Stabilization of Stochastic Differential Equations Driven by G-Brownian Motion with Aperiodically Intermittent Control

: The paper is devoted to studying the exponential stability of a mild solution of stochastic differential equations driven by G-Brownian motion with an aperiodically intermittent control. The aperiodically intermittent control is added into the drift coefﬁcients, when intermittent intervals and coefﬁcients satisfy suitable conditions; by use of the G-Lyapunov function, the p -th exponential stability is obtained. Finally, an example is given to illustrate the availability of the obtained results.


Introduction
In this paper, stochastic differential equations driven by G-Brownian motion (G-SDEs) are considered as follows where the coefficients f , g, σ : R + × R n → R n , x(0) = x 0 ∈ R n . B(t) is G-Brownian motion, B (t) is usually called the quadratic variation process of G-Brownian motion. Due to the nonlinear properties of expectation with G-Brownian motion, G-SDEs are more general than classical SDEs driven by Brownian motion, and can be widely used in many fields. With the development of G-theory and related stochastic calculus ( [1]), many interesting results of G-SDEs have been obtained, for instance, existence, uniqueness, boundedness ( [2][3][4][5][6][7] and the references therein). As we know, stability is one of the most interesting topics in dynamic behaviors. Regarding SDEs, many interesting works have been obtained on this issue (one can see [8,9]). Similarly, a lot of researchers have made great efforts on the subject of G-SDEs, for instance, exponential stabilization and quasi-sure exponential stabilization ( [10]). However, the most relevent is how to make an unstable system stable. Recently, an aperiodically intermittent control has been presented to make systems stable ( [11,12]). In particular, Yang et al. [13] investigated the stability of a solution of G-SDEs by constructing an aperiodically intermittent control which is set in diffusion coefficient. Meanwhile, based on the stability of G-SDEs, the stabilization of a stochastic Cohen-Grossberg neural networks driven by G-Brownian motion was established. A natural problem is whether one can stabilize the G-SDEs when an aperiodically intermittent control is added into the drift coefficients. As far as we know, there is no result on this topic. Taking the issue under consideration, we will investigate the stability of (1) with an aperiodically intermittent control added into the drift coefficient where the aperiodically intermittent control Differing from Yang et al. [13], we investigate the stabilization problem of G-SDEs, whose drift coefficients are added with an aperiodically intermittent control. The main innovations and contributions of this paper are highlighted as follows. • A new aperiodically intermittent control is designed to stabilize this class stochastic system, driven by G-Brownian motion. Moreover, the aperiodically intermittent control is added to the drift coefficient. • The aperiodically intermittent interval satisfies By the Lyapunov function satisfying suitable conditions, the p-th exponential stability is obtained. When p = 2, it is the exponential stability in mean square. Finally, an example is presented to show the efficiency of the obtained result .
The rest of the paper is arranged as follows. In the next section, some basic notions, preliminaries and lemmas are provided. In Section 3, we prove exponential stability for the solution of G-SDEs, whose drift coefficients are added to an aperiodically intermittent control. Finally, an example is presented to show the efficiency of the result.

Notations
In this section, some notations, with respect to G-Brownian motion and related stochastic calculus, are introduced. Ω denotes the collection of all continuous functions ω on R n with ω 0 = 0, and the distance in Ω is given by as the space of all bounded Lipschitz continuous functions on R n , and is the viscosity solution of the following nonlinear heat equation Definition 2. The canonical process B(t) t≥0 is called G-Brownian motion, if the following properties are verified Furthermore, the sublinear expectation E is called the G-expectation.
In the following part, we introduce the Itô integral with respect to the G-Brownian motion. Firstly, some space notations are prsented.
moreover, the quadratic variation process of the G-Brownian motion B(t) is defined by

Main Results
Definition 5. Suppose there exist positive constants λ and C, such that the solution X(t) of (1) satisfies E X(t) p ≤ CE|X(0)| p e −λt , for any initial value X(0), p ≥ 2, then, the mild solution X(t) is said to be p-th exponentially stable.
If V(t, x) ∈ C 1,2 (R + × R n ; R n ), the Lyapunov operator L : R + × R n → R n associated to the G-SDEs (1) is defined as below Theorem 1. Assume that the function V(t, x) associated with (2) is in C 1,2 (R + × R n ; R n ) and there exist positive constants c 1 , c 2 , a 1 , a 2 , γ, M such that and LV(t, y(t)) ≤ a 2 V(t, y(t)), t ∈ [s i , t i+1 ).
It follows from (3) that Setting M = c 2 c 1 e (a 1 +a 2 )ψυ , we get the desired result.

Conclusions
The paper studied the p-th exponential stability for the mild solution of stochastic differential equations driven by G-Brownian motion. By using an aperiodically intermittent control, added to the drift coefficients and G-Lyapunov function, the desired result is obtained under suitable conditions. Moreover, the length of intermittent intervals is given. Finally, an example is presented to introduce the effectiveness of the results.