EXPONENTIAL STABILITY OF SDES DRIVEN BY G -BROWNIAN MOTION WITH DELAYED IMPULSIVE EFFECTS: AVERAGE IMPULSIVE INTERVAL APPROACH

. In this article, we discuss a class of impulsive stochastic function diﬀerential equations driven by G -Brownian motion with delayed impulsive ef- fects ( G -DISFDEs, in short). Some suﬃcient conditions for p -th moment exponential stability of G -DISFDEs are derived by means of G -Lyapunov function method, average impulsive interval approach and Razumikhin-type conditions. An example is provided to show the eﬀectiveness of the theoretical results.


1.
Introduction. Impulsive dynamical systems have been widely used in many branches of science and technology such as in the transmission of the impulse information, control systems with communication constraints etc. Recently, special interest has been focused on the issues of delayed impulsive. In this case, the impulsive states are related not only to the current states but also to the past states. Consequently, it is more realistic to impulsive systems with delayed impulsive effects, one can see Cheng et al. [1], Li et al. [9] and Yao et al. [22] for more details.
Recently, Peng [12] established the fundamental theory of time-consistent nonlinear G-expectation. Under the framework of the nonlinear G-expectation, Peng [12,13] introduced the G-Gaussian distribution and the G-Brownian motion, which have very rich and interesting new structures which nontrivially generalize the classical ones. Since these notions were introduced, many investigators have studied the properties on G-Brownian motion (see Hu et al. [4,5,6]) and stochastic differential equation driven by G-Brownian motion (G-SDEs, in short). Gao [3] and Peng [15] have proved the existence and uniqueness of the solution for G-SDEs. Since then, this kind of G-SDEs have generated lots of developments. For more details, we refer the reader to Ren et al. [18,19,20,21], Yin and Ren [23], Zhang and Chen [24,25] and the references therein. For the updated developments on G-stochastic analysis and G-SDEs, one can see the survey paper by Peng [16].
Motivated by the aforementioned works, we aim to study the stability problem of stochastic function differential equations driven by G-Brownian motion with delayed impulsive effects (G-DISFDEs, in short). Some sufficient conditions for p-th moment exponential stability of G-DISFDEs are derived by means of G-Lyapunov function method, average impulsive interval approach and Razumikhin-type conditions. It should be mentioned that Ren et al. [19] derived some sufficient conditions for p-th moment exponential stability and quasi sure exponential stability of solutions to impulsive G-SDEs. However, their results are limited to the supremum or infimum of impulsive interval to some degree. Different from their works, delay and average impulsive interval method are considered in this paper, which are less conservative from the view of impulsive stabilization.
The paper is organized as follows. In Section 2, we introduce some preliminaries and notations of G-stochastic analysis for further discussion. In Section 3, we discuss the exponential stability of G-SDE with delayed impulsive effects by means of G-Lyapunov function, average impulsive interval approach and Razumikhin-type conditions. In the last Section, we apply the results to illustrate the stability of G-SDE with linear impulsive sequence and propose an example to verify the effectiveness of the obtained results.

2.
Preliminaries. In this part, we recall some useful notions. For more details, one can refer to Denis et al. [2] and Peng [13,15].
Let Ω denote the space of all R d -valued continuous functions It is easy to notice (Ω, ρ) is a metric space.
Definition 2.1. E : Lip(R d ) → R is called a sublinear expectation. If the following properties are satisfied.
For each monotonic and sublinear function G : S d → R by where Λ ⊂ S d + is a bounded, convex and closed set. Peng [12] constructed a sublinear expectation space (Ω, L ip (Ω), E, (E t ) t≥0 ) called G-expectation space. To be exactly, for each ξ ∈ L ip (Ω) with the form of we define the conditional G-expectation by for each t ∈ [t i−1 , t i ), i = 1, · · · , k. Here, the function u i (t, x; x 1 , · · · , x i−1 ) parameterized by (x 1 , · · · , x i−1 ) ∈ R d×(i−1) is the viscosity solution of the following G-heat equation: In this space the corresponding canonical process B t is called G-Brownian motion. The Itô integral with respect to the G-Brownian motion is discussed as follows. We first consider the following set of step processes: Now, we propose the definition of the G-Itô integral.
Furthermore, the mutual variation process of B a and Bā is defined by where a = (a 1 , · · · , a d ) T ,ā = (ā 1 , · · · ,ā d ) T . such that where E P is the linear expectation with respect to P . For this P, the associated capacity is defined by 3. System description. Throughout this paper, unless otherwise specified, we let (Ω, H, E) be a sublinear space, R n stands for the n-dimensional Euclidean space, R + denotes the set of all positive real numbers, N denotes the set of positive integers.
In this paper, we consider the following G-DISFDES is the mutual variation process of the (B)(·).
As a standing hypothesis, we assume that f, h ij , σ j , I k , J k satisfy locally Lipschitz and linear growth conditions. We can prove the existence and uniqueness of the system (2) by applying the similar methods as in Hu and Ren [7]. In order to discuss the stability of system (2), we assume that Remark 3. In this paper, we use the Einstein convention, i.e. the above repeated indices of i and j imply the summation, i.e., where V x (t, x(t)), h(t, ϕ) + V xx (t, x(t))σ(t, ϕ), σ(t, ϕ) is the symmetric in S d (R), with the form (2) is said to be p-th moment exponentially stable, if for any initial ξ, there exist two positive constants λ and M such that where N (t, s) denotes the number of impulsive times of the impulsive sequence {t k } k∈N on the interval (s, t].
Letting J k ≡ 0 in the system (2), we have (29) For the system (29), we have the following result.
Corollary 1. Assume that there exists a function V ∈ υ 0 and some constants e N 0 lnd . Then, when d < 1, the impulse times N (t, s) ≥ t−s − N 0 , when d > 1, the impulse times N (t, s) ≤ t−s + N 0 . Thus the trivial solution of system (29) is p-th moment exponentially stable.
Proof. We need to apply Theorem (4.1) with d 1 = d, d 2 = 0. 5. Applications and example. In this section, we consider the following system where x(t) = (x 1 (t), · · · , x n (t)) T and the mappings f, h ij , σ j : is the mutual variation process of the (B)(·). For simplicity, we just consider an appropriate linear impulsive sequence as follows To establish the sufficient conditions ensuring the exponential stability of system (30), we need to the following hypothesis.
Theorem 5.2. Under the above assumption, there exists a symmetrical positive matrixes Q and some constants d 1 > 0, d 2 ≥ 0, η 2 ≥ 0, > 0, N 0 ∈ N and η 1 ∈ R such that (a) the following matrix inequalities hold: Then the solution of system (30) is exponentially stable in the mean square and λ is the unique positive solution of the following equation λ + η 1 + ln(d 1 + γ 1 d 2 ) + αη 2 e λτ = 0.
Hence, by using (35) and (38), we derive that Consequently, the conclusion follows from Theorem 4.1, directly, which completes the proof.