumption 1.3 holds. Up to now, all conditions in Theorem 1 have been checked. Hence, system (2) has a stationary distribution. This completes the proof.  Remark 6. In fact, Theorem 1 is given in terms of Lyapunov functions of subnetworks. However, the construction of function Ml (zl , i) is sometimes not an easy work. Therefore, in Theorem 2, we establish some suitable conditions in the form of coefficients of the system based on Theorem 1 for the existence of a stationary distribution to verify the availability  p of Theorem 1. Specifically, we denote Ml (zl , i) = βl (i) zlT Bzl 2 , in which constant β l (i) and matrix B are not complicated to find. Remark 7. Through the proof of Theorems 1 and 2, we can conclude that the existing area of a stationary distribution is related to ρ, which has a close relationship with many factors in this paper, including stochastic disturbance, Markov chain, coupling strength, the dimension of coupled networks, control duration rate and control strength. When ql (i) = 0 and Pl (0, i) = Gl (0, i) = 0, there exists a trivial solution in system (2). Based on this, a corollary is shown as below. Corollary 1. For any l ∈ M, zl ∈ RS , if conditions 2.1 and 2.2 hold, the trivial solution of system (2) is stable. The proof of Corollary 1 can be completed in a similar way to that of Theorems 1 and 2, thus, it is omitted here. 3.2. pth moment exponential stability for HMWCN When in the absence of stochastic disturbance for system (2), in other words, stochastic disturbance gets to zero, system (2) can achieve exponential stability in pth moment. Theorem 3. Suppose digraph (G, (aln )m×m ) is strongly connected and there is p ≥ 0 satisfying 3.1. There exist constants η(1) (i), η(2) (i) and a positive definite matrix B such that zlT BPl (zl , i) ≤ η (1) (i)zlT Bzl ,

zlT BGl (zl , i) ≤ η (2) (i)zlT Bzl .

3.2. There is β l (i) > 0 and minl∈M {ψl(1) (i)} > 0, in which   m s   p p  (1)  2p (1) (2) 2 2 ψl (i) = −βl (i ) p η (i ) + η (i) λmin (B) + kλmax (B) aln − λmax (B) ϒih βl (h). n=1

h=1

7276

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

¯ where 3.3. There is β − ρψ ˆ > 0 and ρˆ = β + β, $ % ψl(1) (i) tm˜ −1 − xm˜ ψ = lim sup , β = min , p i∈I,l∈M 2 m˜ →∞ tm˜ −1 − tm˜ βl (i)λmax (B) and

'

ψl(2) (i)



= βl (i ) p η (1) (i ) + k

m 

 aln +

n=1

s 

$ β¯ = max

i∈I,l∈M

ψl(2) (i)

%

p

2 βl (i)λmin (B)

( p

2 ϒih βl (h) λmax (B).

h=1

Then, the trivial solution of system (2) is pth moment exponentially stable. Proof. Using the same theoretical technique of Theorem 2 and making Ml (zl , i) =  p βl (i) zlT Bzl 2 , we have p

p

2 2 βl (i)λmin (B)|zl | p ≤ Ml (zl , i) ≤ βl (i)λmax (B)|zl | p .

When tm˜ ≤ t < xm˜ , from condition 3.1, it follows that

 m  k   T  2p −1 T zl B Pl (zl , i) + LMl (zl , i) = βl (i) p zl Bzl α j (i )Clnj (i )N j (i )zn n=1 j=1

 +

+ Vl (zl , i)

s 

 p ϒih βl (h) zlT Bzl 2

h=1

 

m  p  (1)  2p (2) 2 ≤ βl (i) η (i) + η (i) λmin (B) + kλmax (B) p aln p 2

+ λmax (B)

s 

n=1



p 2

ϒih βl (h) + βl kλmax (B)

m 

aln (|zn | p − |zl | p )

n=1

h=1

 − ψl(1) (i)|zl | p +

m 

Aln Qln (zl , zn )

n=1

and from condition 3.3, we can compute −ψl(1) (i) < 0. When xm˜ ≤ t < tm˜ +1 , it shows that '   ( m s   p (1) 2 LMl (zl , i) ≤ λmax (B) βl (i) p η (i) + k aln + ϒih βl (h) |zl | p n=1 p 2

+ βl kλmax (B)

m 

h=1

aln (|zn | p − |zl | p )

n=1

 ψl(2) (i)|zl | p + Letting M(z, i) = p 2

m l=1

m 

μl Ml (zl , i), it is obvious to see that p

λmin (B) min {μl βl (i)}m (1− 2 ) i∈I,l∈M

Aln Qln (zl , zn ).

n=1



0

p

p

2 |z| p ≤ M(z, i) ≤ λmax (B) min {μl βl (i)}m (1− 2 )

i∈I,l∈M



0

|z| p .

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7277

Hence, when tm˜ ≤ t < xm˜ , LM(z, i) ≤ −

m 

μl ψl(1) (i)|zl | p +

l=1

%

ψl(1) (i) p

2 βl (i)λmax (B)

i∈I,l∈M

μl Aln Qln (zl , zn )

l=1 n=1

$

≤ − min

m  m 

M(z, i)  −βM(z, i).

(13)

When xm˜ ≤ t < tm˜ +1 , LM(z, i) ≤

m 

μl ψl(2) (i)|zl | p +

l=1

≤ max

i∈I,l∈M

m  m 

μl Aln Qln (zl , zn )

l=1 n=1

$

ψl(2) (i) p 2

βl (i)λmin (B)

%

¯ M(z, i)  βM(z, i).

Combining Eqs. (13) with (14) gives that  + D EM(z(t ), r(t )) ≤ −βEM (z(t ), r(t )), ¯ M(z(t ), r(t )), D+ EM(z(t ), r(t )) ≤ βE

(14)

tm˜ ≤ t < xm˜ , xm˜ ≤ t < tm˜ +1 .

Write Y (t ) = EM(z(t ), r(t )), W (t ) = Y (t ) exp (βt ), M0 = Y (0) = 0, in which Y(0) is the initial value when t = 0. ˜ ˜ For 0 ≤ t < x0 , making  Q (t ) = W (t ) − M0 yields that Q (t ) ≤ 0. For x0 ≤ t < t1 , denote ˆ , ρˆ = β + β¯ and we need to prove H(t) ≤ 0. Otherwise, there is H (t ) = W (t ) − M0 exp ρt P1 ∈ [x0 , t1 ) such that D+ H (P1 ) = D+W (P1 ) − ρW ˆ (P1 ) ≤ 0,   ˆ 1 . In conclusion, we which is contradictory, so it can be obtained that W (t ) ≤ M0 exp ρt get   ˆ 1 , 0 ≤ t < t1 . W (t ) ≤ M0 exp ρt   ˆ 1 − x0 ) , we need to prove H¯ (t ) ≤ 0. OtherFor t1 ≤ t < x1 , write H¯ (t ) = W (t ) − M0 exp ρ(t wise, there is P2 ∈ [t1 , x1 ) such that H¯ (P2 ) = 0, D+ H¯ (P2 ) > 0, which leads to a contradiction with H (P1 ) = 0,

D+ H (P1 ) > 0,

D+ H¯ (P2 ) = D+W (P2 ) ≤ βW (P2 ) − βW (P2 ) = 0.   ˆ 1 . For x1 ≤ t < t2 , similarly, we have So we can know that W (t ) ≤ M0 exp ρt   ˆ 1 − x0 ) + (t − x1 )) . W (t ) ≤ M0 exp ρ((t Applying induction, we can drop that when tm˜ ≤ t < xm˜ , ⎛ ⎞ m˜  W (t ) ≤ M0 exp ⎝ρˆ (t j˜ − x j˜−1 )⎠, j˜=1

when xm˜ ≤ t < tm˜ +1 , ⎛ ⎛ ⎞⎞ m˜  W (t ) ≤ M0 exp ⎝ρˆ ⎝ (t j˜ − x j˜−1 ) + (t − xm˜ )⎠⎠. j˜=1

7278

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

For tm˜ ≤ t < xm˜ , there is ⎛ ⎞ ⎛ ⎞ m˜ m˜     ˆ . W (t ) ≤ M0 exp ⎝ρˆ (t j˜ − x j˜−1 )⎠ ≤ M0 exp ⎝ρψ ˆ (t j˜ − t j˜−1 )⎠ ≤ M0 exp ρψt j˜=1

j˜=1

For xm˜ ≤ t < tm˜ +1 , there is ⎛ ⎞ m˜  W (t ) ≤ M0 exp ⎝ρˆ (t j˜ − x j˜−1 ) + (t − xm˜ )⎠ ⎛

j˜=1

≤ M0 exp ⎝ρψ ˆ

m˜  j˜=1

⎞ tm˜ +1 − xm˜ (t j˜ − t j˜−1 ) + (t − tm˜ )⎠ tm˜ +1 − tm˜

  ˆ . ≤ M0 exp ρψt

  ˆ , which indicates that For any t ≥ 0, it can be concluded that W (t ) ≤ M0 exp ρψt   ˆ )t . EM(z(t ), r(t )) ≤ M0 exp (−β − ρψ On the other hand, because p

p

2 λmin (B) min {μl βl (i)}m (1− 2 )



i∈I,l∈M

0

  ˆ )t , E|z (t )| p ≤ EM(z (t ), r(t )) ≤ M0 exp −(β − ρψ

we acquire E|z(t )| p ≤

M0 p 2

λmin (B) mini∈I,l∈M {μl βl (i)}m

 (1− 2p ) 0

  ˆ )t , exp −(β − ρψ

which further implies that lim supt→∞ 1t ln (E|z(t )| p ) ≤ −(β − ρψ ˆ ). Condition 3.3 tells that β − ρψ ˆ > 0, as a result of which the trivial solution of system (2) is pth moment exponentially stable. This completes the proof.  Remark 8. In Theorem 3, the stability is dependent on control strength η(2) and coupling strength aln according to conditions 3.1 and 3.2. When |η (1) + η (2) | is bigger while aln is smaller, then condition 3.2 can be satisfied easily. In the real world, owing to the economical cost, many significant problems are related to the supremum of rest width rate as ψ in AIC, which is shown in condition 3.3. It is obvious that 0 ≤ ψ < 1. If ψ = 0, it reduces to the usual continuous control. Therefore, without loss of generality, we always assume that 0 < ψ < 1 in the following. 4. Application results Coupled oscillators are intensively appearing in power systems and other fields. The stability of coupled oscillators has been widely studied by researchers (see [18,40]). While for deterministic coupled oscillators, multi-weight should be considered in virtue of its potential advantages in depicting the real world. Besides, it should be pointed out that environmental noise, such as stochastic disturbance and Markovian switching, may cause the existence of the equilibrium losing, which means the importance to survey the stationary distribution in this situation. Nevertheless, to our knowledge, few researchers have dealt with the stationary distribution for hybrid multi-stochastic-weight coupled oscillators (HMSWCO) and the exponential stability for hybrid multi-weight coupled oscillators (HMWCO).

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7279

Therefore, in this section, as an application to our main results, we consider a model of HMSWCO on a digraph (G, (aln )m×m ) with m (m ≥ 2) vertices. By network split method, coupled oscillators network can be split into k sub-networks with different weights. For any l ∈ M, each independent vertex is assigned by the following second order oscillator with Markovian switching ˜ l (zl , r(t ))z˙l + zl + z¨l + 

m  k 

α j (r(t ))Clnj (r(t ))N j (r(t ))zn = 0,

(15)

n=1 j=1

where for the jth sub-network, coupling, Nj (r(t)) indicates the cient, in which   (z , r(t )), ˜ l (zl , r(t )) = ¯ l l l (zl , r(t )),

α j (r(t)) stands for the coupling strength, Clnj (r(t )) is the outer ˜ l : R1 × I → R1 stands for damping coeffiinner coupling,  t ∈ [tm˜ , xm˜ ), t ∈ [xm˜ , tm˜ +1 ).

Making a transformation z˜l = z˙l + l zl with constant ϖl > 0 and considering stochastic disturbance and AIC into system (15), for any l ∈ M, we come up with HMSWCO via AIC as ⎧ . ⎪ dzl = z˜l − l zl + ul(1) (zl , r(t )) dt + fl(1) (r(t ))dwl(1) , ⎪ ⎪ ⎪ ⎪ ⎪

 ⎪    ⎪ ⎨ ˜ l (zl , r(t ) z˜l + l  ˜ l (zl , r(t )) −  2 − 1 zl + u(2) (z˜l , r(t )) dz˜l = l −  l l (16) ⎪ ⎪  m  k ⎪  ⎪ ⎪ ⎪ ⎪ − α j (r(t ))Clnj (r(t ))N j (r(t ))zn dt + fl(2) (r(t ))dwl(2) , ⎪ ⎩ n=1 j=1

where fl(1) , fl(2) represent the stochastic disturbance intension existing in the lth vertex and ul(1) , ul(2) : R1 × I → R1 stand for AIC. For brevity, we make Zl = (zl , z˜l )T , Ul (Zl , r(t )) =  T ul(1) (zl , r(t )), ul(2) (z˜l , r(t )) and  −Ul (r(t ))Zl , t ∈ [tm˜ , xm˜ ), Ul (Zl , r(t )) = 0, t ∈ [xm˜ , tm˜ +1 ). In addition, we define the supremum of rest width rate as ψ, in which ψ = −xm˜ lim supm˜ →∞ ttmm˜˜−1 . −1 −tm˜ 4.1. Stationary distribution for HMSWCO To start with, we give a theorem to illustrate it has a stationary distribution for system (16). Theorem 4. Let aln = max1≤ j≤k,i∈I {α j (i)Clnj (i)N j (i)} and assume that digraph (G, (aln )m×m ) is strongly connected. For any i ∈ I, l ∈ M, if the following conditions hold   4.1. There are positive constants M˜ l (i), M¯ l (i), ml (i) such that M¯ l (i) − l l ≤ 1 and 2l ≤ l (zl , i ) ≤ M˜ l (i ),

¯ l (zl , i ) ≤ M¯ l (i ). 2l ≤ ml (i ) ≤ 

7280

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

4.2. It holds that 1 l (M˜ l (i) − l ) − Ul (i) < 0. 2 4.3. Let p ≥ 2 and suppose that Al = −diag{Al (1), Al (2), . . . , Al (s)} −  is a nonsingular M-matrix, where  m  1 / p   1  Al (i) = −p max Ul (i) − l M˜ l (i) − l , l + k p aln + − 1 (p − 1). 2 2 2 n=1 Then, it has a stationary distribution for system (16). Proof. Let B be an identity matrix and denote      T ˜ l (zl , i) z˜l + l  ˜ l (zl , i) − l2 − 1 zl . Pl (Zl , i) = z˜l − l zl , l −  When tm˜ ≤ t < xm˜ , making use of condition 4.1 yields that ZlT (Pl (Zl , i) + Ul (Zl , i) ) = zl z˜l − l zl2 − Ul |Zl |2 + (l − l (zl , i) )z˜l2   + l l (zl , i) − l2 − 1 zl z˜l ≤ −l zl2 − Ul (i)|Zl |2 + (l − l (zl , i))z˜l2 & & 1& 1& + &l l (zl , i) − l2 &zl2 + &l l (zl , i) − l2 &z˜l2 2  2 1 ≤ l (M˜ l (i) − l ) − Ul (i) |Zl |2 . 2 From condition 4.2, we get 21 l (M˜ l (i) − l ) − Ul (i) < 0. When xm˜ ≤ t < tm˜ +1 , by utilizing condition 4.1, it holds that     ¯ l (zl , i) z˜l2 + l  ¯ l (zl , i) − l2 − 1 zl z˜l ZlT (Pl (Zl , i) + Ul (Zl , i)) = zl z˜l − l zl2 + l −  & &   ¯ l (zl , i) z˜l2 + 1 &l  ¯ l (zl , i)) − l2 &εl2 zl2 ≤ −l zl2 + l −  2 & 1 && ¯ 2& 2 + 2 l l (zl , i) − l z˜l . 2εl Then making εl2 = l shows that      2 2   1 ¯ l 2 T ¯ Zl (Pl (Zl , i) + Ul (Zl , i)) = −l 1 − l (zl , i) − l εl zl − l (zl , i) − l 1 − 2 z˜l 2 2ε     l  1 l ≤ −l 1 − M¯ l (i) − l l zl2 − (ml (i) − l ) 1 − 2 z˜l2 2 2εl 1 ≤ − l | Zl | 2 , 2 which means condition 2.1 holds. Then, we can easily verify that condition 2.2 holds by condition 4.3. Therefore, we have checked all conditions in Theorem 2, which indicates that system (16) has a stationary distribution. This completes the proof. 

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7281

4.2. pth moment exponential stability for HMWCO When fl(1) = fl(2) = 0, the exponential stability in pth moment for system (16) is considered. Before giving the main theorem, we need to introduce the following assumptions. 5.1. There exists p ≥ 2 and minl∈M {ψl(1) (i)} > 0, in which   & 1& ˜ 2& ˜ ψl(1) (i) = −βl (i) p max − l + &l  l (zl , i) − l , l − l (zl , i) 2 /   m s  & 1& ˜ 2& − Ul (i) + k + &l  aln − ϒih βl (h). l (zl , i) − l 2 n=1 h=1 β − ρψ ˆ >0 " (2) # ψl (i) maxi∈I,l∈M βl (i) and

5.2. There

is

and

¯ ρˆ = β + β,

where

β = mini∈I,l∈M

"

ψl(1) (i) βl (i)

#

, β¯ =

  & 1& ˜ 2& ˜ ψl(2) (i) = βl (i) p max − l + &l  l (zl , i) − l , l − l (zl , i) 2 /   m s  & 1& ˜ 2& + + + &l  (z , i) −  k a ϒih βl (h). l l ln l 2 n=1 h=1 Theorem 5. Let digraph (G, (aln )m×m ) be strongly connected. If Assumptions 5.1 and 5.2 hold, then the trivial solution of system (16) is pth moment exponentially stable. Proof. Use the notations which are similar to Theorem 4 and set B as an identity matrix, we can deduce that     ˜ l (zl , i) z˜l2 + l  ˜ l (zl , i) − l2 − 1 zl z˜l ZlT BPl (Zl , i) = zl z˜l − l zl2 + l −      & 2 & 2 1& ˜ 1 && ˜ 2& 2& ˜ z   ≤ −l + &l  (z , i) −  + −  (z , i) +  (z , i) −  z˜l l l l l l l l l l l l 2 2 /  & & 1& ˜ 1 && ˜ 2& 2& ˜ ,  | Zl | 2 ≤ max − l + &l  (z , i) −   −  (z , i) +  (z , i) −  l l l l l l l l l l 2 2 and ZlT BUl (Zl , i) = −Ul (i)|Zl |2 , which indicates condition 3.1 holds. Ultimately, we can estimate that conditions 3.2 and 3.3 are also satisfied in the view of Assumptions 5.1 and 5.2. Hence, we show all conditions in Theorem 3 have been checked, as a result, the trivial solution of system (16) is pth moment exponentially stable. The proof is completed.  Remark 9. In [5], Zhang and Chen presented some sufficient conditions to ensure the moment exponential stability and almost surely exponential stability for stochastic coupled networks. In this paper, when considering the Markovian switching and white noise at the same time, the existence of the equilibrium is destroyed, which may lead to the existence of a stationary distribution. As a special case, in the absence of stochastic perturbations, coupled networks may not achieve stability while taking the AIC strategy, the trivial solution is pth moment exponentially stable.

7282

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

Fig. 1. A coupled oscillators network with four oscillators and four sub-networks after split.

Remark 10. When tacking stochastic coupled networks with multi-weights and Markovian switching, we cannot merely combine the sides with different natures to one side. One valid way is that the network links with different natures are split into different sub-networks. Taking a coupled oscillators network with four oscillators as an example which will be presented in Section 5 (see Fig. 1), there are two Markov chain states and two different natures for each state. After splitting, we can get four sub-networks and the common method for dealing with the coupled networks can be applicable. We refer the reader to [6] to get more details about the network split. In addition, each sub-network may be connected or disconnected, which means sub-networks we garner are not acquired to be strongly connected. While in multi-group coupled networks [34,35], each outer coupling matrix (Clnj )S×S is irreducible. This condition is weakened in our model. 5. Numerical simulations In this section, we present two numerical examples to illustrate the effectiveness of our results in this article. Example A. Stationary distribution for HMSWCO We tend to show it deed has a stationary distribution for system (16) in Theorem 4. For brevity, consider a digraph with m vertices, where m = 4 and it is assumed that Markov chain

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7283

r(t) taking values in I = {1, 2} with generator   2.4 −2.4 .  = (ϒih )2×2 = 2.2 −2.2 By the method of network split, we can get Fig. 1 (a coupled oscillators network with four oscillators and four sub-networks after split), in which there are two different kinds of weights, i.e., k = 2 for each Markovian switching. Then we select ⎛ ⎞ ⎛ ⎞ 0 0 0 0 0 0.20 0 0 ⎜0.15 ⎜ 0  1    0 0.05 0⎟ 0 0 0 ⎟ ⎟, C 2 (1) ⎜ ⎟, Cln (1) 4×4 = ⎜ = ln 4×4 ⎝ 0 ⎝ 0 0.05 0 0⎠ 0 0 0.10⎠ 0 0 0.15 0 0.10 0 0 0 ⎛ ⎞ ⎛ ⎞ 0 0 0 0 0 0. 0 0.20 ⎜ 0 ⎜ 0  1    0 0 0⎟ 0 0.20 0 ⎟ ⎟, C 2 (2) ⎟. Cln (2) 4×4 = ⎜ =⎜ ln 4×4 ⎝0.15 ⎝0.20 0 0 0 0⎠ 0 0.10⎠ 0 0.20 0.10 0 0 0 0 0 At the same time, take N1 (1) = 0.5, N2 (1) = 1, N1 (2) = 2, N2 (2) = 2, α1 (1) = 0.4, α2 (1) = 0.1, α1 (2) = 0.1, α2 (2) = 0.05. According to aln = max j∈{1,2},i∈{1,2} {α j (i)Clnj (i)N j (i)}, we can get ⎛ ⎞ 0 0.02 0 0.02 ⎜0.03 0 0.03 0 ⎟ ⎟, (aln )4×4 = ⎜ ⎝0.03 0.01 0 0.01⎠ 0.01 0.04 0.03 0 which means (G, (aln )4×4 ) is strongly connected. For l ∈ {1, 2, 3, 4}, choose ¯ l (1) = 2.3–1.5 sin zl ,  ¯ l (2) = 2.6–1.8 sin zl , l (2) = 1.3 − 0.8 sin zl ,  l (1) = 1 − 0.5 sin zl ,

and Ul (1) = 0.2, Ul (2) = 0.3. By simple calculation, for l ∈ {1, 2, 3, 4}, it is not complicated to get that l = 0.25, M˜ l (1) = 3.5, M¯ l (1) = 1.5, ml (1) = 0.7, M˜ l (2) = 4.1, M¯ l (2) = 2.1, ml (2) = 0.8. Furthermore, we can verify   M¯ l (1) − l l ≤ 1,   M¯ l (2) − l l ≤ 1,

1  (M˜ l (1) 2 l

− l ) − Ul (1)

< 0,

1  (M˜ l (2) 2 l

− l ) − Ul (2)

<0

are satisfied, which shows that conditions 4.1 and 4.2 hold. Besides, we let p = 3, by calculation based on the data above, we can attain the values of Al (i) shown as A1 (1) = 0.8650, A2 (1) = 0.9850, A3 (1) = 0.9250, A4 (1) = 1.1050, A1 (2) = 0.7337, A2 (2) = 0.8537, A3 (2) = 0.7937, A4 (2) = 0.9737. Then, we have

7284

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289 10 -3 15

0.06 0.05 0.04

10

0.03 0.02

5

0.01 0 10

5

0

-5

y1(t)

-10

-10

0

-5

5

15

10

x1(t)

0 15

10

5

0

y2(t)

-5

-10

-15

-20

-20

-10

0

20

10

x2(t)

0.06

0.08

0.05 0.06

0.04 0.03

0.04

0.02 0.02 0.01 0 10

0 5

0

y3(t)

-5

-10

-15

-20

-10

0

10

x3(t)

20

5

0

y4(t)

-5

-10

-5

0

5

x4(t)

Fig. 2. The stationary distribution of system (16) with the initial values (17) when ϕ = 0.7.

 1.535 A1 = −1  1.295 A4 = −1

  −1 1.415 , A2 = 1.4663 −1  −1 , 1.2263

  −1 1.475 , A3 = 1.3463 −1

 −1 , 1.4063

which indicates that the matrices above are all nonsingular M-matrices. Therefore, condition 4.3 holds. Up till now, all the conditions in Theorem 4 have been checked. Besides, the stochastic perturbation intension can be selected as follows f1(1) (1) = 2.5, f1(2) (1) = 4, f1(1) (2) = 3, f1(2) (2) = 3.5, f2(1) (1) = 4.8, f2(2) (1) = 5.5, f2(1) (2) = 4.6, f2(2) (2) = 6.4, f3(1) (1) = 5, f3(2) (1) = 6.9, f3(1) (2) = 5.8, f3(2) (2) = 4.5, f4(1) (1) = 1.1, f4(2) (1) = 2.3, f4(1) (2) = 1.5, f4(2) (2) = 3.8. Let z1 (0) = x1 (0), z˜1 (0) = y1 (0), z2 (0) = x2 (0), z˜2 (0) = y2 (0), z3 (0) = x3 (0), z4 (0) = x4 (0), z˜4 (0) = y4 (0) and the initial values be chosen as x1 (0) = 15, y1 (0) = −2; x2 (0) = 10, y2 (0) = 2; x3 (0) = −4, y3 (0) = −1.5; x4 (0) = −15, y4 (0) = 4.

(17)

In order to further verify the effects of AIC for the existing area of a stationary distribution in Theorem 4, we firstly choose ϕ = 0.7, then the possibility density with the initial values (17) is shown in Fig. 2. Secondly, we choose ϕ = 0.1. The possibility density with the initial values (17) is shown in Fig. 3. In fact, we can easily observe that system (16) has a stationary distribution through Figs. 2 and 3. On the other hand, compared Fig. 2 with Fig. 3, we can find that the possibility density is variable with different ϕ. Furthermore, the larger ϕ we select, the smaller existing area of a stationary distribution will be, which indicates the importance of AIC. Therefore, the numerical example shows the effectiveness and feasibility of the developmental results. Example B. pth moment exponential stability for HMWCO In the following example, consider HMWCO consisting of 28 oscillators when stochastic disturbance tends to 0 and generator is chosen as follows   0.024 −0.024 .  = (ϒih )2×2 = 0.018 −0.018

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7285

10 -3 15

0.06 0.05 0.04

10

0.03 0.02

5

0.01 0 10

5

0

-5

y1(t)

-10

-10

0

-5

5

10

0 15

15

10

5

0

x1(t)

y2(t)

-5

-10

-15

-20

-10

-20

20

10

0

x2(t)

0.06 0.08 0.05 0.06

0.04 0.03

0.04

0.02 0.02

0.01 0

0 10

5

0

y3(t)

-5

-10

-15

-20

-10

0

10

20

5

x3(t)

0

-5

y4(t)

-10

-5

0

5

x4(t)

Fig. 3. The stationary distribution of system (16) with the initial values (17) when ϕ = 0.1. Table 1 The values of Clnj (1) ( j = 1, 2) for two kinds of weights in state 1. 1 (1) Cln

C31,1 (1) C51,4 (1) 1 (1) C2, 6 1 C27 ,10 (1) 1 C15 ,16 (1) 1 C22, 23 (1)

Values 1 (1) C4, 1 C11,4 (1) 1 (1) C11 ,6 C51,11 (1) 1 C18 ,17 (1) 1 C24, 23 (1)

C51,1 (1) C91,4 (1) C61,7 (1) 1 C13 ,11 (1) 1 C16 ,18 (1) 1 C23 ,25 (1)

C61,1 (1) C11,5 (1) 1 (1) C2, 8 1 C11 ,12 (1) C81,19 (1) 1 C10, 26 (1)

C71,2 (1) C61,5 (1) C31,9 (1) 1 C12, 13 (1) 1 C19 ,20 (1) 1 C26 ,27 (1)

C11,3 (1) 1 (1) C10, 5 1 (1) C23 ,9 1 C13 ,14 (1) 1 C20, 21 (1) 1 C28 ,27 (1)

1 (1) C4, 3 C11,6 (1) 1 (1) C4, 10 C71,15 (1) C91,22 (1) 1 C27 ,28 (1)

2 (1) Cln

0.06 0.013 0.025 0.04 0.036 0.048 values

2 (1) C2, 1

C32,1 (1)

C52,1 (1)

C62,1 (1)

C32,2 (1)

C12,2 (1)

C72,2 (1)

0.02

C82,3 (1)

C52,4 (1)

C12,4 (1)

C92,4 (1)

C12,5 (1)

C62,5 (1)

2 (1) C10, 5

0.025

C12,6 (1) 2 (1) C4, 10 2 C13 ,14 (1) 2 C21 ,20 (1)

2 (1) C2, 6 2 C27 ,10 (1) C72,15 (1) C92,22 (1)

2 (1) C11 ,6 C52,11 (1) 2 C15 ,16 (1) 2 C22, 23 (1)

C62,7 (1) 2 C13 ,11 (1) 2 C17 ,16 (1) 2 C24, 23 (1)

2 (1) C16 ,7 2 C11 ,12 (1) 2 C18 ,17 (1) 2 C25 ,24 (1)

2 (1) C2, 8 2 C12, 13 (1) C82,19 (1) 2 C26 ,27 (1)

2 (1) C20, 8 2 C14, 13 (1) 2 C19 ,20 (1) 2 C28 ,27 (1)

0.01 0.032 0.015 0.023

Suppose that Markov chain r(t) taking values in I = {1, 2} and there are two different kinds of weights, i.e., k = 2 for each network switching. Then we select N1 (1) = 0.2, α1 (1) = 0.1,

N2 (1) = 0.25, α2 (1) = 0.2,

N1 (2) = 0.1, α1 (2) = 0.25,

N2 (2) = 0.05, α2 (2) = 0.2.

Some values of Clnj (i), in which j ∈ {1, 2}, i ∈ {1, 2} can be seen in Tables 1, 2 and others are all zero. Therefore, we can see that digraph (G, (aln )28×28 ) is strongly connected apparently, which is shown in Fig. 4. For l ∈ {1, 2, . . . , 28}, we let p = 2 and other parameters are taken as βl (1) = 1, βl (2) = 1.05,

l (1) = 0.008 − 0.003 sin zl , l (2) = 0.006 − 0.001 sin zl ,

¯ l (1) = 0.0075 − 0.0025 sin zl ,  ¯ l (2) = 0.007 − 0.0025 sin zl . 

7286

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

Table 2 The values of Clnj (2) ( j = 1, 2) for the two kinds of weights in state 2. 1 (2) Cln 1 (2) C2, 1 C11,3 (2) C11,6 (2) C31,9 (2) 1 C15 ,16 (2) 1 C9,22 (2)

Values C31,1 (2) 1 (2) C4, 3 1 (2) C2, 6 1 (2) C23 ,9 1 C17 ,16 (2) 1 C24,23 (2)

1 (2) C4, 1 C81,3 (2) 1 (2) C11 ,6 C51,11 (2) 1 C18 ,17 (2) 1 C25,24 (2)

C61,1 (2) C91,4 (2) C61,7 (2) 1 C13 ,11 (2) 1 C16 ,18 (2) 1 C23,25 (2)

C31,2 (2) C11,5 (2) 1 (2) C16 ,7 1 C12, 13 (2) 1 C19 ,20 (2) 1 C10,26 (2)

C11,2 (2) C61,5 (2) 1 (2) C2, 8 1 C14, 13 (2) 1 C21 ,20 (2) 1 C26,27 (2)

C71,2 (2) 1 (2) C10, 5 1 (2) C20, 8 1 C13 ,14 (2) 1 C20, 21 (2) 1 C27,28 (2)

2 (2) Cln

0.04 0.056 0.08 0.05 0.02 0.034 values

2 (2) C2, 1

C32,1 (2)

2 (2) C4, 1

C52,1 (2)

C62,1 (2)

C12,2 (2)

C32,2 (2)

0.02

C72,2 (2)

C12,3 (2)

2 (2) C4, 3

C82,3 (2)

C52,4 (2)

C12,4 (2)

2 (2) C10, 5

0.016

2 C16 ,07 (2)

2 (2) C2, 8

2 (2) C20, 8

C32,9 (2)

2 (2) C23 ,9

2 (2) C4, 10

C52,11 (2)

0.075

2 C13 ,11 (2) 2 C16 ,18 (2) 2 C24, 23 (2)

2 C11 ,12 (2) C82,19 (2) 2 C25 ,24 (2)

2 C12, 13 (2) 2 C19 ,20 (2) 2 C23 ,25 (2)

2 C14, 13 (2) 2 C21 ,20 (2) 2 C10, 26 (2)

C72,15 (2) 2 C20, 21 (2) 2 C26 ,27 (2)

2 C15 ,16 (2) C92,22 (2) 2 C28 ,27 (2)

2 C17 ,16 (2) 2 C22, 23 (2) 2 C27 ,28 (2)

0.025 0.043 0.038

Fig. 4. A strongly connected digraph (G, (aln )28×28 ) with 28 vertices.

When we do not take control strategy into system (16), we can get the mean square figure of system (16), shown in Fig. 5, which indicates that system (16) can not achieve stability. In order to make system (16) achieve stability, we introduce AIC. Specifically, we choose Ul (1) = 0.2, Ul (2) = 0.3, for l ∈ {1, 2, . . . , 28} and the supremum of rest width rate ψ = 0.5. Subsequently, we can derive that " # " # min ψl(1) (1) = 0.5753, min ψl(1) (2) = 0.8077, β = 0.3753, β¯ = 0.0307, ρˆ = 0.4060. l∈M

l∈M

As a result, it follows easily that minl∈M {ψl(1) (1)} > 0, minl∈M {ψl(1) (2)} > 0, β − ρψ ˆ > 0.

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7287

E|zl|2 , l=1,2,...,28

800

600

400

200

0 0

1

2

3

4

5

6

7

8

9

10

9

10

t Fig. 5. The mean square figure of system (16) without control. 300

E|zl|2 , l=1,2,...,28

250 200 150 100 50 0 0

1

2

3

4

5

6

7

8

t Fig. 6. The mean square figure of system (16) with control.

Till now, all conditions in Theorem 5 are satisfied. Then the mean square figure of system (16) is shown in Fig. 6. Through adapting the AIC scheme, we can know that the trivial solution in system (16) can realize stability. 6. Conclusions In this paper, the stationary distribution for HMSWCN was achieved and as a special case, the exponential stability in pth moment for HMWCN was also investigated, when stochastic disturbance got to zero. Based on the Lyapunov method, M-matrix and Kirchhoff’s Matrix Tree Theorem in the graph theory, several theorems were given to ensure the existence of a stationary distribution and exponential stability. Sufficient conditions reflected that the existing area of a stationary distribution was related to the topological structure of coupled networks and AIC. Besides, compared with [34,35], it was unnecessary to require every subnetwork to be strongly connected. Furthermore, an application to HMSWCO was presented to show the feasibility of theoretical results. In our follow-on work, we will try to remove the assumption

7288

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

of the strong connectedness of the maximum subnetwork, which can be found in [41,42], and apply the developmental results to more practical problems. Acknowledgments The authors thank the reviewers and the editor for their valuable comments. This work was supported by Shandong Province Natural Science Foundation (Nos. ZR2018MA005, ZR2018MA020, ZR2017MA008); the Key Project of Science and Technology of Weihai (No. 2014DXGJMS08), the Innovation Technology Funding Project in Harbin Institute of Technology (No. HIT.NSRIF.201703) and the Project of Shandong Province Higher Educational Science and Technology Program of China (No. J18KA218). References [1] A. Kikkawa, Random matrix analysis for gene interaction networks in cancer cells, Sci. Rep. 8 (2018) 10607. [2] A. Zakari, S. Lee, C. Chong, Simultaneous localization of software faults based on complex network theory, IEEE Access 6 (2018) 23990–24002. [3] M. Gelbrecht, N. Boers, J. Kurths, A complex network representation of wind flows, Chaos 27 (2017) 035808. [4] N. Katchinskiy, H. Goez, I. Dutta, R. Godbout, A. Elezzabi, Novel method for neuronal nanosurgical connection, Sci. Rep. 6 (2016) 20529. [5] C. Zhang, T. Chen, Exponential stability of stochastic complex networks with multi-weights based on graph theory, Phys. A 496 (2018) 602–611. [6] N. Li, H. Sun, X. Jing, Q. Zhang, Exponential synchronisation of united complex dynamical networks with multi-links via adaptive periodically intermittent control, IET Contr. Theory Appl. 7 (2013) 1725–1736. [7] M. Luo, X. Liu, S. Zhong, J. Cheng, Synchronization of multi-stochastic-link complex networks via aperiodically intermittent control with two different switched periods, Phys. A 509 (2018) 20–38. [8] P. Wang, W. Jin, H. Su, Synchronization of coupled stochastic complex-valued dynamical networks with time– varying delays via aperiodically intermittent adaptive control, Chaos 4 (2018) 043114. [9] Z. Chao, W. Feng, X. Wen, L. Zu, Stationary distribution of a stochastic predator-prey model with distributed delay and higher order perturbations, Phys. A 521 (2019) 467–475. [10] J. Yu, M. Liu, Stationary distribution and ergodicity of a stochastic food-chain model with Lévy jumps, Physi. A 482 (2017) 14–28. [11] W. Guo, Y. Cai, Q. Zhang, W. Wang, Stochastic persistence and stationary distribution in an SIS epidemic model with media coverage, Phys. A 492 (2018) 2220–2236. [12] V. Sukumaran, U. Mukherji, Asymptotic bounds on the power-delay tradeoff for fading point-to-point links from geometric bounds on the stationary distribution of the queue length, IEEE Trans. Inf. Theory 61 (2015) 6145–6167. [13] J. Heninger, D. Lippolis, P. Cvitanovi´c, Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system, Phys. Rev. E 92 (2015) 062922. [14] B. Hassibi, M. Hansen, A. Dimakis, H. Alshamary, W. Xu, Optimized Markov chain Monte Carlo for signal detection in MIMO systems: an analysis of the stationary distribution and mixing time, IEEE Trans. Signal Process. 62 (2014) 4436–4450. [15] X. Guo, J. Luo, Stationary distribution and extinction of SIR model with nonlinear incident rate under Markovian switching, Phys. A 505 (2018) 471–481. [16] C. Zhu, G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim. 4 (2007) 1155–1179. [17] X. Zhang, D. Jiang, A. Alsaedi, T. Hayat, Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching, Appl. Math. Lett. 59 (2016) 87–93. [18] M.Y. Li, Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ. 248 (2010) 1–20. [19] H. Guo, M.Y. Li, Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Am. Math. Soc. 136 (2008) 2793–2802. [20] Y. Liu, W. Li, J. Feng, Graph-theoretical method to the existence of stationary distribution of stochastic coupled systems, J. Dyn. Differ. Equ. 30 (2018) 667–685.

Y. Liu, H. Xu and W. Li / Journal of the Franklin Institute 356 (2019) 7263–7289

7289

[21] W. Huang, M. Ji, Z. Liu, Y. Yi, Steady states of Fokker–Planck equations: I. existence, J. Dyn. Differ. Equ. 27 (2015) 721–742. [22] I. Chen, C. Li, T. Huang, Global stabilization of memristor-based fractional-order neural networks with delay via output-feedback control, Mod. Phys. Lett. B 31 (2017) 1750031. [23] X. Yang, J. Cao, J. Lu, Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control, IEEE Trans. Circuits Syst. I Regul. Pap. 59 (2012) 371–384. [24] X. Yang, J. Cao, J. Qiu, pth moment exponential stochastic synchronization of coupled memristor-based neural networks with mixed delays via delayed impulsive control, Neural Netw. 65 (2015) 80–91. [25] R. Samidurai, S. Senthilraj, Q. Zhu, R. Raja, W. Hu, Effects of leakage delays and impulsive control in dissipativity analysis of Takagi–Sugeno fuzzy neural networks with randomly occurring uncertainties, J. Frankl. Inst. Eng. Appl. Math. 354 (2017) 3574–3593. [26] J. Mei, M. Jiang, Z. Wu, Periodically intermittent controlling for finite-time synchronization of complex dynamical networks, Nonlinear Dyn. 79 (2015) 295–305. [27] L. Li, Z. Tu, J. Mei, Finite-time synchronization of complex delayed networks via intermittent control with multiple switched periods, Nonlinear Dyn. 85 (2016) 375–388. [28] B. Zhang, F. Deng, S. Peng, S. Xie, Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system, J. Frankl. Inst. Eng. Appl. Math. 355 (2018) 3829–3852. [29] B. Guo, Y. Xiao, C. Zhang, Graph-theoretic approach to exponential synchronization of coupled systems on networks with mixed time-varying delays, J. Frankl. Inst. Eng. Appl. Math. 354 (2017) 5067–5090. [30] L. Cheng, X. Chen, J. Qiu, J. Lu, J. Cao, Aperiodically intermittent control for synchronization of switched complex networks with unstable modes via matrix omega-measure approach, Nonlinear Dyn. 92 (2018) 1091–1102. [31] X. Liu, T. Chen, Synchronization of complex networks via aperiodically intermittent pinning control, IEEE Trans. Autom. Control 60 (2015a) 3316–3321. [32] X. Liu, T. Chen, Synchronization of nonlinear coupled networks via aperiodically intermittent pinning control, IEEE Trans. Neural Netw. Learn. Syst. 26 (2015b) 113–126. [33] Q. Gan, Exponential synchronization of generalized neural networks with mixed time-varying delays and reaction-diffusion terms via aperiodically intermittent control, Chaos 27 (2017) 013113. [34] M.Y. Li, Z. Shuai, C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl. 361 (2010) 38–47. [35] Y. Guo, Y. Wang, X. Ding, Global exponential stability for multi-group neutral delayed systems based on Razumikhin method and graph theory, J. Frankl. Inst. Eng. Appl. Math. 355 (2018) 3122–3144. [36] D.B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. [37] X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. [38] Y. Cai, J. Jiao, Z. Gui, Y. Liu, W. Wang, Environmental variability in a stochastic epidemic model, Appl. Math. Comput. 329 (2018) 210–226. [39] M. Liu, Y. Zhu, Stationary distribution and ergodicity of a stochastic hybrid competition model with Lévy jumps, Nonlinear Anal. Hybrid Syst. 30 (2018) 225–239. [40] Y. Wu, S. Yan, M. Fan, W. Li, Stabilization of stochastic coupled systems with Markovian switching via feedback control based on discrete-time state observations, Int. J. Robust Nonlinear Control 28 (2018) 247–265. [41] Y. Liu, W. Li, J. Feng, The stability of stochastic coupled systems with time-varying coupling and general topology structure, IEEE Trans. Neural Netw. Learn. Syst. 29 (2018) 4189–4200. [42] H. Zhou, W. Li, Synchronisation of stochastic-coupled intermittent control systems with delays and Lévy noise on networks without strong connectedness, IET Contr. Theory Appl. 13 (2019) 36–49.