ALMOST SURE EXPONENTIAL STABILIZATION AND SUPPRESSION BY PERIODICALLY INTERMITTENT STOCHASTIC PERTURBATION WITH JUMPS

. The main aim of this article is to examine almost sure exponential stabilization and suppression of nonlinear systems by periodically intermittent stochastic perturbation with jumps. On the one hand, some suﬃcient crite-ria ensure almost sure stabilization of the unstable deterministic system by applying exponential martingale inequality with jumps. On the other hand, suﬃcient conditions of destabilization are provided under which the system is stable by the well-known strong law of large numbers of local martingale and Poisson process. Both the sample Lyapunov exponents are closely related to the control period T and noise width θ . As for applications, the well-known Lorenz chaotic systems and nonlinear Li´enard equation with jumps are dis- cussed. Finally, two simulation examples demonstrating the eﬀectiveness of the results are provided.


1.
Introduction. Applications of stochastic noise feedback have emerged in a wide range of fields such as plane model [23], engineering [12,19], finance [9], multiagent systems [37]. In a string of previous works, the literature on stabilization and destabilization of the Brownian noise has fully spelled out (see, e.g., Appleby [5], Deng et al. [7], Scheutzow [23], Mao [18,20], Mao et al. [19], Zhang et al. [35]). On the one hand, the approach of input feedback control with white noise has successfully stabilized a deterministic system. On the other hand, some authors have investigated almost sure stabilization and suppression of stochastic systems in terms of coefficient condition (see, e.g., [16,17,27,28,36]) and the references therein. It should be mentioned that these works in the literature we presented concentrate on the case of SDEs driven by classical Brownian motion. Put differently, the development of the general Lévy noise stabilization and destabilization is rarely tracked synchronously.
Recent years have witnessed Lévy process theory (see, e.g., [1]) as an important branch of modern probability theory, and a great deal of rapid development in both theory and applications. We are referring here to Lévy processes, which include continuous Brownian motion and Poisson jump process as special cases. In addition, Lévy processes also have many aliases (including almost analogous processes), additive processes, independent incremental processes, infinitely divisible processes, and their associated distributions and so on. Under the real circumstance, one often encounters obstacles influenced by event-driven uncertainties, which can be captured by features of these systems. It has been widely applied in basic mathematics, statistics, economics, finance, insurance, operations research, physics, engineering and other fields. In order to describe stochastic abrupt phenomena, it is quite suitable to introduce SDEs with Lévy noise. For instance, the continuous and discrete coexistence models have attracted widespread attention. From the perspective of modeling, Cont and Tankov [6] introduced financial modeling with jump processes. Patel and Kosko [22] investigated stochastic resonance in continuous and spiking neuron models with Lévy noise. Applebaum [2] further developed stochastic resonance for neuron models to general Lévy noise. Very recently, Gao and Wang [10] were concerned with a stochastic mutualism model under regime switching with Lévy jumps. From the viewpoint of stability, Zhu [32,33,34] revealed asymptotic stability in the pth moment and Li et al. [15] studied almost sure stability linear SDEs with jumps. In view of stabilization, Applebaum and Siakalli [4] used Lévy noise to establish stochastic stabilization of dynamical systems. Furthermore, Zong et al. [38] examined almost sure and pth-moment stability and stabilization of regime-switching jump diffusion systems. For more details, one can refer to [29,30] and the references therein. Therefore, it is fairly interesting to investigate this problem driven by Lévy process.
Let us turn to the topic of intermittent control. In terms of saving costs, intermittent periodically control, which includes control time and rest time, has gained wide popularity in the control theory. Recently, great progress has been made not only in theory but also in practice. Especially in complex systems, Li and Cao [13] applied the periodically intermittent control to switched networks and switched interval coupled networks respectively. Pinning controllability scheme of directed complex delayed dynamical networks via periodically intermittent control is investigated in [14]. Wan and Cao [25] studied distributed robust stabilization of linear multi-agent systems with intermittent control. For the latest development, one can refer the reader to [8,11,21,26] and the references therein. In this paper, in order to extend the role of intermittent control for stochastic stabilization and suppression, we introduce the continuous Brownian motion and the compensated Poisson integral and periodically intermittent control. We called it periodically intermittent stochastic perturbation with jumps.
Along with the above concerns, it has been well recognized that the nonlinearities and random fluctuation phenomenon are unavoidable in almost all the practical systems and it can not be completely eliminated. In this paper, we further the works on stabilization of nonlinear systems induced by Lévy noise. More specifically, we are interested in periodically intermittent stochastic perturbation with jumps. Since the quantitative property of nonlinear SDEs with periodically intermittent stochastic noise feedback with jumps is rarely available and still remains open for a while, and recent studies of periodically intermittent stochastic feedback stabilization in [35] give us several motivations, we establish almost sure exponential stabilization and destabilization by adopting the means of exponential martingale inequality, segmented Lyapunov operator, the well-known law of large numbers of Poisson process and Brownian motion, and by using the method of intermittent stochastic perturbation with jumps. The main highlights of this article are as follows: • One novel aspect of our methods is that periodically intermittent control, Brownian motion and a compensated Poisson integration (small jumps) are unified. For almost sure stabilization, periodically intermittent stochastic perturbation induced by Brownian motion and the compensated Poisson integral contributes to the stabilization of the system. In the case of destabilization, periodically intermittent stochastic perturbation induced by Brownian motion serves as a destructive factor. • The well-known Lorenz chaotic systems were stabilized via periodically intermittent stochastic perturbation induced by Brownian motion and the compensated Poisson integral, and a stable nonlinear Liénard equation with jumps was suppressed by periodically intermittent stochastic perturbation induced by Brownian motion. This paper is organized as follows. Section 2 begins with an overview of Lévy process and model descriptions. The objective of Section 3 is to present some useful lemmas and to show almost sure exponential stabilization and suppression of nonlinear systems by periodically intermittent stochastic perturbation with jumps. Section 4 shows two applications and numerical results are finally reported. Section 5 summarizes the full paper and puts forward future works as the end of this paper.
Notations. Throughout the text for each x ∈ R n , we denote B c (x) = {y ∈ R n : . For a matrix D, denote by D the Hilbert-Schmidt norm, i.e. D = trace(D T D). Let C 2 (R n ; R + ) be the family of all nonnegative functions V (x) on R n that have continuous partial derivatives w.r.t. x up to the second order. The function f : R + → R n is said to be càdlàg if it is right continuous and left limits with f (t−) = lim s↑t f (s).

2.
Preliminaries. Let (Ω, F, (F t , t ≥ 0), P ) be a filtered probability space satisfying the usual condition (i.e., it is right continuous and completeness) which is defined an m-dimensional standard F t -adapted Brownian motion B t . Let ϕ(t) be an F t -adapted Lévy process with Lévy measure ν(·). Define The compensatorÑ of N is defined by N (dt, dz) = N (dt, dz) − ν(dz)dt, where a measure ν(dz), called a Lévy measure on Z, satisfies R n −{0} (|z| 2 ∧ 1)ν(dz) < ∞ and Z is a Borel subset of (R n − {0}). Assume that Brownian motion and Poisson random process are mutually independent.
Assumption 2.2. There exists a positive constant ρ k (k ≥ 1) such that for x, y ∈ R n with |x| ∨ |y| ≤ k.
Assumption 2.3. There exists a nonnegative Lyapunov function V ∈ C 2 (R n ; R + ), such that inf |y|>k V (y) → ∞, as k → ∞, and for some constant C 0 and all y ∈ R n , LV (y) ≤ C 0 V (y).

Remark 1.
Under the Assumptions 2.2 and 2.3, it is easy to conclude that system (1) has a unique global solution y(t) (see, Lemma 4.7, [38]). We will denote the solution by y(t; t 0 , y 0 ) when we need to emphasize the initial data y 0 at time t 0 . Suppose that f (0) = 0, g(0) = 0, and H(0, z) = 0 for all |z| < c, then (1) has a unique solution y(t) = 0 for all t > t 0 corresponding to the initial value y(t 0 ) = 0, which is called the trivial solution.
Definition 2.4. The trivial solution of (1) is called to be almost surely (a.s. for short) exponentially stable if there exists a constant λ > 0 such that lim sup for all y 0 ∈ R n .
3.1. Stabilization of noise. In this paper, let us first recall some results of [4] which will be useful in the forthcoming Theorem. The two auxiliary statement is Lemma 3.1-3.2 below and we put forward it without proof as it is straightforward.
Define, for any α ∈ (0, 1) and z ∈B c , Proof. For any z ∈B c , let By Lemma 3.1 and taking a = log V (y+H(y,z)) On the other hand, using Lemma 3.2, we can verify that It follows from dominated convergence theorem and Lemma 3.3 (see Applebaum and Siakalli [3]) that Therefore, we obtain the desired assertion.
We now give general hypothesis which enables us to prove almost sure exponential stability results.
Assumption 3.4. Assume that a function V ∈ C 2 (R n ; R + ) and there exist some Remark 2. From the logarithmic inequality log(y) ≤ y − 1 for y > 0, we conclude that the (iv) in Assumption 3.4 is reasonable.
After above preparations, we now are in a position to state the main results in this section.
As a consequence, the required result holds.
Remark 3. The main results of Theorem 3.5 are to apply the periodically intermittent stochastic perturbation induced by Brownian motion and the compensated Poisson integral to stabilize a nonlinear system, and establish a class of theories on almost surely exponential stabilization based on Mao's ideal (see, [18]). Of course, it makes the analysis more difficult owing to the discontinuity of its sample paths. And quite remarkably, the results in [35] are covered by Theorem 3.5.

Remark 5.
When θ → 0, the system (1) will reduce to ODE system. By similar analysis, we conclude that the ODE is exponentially attractive if k 2 < 0.
Proof. Let V (y) = |y| 2 . With assumptions given by (12), (13) and (14), we have In views of Theorem 3.5, it is easy to deduce that lim sup This completes this proof.
Now the stability of system (11) are discussed as follows: Case 4.1. Stabilization from only periodically intermittent stochastic perturbation induced by the Brownian motion if Remark 7. In Corollary 1, it is worth mentioning that Case 4.3 showcases that an unstable system is stabilized by a mixture of the compensated Poisson jumps and Brownian motion. However, Zhang et al. [35] studied stabilization via the periodically intermittent stochastic perturbation induced by Brownian motion, and it is regarded as a special case of these results.

3.2.
Suppression of noise. From Case 4.2, an unstable system can be stabilized by a compensated Poisson intergral, i.e.
It is well known that Brownian motion can destabilize a stable ODE system.
Question. Can we use periodically intermittent stochastic perturbation induced by Brownian motion to suppress a stable stochastic system with jumps? This is the main task of this section. A single compensated Poisson process enables the readers to chooseπ flexibly according to their needs from a wide range (0, ∞). In what follows, we demonstrate that a stochastic system with jumps is destabilized by periodically intermittent stochastic perturbation induced by Brownian motion: where C is a symmetric positive definite matric, (Ñ (t) t≥0 ) is the compensated Poisson process withÑ (t) = N (t) −πt andπ is the intensity of the Poisson process (N (t) t≥0 ). Assume that B t and N (t) are independent. We propose the following assumption.
Corollary 2. Assume that there exist three constants ξ ∈ R, η 1 > 0 and for all y ∈ R n . Then the solution of system (23) satisfies lim inf for any y 0 ∈ R n , where λ min and λ max denote the minimum and maximum eigenvalues of the matrix C, respectively. Particularly, if Λ > 0 holds, then the trivial solution of system (23) is almost surely exponentially unstable.
Hence, Assumption 3.7 is easily examined and the required assertion follows from Theorem 3.8.
Remark 10. The Corollary 2 manifests that the stochastic noise (Brownian motion) plays the dominating role in determining the almost surely exponentially unstable. To date, we have not been able to determine whether Poisson noise can destabilize the ODE system. But, Poisson noise contributes to stabilize an unstable system (see, Example 4.1) and to make a system more stable when it is already stable. Here is an example.
It is not difficult to check that D is stable and G(x) = o(|x|) as |x| → 0. Consequently, the equation (29) is uniformly asymptotic stability and the state x(t) of (29) is shown Figure   follows: Takingπ = 4, C = 2.7182 0 0 1.7182 . From Theorem 7.1 (see [4]), it is easy to know system (30) is almost sure exponential stability (see, Figure 5).
Remark 11. Note that how to choose a constant matrix A i will be the key question.

5.
Conclusions and future works. This paper is devoted to study almost sure exponential stabilization and suppression with Brownian motion, the compensated Poisson jumps process and periodically intermittent control, respectively. We called it periodically intermittent stochastic perturbation with jumps. Exponential martingale inequality have been used to derive sufficient conditions for almost sure stabilization and strong law of large numbers of local martingale and Poisson process for almost sure exponential destabilization. As for application, two classes of nonlinear stochastic systems induced by periodically intermittent stochastic noise have been discussed. Finally, some examples are given to support the results obtained. Based on analysis above, the following problems will be of interest in the future.
Since Lévy noise plays a significant role in practical systems whose structure is subject to the fluctuations and large disasters, hence, complex systems (including multi-agent networks, ecosystems) by the periodically intermittent stochastic noise with jumps will be our future work.