Razumikhin Theorems on Polynomial Stability of Neutral Stochastic Pantograph Differential Equations with Markovian Switching

: This paper investigates the polynomial stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEsMS). Firstly, under the local Lipschitz condition and a more general nonlinear growth condition, the existence and uniqueness of the global solution to the addressed NSPDEsMS is considered. Secondly, by adopting the Razumikhin approach, one new criterion on the q th moment polynomial stability of NSPDEsMS is established. Moreover, combining with the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure polynomial stability of NSPDEsMS is examined. The results derived in this paper generalize the previous relevant ones. Finally, two examples are provided to illustrate the effectiveness of the theoretical work.


Introduction
Due to the existence of random disturbances, neutral stochastic differential equations (NSDEs) can be utilized to characterize those complicated systems such as population system, chemical reaction process, heating control systems, complex networks and other systems [1][2][3][4][5][6][7]. The structures and parameters of some systems may encounter unpredictable variations, so Markovian jump systems are introduced to depict these phenomena. During the past several decades, many scholars have been absorbed in the neutral stochastic differential equations with Markovian switching (NSDEsMS), and large amounts of interesting results have been acquired [8][9][10].
The pantograph system was presented by Ockendon and Tayler in 1971 [11], which could be seen as one important class of systems with unbounded delays. Recently, network systems with pantograph delays as one class of pantograph systems have received extensive attention. Particularly, in [12], the exponential stability of switching neural networks with pantograph delays was discussed by adopting the average dwell-time (ADT) technique and Lyapunov stability approach. In [13], global h-stability criteria for pantograph delay high-order inertial neural networks were examined by utilizing the non-reduced order method. In [14,15], periodic solutions and anti-periodic solutions of neural networks with pantograph delays were analyzed by means of differential inequality techniques. In [16,17], based on the comparison principle and some analysis techniques, control issues, such as the synchronization and passivity of neural networks with pantograph delays were investigated. On the other hand, by employing the stochastic Lyapunov method, the stability of linear or highly nonlinear stochastic pantograph equations were extensively investigated [18][19][20]. Moreover, the referent results were generalized to the stochastic pantograph differential equations (SPDEs) or neutral stochastic pantograph differential equations with Markovian switching (NSPDEsMS) [21][22][23].
The Razumikhin approach is one effective tool to deal with the stability issue of the time delayed system. This approach was initiated in [24,25] and it was developed in various different systems, including discrete systems, impulsive systems and stochastic systems, and many publications have been reported [26][27][28][29][30][31][32][33][34][35]. In particular, the Razumikhin technique was also extensively applied to NSDEs. For instance, Mao [28] adopted the Razumikhin technique to investigate the mean-square moment exponential stability of NSDEs. Subsequently, the theory wa sfurther extended to analyze the pth moment stability of NSFDEs in [29]. Huang and Deng [30] used the Razumikhin technique to examine the asymptotic stability of NSDEs. By incorporating the stability with general decay rate, Pavlović and Janković [31] established new Razumikhin theorems, which may be specialized on the different types of stability. Moreover, Razumikhin techniques were generalized to NSDEs with Markovian switching [32,33] and NSDEs with unbounded delays [34]. For NSPDEs, Yu [35] constructed the criterion on Razumikhin-type pth moment asymptotic stability and discussed the stability of the numerical solutions in virtue of the backward Euler method.
In addition, different from exponential stability, polynomial stability is also one class important stability. In [36], Mao considered the almost sure polynomial stability of the stochastic systems by using the semimartingale theory. Inspired by several practical examples, Liu [37] investigated moment stability with general decay speeds. Lan et al. [38] proposed one modified truncated Euler-Maruyama (MTEM) approach and explored the almost sure and mean square polynomial stability of the numerical technique. For SPDEs, many scholars [39][40][41][42][43] analyzed the polynomial stability by using the stochastic Lyapunov function method and some numerical algorithms. More recently, Mao et al. [44] constructed the novel Razumikhin theorems on the pth moment polynomial stability of the SPDEs. For NSPDEs, it can be observed that the references listed above focus on two aspects. One is the pth moment exponential stability [20][21][22], the other one is the pth moment stability with general decay rate [23]. Meanwhile, all the results in Refs. [20][21][22][23] required that the coefficients of delayed terms keep time varying. Therefore, it is necessary to develop other stabilities, such as the polynomial stability of NSPDEs with constant coefficients and generalize the theory in [35,44] to NSPDEsMS.
Inspired by the aforementioned discussions, this paper will investigate the polynomial stability of NSPDEsMS by virtue of the Razumikhin method and several stochastic analysis techniques. The contributions of our article are listed below. Firstly, the existence and uniqueness of the solutions to NSPDEsMS are analyzed, where the condition on upper bound of the operator LU is relaxed. Secondly, the Razumikhin theorem on the qth polynomial stability of NSPDEsMS is established, and the drift term does not need to meet the linear growth condition. Moreover, based on some stochastic theories, the criterion on almost sure polynomial stability of NSFDEsMS is provided. Thirdly, all the existing stability results [20][21][22][23] require that coefficients of the delay term be time-varying, but the restriction in this paper is removed and the coefficients may keep constant. This paper also generalizes the theory in [35,44] to NSFDEsMS. The structure of this article is arranged appropriately. In Section 2, standard notations are introduced, and several importance assumptions are proposed. In Section 3, the existence and uniqueness of the global solutions to NSPDEsMS are considered. Furthermore, some criteria on polynomial stability are constructed by utilizing the Razumikhin approach and stochastic analysis techniques. Section 4 illustrates the validity of the theoretical work through two concrete examples, and a full summarization is made in the last part.
Consider the following NSPDEsMS In order to discuss the polynomial stability of Equation (1), we suppose that the initial value ζ ∈ L q F t 0 , and F(0, 0, t, Υ(t)) = 0, Obviously, this means that Equation (1) has one trivial solution.
To acquire our main results, the following definitions and assumptions on the addressed system are imposed. (1)

Main Results
In this section, the existence and uniqueness of the global solutions to NSPDEsMS are considered. Furthermore, some criteria on polynomial stability are constructed by utilizing the Razumikhin approach and stochastic analysis techniques.
Proof. According to Assumption 3, we can obtain that When t 1 ∈ [t 0 , t 0 δ ], on the basis of Assumption 3, we obtain that It implies that By applying the Gronwall inequality, we have that According to the elementary inequality |l 1 In particular, when Applying the Gronwall inequality yields that Similarly, we also have that In particular, when t 1 = t 0 δ 2 , we have that Repeating this procedure, we can show that P{θ ∞ > t 0 δ j } = 1 for any integer j ≥ 1. Letting j → ∞ yields that θ ∞ = +∞ a.s. It means that the above conclusion holds.

Remark 3.
It is noted that all the existing stability results [20][21][22][23] require that the coefficients of the delay term be time varying, but the restriction in this paper is removed and the coefficients may keep constant. Theorem 2 also generalizes the theory in [35,44] to NSFDEsMS. In [45], an efficient method based on the generalized hat functions for solving nonlinear stochastic differential equations driven by the multi-fractional Gaussian noise was proposed, and the theory was applied to some stochastic population models. Moreover, dynamic properties of stochastic pantograph systems with multi-fractional Gaussian noise are worthy of exploration.
Let µ = 1.5. Noting that 1 ) q = min{1.5, 0.8737} = 0.8737 < 1. Hence, according to Theorems 2 and 3, we can conclude that the above system is polynomially stable in mean square rather than almost surely polynomially stable.

Conclusions
In this paper, the new Razumikhin theorem on the qth moment polynomial stability of NSPDEsMS is established. Furthermore, combining with several stochastic analysis techniques, the almost sure polynomial stability of NSPDEsMS is explored. In the end, the effectiveness of the main results is demonstrated through two concrete examples. In years to come, our theoretical work can be further generalized to the SPDEs with Lévy noise or neural network systems.