Synchronization for stochastic switched networks via delay feedback control

1. INTRODUCTION

Complex networks are ubiquitous in society and nature because they can abstractly describe almost all actual complex systems, such as the relationship between bacteria and cells, cooperation among academia, intelligent systems, Internet communication, biological engineering, power systems, Brownian motion, and others (see, e.g., ^[1–3] and reference therein). Therefore, they have become an irreplaceable framework and natural phenomenon in many complex social studies, which provides us with a new perspective and method of complexity research^[4,5]. So, the research on complex networks is of great significance.

At present, the research content of complex networks mainly focuses on the following aspects: the formation mechanism of the network, the geometric nature of the network, the nature of the network model, the stability of the network, the synchronization and consistency of the network, and other issues^[6–12]. Synchronization is a common collective behavior phenomenon in nature. In recent years, the synchronization theory and application research of complex networks have received extensive attention from domestic and foreign scholars. In terms of application research, synchronization mechanisms are applied to issues such as secure communications, nervous systems, superconducting materials, transportation networks, the Internet, and so on^[13–17]. The most common synchronization problems in life are the glow of fireflies, the behavior of fish swimming in the ocean, and the gradual regular applause when the audience applauds. Therefore, the synchronization study of complex networks is of great significance in revealing the universal laws of network dynamics.

In the existing literature, people have studied various synchronization problems of the network model. In most of the literature, a commonly used and taken-for-granted method is to design a controller to synchronize the network. However, a more practical problem is the difference between the time to observe the state and the time to control the system. The time interval is greater than zero, so a more reasonable explanation is to set a controller with a time delay for the system. The paper^[18] first proposed feedback control with time delays and applied it in the study of the stabilization of stochastic differential equations (SDEs). As we know, the delay feedback has not been used in the study of the synchronization issue for Markov switched stochastic networks. However, there are a large number of time-delay feedbacks in reality, such as monetary policy in economic systems, control systems in industrial processes, and neural systems in biology. Therefore, we study time-delay feedback control in Markov switching stochastic networks.

In addition, the existing research on Markov switching networks mainly focuses on the exponential mean square synchronization, such as ^[19]. Although exponential mean square synchronization provides convergence characteristics, it cannot guarantee that every orbit can be synchronized. In contrast, almost sure exponential synchronization has more advantages, because it not only ensures that all orbits are synchronized but also enables the synchronization speed to be faster in comparison. We not only study almost sure exponential synchronization but also investigate asymptotical synchronization in mean square and $$H_{\mathrm{\infty }}$$-synchronization.

Based on the theory of SDEs, the properties of Markov processes, and the It$$\hat{o}$$ formula by designing a delay feedback controller, the almost sure exponent is established with stochastic synchronization of noisy complex networks. Next, we will introduce the contribution of this article:

1.We design a suitable controller with time delay to achieve almost sure exponential synchronization, asymptotical synchronization in mean square and $$H_{\mathrm{\infty }}$$-synchronization in the Markov switched stochastic networks, which is more consistent with the actual situation and differs from the previous control strategies.

2.From a practical point of view, almost sure exponential convergence is almost certainly more effective, because it can ensure that each orbit of the stochastic process reaches convergence. The synchronization problem of complex networks with random noise and Markov switching under time-delay feedback control discussed in this paper is an infinite-dimensional problem, which is more difficult than a finite-dimensional problem.

3.Developing almost sure convergence in network synchronization is inherently challenging due to the need for estimating the time tail probability. As far as our current knowledge extends, there has not been any study addressing the almost sure convergence for complex networks involving a controller with time delay in the existing literature.

2. PROBLEM FORMULATION AND PRELIMINARIES

The notion diag {$$p_1,p_2,\cdots ,p_n$$} denotes the diagonal matrix with entries $$p_1,p_2,\cdots ,p_n$$ on the diagonal. $$I_N$$ stands for the identity matrix with dimension $$N$$. $$1_n$$ is a $$n$$-dimension vector whose entries are $$1$$. The notions $${\lambda }_{min}(\cdot )$$, $${\lambda }_{max}(\cdot )$$ represent the minimum and maximum eigenvalue of a given matrix, respectively. For a vector $$x$$, let $$x^{\mathrm{\top }}$$ denote the transpose vector and $$\Vert x\Vert$$ denote $$L_2$$-vector norm. Let $$\mathbb{R}$$ be the set of real numbers, $$\mathbb{N}=\{1,2,3,\cdots \}$$ be the set of natural numbers, $${\mathbb{R}}^n$$ be the n-dimensional Euclidean space and $${\mathbb{R}}^{n\times n}$$ be the set of all $$n\times n$$ real matrices. The symbol $$\otimes$$ denotes the standard Kronecker product.

Let $$B_i(t)=[B_1(t),B_2(t),\dots ,B_m(t){]}^T$$ be an $${\mathcal{F}}_t$$ -adapted Brownian motion and $$\{\sigma (t),t\ge 0\}$$ be a continuous time Markov process, where the state space of $$\sigma (t)$$ is $$\mathbb{S}=\{1,2,\cdots ,m\}$$. The generator $$Q=(q_{ij}{)}_{(m\times m)}$$ of Markov process is given by

where $$\underset{\mathrm{\Delta }t\to 0^{+}}{lim}o(\mathrm{\Delta }t)/\mathrm{\Delta }t=0$$, $$0\le q_{r\ell },(\ell \ne r)$$, and $$q_{\ell \ell }=-\sum _{r=1,r\ne \ell }^m{q}_{r\ell }$$.

Assume that $$\sigma (\cdot )$$ and $$B(\cdot )$$ are independent of each other. By using a delay pinning feedback controller, we study the Markovian switched stochastic network as follows:

where $$i=1,2,3,\cdots ,M,x_i(t)\in {\mathbb{R}}^n$$ represents the state vector of the $$i$$th node; $$h(x_i(t),{\sigma }_t)$$ and $$r(x_i(t),{\sigma }_t,t)$$ are continuous functions, and they separately describe the dynamics and noise intensity. The coupling matrix $$A({\sigma }_t)=(a_{ij}({\sigma }_t){)}_{N\times N}$$ is irreducible, which also satisfies: $$a_{ij}({\sigma }_t)\ge 0$$, $$a_{ij}({\sigma }_t)=a_{ji}({\sigma }_t)$$, $$i\ne j$$ and $$a_{ii}({\sigma }_t)=-\sum _{j=1,j\ne i}^M{a}_{ij}({\sigma }_t)$$; where $$d_i({\sigma }_t)=I_{i\in \mathcal{D}({\sigma }_t)}$$ is the indication function for the pinned node subset $$\mathcal{D}({\sigma }_t)\subset \{1,2,\cdots ,N\}$$, $$\rho$$ is the control gain, $$u(0,i,t)\equiv 0$$. The initial data are $$\{x(t):-\tau \le t\le 0\}=\xi \in C([-\tau ,0];R^n)$$ and $$\sigma (0)={\sigma }_0\in S$$.

Denote $$s(t)$$ as a desirable state solution that satisfies:

The set $$s(t)\times 1_n$$ is used as the synchronization manifold. The initial value of $$s(t)$$ is given by $$s(t)=\psi (t)\in C_{{\mathcal{F}}_0}^b([-\tau ,0],{\mathbb{R}}^n)$$. The error system of this network can be written as follows:

where: $$e(t)=(e_1^{\mathrm{\top }}(t),\cdots ,e_N^{\mathrm{\top }}(t){)}^{\mathrm{\top }},$$$$\hat{H}(e(t),{\sigma }_t)=H(x(t),{\sigma }_t)-H(s(t),{\sigma }_t),$$$$\hat{R}(e(t),{\sigma }_t)=R(x(t),{\sigma }_t)-R(s(t),{\sigma }_t)$$. $$H(x(t),{\sigma }_t)=(h(x_1(t),{\sigma }_t),\cdots ,h(x_N(t),{\sigma }_t){)}^{\mathrm{\top }},$$$$R(x(t),{\sigma }_t)=(r(x_1(t),{\sigma }_t),\cdots ,r(x_N(t),{\sigma }_t){)}^{\mathrm{\top }},$$$$H(s(t),{\sigma }_t)=(h(s(t),{\sigma }_t)\times 1_N{)}^{\mathrm{\top }},$$$$R(s(t),{\sigma }_t)=(h(s(t),{\sigma }_t)\times 1_N{)}^{\mathrm{\top }},$$$$D({\sigma }_t)=$$diag$$[d({\sigma }_t),d({\sigma }_t),\cdots ,d({\sigma }_t)].$$

Definition 1.^[20] ($$H_{\mathrm{\infty }}$$-Synchronization) The network (4) achieves $$H_{\mathrm{\infty }}$$-synchronization, if we have

Definition 2.^[20] (Synchronization in Mean Square) The solution of the network (4) satisfies

for $$\mathrm{\forall }$$$$\psi (t)$$, namely, the network (1) achieves synchronization in mean square.

Definition 3.^[16] (Exponential Synchronization) System (1) is said to achieve exponential synchronization, if there exist constants $$\epsilon >0$$ and $$G>0$$ such that

Definition 4.^[16] (Almost Surely Exponentially Synchronization) The solution of the network (1) with the initial has the property that:

That is, the network (1) is almost surely exponentially synchronization.

Definition 5. (QUAD Condition) The function $$h(x,\ell )$$ is said to satisfy the $$QUAD$$ condition, denoted as $$h(x,\ell )\in QUAD(P_{\ell },{\mathrm{\Delta }}_{\ell },{\phi }_{\ell })$$, if we can find positive define diagonal matrices $$P_{\ell }$$ and $${\mathrm{\Delta }}_{\ell }$$, for $$\ell =1,2,\cdots ,m$$ and a positive constant $${\phi }_{\ell }$$, such that for any $$x,y\in {\mathbb{R}}^n$$, the following condition holds:

Assumption 1.^[21] If we can find $${\eta }_{\ell }>0$$ and $${\alpha }_{\ell }>0$$ such that

Assumption 2.^[9,18]} The function $$r(x,\ell )$$ satisfies the Lipschitz condition and we can find $${\omega }_l>0$$ such that for $$l=1,2,\dots ,N$$,

Assumption 3.^[20] If we can find $$\gamma >0$$ such that

for all $$(x,\ell )\in R^n$$ and $$t\ge 0$$. This assumption, together with $$u(0,\ell )\equiv 0$$, implies

Assumption 4.^[22] If we can find $$K>0$$ such that

for all $$(x,\ell )\in R^n\times S.$$

Assumption 5.^[20] If we can choose functions $$W$$ and $$\mathrm{\Lambda }$$, as well as a positive number $$c$$ and $$q\ge 2$$, such that

and

for all $$(x,\ell )\in R^n\times S$$, $$W\in C^{2,1}(R^n\times S;R_{+})$$, $$\mathrm{\Lambda }\in C(R^n\times [-\tau ,\mathrm{\infty }))$$.

3. SYNCHRONIZATION ANALYSIS

For studying the problem of the synchronization of the controlled complex network system (4), we define two segments: $${\hat{e}}_t:=e(t+s):-\tau \le s\le 0$$ and $${\hat{\sigma }}_t:=\sigma (t+s):-\tau \le s\le 0$$ for $$t\ge 0$$. For $${\hat{e}}_t$$ and $${\hat{\sigma }}_t$$ to be well defined for $$0\le t\le \tau$$, we set $$e(s)=e_0$$ and $${\sigma }_s={\sigma }_0$$ for $$s\in [-\tau ,0)$$.

We choose the Lyapunov-Krasovskii function as follows:

where for $$t\ge 0$$, $$U(e(t),{\sigma }_t)=\frac{1}{2}{e}^{\mathrm{\top }}(t)\mathbf{\text{P}}({\sigma }_t)e(t)$$, $$\mathbf{\text{P}}({\sigma }_t)=I_N\otimes P({\sigma }_t)$$.

We claim that $$V({\hat{e}}_t,{\hat{\sigma }}_t)$$ is an It$$\hat{o}$$ process on $$t\ge 0$$, In fact, according to the generalized It$$\hat{o}$$ formula^[22], we have

For $$t\ge 0$$, where $$M(t)$$ is a martingale, with the initial value is $$0$$, and

where $$I(t)=U_x(e(t),{\sigma }_t)$$$$[u(e(t-\tau ),{\sigma }_t)-U(e(t),{\sigma }_t)]$$$$+\frac{2{\gamma }^2}{\beta }\tau [\tau \Vert \hat{H}(e(t),{\sigma }_t)$$$$+cA({\sigma }_t)e(t)$$$$-\rho D({\sigma }_t)e(t-\tau ){\Vert }^2+\Vert \hat{R}(e(t),{\sigma }_t){\Vert }^2]$$$$-\frac{2{\gamma }^2}{\beta }{\int }_{t-\tau }^t[\tau \Vert \hat{H}(e(v),{\sigma }_v)+cA({\sigma }_v)e(v)$$$$-\rho D({\sigma }_v)e(v-\tau ){\Vert }^2+\Vert \hat{R}(e(v),{\sigma }_v){\Vert }^2]dv.$$

Lemma 1. Under Assumptions (1)-(6), then we have the following inequality:

where $${\lambda }_1={\lambda }_{min}[\phi \mathbf{\text{P}}({\sigma }_t)-I_N\otimes P({\sigma }_t)\mathrm{\Delta }]$$.

Proof: Step 1, for $$u(x,\ell )$$, we will let $$U_t(x,\ell )=\frac{\partial u(x,\ell )}{\partial t}$$, $$U_x(x,\ell )=(\frac{\partial u(x,\ell )}{\partial x_1},\cdots ,\frac{\partial u(x,\ell )}{\partial x_n})$$, and $$\mathcal{L}U$$:$$R^n\times S⟶R$$ is defined by

Step 2, we compute the $$\mathcal{L}{U}_1(t)-\mathcal{L}{U}_4(t)$$ respectively. For the $$\mathcal{L}{U}_1(t)$$, $$U_x(e(t),{\sigma }_t)=e^{\mathrm{\top }}(t)\mathbf{\text{P}}({\sigma }_t)$$, $$U(e(t),{\sigma }_t)=-\rho D({\sigma }_t)e(t)$$, we obtain that

where $$\overline{p}={\lambda }_{max}\mathbf{\text{{}}P({\sigma }_t)\}$$, $$\underset{―}{p}={\lambda }_{min}\mathbf{\text{{}}P({\sigma }_t)\}$$, $$\overline{\lambda }={\lambda }_{max}\mathbf{\text{{}}D({\sigma }_t)\}$$, $$\underset{―}{\lambda }={\lambda }_{min}\mathbf{\text{{}}D({\sigma }_t)\}$$. For the $$\mathcal{L}{U}_2(t)$$, one can see

From the definition of $$A(r)$$, we know that $$\lambda (A(r))\le 0$$, which yields

According to Assumption 2, we obtain

where $${\lambda }_1={\lambda }_{min}[\phi \mathbf{\text{P}}({\sigma }_t)-I_N\otimes P({\sigma }_t)\mathrm{\Delta }]$$. By substituting (14), (15) into (13), we then obtain

Furthermore, by Assumption 3, one can see

where $$\omega =max\{{\omega }_{\ell }\}$$. Based on the properties of the Markov process, we can obtain

where $$\pi =\frac{1}{2}\sum _{l=1,l\ne {\sigma }_t}^m\overline{p}{q}_{{\sigma }_{t}l}+\underset{―}{p}{q}_{ll}.$$

Step 3, substituting (12)-(18) into (11), we can get;

where $$c=[\rho \overline{p}\underset{―}{\lambda }+{\lambda }_1-\frac{\overline{p}\omega }{2}-\pi ].$$

The proof of Lemma 1 is, therefore, completed.

Lemma 2.

Given that Assumptions (5) and (6) hold, the solution of the complex (4) satisfies

Theorem 1. Under Assumptions (1)-(6), if $$U_x(e(t),{\sigma }_t)$$ satisfies that $$U_x(e(t),{\sigma }_t)\le \Vert e(t){\Vert }^2$$, the delay pinning feedback control and $$d>0$$, the coupled network (1) can achieve $$H_{\mathrm{\infty }}$$-synchronization.

Proof: Step 1: According to (19) and (10), one can see that

By Assumption 4, it is easy to see that

According to the inequality $$(a+b{)}^2\le 2(a^2+b^2)$$ and Assumption 1,

where $$\eta =max\{{\eta }_1,{\eta }_2,\cdots ,{\eta }_N\}$$, $$\alpha =max\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_N\}$$, $$\ell =1,2,\cdots ,N$$, $${\eta }_{\ell }$$ and $${\alpha }_{\ell }$$ are defined in Assumption 1. Noting $$\tau \le \sqrt{\frac{\beta }{16{\rho }^2{\bar{\lambda }}^2}}$$, we have

Substituting (22), (23), (24) into (21), we can obtain

where $$\chi =[c-\frac{\beta {\overline{p}}^2}{2}-\frac{8{\gamma }^2{\tau }^2{\rho }^2{\bar{\lambda }}^2}{{\beta }^2}-\frac{2{\gamma }^2}{\beta }(2{\tau }^2\eta +\tau \alpha )]$$.

Follows from the error system (4) that, for $$t\ge \tau$$

Substituting (26) into (25), we can obtain

Step 2: Using a similar method in Theorems 3 and 4^[23], we define the stopping time as follows

According to Lemma 1, $$U(e(t),{\sigma }_t)$$ satisfied assumption 6, and $$h(x,\ell )$$ satisfied assumption 5; by lemma 2, $${\zeta }_k$$ is increasing to infinity almost surely as $$k\to \mathrm{\infty }$$. Where $$t\ge 0$$, $$H\in C^{2,1}(R^n\times S;R_{+})$$.

Then, we can obtain

for any $$t\ge 0$$ and $$k\ge k_0$$, we can let $$k\to \mathrm{\infty }$$ and then apply the Fubini theorem to get

for any $$t\ge 0$$, substituting (27) into (30), we obtain

We can get from (31)

Step 3: Noting that $$\chi >0$$, we see from the above inequality that

Letting $$t\to \mathrm{\infty }$$, we obtain that

The proof is, therefore, complete.

Theorem 2. Under Assumptions (1)-(6), the solution of the controlled network (4) for any given initial data, the controlled system (1) is asymptotical synchronization in mean square.

Proof: For any $$0\le t_1<t_2<\mathrm{\infty }$$, according to Assumptions 4 and 5, we can apply the It$$\hat{o}$$ formula to show

where $$\mathrm{\Theta }$$ is a constant independent of $$t_1$$ and $$t_2$$. That is, $$\underset{t\to +\mathrm{\infty }}{lim}E\Vert e(t){\Vert }^2=0.$$

Lemma 3. (Hanalay inequality) Let w(t) be a nonnegative function defined on the interval $$[t_0-\tau ,\mathrm{\infty })$$, and be continuous on the subinterval $$[t_0,\mathrm{\infty })$$. If there exist two positive constants $$a,b$$ satisfying $$a>b$$, such that:

then $$w(t)\le w_{t_0}{e}^{-\gamma (t-t_0)}$$, $$t\ge t_0$$, there $$w_{t_0}=\underset{t_0-\tau \le t\le t_0}{sup}w(t).$$$$\gamma >0$$ is the smallest real root of the equation $$a-\gamma -b{e}^{\gamma \tau }=0.$$

Theorem 3. Under Assumptions (1)-(6), and the delay feedback pinning control given by (2). $$\epsilon \tau \le \frac{1}{2}$$ exists; the coupled network (1) can achieve exponential synchronization.

Proof: We choose Lyapunov function $$V({\hat{e}}_t,{\hat{\sigma }}_t)$$ as defined by (8). Similar to the proof in Step 2 of Theorem 1 and according to the formula (30), we can show that

Let $$h_1=\underset{\ell \in S}{min}{\underset{―}{p}}_{\ell },h_2=\underset{\ell \in S}{max}{\overline{p}}_{\ell }$$. From (8), we can get

where

Similar to what we did in Theorem 1, we can show that:

where: $$J_2=\frac{2{\gamma }^2}{\beta }{\int }_{s-\tau }^s(\tau \Vert \hat{H}(e(v),{\sigma }_v)+cA({\sigma }_v)e(v)-\rho D({\sigma }_v)e(v-\tau ){\Vert }^2+\Vert \hat{R}(e(v),{\sigma }_v){\Vert }^2)dv,$$$$\mathrm{\Phi }=[c-\frac{\beta {\overline{p}}^2}{2}-\frac{2{\gamma }^2}{\beta }(2{\tau }^2\eta +\tau \alpha )].$$ For all $$t\ge 0$$, where $$\epsilon$$ is a sufficiently small positive number to be determined later. Substituting (37) and (38) into (36), then we have