Stabilization of nonlinear systems via aperiodic intermittent stochastic noise driven by G-Brownian motion with application to epidemic models

*Correspondence: oyhb1987@163.com 1School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou, Guangdong 510006, China Full list of author information is available at the end of the article Abstract To stabilize a nonlinear system dx(t) = f (t, x(t))dt, we stochastically perturb the deterministic model by using two types of aperiodic intermittent stochastic noise driven by G-Brownian motion. We demonstrate quasi-sure exponential stability for the perturbed system and give the convergence rate, which is related to the control intensity. An application to SIS epidemic model is presented to confirm the theoretical results.


Introduction
Since Khas'minskii [1] used two white noise sources to stabilize a system, a wide range of works have appeared on stochastic stabilization problems. Arnold et al. [2] obtained stabilization results by using noisy terms in Stratonovich sense. Mao [3] presented a general theory on the stabilization by Brownian motion. Huang [4] further developed the general theory by Mao and revealed a more fundamental principle. Zhao et al. [5] established a new type of stability theorem which generalized local Lipschitz and one-sided linear growth conditions. From the considerations of reducing control cost and time, discontinuous controllers have been designed to stabilize a given system, such as discrete-time feedback control [6,7], pinning control [8], impulsive control [9], adaptive control [10], intermittent control [11], etc. As for intermittent control, the control time is divided into periodic and aperiodic type. Periodically intermittent control has been studied by many authors, especially in synchronization problems. Zhang et al. [11] considered a periodic intermittent Brownian noise perturbation to stabilize and destabilize a given nonlinear system, the obtained criteria are different. Recently, Liu et al. [12] investigated the aperiodically intermittent control which has good performance to quasi-synchronize nonlinear coupled networks [13].
Motivated by the idea of stochastic stabilization via intermittent stochastic noise driven by Brownian motion, we are interested in analyzing whether the presence of intermittent stochastic perturbation driven by G-Brownian motion can stabilize a nonlinear system, since G-Brownian motion has powerful applications in modeling uncertainties. It is necessary to mention the pioneering work by Peng [14] who set up the G-framework. He pointed out that G-Brownian motion has independent increments and can be consistent with the classical Brownian motion in the sense of no volatility uncertainty. Many works have been done on G-Brownian motions [15][16][17][18][19][20], in particular existence and uniqueness theory for stochastic differential equations driven by G-Brownian motion (G-SDEs), as well as stability behavior and control theory, has been developed. Fei [16] investigated the exponential stability of paths for a G-SDE. Ren [19] designed a feedback control based on discrete-time observations to stabilize a G-SDE system. In [18], the aperiodically intermittent control has been embedded into the drift part, the authors obtained a set of piecewise Lyapunov-type conditions for the moment exponential stability theory.
As far as we know, there is hardly any literature about stochastic stabilization of deterministic systems via aperiodic intermittent stochastic perturbation driven by G-Brownian motion. In the present paper, we add two aperiodic intermittent stochastic perturbations driven by G-Brownian motion into a general deterministic nonlinear system. Those stochastic perturbations can stabilize the nonlinear system. The main contributions are summarized as follows: • The control itself is a stochastic perturbation driven by G-Brownian motion, which contains mean and volatility uncertainties, therefore, expands the general deterministic intermittent control and the stochastic intermittent control which is driven by classical Brownian motion. • The control time is aperiodically intermittent, which improves flexibility to time nodes and length. The acquired criteria consist of the work and rest width, we can control the steady rate autonomously by adjusting the work and rest width. In Sect. 2, we establish the aperiodic intermittent stochastically perturbed system (2.2) driven by G-Brownian motion, present four notions, two lemmas, and one definition which will be used in the next section. Stabilization analysis is carried out in Sect. 3. In Sect. 4, we provide an application on stabilizing an SIS epidemic model by adding a special aperiodic intermittent stochastic perturbation driven by G-Brownian motion. This example clearly shows the power of stabilization by aperiodic intermittent stochastic perturbation driven by G-Brownian motion.

Preliminaries
Consider a nonlinear system with initial value x(t 0 ) = x 0 ∈ R n . We add two aperiodic intermittent stochastic perturbations driven by G-Brownian motion to the nonlinear system, then the system becomes is the quadratic variation process of the G-Brownian motion, which is also a continuous process with independent and stationary distribution, thus can still be regarded as a Brownian motion. Under the perturbation of h type, the time For the aperiodically intermittent perturbation strategy, the start time and the noise width might be different, but the total perturbation time ratio should be fixed in the long term. Mathematically, we assume there exist two positive scalars ω h , ω σ such that the above time nodes satisfy the following assumptions: We call ω h the h-type perturbation time ratio and ω σ the σ -type perturbation time ratio.

Definition 2.1
The trivial solution of the intermittent G-stochastic system (2.2) in R n is said to be quasi-sure exponentially stable, if for any x 0 = 0 and t ≥ t 0 , lim sup Proof The global existence of a unique solution follows from Theorem 4.5 in Li et al. [21], the nonzero property follows from the same method as in Mao [3] (see Lemma 3.2, p. 120).

Lemma 2.2
Let N(t) be G-Ito stochastic integral, τ n be a sequence of positive numbers with τ n → ∞. Then for all ω ∈ there exists a random integer n 0 (ω) such that for all n ≥ n 0 , Proof According to Lemma 2.6 in Fei et al. [16], We choose γ n = ε, θ = 2, g(n) = n, and the conclusion of Lemma 2.2 can be obtained naturally.
Remark 2.1 If t i+1t i = T, c i = δ for all i ∈ N , andδ = δ, then the system (2.2) becomes a periodic intermittent system. This agrees with system 1 in Zhang et al. [11]. Our results can be regarded as a generalization of Zhang et al. [11].

Main results
In this section, we will establish the quasi-sure exponential stability theorem based on aperiodic intermittent stochastic noise driven by G-Brownian motions. Since x 0 = 0 implies x(t; t 0 , 0) = 0, we only need to concentrate on x 0 = 0.
Case 2. For all ω ∈ and n > n 0 , t ∈ [t h n 1 , t h n 1 + c h n 1 ) ∩ [t σ n 2 + c σ n 2 , t σ n 2 +1 ), the integral interval length of σ (t, x(t)) has changed compared to Case 1. Hence we have By conditions (ii), (iii), (iv), and (v), we obtain Using the same method as in Case 1, we conclude lim sup q.s.
Case 3. For all ω ∈ and n > n 0 , t ∈ [t h n 1 + c h n 1 , t h n 1 +1 ) ∩ [t σ n 2 , t σ n 2 + c σ n 2 ) for all ω ∈ and n > n 0 . This case is similar to Case 1 except for the additional time interval [t h n 1 + c h n 1 , t h n 1 +1 ) of h(t, x(t)). Since h(t, x(t)) = 0, t ∈ [t h n 1 + c h n 1 , t h n 1 +1 ), log V (t, x(t)) can be written as Thus we claim lim sup q.s.
From the above four cases, for all ω ∈ and t ∈ [t h n 1 , t h n 1 +1 ] ∩ [t σ n 2 , t σ n 2 +1 ], the following inequality always holds lim sup The proof is complete.
This agrees with Theorem 1 in Zhang et al. [11]. Our results can be regarded as a generalization of Zhang et al. [11].

Application to an epidemic system
In this section, we study an application of our theoretical results in Sect. 3 on SIS epidemic model. A classical deterministic SIS epidemic model partitions the host population into the susceptible compartment S and the infectious compartment I. Ordinary differential equations (ODEs) that describe the change of size in compartments S and I can be written as (4.1) Since S, I ≥ 0 and S + I = A d , the above two ODEs can be rewritten as The dynamics of the SIS epidemic model is completely determined by the basic reproduction number .
If R 0 ≤ 1, the disease-free equilibrium P 0 = ( A d , 0, 0) is globally asymptotically stable and the disease always dies out; if R 0 > 1, then P 0 is unstable and an endemic equilibrium exists which means the disease will persist. Now, we aim to control the number of infectious even if R 0 > 1.