Stochastic stabilization of switching diffusion systems via an intermittent control strategy with delayed and sampled-data observations

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Abstract

This paper investigates the intermittent stochastic stabilization of switching diffusion systems (SDSs) via sampled data with time delays of state and mode. First, by virtue of analysis techniques for Markov chains, some estimates of the solutions for SDSs are derived. Second, with the aid of iteration by steps, a sufficient criterion is proposed for intermittent stochastic stabilization via sampled data with time delays of state and mode. Finally, a simulation example verifies the effectiveness of the theoretical results.

Introduction

It is well known that noise can destabilize a stable system. Moreover, it can stabilize an unstable system. Khasminskii [1] first demonstrated that an unstable system can be stabilized by white noise. Subsequently, Mao [2] investigated stochastic stabilization by Brownian motion under a linear growth condition. Since then, stochastic stabilization theory has been further developed. Appleby et al. [3] and Appleby and Mao [4] lifted the linear growth condition and studied stochastic stabilization. Stochastic stabilization theory has been generalized to stochastic suppression [5], [6], [7].

The switching diffusion systems (SDSs) modulated by Markov chains have been used to describe the practical systems where there are abrupt changes in structure and parameters. In the past several decades, stabilization of SDSs has been an important issue in control theory [8], [9], [10], [11], [12], [13], [14]. Mao and his collaborators obtained several results on the stochastic stabilization of SDSs [15], [16], [17], [18]. For more references on this topic, readers can refer to [19], [20].

To improve the practicability and efficiency of controllers, increasing numbers of scholars have considered novel control strategies, such as the sampled-data strategy and the delayed observation strategy [9], [10], [13], [21], [22], [23], [24], [25]. Mao [22] first proposed the sampled-data strategy of state and obtained mean-square exponential stabilization. This was then generalized to almost sure exponential stabilization by You et al. [25]. For the delayed observation strategy, Hu et al. [9] designed a delayed feedback control and discussed the connection between the delayed feedback control and the control function without time delays. Mao and his collaborators integrated the delayed observation strategy and the sampled-data strategy into stochastic stabilization in [21] and [23], respectively. Note that mode might not be observed continuously in practical models. However, most existing controllers are based on sampled state data or delayed state observations. Only a few studies have considered sampled mode data or delayed mode observations [18], [26], [27], [28], [29], [30]. Especially, Li et al. [26] studied the stabilization of SDSs via sampled data with time delays of state and mode. It is meaningful to further introduce the sampled-data strategy with time delays of state and mode into the stochastic stabilization of SDSs.

Moreover, an intermittent control strategy can avoid the continuous operation of controllers and extend controllers’ working life, and it is widely used in stabilization theory [16], [31], [32], [33], [34], [35]. In [35], Deng and his collaborators analyzed the stabilization of a nonlinear system by an intermittent Brownian noise perturbation. In [32], Cao and his collaborators investigated intermittent stochastic stabilization through the sampled-data strategy or the delayed observation strategy. Especially, Liu and Wu [31] studied the intermittent stochastic stabilization via sampled data with a time delay.

Motivated by the above-mentioned literature, this paper is devoted to stabilizing an unstable SDS by an intermittent stochastic feedback control based on sampled data with time delays of state and mode. The contributions are summarized in two aspects:

  • (1)

    With the aid of analysis techniques for Markov chains and iteration by steps, the difficulty arising from the simultaneous existence of an intermittent regime and Markovian switching is overcome.

  • (2)

    A sufficient criterion is proposed that ensures the stochastic stabilization via integrating the intermittent control strategy, the sampled-data strategy and the delayed observation strategy.

The reminder of this paper is organized as follows. Basic concepts are introduced in Section 2. Main results are presented in Section 3, including almost sure exponential stabilization. A simulation example is given to illustrate the effectiveness of our methods in Section 4. Concluding remarks are presented in Section 5.

Section snippets

Preliminaries

Let (Ω,F,{Ft}t0,P) be a complete probability space with {Ft}t0 satisfying the usual conditions. Denote by E() the mathematical expectation with respect to probability measure P. Let B(t) be a scalar Brownian motion on the probability space. Let r(t) be a right-continuous Markov chain on the probability space taking values in a finite space S={1,2,,N} with generator Γ=(γij)N×N given by P{r(t+Δ)=j|r(t)=i}=γijΔ+o(Δ)if ij,1+γiiΔ+o(Δ)ifi=j,where Δ>0. The transition rate from i to j is γij0 if i

Main results

To investigate the stability of SDS (2.2), we first provide an auxiliary system as follows: dι(t)=f(ι(t),r(t))dt+h(t)g(ι(t),r(t))dB(t).In SDS (2.2), we consider initial data x0=ξLF02([τ1,0];Rn) and r0=i0S. In this section, we choose the parameters p(0,1] and c=(c1,,cN)T0 satisfying Assumption 2.2.

In SDSs (2.2), (3.1), the global Lipschitz conditions on f and g ensure the existence and uniqueness of the global solution on [0,+), and the conditions f(0,i)=0 and g(0,i)=0 ensure the

Numerical simulation

A feedback control is applied to a Hopfield neural network with Markovian switching in the following Example 4.1.

Example 4.1 Hopfield Neutral Network

Consider the following Hopfield neural network: dx(t)=D(r(t))x(t)+W(r(t))F(x(t))dt,where x(t)=(x1(t),x2(t))T is the state vector of neurons at time t and S={1,2}. Let generator matrix Γ, passive decay rates D(i) and connection weight matrices W(i) be Γ=3311,D(1)=0.45000.45,D(2)=0.55000.55,W(1)(wij1)2×2=0.050.10.10.05,W(2)(wij2)2×2=0.150.10.10.15.Let activity function

Conclusion

In this paper, we have investigated the intermittent stochastic stabilization of SDSs via sampled data with time delays of state and mode. First, some estimates of solutions for SDSs were established with the aid of analysis techniques for Markov chains. Second, by using the iteration by steps in Theorem 3.1, the difficulty arising from the simultaneous existence of Markovian switching and an intermittent regime was overcome. A sufficient criterion was given to ensure the intermittent

CRediT authorship contribution statement

Runyu Zhu: Methodology, Writing – original draft. Lei Liu: Conceptualization, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The project reported here is supported by the National Science Foundation of China under Grant 61773152, the Fundamental Research Funds for the Central Universities under Grant 2019B19214, the Qing Lan Project of Jiangsu Province, China .

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