Stabilization of stochastic regime-switching Poisson jump equations by delay feedback control

This paper is concerned with the stabilization of stochastic regime-switching Poisson jump equations (also known as stochastic differential equations with Markovian switching and Poisson jumps, abbreviated as SDEwMJs). The aim of this paper is to design a feedback controller with delay δ (δ > 0) to make an unstable SDEwMJ become stable. It is proved that the delay δ is bounded by a constant δ̄. Moreover, an implicit lower bound for δ̄, which can be computed numerically, is provided. As a product, the almost sure exponential stability of the controlled SDEwMJ is obtained. Besides, an example is given to demonstrate the theoretical results.


Introduction
Stochastic hybrid systems have been widely used to model such systems where they may experience abrupt changes in their structure and parameters (see [1]). One of the important types of stochastic hybrid systems is the SDEs with Markovian switching (SDEwMSs). In the study of the SDEwMSs, stability analysis has received a great deal of attention (see [2][3][4][5][6][7] and the references therein).
As we know, a Brownian motion is a continuous stochastic process. However, in real life, some systems may suffer from jump-type stochastic abrupt perturbations, such as financial crisis, earthquakes, and hurricanes. In these cases, employing the Brown motions to depict these systems cannot meet the needs of reality. At the same time, it is found that Poisson jumps can describe these jump-type phenomena. It is, therefore, reasonable to use Poisson jump processes to cope with such jump-type discontinuous systems (see, e.g., [8][9][10][11][12][13]). Results on the stability of stochastic systems with the Markovian switching and Poisson jumps have been obtained, see, e.g., [14,15].
Given an unstable SDEwMJ

dx(t) = f x(t), t, r(t) dt + g x(t), t, r(t) dω(t) + h x(t), t, r(t) dN(t),
where x(t) ∈ R n is the state. The details of (1) can be found in (4) below. As we know, it is classical to design a feedback control u(x(t), t, r(t)) (based on the current state x(t)) to make the controlled system dx(t) = f x(t), t, r(t) + u x(t), t, r(t) dt + g x(t), t, r(t) dω(t) stable. When h ≡ 0, (2) becomes an SDEwMS. Many results on the stabilization problem of SDEwMSs by state feedback controls without delay have been well studied (see [1,16,17] and the references therein). However, due to the fact that there exists a time lag δ between the time when the observation of the state is made and the time when the feedback control reaches the system; therefore, the feedback control should be designed depending on a past state x(tδ) (see, e.g., [18][19][20]). Then the controlled system can be described by Paper by Mao et al. [18] was the first one to design a delay feedback control in the drift part for an SDEwMS to make the unstable system become stable by using the linear matrix inequalities and Lyapunov functionals. Based on such a feedback control, this paper is concerned with the exponential mean-square stabilization of SDEwMSs. Fei et al. [21] focused on the stabilization problem for a class Markovian jump linear systems with time delay by the linear matrix inequalities. Along this line, Chen et al. [19,20,22] investigated the neutral stochastic delay differential equations with Markovian switching by the Lyapunov functions technique and the linear matrix inequalities. Hu et al. [23] investigated the stabilization of SDEwMSs by a new theorem. Authors in [24][25][26] successfully used the method of the Lyapunov functionals to study the stabilization by delay feedback controls of highly nonlinear neutral stochastic delay differential equations with the Markovian switching. Subsequently, Mei et al. [27] studied the exponential stabilization by delay feedback controls for highly nonlinear hybrid stochastic functional differential equations with infinite delay, which generalizes the results of [24][25][26].
It is a pity that the above-mentioned work did not discuss the stabilization by delay feedback controls for SDEwMJs. In reality, because of the movement of a Markov chain, a stochastic system including several regimes may switch from one to the others. Moreover, the increment of a Brownian motion is a zero mean, but for a Poisson jump, its increment is a nonzero mean. Therefore, due to the difficulty stemming from the presence of the Markovian switching and Poisson jump, more techniques are needed to study the stability of SDEwMJs. The aim of this paper is to investigate the exponential stabilization of SDEwMJs by designing a feedback control in the drift term with the delay δ. Moreover, it is not only illustrated that the δ is bounded byδ but also presented an implicit lower bound forδ.
The remainder of this paper is arranged as follows: Sect. 2 introduces some preliminaries. Section 3 is devoted to presenting the main results. In Sect. 4, an illustrative example is provided to show the effectiveness of the obtained theory.

Preliminaries
Throughout this paper, unless otherwise specified, the following notations are used. Let R = (-∞, +∞), R + = [0, +∞). R n represents the n-dimensional Euclidean space, and |x| denotes its norm of a vector x ∈ R n . ( , F, P) indicating a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions. For δ > 0, C([-δ, 0]; R n ) represents the family of all continuous R n -valued functions ϕ defined on [-δ, 0]. Denote by C b F 0 ( ; R n ) the family of all F 0 -measurable bounded C([-δ, 0]; R n )-valued random variables equipped with the norm ϕ = sup -δ≤θ≤0 |ϕ(θ )|. For ∀x, y ∈ R n , x, y or x T y represents the inner product. Let {r(t)} t≥0 be a right-continuous Markov chain on the complete probability space taking values in a finite state space S = {1, 2, . . . , N} with the generator = (γ ij ) N×N given by [1]). Consider the following n-dimensional unstable stochastic regime-switching jump equation, that is, the n-dimensional SDEwMJ on t ≥ t 0 ≥ 0, with the initial data where f , g, h : R n × R + × S → R n , ω(t) is a scalar Brownian motion, and N(t) is a scalar Poisson process with intensity λ > 0.Ñ(t) = N(t)λt is a compensated Poisson process. Moreover, B(t), N(t), and r(t) are assumed to be mutually independent of each other. To achieve the goal of the stability, it is assumed that The aim of this paper is to design a control function u : R n × R + × S → R n making the unstable SDEwMJ (4) become stable. The form of the corresponding controlled SDEwMJ is described by on t ≥ t 0 , with the initial value The coefficients of SDEwMJ (4) are assumed to satisfy the following global Lipschitz condition.
(A1) There are four positive constants L i (i = 1, 2, 3) and K such that for all , one can thus obtain the following linear growth condition Remark 1 Under Assumption (A1), the controlled SDEwMJ (6) admits a unique solution x(t) satisfying E|x(t)| 2 < ∞ for all t ≥ 0 corresponding to the initial value (7). The proof is standard, and one thus omits it.
The auxiliary controlled SDEwMJ is presented as follows: Similar to the proof of Theorem 3.8 in [1], one can obtain that under (A1), the auxiliary controlled SDEwMJ (10) with the initial data y(t 0 ) ∈ C b F t 0 (R n ) and r(t 0 ) ∈ S has a unique solution y(t) (t ≥ t 0 ≥ 0), which satisfies E|y(t)| 2 < ∞ for all t ≥ t 0 ≥ 0.
Let C 2,1 (R n × R + × S; R + ) be the family of non-negative functions V (x, t, i) defined on (x, t, i) ∈ R n × R + × S, which are continuously twice differentiable in x and once in t. For V ∈ C 2,1 (R n × R + × S; R + ), define LV : where Next, according to the Theorem 5.8 in Mao and Yuan [1], one can acquire the following lemma in parallel, which will be used in the sequel. Lemma 2.1 Let (A1) hold. Let p, c 1 , c 2 , and λ 1 be positive constants. For all (y, t, i) ∈ R n × R + × S, assume that there is a function V ∈ C 2,1 (R n × R + × S; R + ) satisfying and LV (y, t, i) ≤ -λ 1 |y| p . Then for all y(t 0 ) ∈ C b F t 0 (R n ), (t 0 , r(t 0 )) ∈ R + × S. In other words, the trivial solution of the auxiliary controlled SDEwMJ (10) is the pth moment exponentially stable.
From Lemma 2.1, one can obtain the auxiliary controlled SDEwMJ (10) is the pth moment exponentially stable. While the pth moment exponential stability of SDEwMJ (10) was well studied (see, e.g., [1,17]). Therefore, one can assume that (10) is the pth moment exponentially stable in this paper, which is given as follows.
(A2) Let p > 0. Assume that there is a pair of positive constants M and γ such that the solution of the auxiliary controlled SDEwMJ (10) satisfies for all y(t 0 ) ∈ C b F t 0 (R n ), (t 0 , r(t 0 )) ∈ R + × S.

Main results
The aim of this paper is to show the controlled SDEwMJ (6) is almost surely exponentially stable. Before proving this result, a number of lemmas should be proved.

s, r(s) + u x(sδ), s, r(s) ds
Using the Hölder inequality and (A1), one gains By the Doob martingale inequality, one can gain

s, r(s) ds
The proof is complete.
Remark 3 It follows from Theorem 3.3 that an unstable SDEwMJ can be stabilized by a delay feedback control. It is well known that SDEs, SDEwMSs, and SDEs with Poisson jumps are the special cases of SDEwMJs. Though [1,16,17] well studied the stabilization problem of SDEwMSs by state feedback controls, authors in these papers neither designed the feedback controls with delay nor took Poisson jumps into account. Authors in [18][19][20] designed the feedback controls depending on a past state, but they did not consider the Poisson jumps. Therefore, the results in this paper cover part of the results in [1,[16][17][18][19][20].

An example
In this section, an example is provided to illustrate the obtained results. Before giving an example, a lemma is presented as follows.
It follows from Lemma 4.1 that this Markov chain r(t) is irreducible and π = (1/2, 1/2). Consider the following 1-dimensional Black-Scholes model with the Markovian switching and Poisson jumps It is easy to check that L 1 = 0.12, L 2 = 0.2, L 3 = 0.02. It follows from Theorem 3.1 in [28] that one can get That is, the system (62) is not almost surely exponentially stable.
Let us now design a delay feedback control to stabilize the system (62). Consider a linear delay feedback controller of the form u(x, t, i) = d(i)x (i = 1, 2). Therefore, the controlled system is given by where d(1) = -1.3 and d(2) = -1.2. So, the condition (A1) is satisfied with L 1 = 0.12, L 2 = 0.2, L 3 = 0.02 and K = 1.3. The auxiliary controlled system is of the form

Conclusion
So far, there are few results on the stabilization analysis of SDEwMJs. This paper discussed that an unstable SDEwMJ could be stabilized by a delayed feedback control u(x(tδ), t, r(t)) designed in the drift term, and the delay δ is bounded by a constantδ. Moreover, the implicit lower bound forδ can be computed numerically. As a product, the almost sure exponential stability of the controlled SDEwMJ is obtained. An example is also given to verify the obtained results.