Aperiodically intermittent control for stabilization of random coupled systems on networks with Markovian switching

Neurocomputing 373 (2020) 1–14

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Aperiodically intermittent control for stabilization of random coupled systems on networks with Markovian switching Mengxin Wang, Jia Guo, Wenxue Li∗ Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, PR China

a r t i c l e

i n f o

Article history: Received 20 February 2019 Revised 16 June 2019 Accepted 8 September 2019 Available online 24 September 2019 Communicated by Dr. Lei Zou Keywords: Aperiodically intermittent control Random coupled systems Exponential stability in pth moment Global asymptotical stability in probability Markovian switching

a b s t r a c t Throughout this paper, the stabilization problem of random coupled systems on networks with Markovian switching (RCSNMS) is considered via aperiodically intermittent control. It is worth noting that aperiodically intermittent control is firstly utilized to stabilize random systems which are derived by more general noise. On the basis of Lyapunov method, graph theory together with some new stochastic analysis techniques, several stability criteria are derived. Different from most existing stability results on random systems in the literature which are mainly noise-to-state stability, we consider global asymptotical stability in probability and exponential stability in pth moment of RCSNMS. Then an application of the obtained results for a class of random coupled oscillators with Markovian switching via aperiodically intermittent control is presented. Finally, two numerical examples are concerned to demonstrate the validity and feasibility of the theoretical results.

1. Introduction In recent years, coupled systems on networks (CSN), as a special kind of complex systems, have attracted a great quantity of attention due to their wide range of applications in many fields, including physics, epidemiology, neural networks and so on (see, e.g., [1–3]). Thus, it is feasible to use CSN to model systems in real world [4–6]. However, in view of the complexity and variability of the external environment, such as component failures or repairs and so on, many practical systems may experience abrupt changes in their parameters or structures. Continuous-time Markov chains can be used to model such abrupt changes. Up to now, Markovian switching factors have been widely considered to ecosystems, power systems, economic systems and so on. Taking the economy model [7] as an example, assume that the economy state could be roughly lumped into three operation modes (slump, normal and boom), then the switching among them is modeled as a Markov chain. Therefore, based on the real situation, it is very meaningful to study CSN with Markovian switching [8,9]. It is well known that the stability of systems has become an active research topic, and readers can refer to [10–12] and references therein. The practical systems are often affected by stochastic perturbations due to various types of environmental noises, which may affect or even destroy the stability of systems. For example, in ∗

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

https://doi.org/10.1016/j.neucom.2019.09.036 0925-2312/© 2019 Elsevier B.V. All rights reserved.

© 2019 Elsevier B.V. All rights reserved.

power systems, stochastic perturbations resulting from small signal disturbances greatly affect the stability of power systems [13]. In biology, epidemic diseases can be influenced by stochastic perturbations under media coverage [14]. In general, stochastic systems are described by stochastic differential equations (consider perturbations as white noise). And so far, noticeable results have been obtained for the stability of stochastic systems (see [10,13] and references therein). However, it was pointed out in [15] that stochastic systems are not adequate to describe all practical systems because lots of noises under the particular environment are nonwhite. For instance, in a power system with power noise filter, it is much more reasonable to describe the effects of stochastic perturbations of some electric elements through stationary stochastic processes than white noise. Stationary stochastic processes can describe random perturbations, and this kind of disturbed systems are called random systems (consider perturbations as general noise). In [15], Wu considered the stability of random nonlinear systems with second-order stationary processes for the first time. Following the line of [15], the noise-to-state stability (NSS) of random switched systems was studied in [16–21]. Nevertheless, except for [20,21], there are few papers considering random systems with Markovian switching. And the work in [20,21] both focused on the NSS of random systems with Markovian switching. Therefore, there is still a huge gap about the stability of random systems with Markovian switching, such as the global asymptotical stability in probability (GAS-P) and the exponential stability in pth moment (ES-p-M).

2

M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

In fact, it is quite difficult for a system to realize selfstabilization because of the existence of Markovian switching and random perturbations. An effective way to stabilize a system is to utilize appropriate control schemes. Recently, intermittent control has been widely adopted in engineering fields, such as manufacturing, communication and transportation, due to the fact that it is easy to be implemented in engineering control [22–25]. Compared with periodically intermittent control, the control width of aperiodically intermittent control is more general and more flexible, because it does not require the periodicity of control interval. For example, in the real world, the generation of wind power in smart grid is aperiodically intermittent [26]. Therefore, no matter in the theoretical analysis or in the real applications, it is necessary to consider aperiodically intermittent control. Until now, there have existed some papers discussing the dynamical properties of systems via aperiodically intermittent control (see [27,28] and references therein). The cluster synchronization of colored community networks can be achieved via aperiodically intermittent pinning control [27]. In [28], the exponential cluster synchronization for directed community networks was considered via adaptive nonperiodically intermittent pinning control. With the development of stochastic control theory, aperiodically intermittent controllers have been widely used in stochastic systems [29–31]. In [30], Wang et al. studied the synchronization of coupled stochastic complex-valued drive-response networks with time-varying delays via aperiodically intermittent adaptive control. And synchronization of neural networks with stochastic perturbations was considered via aperiodically intermittent control in [31]. It is worth noting that most existing results are about the stabilization problem of stochastic systems via aperiodically intermittent control. There are no relevant results on the random systems via aperiodically intermittent control. At the same time, most of the existing results are about the mean square stability, other stability also need to be further considered. Motivated by the above discussions, the aim of this paper is to study the stabilization problem of random coupled systems on networks with Markovian switching (RCSNMS) via aperiodically intermittent control. And the Lyapunov-type theorem and the coefficients-type theorem about GAS-P and ES-p-M are considered in detail. Compared with the results on NSS of random systems (see [16,18–21]), the stability criteria in this paper are more general. Noting that the above results on the system with Markovian switching via aperiodically intermittent control often thought that Markovian switching could not happen when systems switch between controlled state and uncontrolled state. Actually, Markovian switching happens randomly. In this paper, we do not limit the switching time. And the transforming constants ϑij are introduced to help us solve this problem, which will be introduced in detail below. Furthermore, we apply the results to a class of random coupled oscillators with Markovian switching. And two numerical examples are discussed to ensure the effectiveness of the derived results. The primary contributions of our findings can be summarized as the following three aspects. • We study RCSNMS, which is driven by general noise that is more general than white noise. • Aperiodically intermittent control is used to stabilize this kind of random systems for the first time. • Compared with existing results about stabilization of systems via aperiodically intermittent control [23,26], the GAS-P and ESp-M of random systems via aperiodically intermittent control are firstly studied. The remainder of the paper is arranged as follows. In Section 2, we introduce some preliminaries as well as the model formulation. In the following, two relevant stability criteria, that is, GAS-P and ES-p-M, are presented in Section 3. Next, the application to cou-

pled oscillators is investigated in Section 4. Finally, two numerical examples are given in Section 5 and the paper ends with the conclusions in Section 6.

2. Preliminaries and model formulation This section contains some common notations and basic concepts of graph theory. Then, our model, some essential definitions and assumptions will also be given for the sake of better comprehension. To begin with, write (, F, {Ft }t≥0 , P ) as a complete probability space with filtration {Ft }t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all Pnull sets). And E(·) stands for the mathematical expectation with respect P. Write r(t) as a right-continuous homogeneous Markov process on the probability space (, F, {Ft }t≥0 , P ) taking values in a finite state space S = {1, 2, · · · , S} with generator  = (πi j )S×S given by

P{r (t + ) = j|r (t ) = i} =

 πi j  + o(), if i = j, 1 + πii  + o(), if i = j,

where  > 0. Furthermore, π ij indicates the transition rate from i  to j, where π ij ≥ 0 if i = j and πii = − Sj=1, j=i πi j . In addition, notations N = {1, 2, · · · , n}, N = {0, 1, · · · } and R+ = [0, +∞ ) are defined. Let Rm be the set of m-dimensional vectors and Rn×m be the set of all n × m matrixes. For a vector or matrix A, its transpose is denoted as AT . Also, the Euclidean norm of vectors and 2-norm of matrixes are defined as | · | and || · ||, respectively. For any a, b ∈ R1 , we denote the maximum of a and b by a∨b, and the minimum of a and b by a∧b. K is the set of functions φ : R+ → R+ which are continuous, strictly increasing and satisfy φ (0 ) = 0; K∞ is the set of functions which are of class K and unbounded; then, KL is the set of functions ι(s, t ) : R+ × R+ → R+ which are of class K for each fixed t, and decrease to zero as t → ∞ for each fixed s. For function ν : R+ → R+ , define ν (t − ) = lims→0− ν (t + s ). Besides, some basic concepts of graph theory will be introduced in the following. Denote H = (N , B, A ) as a weighted digraph, where N and B are the sets of vertices and arcs, respectively; A is defined as the weighted matrix of H, where A = [akh ]n×n and akh > 0 is the weight of arc (h, k). In order to avoid self-loops, it is assumed that akk = 0 for any k ∈ N . For convenience, the digraph is also written as (H, A ). The Laplacian matrix of (H, A ) is represented as LA = [lkh ]n×n , where lkh = −akh for k = h and lkk = n i=1,i=k aki . The digraph H is called to be strongly connected if for any pair of distinct vertices, there exists a directed path from one to the other, where a directed path is defined as a sequence of arcs in a digraph. More details about graph theory can be found in [32]. Consider the RCSNMS on a digraph H with n (n ≥ 2) vertices

⎧ n 

 ⎪ (r (t )) ⎪ akh Hkh xk (t ), xh (t ), t, r (t ) ⎨x˙ k (t ) = fk xk (t ), t , r (t ) + (1)  h=1 ⎪ + gk xk (t ), t, r (t ) σ (t ), t ≥ t0 , ⎪ ⎩ xk (t0 ) = xk0 , r (t0 ) = r0 , k ∈ N ,

in which xk (t ) ∈ Rm represents the state vector of the kth vertice and fk : Rm × R+ × S → Rm , gk : Rm × R+ × S → Rm×N and Hkh : Rm × Rm × R+ × S → Rm all indicate the Borel measure functions, (r (t )) where Hkh is the coupling function; akh is the coupling strength N at time t; and σ (t ) ∈ R stands for an N-dimensional stochastic process defined on (, F, {Ft }t≥0 , P ). Assume that Markov process r(t) and stochastic process σ (t) are independent of each other. It is worth noting that Markovian switching and random perturbations could make system (1) unstable. In order to stabilize it, take aperiodically intermittent control into consideration, then we

M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

3

Assumption 3 [22]. For the aperiodically intermittent control, there exist positive constants λ and T∗ satisfying 0 < λ < T∗ < ∞, and

 Fig. 1. The figure of aperiodically intermittent control uk (t, i).

have

⎧ n 

 ⎪ (r (t )) ⎪ akh Hkh xk (t ), xh (t ), t, r (t ) ⎨x˙ k (t ) = fk xk (t ), t , r (t ) + (2)  h=1 ⎪ +uk (t, r (t )) + gk xk (t ), t, r (t ) σ (t ), t ≥ t0 , ⎪ ⎩ xk (t0 ) = xk0 , r (t0 ) = r0 , k ∈ N , where for r (t ) = i ∈ S, aperiodically intermittent controller uk (t, i) is given as follows



uk (t, i ) =

−Uk(i ) xk (t ), t ∈ [tl , sl ), 0, t ∈ [sl , tl+1 ),

where l ∈ N, t0 = 0 and Uk(i ) > 0 is the control gain. For any time span [tl , tl+1 ), [tl , sl ) is the control time span and [sl , tl+1 ) is the rest time span. Meanwhile, the figure of aperiodically intermittent control is presented in Fig. 1 for the sake of better comprehension. Definition 1 [20]. An mn-valued stochastic process x(t ) = (xT1 (t ), xT2 (t ), · · · , xTn (t ))T on [t0 , T) is called a solution to system (2), for k ∈ N , T < ∞, if (i) xk (t) is continuous for any t ∈ [t0 , T), (ii) xk (t) is Ft -adapted, (iii) xk (t) satisfies the following equation

⎧  t ⎪ ⎪ xk (t ) = xk0 fk (xk (s ), s, r (s )) ⎪ ⎪ ⎪ t0 ⎪

⎪ n ⎪

⎨ (r (s )) + akh Hkh (xk (s ), xh (s ), s, r (s )) + uk (s, r (s )) ds ⎪ ⎪ h=1t  ⎪ ⎪ ⎪ + gk xk (s ), s, r (s ) σ (s )ds, ⎪ ⎪ ⎪ t0 ⎩ xk (t0 ) = xk0 , r (t0 ) = r0 , k ∈ N .

Then x(t) (t ∈ [t0 , T)) is said to be unique, if there exists any other solution x˜(t ) (t ∈ [t0 , T )) such that

P{x(t ) = x˜(t ), t ∈ [t0 , T )} = 1. In order to promise the existence and uniqueness of solution and get our main results, the following assumptions are required. Assumption 1 [15]. Stochastic process σ (t ) ∈ RN satisfies that it is Ft -adapted, piecewise continuous and there exists a constant D > 0 such that

sup E|σ (t )|2 < D. t≥t0

Assumption 2. For any t ≥ t0 , k, h ∈ N , i ∈ S, functions fk (xk , t, i), gk (xk , t, i) and Hkh (xk , xh , t, i) are piecewise continuous in t, and satisfy Lipschitz condition with respect to xk and xh , which means (i ) there exist positive constants Pk(i ) , Qk(i ) and Rkh such that for any m xk , xh , yk , yh ∈ R ,

| fk (xk , t, i ) − fk (yk , t, i )| ≤ Pk(i) |xk − yk |, ||gk (xk , t, i ) − gk (yk , t, i )|| ≤ Qk(i) |xk − yk |, (i ) |Hkh (xk , xh , t, i ) − Hkh (yk , yh , t, i )| ≤ Rkh (|xk − yk | + |xh − yh |).

Besides, functions fk (xk , t, i), gk (xk , t, i) and Hkh (xk , xh , t, i) vanish at the origin, i.e., fk (0, t, i ) = 0, gk (0, t, i ) = 0 and Hkh (0, 0, t, i ) = 0.

infl∈N (sl − tl ) = λ, supl∈N (tl+1 − tl ) = T ∗ .

Noting that the time span of all control width should be larger than λ, while the sum span of control and rest width should be less than T∗ . In other words, the span of rest width should not be larger than T ∗ − λ. It is noteworthy that Assumption 1 is a standard assumption for stochastic process. Stochastic processes such as strictly stationary processes, widely periodic process and widely stationary process all satisfy this assumption, and similar assumption can be found in [15]. In addition, Assumption 2 is a common assumption to guarantee the existence and uniqueness of a solution, and similar assumption can be found in [9,10,23,29]. With the help of Assumptions 1, 2 and Lemma 1 in [20], system (2) is said to have a globally unique solution, if it has a unique solution on the finite interval [t0 , T) for any T > t0 . And based on Assumption 3, for convenience, define

κ = lim inf l→∞

sl − tl , tl+1 − tl

which is the minimum proportion of control width in the time span. Obviously, κ ∈ (0, 1], and aperiodically intermittent control becomes continuous control when κ = 1. In this paper, without loss of generality, we always assume that κ ∈ (0, 1). Remark 1. Obviously, when sl → tl , intermittent control becomes impulsive control; similarly, when sl → tl+1 , intermittent control becomes continuous control. Here, we pay no attention to impulsive control and continuous control due to their non generality and waste of resources, respectively. Besides, aperiodically intermittent control degenerates into periodically intermittent control when tl ≡ lT ∗ , sl ≡ (l + ψ )T ∗ , where 0 < ψ < 1, which was applied in [23,24]. It is easy to check that the requirement that periodically intermittent control needs is quite restricted. However, for aperiodically intermittent control, its control width is very flexible. Hence, it is highly desirable to discuss aperiodically intermittent control to develop some less conservative results. Two essential definitions and lemmas will be stated in the following. Definition 2 [15]. For system (2), the equilibrium solution x(t) ≡ 0 is called globally asymptotically stable in probability, if for any ε > 0, there exists a class KL function  ( · , · ) such that for any t ≥ t0 , x0 ∈ Rmn \ {0},





P |x(t )| ≤  |x0 |, t − t0



≥ 1 − ε.

Definition 3 [15]. For system (2), the equilibrium solution x(t) ≡ 0 is called exponentially stable in pth moment, if there exist parameters k1 , k2 > 0 such that for any t ≥ t0 , x0 ∈ Rmn \ {0},

E|x(t )| p < k1 |x0 | p exp(−k2 (t − t0 )). Definition 4 [15]. For any ε > 0, δ > 0, there is a constant L(ε , δ ) > t0 satisfying that for any t ≥ L(ε , δ ),

    1 t  P  |σ (s )|ds − E|σ (t )| ≥ δ ≤ ε . t − t0 t0

(3)

Then, the stochastic process |σ (t)| satisfies the law of large numbers. Lemma 1 [33]. Function y(t) is absolutely continuous for t ≥ t0 , and its derivative satisfies the inequality

y˙ (t ) ≤ p(t )y(t ) + b(t )

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M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

for almost all t ≥ t0 , where p(t) and b(t) are almost everywhere continuous functions integrable over every finite interval. Then, for t ≥ t0 , we have

y(t ) ≤ y(t0 ) exp



t

t0



p( s ) d s



+



t

exp t0

s

t

and



p(v )dv b(s )ds.

   ∂ Vk (xk , t, i )   gk (xk , t, i ) ≤ ξk(i )Vk (xk , t, i ).  ∂ xk

(6)

3.1. GAS-P

Lemma 2 [2]. Assume n ≥ 2. The following identity holds: n

dk a¯ kh Fkh (xk , xh ) =



In this subsection, two sufficient criteria will be given to illustrate the GAS-P of RCSNMS. Firstly, we introduce the Lyapunovtype theorem as below.

Frs (xr , xs ).

(s,r )∈B(CQ )

Q∈Q

k,h=1



W (Q )

Here, dk denotes the cofactor of the kth diagonal element of Laplacian ¯ ). And Fkh (xk , xh ), k, h ∈ N , are arbitrary functions, matrix of (H, A ¯ ), W(Q) is the Q is the set of all spanning unicyclic graphs of (H, A weight of Q and CQ represents the directed cycle of Q. In particular, ¯ ) is strongly connected. for k ∈ N , dk > 0, if (H, A 3. Main results Through this section, the stability of system (2) will be analyzed. Specifically, the GAS-P of system (2) will be proved in Section 3.1. Then, in Section 3.2, we will discuss the ES-p-M of system (2). To get our main results, the following definition is required. According to [20], we obtain that for Vk (xk , t, i ) ∈ C 1,1 (Rm × R+ × S; R+ ), which denotes that Vk is continuously once differential in xk and t, operator LVk (xk , t, i ) with respect to the kth vertice of system (2) is defined as LVk (xk , t, i )

 

= lim



Theorem 1. Suppose that Assumption 1–3 hold, |σ | satisfies the law of large numbers and system (2) has vertex-Lyapunov func¯ tions Vk (xk , t, i). The digraph √ (H, A ) is strongly connected, where ¯ = [a¯ kh ]n×n . If μ/T ∗ > 2ξ D, where μ = ηλ − ρ (T ∗ − λ ), η = A mini∈S,k∈N {ηk(i ) }, ξ = maxi∈S,k∈N {ξk(i ) } and ρ = maxi∈S,k∈N {ρk(i ) }, then the equilibrium solution of system (2) is globally asymptotically stable in probability.  Proof. Define V (x, t, i ) = nk=1 dkVk (xk , t, i ), where dk is the co¯ ) is factor of the kth diagonal element of LA¯ . Because (H, A strongly connected, by Proposition 2.1 in [2], we know that dk > 0 for any k ∈ N . Let d = mink∈N {dk }, D∗ = maxk∈N {dk }. For any y ≥ 0, denote α∗ (y ) = min{α1 (y ), α2 (y ), · · · , αn (y )}, β ∗ (y ) = max{β1 (y ), β2 (y ), · · · , βn (y )}. Combining this with (4), it is easy to reach

V (x, t, i ) ≥





+ uk (t, i ) + gk xk , t, i





σ (t ) +

S

πi jVk (xk , t, j )

n

dk βk (|xk | ) ≤ D∗

k=1

+

n

n

β ∗ (|x| ) = nD∗ β ∗ (|x| ).

α (|x| ) ≤ V (x, t, i ) ≤ β (|x| ).



akh Hkh xk , xh , t, i + uk (t, i )

n

dk LVk (xk (t ), t , r (t ))

k=1

πi jVk (xk , t, j ).

(7)

Meanwhile, according to (5) and (6), for t ∈ [tl , sl ), LV (x(t ), t , r (t )) =

=

n

 dk L∗Vk (xk (t ), t , r (t ))

k=1

j=1

Definition 5. Functions Vk (xk , t, i ) ∈ C 1,1 (Rm × R+ × S; R+ ), where k ∈ N , i ∈ S, are called vertex-Lyapunov functions for system (2) if they satisfy the following conditions A1. There exist class K∞ functions α k , β k such that

αk (|xk | ) ≤ Vk (xk , t, i ) ≤ βk (|xk | ).

∂ Vk (xk (t ), t , r (t )) gk (xk (t ), t , r (t ))σ (t ) ∂ xk (t )  n

≤ dk − ηk(r (t ))Vk (xk (t ), t , r (t )) k=1

(4)

A2. There exist positive constants ϑij satisfying Vk (xk , t, i ) = ϑi jVk (xk , t, j ). A3. There exist positive constants ηk , ρk , ξk , nonnegative constants a¯ kh and functions Fh (xh , t, i) such that

+

(i )

(i )

(i )

L∗Vk (xk , t, i )

⎧ n

(i ) ⎪ ⎪ ⎨−ηk Vk (xk , t, i ) + a¯ kh [Fh (xh , t, i ) − Fk (xk , t, i )], t ∈ [tl , sl ), h=1

(5)

n

⎪ ⎪ ⎩ρk(i)Vk (xk , t, i ) + a¯ kh [Fh (xh , t, i ) − Fk (xk , t, i )], t ∈ [sl , tl+1 ), h=1



+

+

≤

 |x| 

k=1



h=1 S



Define α (|x| )  dα∗ |nx| , β (|x| )  nD∗ β ∗ (|x| ). Obviously, α and β are class K∞ functions, which yields

∂ Vk (xk , t, i ) ∂ Vk (xk , t, i ) + ∂t ∂ xk  n  (i ) 

× fk xk , t, i +

≥ d max{α∗ (|xk | )} ≥ dα∗

V (x, t, i ) ≤

j=1

L∗Vk (xk , t, i ) =

α∗ (|x1 | ) + · · · + α∗ (|xn | )

and

∂ Vk (xk , t, i )   L∗Vk (xk , t, i ) + gk xk , t, i σ (t ), ∂ xk where



k∈N

n

 ∂ Vk (xk , t, i ) ∂ Vk (xk , t, i )   (i ) = + fk xk , t, i + akh Hkh xk , xh , t, i ∂t ∂ xk h=1



dk αk (|xk | ) ≥ d

k=1

E Vk xk (t + ), t + , r (t + ) |xk (t ), r (t ) = i − Vk (xk (t ), t , i )

→0

n

n



a¯ kh Fh (xh (t ), t , r (t )) − Fk (xk (t ), t , r (t ))



h=1 n

dk ξk(r (t )) |σ (t )|Vk (xk (t ), t , r (t ))

k=1





≤ − η − ξ |σ (t )| V (x(t ), t , r (t )) +

n





dk a¯ kh Fh (xh (t ), t , r (t )) − Fk (xk (t ), t , r (t )) .

k,h=1

(8) Analogously, for t ∈ [sl , tl+1 ),

M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

LV (x(t ), t , r (t )) ≤

n

     tltl∗ , tltl∗ , r tltl∗



V (x(t ), t , r (t )) ≤ V x

dk ρk(r (t ))Vk (xk (t ), t , r (t ))

k=1

+





n

+

l

ξ

exp



 



t

tlt∗

|σ ( s )|ds

  −  tltl∗ , tltl∗ , r tltl∗ l l    t    lt × exp −η t − t ∗ exp ξ l |σ (s )|ds . l

k=1

≤ (ρ + ξ |σ (t )| )V (x(t ), t, r (t )) +





l

= ϑr (lt )r (lt )V x t∗ t ∗ −1

dk ξk(r (t )) |σ (t )|Vk (xk (t ), t , r (t ))

n



× exp −η t − tlt∗

a¯ kh (Fh (xh (t ), t , r (t )) − Fk (xk (t ), t , r (t )))

h=1 n

5

tt∗ l

dk a¯ kh (Fh (xh (t ), t , r (t )) − Fk (xk (t ), t , r (t ))).

(13)

k,h=1

(9) Continuously using Lemma 1 on interval [tt∗ −1 , tt∗ ) and (12), we l

get that

Based on Lemma 2, it is not difficult to see that n

=



W (Q )



  −  tltl∗ , tltl∗ , r tltl∗         ≤ V x tlt∗ −1 , tlt∗ −1 , r tlt∗ −1 exp −η tlt∗ − tlt∗ −1 l l l l l   l

V x

(Fr (xr , t, i ) − Fs (xs , t, i )).

(s,r )∈B(CQ )

Q∈Q

× exp Since 

CQ

is a directed cycle, it follows (s,r )∈B(CQ ) (Fr (xr , t, i ) − Fs (xs , t, i )) = 0. Hence,

n

l

 

dk a¯ kh (Fh (xh , t, i ) − Fk (xk , t, i ))

k,h=1

l

l

easily

that

ξ

  t∗ t l

tlt∗ −1

|σ ( s )|ds

  −  tlt∗ −1 , tlt∗ −1 , r tlt∗ −1 l l l l l   l     tt∗ l lt lt × exp −η t ∗ − t ∗ −1 exp ξ l |σ (s )|ds . l l

 

= ϑr (lt )r (lt )V x t ∗ −1 t ∗ −2

dk a¯ kh (Fh (xh (t ), t , r (t )) − Fk (xk (t ), t , r (t ))) = 0.

(10)

tt∗ −1

l

k,h=1

(14)

l

Substitute (10) into (8) and (9), respectively, which follows

  − η − ξ |σ (t )| V (x(t ), t , r (t )), t ∈ [tl , sl ), LV (x(t ), t , r (t )) ≤  ρ + ξ |σ (t )| V (x(t ), t , r (t )), t ∈ [sl , tl+1 ). (11)

Substituting (14) into (13) and repeating the above process until 0lt , it is straightforward to show that

V (x(t ), t , r (t )) ≤ ϑr (lt )r (lt ) ϑr (lt )r (lt ) t∗ t ∗ −1 t ∗ −1 t ∗ −2

l l  l  l  −  tlt∗ −1 , tlt∗ −1 , r tlt∗ −1 l l l      lt × exp −η t − t ∗ −1 exp ξ

×V x With the help of condition A2, one can get that ϑi j ϑ js = ϑis and ϑss = 1. Then, combine this with the definition of V(x, t, i), which gives

l

t

tlt∗ −1

 |σ ( s )|ds

l

≤···

V (x, t, i ) =

n

k=1

dkVk (xk , t, i ) =

n

dk ϑi jVk (xk , t, j ) = ϑi jV (x, t, j ).

k=1

(12) For any t ∈ [tl , sl ), the switching moment of the Markov process r(t) l l l l is defined as tl ≤ 0t < 1t < 2t < · · · < tt∗ < sl , where l

   mlt ∗ +1 = inf t > mlt ∗ : r (t ) = r mlt ∗ , m∗ = 0, 1, · · · , tl∗ . For any t ∈ [tl , sl ), there exists mt ∗ such that t ∈ [mt ∗ , mt ∗ +1 ). It is easy to check that the solution to system (2) is absolutely l l continuous on [mt ∗ , mt ∗ +1 ) with the aid of Theorem 1.7 in [33]. For any fixed i ∈ S and Vk (xk , t, i ) ∈ C 1,1 (Rm × R+ × S; R+ ), V(x(t), t, l r(t)) is also absolutely continuous on [tt∗ , sl ). Obviously, combining l

Lemma 1, (11) with (12), one has

l

l

l

   −  1lt , 1lt , r 1lt  t | σ ( s ) | d s 1lt   lt lt  lt ≤ ϑr (lt )r (lt ) · · · ϑr (lt )r (lt )V x 0 , 0 , r 0 1 0 t∗ t ∗ −1 l l  t    × exp −η t − 0lt exp ξ l |σ (s )|ds 0t   lt lt = ϑr (lt )r (t )V x 0 , 0 , r (tl ) l t∗ l  t    lt × exp −η t − 0 exp ξ l |σ (s )|ds ≤ ϑr (lt )r (lt ) · · · ϑr (lt )r (lt )V x 1 0 t∗ t ∗ −1 l l     × exp −η t − 1lt exp ξ

0t

≤ ϑr (lt )r (t )V (x(tl ), tl , r (tl )) exp (−η (t − tl ) ) t∗ l

l

 t  × exp ξ |σ ( s )|ds . tl

(15)

6

M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

V (x(t2 ), t2 , r (t2− )) ≤ ϑr (t2− )r (0)V (x0 , t0 , r0 )

When t = s− , we have l

V ( x ( sl ), sl , r ( s− )) ≤ ϑr (s−l )r (tl )V (x(tl ), tl , r (tl ) ) l  s  l  × exp − η (sl − tl ) exp ξ |σ ( s )|ds .

× exp(−μ − ηλ + ρ (T ∗ − λ ))  t  2 × exp ε |σ ( s )|ds

(16)

0

tl

= ϑr (t2− )r (0)V (x0 , t0 , r0 ) exp(−2μ )  t  2 × exp ε |σ ( s )|ds .

Similarly, for t ∈ [sl , tl+1 ), it follows easily that

V (x(t ), t , r (t )) ≤ ϑr (ls )r (s )V (x(sl ), sl , r (sl ) ) s∗ l

l

0

 t  × exp ρ (t − sl ) exp ξ |σ ( s )|ds . 



(17)

sl

V (x(tl ), tl , r (tl− )) ≤ ϑr (t − )r (0)V (x0 , t0 , r0 )  l t  l × exp(−l μ ) exp ε |σ ( s )|ds .

− When t = tl+1 , it is obvious that − − V (x(tl+1 ), tl+1 , r (tl+1 )) ≤ ϑr (tl+1 )r (sl )V (x (sl ), sl , r (sl ) )  t  l+1  × exp ρ (tl+1 − sl ) exp ξ |σ ( s )|ds .

0

(18)

sl

Now, we carry out the above process to V(x(t), t, r(t)). For t ∈ [t0 , s0 ), where t0 = 0, using (15), we can obtain

× exp(−ηt ) exp When t =

s− , 0

For any t > 0, there always exists l such that t ∈ [tl , tl+1 ). We can obtain from tl ≤ t < tl+1 that −l μ < (−t/T ∗ + 1 )μ. If t ∈ [tl , sl ), we have

V (x(t ), t , r (t )) ≤ ϑr (lt )r (t )V (x(tl ), tl , r (tl )) t∗ l

 t  ε |σ ( s )|ds .

tl

≤ ϑr (lt )r (0)V (x0 , t0 , r0 ) exp(−l μ )

0

t∗ l

 t  × exp ε |σ ( s )|ds

we get

− V ( x ( s0 ), s0 , r ( s− 0 )) ≤ ϑr (s0 )r (0 )V (x0 , t0 , r0 )

  × exp(−ηs0 ) exp ε

s0

0

0



≤ ϑr (lt )r (0)V (x0 , t0 , r0 ) exp(μ )

|σ ( s )|ds .

t∗ l

V (x(t ), t , r (t )) ≤ ϑr (0∗s )r (s ) ϑr (s0 )r (s−0 )V (x(s0 ), s0 , r (s− 0 )) 0 s 0  t  × exp(ρ (t − s0 )) exp ε |σ ( s )|ds

s∗ l

l

× exp(ρ (t − sl )) exp

 t  × exp(ρ (t − s0 ) − ηs0 ) exp ε |σ ( s )|ds .

 t  ε |σ ( s )|ds sl

0

≤ ϑr (ls )r (0)V (x0 , t0 , r0 ) exp(−l μ − ηλ s∗ l

 t  ε |σ ( s )|ds 0  μt  = ϑr (ls )r (0)V (x0 , t0 , r0 ) exp − ∗ s∗ T l  t  × exp ε |σ ( s )|ds .

0

According to Assumption 3, when t =

t1− ,

+ ρ (T ∗ − λ )) exp

it can be checked that

)) ≤ ϑr (t1− )r (0)V (x0 , t0 , r0 ) exp(ρ (t1 − s0 ) − ηs0 )  t  1 × exp ε |σ ( s )|ds 0

≤ ϑr (t1− )r (0)V (x0 , t0 , r0 ) exp(ρ (T ∗ − λ ) − ηλ )  t  1 × exp ε |σ ( s )|ds 0

 ϑr (t1− )r (0)V (x0 , t0 , r0 ) exp(−μ )  t  1 × exp ε |σ ( s )|ds .

From the deduction of (19) and (20), denote ϑ = maxi, j∈S {ϑi j } and we can derive that for any t > 0,

V (x(t ), t , r (t )) ≤ ϑ V (x0 , t0 , r0 ) exp(μ )

 μt 

× exp − For any ε > 0, δ ∈ (0,

− V ( x ( s1 ), s1 , r ( s− 1 )) ≤ ϑr (s1 )r (0 )V (x0 , t0 , r0 )  s  1 × exp(−μ − η (s1 − t1 )) exp ε |σ ( s )|ds .

0

And when t = t2− , it follows that

(20)

0

0

√ Since μ/T ∗ > 2ξ D, we have μ > 0. Similarly, when t = s− , we reach 1

(19)

0

V (x(t ), t , r (t )) ≤ ϑr (ls )r (s )V (x(sl ), sl , r (s− )) l

≤ ϑr (0∗s )r (0)V (x0 , t0 , r0 )

V (x(t1 ), t1 , r (

T∗

 t  exp ε |σ ( s )|ds .

If t ∈ [sl , tl+1 ), we reach

s0

t1−

 μt 

× exp −

For t ∈ [s0 , t1 ), based on Lemma 1 and (17), one can get

s

l

 t  × exp(−η (t − tl )) exp ε |σ ( s )|ds

V (x(t ), t , r (t )) ≤ ϑr (0t )r (0)V (x0 , t0 , r0 ) t∗ 0

By repeating the same procedure as above, we get that

A=

T∗

 t  exp ε |σ ( s )|ds .

(21)

0

√ D ), define

    1 t   ≤δ . | σ ( s ) | d s − E | σ ( t ) |  t − t0 t0 

Combining this with (3) follows that there exists a constant L(ε , δ ) > 0 such that for any t ≥ L(ε, δ ),

P (A ) ≥ 1 − ε .

(22)

M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

As a result, it can be checked from the definition of A, (3) and Ho¨ lder’s inequality that



t

t0

|σ (s )|ds ≤ (t − t0 )(E|σ (t )| + δ ) ≤ (t − t0 )((E|σ (t )|2 ) 2 + δ ) √ ≤ 2 D(t − t0 ), w ∈ A. 1

(23)

Substitute (23) into (21), which follows

  μt

V (x(t ), t , r (t )) ≤ ϑ V (x0 , t0 , r0 ) exp(μ ) exp −

T∗

√   − 2ξ D t . (24)

It follows from (7), (22), along with (24) that

P

    μt √   |x(t )| ≤ α −1 ϑβ (|x0 | ) exp(μ ) exp − ∗ − 2ξ D t T

≥ 1 − ε , t ≥ L ( ε , δ ).

(25)

For another, by means of Assumption 1, for above ε , there exists a constant δ ∗ > 0 such that

P{|σ (t )| > δ∗ } ≤

E|σ (t )|2

δ∗2

<

D

δ∗2

= ε.

Combining this with (21) results in



P V (x(t ), t , r (t )) < ϑ V (x0 , t0 , r0 ) exp(μ ) exp



ξ δ∗ −

≥ 1 − ε , t ≤ L ( ε , δ ).

μ T∗

L (ε , δ )



(26)

This, together with (7), implies

    μ P |x(t )| ≤ α −1 ϑβ (|x0 | ) exp(μ ) exp ξ δ∗ − ∗ L(ε , δ ) T

≥ 1 − ε , t ≤ L ( ε , δ ).

(27) √

ς (· )  α −1 (ϑβ (· ) exp(μ ) exp(−( Tμ∗ − 2 Dξ )L(ε, δ ))) + −1 α (ϑβ (· )) exp(μ ) exp((ξ δ∗ − Tμ∗ )L(ε, δ ))). Since α , β ∈ K∞ , it is easy to check that ς ∈ K∞ . Combining (25) with (27), we can Denote

obtain that

P{|x(t )| ≤ ς (|x0 | )} ≥ 1 − ε , t ≥ t0 .

(28)

It follows from (25) and (28) that for every ε > 0, there exists a class KL function  ( · , · ) such that

P{|x(t )| ≤  (|x0 |, t − t0 )} ≥ 1 − ε , t ≥ t0 . This completes the proof.



Remark 2. As periodically/aperiodically intermittent control becomes popular, there exist some papers investigating the synchronization and the stabilization of systems via periodically/aperiodically intermittent control (see [22–25] and references therein). Nevertheless, the combination of Markovian switching and periodically/aperiodically intermittent control makes systems more complex, and this is the reason why few researchers focus on this case in detail. In [23], based on the Lyapunov method, graph theory along with the super martingale convergence theorem, the stability of systems with Markovian switching was studied via periodically intermittent control. Compared with it, our differences are as follows, (i) aperiodically intermittent control is concerned instead of periodically intermittent control; (ii) random systems are considered rather than stochastic systems, which means the systems we consider are more general; (iii) we focus on the GAS-P and ES-p-M. Remark 3. When considering the dynamic properties of systems with Markovian switching via aperiodically intermittent control, it is worth noting that control time and switching moment should be distinguished. And due to interval division of control and

7

rest, Markovian switching should be considered in each interval, which is a complex calculation process. In particular, when systems switch between the controlled state and the uncontrolled state, it is uncertain that whether Markovian switching occurs. The existing results often assume that Markovian switching could not happen when system state is switched. In this paper, we do not limit that. And we introduce transforming constants ϑij to solve this problem, which is detailed in procedure (13)–(18). Remark 4. There exist some papers studying random systems [15–17,20] which are driven by general noise. However, there are few papers considering control into random systems. Among different kinds of control schemes, aperiodically intermittent control is highly favored due to its applicability. Based on the Lyapunov method and graph theory, we investigate the GAS-P of system (2) in Theorem 1 for the first time. This is also the first time that aperiodically intermittent control is utilized to random systems. In the following, a coefficients-type theorem of the GAS-P of system (2) will be presented, which can offer a direct way to verify the sufficient criteria. Remark 5. It is worth noting that compared with the researches on the noise-to-state stability of random systems (see [16,18–21]), the stability criteria in this paper are more general. And we re∂ V (x ,t,i ) quire that | k ∂ xk gk (xk , t, i )| ≤ ξk(i )Vk (xk , t, i ) (see condition A3 k

in Definition 5). This assumption is common and  easily satisfied. In detail, by Assumption 2, we know that gk xk , t, i ≤ Qk(i ) |xk |. Then choosing common Lyapunov function Vk (xk , t, i ) = |xk | p , p ≥ 2, we have

   ∂ Vk (xk , t, i )  gk xk , t, i  = p|xk | p−2 |xTk gk (xk , t, i )| ≤ pQk(i )Vk (xk , t, i ).  ∂x k

While for results in [16,18–21], where the noise-to-state stability of random systems was studied, it requires that

   ∂ Vk (xk , t, i )  gk xk , t, i  ≤ aVk (xk , t, i ).  ∂x k

In this case, only assuming gk (xk , t, i ) ≤ Mk(i ) where Mk(i ) should not be too large and taking Vk (xk , t, i ) = |xk |2 , this assumption is easy to satisfy. If taking Vk (xk , t, i ) = |xk | p , p > 2, we cannot verify this assumption. From this point, the stability criteria in this paper are more general. ˜ ), where A ˜ = [a˜kh ]n×n , is Theorem 2. Suppose that digraph (H, A strongly connected and |σ | satisfies the law of large numbers. The equilibrium solution of system (2) is called to be globally asymptotically stable in probability, if Assumption 1–3 and the following conditions hold. B1. There exist positive constants θk(i ) , such that

xTk fk (xk , t, i ) ≤ θk(i ) |xk |2 . B2. There exist positive constants q(i) and p ≥ 2, such that μ/T ∗ > √ 2ξ D, where μ = ηλ − ρ (T ∗ − λ ), η = mini∈S,k∈N {ηk(i ) }, ξ = maxi∈S,k∈N {ξk(i ) }, ρ = maxi∈S,k∈N {ρk(i ) }, ηk(i ) = pUk(i ) − pθk(i )   ( j) (i ) (i ) − (2 p − 1 ) nh=1 akh Rkh − a˜kh − sj=1 πi j q (i) , ρk(i) = pθk(i) q n  ( j ) (i ) (i ) + 2( p − 1 ) h=1 akh Rkh + a˜kh + sj=1 πi j q (i) and ξk(i ) = pQk(i ) , (i ) (i ) in which a˜kh = maxi∈S {akh Rkh }.

q

Proof. We denote Vk (xk , t, i ) = q(i ) |xk | p . It is obvious that conditions A1, A2 hold with αk (· ) = mini∈S {q(i ) }| · | p , βk (· ) = maxi∈S {q(i ) }| · | p , ϑi j =

q (i ) q( j )

for any k ∈ N .

For any t ∈ [tl , sl ), according to condition B1, it can be derived that

8

M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

L∗Vk (xk , t, i )





n

(i ) T akh xk Hkh (xk , xh , t, i ) − xTk Uk(i ) xk

= q(i ) p|xk | p−2 xTk fk (xk , t, i ) +

h=1

+

s

πi j q ( j ) | x k | p

j=1

≤ q(i ) pθk(i ) |xk | p + q(i ) p|xk | p−1

σ (t ) = a(t ) cos(ωt + ϕ (t )), ∀t > 0,

n

(i ) (i ) akh Dkh (|xk | + |xh | )

where ω and a(t) are a positive constant and an envelope function, respectively; and ϕ (t) is a random phase which is uniformly distributed on the interval [0, 2π ] at any fixed time. It should be pointed out that narrow-band random process can be perceived as narrow-band Gaussian process with zero expectation. This implies that narrow-band random process σ (t) satisfies E|σ (t)|2 < D.

h=1 s

− q(i ) pUk(i ) |xk | p +

πi j q ( j ) | x k | p

j=1

 (i )

n s



q( j ) (i ) (i ) −p akh Rkh − πi j ( i ) q

(i )

= − pUk − pθk

h=1

+ q (i ) p

n

 q ( i ) |xk | p

j=1

(i ) (i ) akh Rkh |xk | p−1 |xh |.

(29)

h=1 ∗ ∗ ∗ Through Young’s inequality |a|w |b|v ≤ ww∗ +v |a|w +v + 1 ∗ where a, b ∈ R , w , v > 0, it easily follows that

q (i ) p

w∗ +v |b|

v

w∗ +v ,

n

(i ) (i ) akh Rkh |xk | p−1 |xh |



≤ q ( i ) |xk | p

Remark 7. It is worth noting that conditions A1-A3 are related to vertex-Lyapunov functions. However, in practical applications, it is difficult for us to promise the existence of above vertex-Lyapunov functions. Thus, it is necessary to construct related Vk (xk , t, i) to verify the effectiveness of Theorem 1, which is also what we do in Theorem 2. Besides, conditions in Theorem 2 are related to the coefficients of system (2), which can offer more convenience in practical verification. 3.2. ES-P-M

h=1 n n



(i ) (i ) (i ) (i ) (i ) (i ) ≤ akh Rkh q ( p − 1 )|xk | p + akh Rkh q |xh | p h=1

cess and so on, were estimated and simulated in [15]. It discussed NSS and asymptotical stability. Inspired by [15] and [34], we choose the narrow-band random process and investigate the GASP of RCSNMS. Narrow-band random process is widely applied in communication systems in view of its feature improving the working characteristics of the receiver. Generally, narrow-band random process is defined as follows

h=1 n

(i ) (i ) akh Rkh ( p − 1 ) + a˜kh

 +

h=1

n

In this subsection, we will prove ES-p-M of RCSNMS according to the same steps as those above.

q(i ) a˜kh (|xh | p − |xk | p ).

h=1

(30)

Theorem 3. Suppose that Assumption 1–3 hold and system (2) has vertex-Lyapunov functions Vk (xk , t, i) with αk (· ) = ak | · | p , βk (· ) = bk | · | p , in which 0 < ak ≤ bk , p > 0. There exists a function δ ( · ) such that for any ε > 0 and t1 > t0 , the following inequality holds

Substituting (30) into (29), we have that (i )

L Vk (xk , t, i ) ≤ −ηk Vk (xk , t, i ) + ∗

n

E exp a¯ kh (Fh (xh , t, i ) − Fk (xk , t, i )),

h=1

where a¯ kh = maxi∈S {a˜kh q(i ) } and Fk (xk , t, i ) = |xk | p . When t ∈ [sl , tl+1 ), one can get

L∗Vk (xk , t, i )



≤

(i )

pθk

n s



q( j ) (i ) (i ) + (2 p − 1 ) akh Rkh + a˜kh + πi j ( i ) q h=1

+

n

(i ) ˜

 q ( i ) |xk | p

j=1

n

t0

 |σ (s )|ds ≤ exp(δ (ε )(t1 − t0 )).

(31)

¯ ) is strongly With the same parameters as those in Theorem 1, if (H, A connected and μ/T∗ > δ (ξ ), then the equilibrium solution of system (2) is exponentially stable in pth moment.  Proof. In a similar way, we define V (x, t, i ) = nk=1 dkVk (xk , t, i ), where dk is defined in Theorem 1. It is easy to observe from [35] that

n ( 1− 2 )∧0 |x| p ≤ p

n

|xk | p ≤ n ( 1− 2 )∨0 |x| p . p

Based on this and ak |xk |p ≤ Vk (xk , t, i) ≤ bk |xk |p , we can easily compute that

p

h=1

= ρk(i )Vk (xk , t, i ) +

t1

k=1

q akh (|xh | − |xk | ) p

  ε

V (x, t, i ) =

a¯ kh (Fh (xh , t, i ) − Fk (xk , t, i ) ).

dkVk (xk , t, i ) ≤

k=1

h=1

(i )

Furthermore, for any t ∈ [t0 , +∞ ), based on ||gk (xk , t, i )|| ≤ Qk |xk | in Assumption 2, the following inequality is established

   ∂ Vk (xk , t, i )   (i )    = q p|xk | p−2 xTk gk (xk , t, i ) ≤ q(i) p|xk | p−1 Q (i) |xk | g ( x , t, i ) k k k   ∂ xk (i )

n

(i )

= q(i ) pQk |xk | p = ξk Vk (xk , t, i ).

˜ ) is strongly Thus, condition A3 is satisfied. Since q(i) > 0 and (H, A ¯ ) is strongly connected. connected, it is easy to check that (H, A Until now, all conditions in Theorem 1 have been fulfilled. Based on it, one can get that the equilibrium solution of system (2) is globally asymptotically stable in probability. The proof is completed now.  Remark 6. Stochastic processes like second-order process, widely stationary process, widely periodic process, strictly stationary pro-

n

dk bk |xk | p ≤ max{dk bk }n(1− 2 )∨0 |x| p p

k∈N

k=1

and V (x, t, i ) =

n

dkVk (xk , t, i ) ≥

k=1

n

dk ak |xk | p ≥ min{dk ak }n(1− 2 )∧0 |x| p . p

k∈N

k=1

(1− 2p )∧0

Let a = mink∈N {dk ak n it is straightforward that

p

} and b = maxk∈N {dk bk n(1− 2 )∨0 }, then

a|x| p ≤ V (x, t, i ) ≤ b|x| p .

(32)

With the same analysis of (8)–(21), it is easy to show that a|x(t )| p ≤ V (x(t ), t , r (t ))



≤ ϑ V (x0 , t0 , r0 ) exp(μ ) exp



−



μt T∗



 t  exp ξ |σ ( s )|ds t0

 t  μ ≤ ϑ exp(μ )b|x0 | p exp − ∗ t exp ξ |σ (s )|ds , (33) T

t0

M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

where μ is the same as that in Theorem 1. Taking the mathematical expectation on both sides of (31), it is known that

 aE|x(t )| p ≤ ϑ exp(μ )b|x0 | p exp

−

μ T



t E exp ∗

 t  ξ |σ ( s )|ds . t0

(34)

Denote Xk (t ) = (xk (t ), yk (t ))T , Ek (Xk (t ), t , r (t )) = (yk (t ), −ζk (xk (t ), r (t ))yk (t ) − hk (xk (t ), t , r (t )))T , Ik (Xk (t ), t , r (t )) = (0, gk (xk (t ), t , r (t )))T and H¯ kh (Xk (t ), Yh (t ), t, r (t )) = (0, Hkh (xk (t ), xh (t ), t, r (t )))T . Therefore, system (37) can be rewritten as the following form

X˙ k (t ) = Ek (Xk (t ), t , r (t )) + Ik (Xk (t ), t , r (t ))σ (t )

In order to stabilize system (38), aperiodically intermittent control is utilized to system (38), which yields

Combining this with (34), we can reach

b exp(μ )|x0 | p exp a





−

μ T∗



X˙ k (t ) = Ek (Xk (t ), t , r (t )) + Ik (Xk (t ), t , r (t ))σ (t )

− δ (ξ ) t .

Theorem 4. Suppose that Assumption 1–3, condition B1 and μ/T∗ > δ (ξ ) hold, where ξ , μ and T∗ are defined in Theorem 1. Then the equilibrium solution of system (2) is exponentially stable in pth ˜ ) is strongly connected, where A ˜ = [a˜kh ]n×n . moment, if (H, A The proof follows the similar lines of the proof of Theorem 2, so we omit it here. Remark 8. According to Theorem 1.16 in [33], it is easy to know that if a Gaussian process σ (t) with zero expectation satisfies

E|σ (t )|2 ≤ D1 ,

∞

t0

E(σ (s ) − E(σ (s )) ) E(σ (t ) − E(σ (t )) )ds ≤ D2 , T

then for any t > 0, we have

  E exp ε

t1

t0



   εD  |σ (s )|ds < exp ε D1 + 2 (t1 − t0 ) .

h=1

(39) Next, for system (39), we give the following stability criteria. Theorem 5. For i ∈ S, k, h ∈ N , hk , gk and Hkh all satisfy Assumption 2, σ (t) satisfies Assumption 1 and Assumption 3 holds. Then the equilibrium solution of system (39) is globally asymptotically stable ˜ ) is strongly connected, where A ˜ = in probability, if digraph (H, A [a˜kh ]n×n , and the following conditions hold. C1. There exist positive constants τk(i ) such that |hk (xk , t, i )| ≤

τk(i) |xk |. C2. The exists a positive constant M such that |ζ k (xk , i)| ≤ M. C3. There exist positive constants q(i) and p ≥ 2 such √ ∗ that μ/T > 2ξ D, where μ = ηλ − ρ (T ∗ − λ ), ηk(i ) =  ( j) (i ) (i ) s 2Uk(i ) − 2θk(i ) − a˜kh − 3 nh=1 akh Rkh − j=1 πi j q (i) , in which (i ) 1+(τ )2 θk = max{ 2k , M + 1},  ( j) (i ) (i ) s 3 nh=1 akh Rkh + j=1 πi j q (i) q

Then the Gaussian process with zero expectation satisfies (31) with δ (ε ) = ε ( D1 + εD2 2 ). 4. Application to coupled oscillators Nowadays, coupled oscillators have been an active research topic in view of their widespread applications [36–38], and our research will discuss the stability of coupled oscillators. To begin with, a second-order oscillator is given by





x¨ (t ) + ζ (x(t ))x˙ (t ) + h x(t ), t = g x(t ), t σ (t ),







= gk xk (t ), t , r (t ) σ (t ) +

≤ ≤

(36) (r (t )) where akh Hkh are the normalized interference functions from oscillator h to oscillator k. By making a transform of yk = x˙ k , system (36) is rewritten as

⎧ ⎨x˙ k (t ) = yk (t ),   y˙ k (t ) = −ζk (xk (t ), r (t ))yk (t ) − hk xk (t ), t , r (t ) + gk xk (t ), t , r (t ) σ (t )   ⎩ (r (t )) + nh=1 akh Hkh xk (t ), xh (t ), t, r (t ) , k ∈ N .

(37)

and ξk(i ) = 2Qk(i ) .

1 2 1 2 xk + y2k + My2k + hk + y2k 2 2



1 + (τk(i ) )2 2

! ≤ max



h=1

ρk = 2θk(i) + a˜kh +

XkT Ek (Xk , t, i ) = xk yk − ζk (xk (t ), i )y2k − hk (xk , t, i )yk

(35)

n

 (r (t )) akh Hkh xk (t ), xh (t ), t, r (t ) , k ∈ N ,

(i )

Proof. Since hk , gk and Hkh satisfy Assumption 2, it is obvious that Ek (Xk , t, i), Ik (Xk , t, i) and H¯ kh (Xk , Yh , t, i ) also satisfy Assumption 2, which means that system (39) exists a unique solution on [t0 , ∞). Through simple calculation based on conditions C1, C2, the following inequality holds

where h : R × R+ → R is the nonlinear restoring force, ζ (x) > 0 is the damping function and g(x(t), t)σ (t) denotes the stochastic driving force driven by a scalar stochastic process σ (t). Considering the coupling factor as well as Markovian switching into system (35), random coupled oscillators with Markovian switching are stated as follows x¨k (t ) + ζk (xk (t ), r (t ))x˙ k (t ) + hk xk (t ), t , r (t )

q

(i )

2



n

(r (t )) ¯ akh Hkh (Xk (t ), Yh (t ), t, r (t )) + uk (t, r (t )), k ∈ N .

+

Since μ/T∗ > δ (ξ ), based on Definition 3, the equilibrium solution of system (2) is exponentially stable in pth moment. Then the proof is completed. 



(38)

h=1

t0

E|x(t )| p ≤ ϑ

n

(r (t )) ¯ akh Hkh (Xk (t ), Yh (t ), t, r (t )), k ∈ N .

+

Based on inequality (31), it can be derived that

 t  E exp ξ |σ (s )|ds ≤ exp(δ (ξ )t ).

9



x2k + (M + 1 )y2k

1 + (τk(i ) )2 2

" ,M + 1

|Xk |2 ,

which implies that condition B1 is satisfied. Until now, all conditions in Theorem 2 are satisfied. Hence, the equilibrium solution of system (39) is globally asymptotically stable in probability on [t0 , ∞). The proof is completed.  Theorem 6. Under Assumption 1, 2 and 3, there exists a function

δ ( · ) such that for any ε > 0, t1 ≥ t0 , stochastic process σ (t) satisfies E exp

  ε

t1

t0

  |σ (s )|ds ≤ exp δ (ε )(t1 − t0 ) .

˜ ) is strongly connected, where A ˜ = [a˜kh ]n×n . If conAssume that (H, A ditions C1, C2 hold and μ/T∗ > δ (ξ ), where μ and ξ are the same as

10

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Fig. 2. The figure of random coupled oscillators at state 1 and state 2 respectively.

those in Theorem 5, then the equilibrium solution of system (39) is exponentially stable in pth moment. The proof of Theorem 6 can be completed by following the same steps as those in Theorem 5. So it is omitted here. Remark 9. Recently, coupled oscillators have become a research subject of great interest because they can describe a variety of biological and physical phenomena (see [9,23,29,39] for example). Winfree [39] modeled each member of the population as a nonlinear oscillator with a globally attracting limit cycle. The oscillators were assumed to be weakly coupled and their natural frequencies were assumed to be randomly distributed across the population, based on which the stability of coupled nonlinear oscillators was studied. In the field of ecology one can imagine two sites each having same prey-predator mechanism which causes the number density of the species to oscillate. If the species are capable of moving from site to site at a proper rate, the two sites may become stable with a constant and different number of species in each. Similar examples can be found in other fields. The examined oscillator was taken to be the Brusselator [40]. It contains only two components, and thus the computations with increasing number of coupled cells are not too difficult. Furthermore, we also consider Markovian switching into coupled oscillators to adapt complexity and variability of the external environment such as component failures or repairs, changing of subsystem interconnections and so on. Therefore, our system enriches the study of coupled oscillators to some extent. 5. Numerical examples In this section, we will give two numerical examples to verify the effectiveness and feasibility of our main results. Consider random coupled oscillators with Markovian switching established on digraphs H with 12 vertices. The right-continuous Markov process r(t) taking values in S = {1, 2} is given with the generator , where



 = (πi j )2×2 =



−2 3

2 . −3

(i ) Digraphs (H, [akh ]12×12 ) are presented in Fig. 2, where i = 1, 2, which are strongly connected obviously. Furthermore, the corresponding weights are as follows. (1 ) (1 ) (1 ) ) State 1. a12 = a27 = a49 = a6(1,11 = 0.08, (1 )

(1 )

(1 )

(1 )

0.2, a23 = a38 = a45 = a56 = 0.09,

(1 ) (1 ) (1 ) a11 = a93 = a12 = ,5 ,6

) (1 ) ) (1 ) (1 ) (1 ) (1 ) a1(1,12 = a34 = a5(1,10 = a61 = 0.1, a71 = a82 = a10 = 0.18, ,4

Table 1 Selection of functions. i

hk (xk , t, i)

ζ k (xk , i)

Hkh (xk , xh , t, i)

gk (xk , t, i)

1 2

0.17xk 0.18xk

0.1 sin xk + 0.051 0.1 sin xk + 0.085

0.5 ( xk + xh ) 0.6 ( xk + xh )

0.1sin xk 0.1sin xk

(2 ) ) (2 ) (2 ) State 2. a43 = a6(2,12 = a71 = a93 = 0.07,

(2 ) (2 ) (2 ) a16 = a10 = a12 = ,9 ,11

(2 ) (2 ) (2 ) (2 ) 0.2, a28 = a65 = a98 = a11 = 0.09, ,5 (2 ) ) (2 ) ) a21 = a4(2,10 = a11 = a7(2,12 = 0.1, ,10 0.19,

(2 ) (2 ) (2 ) a32 = a54 = a87 =

and other weights are 0. Example 1. According to Remark 6, we choose σ (t ) = sin2 t cos(0.02t + U ), where U is a random variable uniformly distributed on [0, 2π ]. And σ (t) is a special narrow-band Gaussian process and satisfies E|σ (t)|2 < D. Then the simulation of σ (t) is given in Fig. 3. Moreover, taking q(1 ) = 1.5, q(2 ) = 1.6, through simple calculation, we obtain a˜kh = 0.2. In addition, for k, h = 1, 2, · · · , 12, we choose the following functions presented in Table 1. Then, based on conditions C1, C2 and Assumption 2, we take τk(1) = 1, τk(2) = 1.1, Rk(1) = 0.5, Rk(2) = 0.6 and M = 0.6. Choose

θk(i) = 1.44,

D = 1 and ξ = 0.2, then we get Qk(i ) = 0.1, i = 1, 2. According to these chosen parameters, conditions C1, C2 in Theorem 5 are satisfied. Taking the corresponding initial values as above, the corresponding numerical simulations for the sample trajectory and the mean square trajectory of solution of system (38) are shown in Fig. 4, which illustrates that system (38) is unstable. In order to stabilize system (38), for i = 1, 2 we take Uk(i ) =

0.6 where k = 1, 2, · · · , 6 and Uk(i ) = 0.7 where k = 7, 8, · · · , 12; making the intermittent control activated on time span # # # # # [0, 0.8 ) [1, 1.82 ) [1.92, 2.8 ) [2.9, 3.71 ) [3.9, 4.73 ) · · · . s −t Then we have κ = lim infl→∞ t l −tl = 0.8. Moreover, it is easy to l+1

l

get that η = 1.46 and ρ = 3.8, which means that condition C3 in Theorem 5 is satisfied. Until now, all conditions in Theorem 5 have been satisfied, which implies that the equilibrium solution of controlled system (39) is globally asymptotically stable in probability. Taking the same initial values, we give the simulation results in Fig. 5. In detail, Fig. 5 shows sample trajectory and the mean square trajectory of the solution of system (39) with control, respectively. The simulation results illustrate the effectiveness and feasibility of our derived results. On one hand, if we only change the control gain Uk(i ) and other parameters remain the same values (make Uk(i ) = 2.5, i = 1, 2,

M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

11

Fig. 3. The figure of narrow-band random process σ (t).

Fig. 4. State evolution of system (38) without control; the left one is the sample trajectory and the right one is the mean square trajectory.

Fig. 5. State evolution of system (39) with aperiodically intermittent control (Uk(i ) = 0.6, κ = 0.8); the left one is the sample trajectory and the right one is the mean square trajectory.

k = 1, 2, · · · , 12, i.e., make control gain larger), it is easy to see that all conditions in Theorem 5 are fulfilled. Taking the same initial values as above, simulation results have been given in Fig. 6. Compared Fig. 6 with Fig. 5, it is obvious to see that larger control gain promises better stability, which conforms the theoretical results. On the other hand, we make other parameters remain the same and control activated all the time (κ = 1 ), which yields that intermittent control becomes continuous control. In this situation, it is clear that all conditions in Theorem 5 are still satisfied, and the

simulation results are presented in Fig. 7. Compared Fig. 7 with Fig. 5, we can see that larger κ also promises better stability. Example 2. Here, take the same values of the parameters and functions as those in Example 1 except for the stochastic process σ (t). And σ (t) is denoted as a Gaussian process with zero expectation and satisfies

E|σ (t )|2 ≤ 0.49

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Fig. 6. State evolution of system (39) with aperiodically intermittent control (Uk(i ) = 2.5, κ = 0.8); the left one is the sample trajectory and the right one is the mean square trajectory.

Fig. 7. State evolution of system (39) with continuous control (Uk(i ) = 0.6, κ = 1); the left one is the sample trajectory and the right one is the mean square trajectory.

Fig. 8. The figure of Gaussian process σ (t).

and



∞

E t0



 σ (s ) − E(σ (s )) E σ (t ) − E(σ (t )) ds < 4, t ≥ t0 .

The numerical simulation for Gaussian process σ (t) with zero expectation is shown in Fig. 8.

Based on Remark 8, we obtain that δ (ε ) = (0.7 + 2ε )ε . By employing the similar analysis of Example 1, one can get that system (38) is unstable without control. According to Example 1, we have ξ = 0.2, η = 1.46. Then through easy calculation, one can get that μ/T∗ > δ (ξ ). Up to now, all conditions in Theorem 6 have been satisfied. As a result, the equilibrium solution of system (39) is exponentially stable in mean square. Finally, the simulation

M. Wang, J. Guo and W. Li / Neurocomputing 373 (2020) 1–14

13

Fig. 9. State evolution of system (38) without control; the left one is the sample trajectory and the right one is the mean square trajectory.

Fig. 10. State evolution of system (39) with aperiodically intermittent control (Uk(i ) = 2.5, κ = 0.8); the left one is the sample trajectory and the right one is the mean square trajectory.

results of system (38) (without control) are shown in Fig. 9, and the simulation results of system (39) (with control gain Uk(i ) = 2.5 and κ = lim infl→∞

sl −tl tl+1 −tl

= 0.8) are shown in Fig. 10, respec-

tively, where i = 1, 2, k = 1, 2, · · · , 12. It is obvious that these numerical simulations all demonstrate the effectiveness of our results. 6. Conclusion In this paper, we studied the stabilization of RCSNMS via aperiodically intermittent control. On the basis of Lyapunov method, graph theory along with stochastic analysis techniques, sufficient criteria were presented to demonstrate the stability of RCSNMS. Distinguished from existing papers, we chose GAS-P and ES-p-M to adapt more practical systems. Based on this paper, some interesting topics for future research are as follows. • There exist undesirable phenomena such as oscillation and instability due to time delay. So, it is meaningful to study systems with time delay. Exploring the stabilization of random coupled systems on networks with Markovian switching and time delay via aperiodically intermittent control is one of our next work. • There exist systems with semi-Markovian switching or uncertain neutral Markovian switching. Subsequently, we will study the stabilization of random coupled systems on networks with semi-Markovian switching via aperiodically intermittent control in the future. • Due to the increasing importance on impulsive control, in the future work, the stabilization of random coupled systems via impulsive control is also one of the topic we are interested in. Declaration of Competing Interest The authors declare that they have no conflict of interest concerning the publication of this manuscript.

Acknowledgements The authors really appreciate the valuable comments of the editors and reviewers. 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) and the Innovation Technology Funding Project in Harbin Institute of Technology (No. HIT.NSRIF.201703).

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[34] V. Sobolev, G. Kashcheeva, F. Zhuravel, Maximum likelihood estimates of the central frequency of narrow-band random normal processes from a minimum number of samples, J. Commun. Technol. Electron. 62 (9) (2017) 990–1003. [35] J. Bao, X. Mao, G. Yin, C. Yuan, Competitive lotka-volterra population dynamics with jumps, Nonlinear Anal.-Theory Methods Appl. 74 (17) (2011) 6601–6616. [36] Y. Wu, B. Chen, W. Li, Synchronization of stochastic coupled systems via feedback control based on discrete-time state observations, Nonlinear Anal.-Hybrid Syst. 26 (2017) 68–85. [37] S. Li, H. Su, X. Ding, Synchronized stationary distribution of hybrid stochastic coupled systems with applications to coupled oscillators and a chua’s circuits network, J. Frankl. Inst.-Eng. Appl. Math. 355 (17) (2018) 8743–8765. [38] H. Zhou, W. Li, Synchronisation of stochastic-coupled intermittent control systems with delays and levy noise on networks without strong connectedness, IET Contr. Theory Appl. 13 (1) (2019) 36–49. [39] A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol 16 (1) (1967) 15–42. [40] I. Prigogine, F. Henin, C. George, Dissipative processes, quantum states, and entropy, Proc. Natl. Acad. Sci. USA 54 (1) (1968) 7–14. Mengxin Wang was born in 1996. She received her B.S. degree in applied mathematics from Ludong University, China, in 2018 and she is currently pursuing the M.S. degree in Harbin Institute of Technology, China. Her current research interests include stability theory of stochastic coupled systems.

Jia Guo was born in 1999. She is currently an undergraduate student in the Department of Mathematics, Harbin Institute of Technology, China. Her current research inter ests include stability of coupled systems.

Wenxue Li was born in 1981. He received his Ph.D. degree from Harbin Institute of Technology, China, in 2009. He is currently an associate professor in Harbin Institute of Technology at Weihai. His current research interests include stability theory for stochastic differential and integral equations.