Analysis on structured stability of highly nonlinear pantograph stochastic differential equations

ABSTRACT This paper investigates the structured stability and boundedness of highly nonlinear hybrid pantograph stochastic differential equations (PSDEs). The main contribution of this paper is to take the different structures into account to establish the structured robust stability and boundedness for highly nonlinear hybrid PSDEs. The theory established in this paper is applicable to hybrid PSDEs which may experience abrupt changes in both structures and parameters.


Introduction
Stochastic delay differential equations (SDDEs) are widely used to model systems which do not only depend on present states but also involves past states. Robust stability and boundedness are two of most popular topics in the area of systems and controls, most of the papers can only be applied to delay systems where their coefficients are either linear or nonlinear but bounded by linear functions (see, e.g. Deng, Fei, Liang, & Mao, 2019;Wu, Tang, & Zhang, 2016). However, the linear growth condition is usually violated in many practical applications. Recently, there are some progress on stability for highly nonlinear stochastic delay systems (see, e.g. Deng, Fei, Liu, & Mao, 2019;Fei, Hu, Mao, & Shen, 2019;Hu, Mao, & Shen, 2013;Liu & Deng, 2017;. Particularly, Hu, Mao, and Zhang (2013) were first to investigate the robust stability and boundedness for SDDEs with Markovian switching without the linear growth condition. Fei, Hu, Mao, and Shen (2017) established stability criteria for delay dependent highly nonlinear hybrid stochastic systems. Pantograph stochastic differential equations (PSDEs) are unbounded delay stochastic differential equations which have been frequently applied in many practical areas, including biology, mechanic, engineering and finance. Baker and Buckwar (2000) established the existence and uniqueness of the solution for the linear stochastic pantograph equation. Shen, Fei, Mao, and Deng (2018) discussed exponential stability CONTACT Mingxuan Shen smx1011@163.com of highly nonlinear neutral PSDEs by Lyapunov functional and M-matrix. Liu and Deng (2018) investigated pth moment exponential stability of highly nonlinear neutral PSDEs which driven by Lévy noise. As we know, the hybrid systems driven by continuous-time Markov chains are often used to model systems that may experience abrupt changes in their structures and parameters caused by phenomena such as component failures or repairs (see, e.g. Mao & Yuan, 2006;Shen, Fei, Mao, & Liang, 2018;Zhou & Hu, 2016). The theory in Hu, Mao, and Zhang (2013) is good at dealing with the hybrid SDDEs that may experience abrupt changes in their parameters. You, Mao, Mao, and Hu (2015) show that a given stable hybrid PSDE can not only tolerate the linear perturbation but also the nonlinear perturbation without loss of the stability, while most of the papers could only cope with the linear perturbation. However, most of references on hybrid systems have dealt with subsystems with the same structures. In fact, stochastic systems may experience changes not only in their coefficients but also in their structures. Fei, Hu, Mao, and Shen (2018) first took the different structures in different modes to develop a new theory on the structured robust stability and boundedness for highly nonlinear hybrid SDDEs. But the theory of Fei et al. (2018) cannot applied directly to highly nonlinear hybrid PSDEs which experience abrupt changes in their structures. Motivated by the above discussion, this paper will study exponential stability of a class PSDEs which experience abrupt changes in their structures.

Notation and assumptions
Throughout this paper, unless otherwise specified, we use the following notation. We denote by |x| the Euclidean norm for x ∈ R n . If A is a vector or matrix, its transpose is denoted by A T . If A is a matrix, its trace norm is denoted by |A| = trace(A T A). If both a and b are real numbers, then a ∨ b = max{a, b} and a ∧ b = min{a, b}. If G is a set, its indicator function is denoted by I G . That is, be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is increasing and right continuous while F 0 contains all P-null sets). Let We assume that the Markov chain r(·) is independent of the Brownian motion B(·).
Consider an n-dimensional hybrid SDDE on t ≥ 0, where the coefficients f : R n × R n × R + × S → R n and g : R n × R n × R + × S → R n×m are Borel measurable and 0 < θ < 1 with initial date x(0) = x 0 ∈ R n . Moreover, assume that f (0, 0, t, i) = 0 and g(0, 0, t, i) = 0 for all (t, i) ∈ R + × S. For the convenience of the reader, let us cite some useful results on M-matrices. For a vector or matrix A, by A > 0 we mean all elements of A are positive. A Z-matrix is a square matrix A = (a ij ) N×N which has non-positive off-diagonal entries.

Lemma 2.1: Let A = (a ij ) N×N be a Z-matrix. Then A is a nonsingular M-matrix if and only if one of the the following statements holds:
(1) A −1 exists and its elements are all nonnegative.
(2) There exists x > 0 in R N such that Ax > 0.
The well-known conditions imposed for the existence and uniqueness of global solution are the local Lipschitz condition and the linear growth condition (see, e.g. Mao, 2007). To be precise, let us state the local Lipschitz condition.
Assumption 2.2: For each integer h ≥ 1 there is a positive constant K h such that for those x, y,x,ȳ ∈ R n with |x| ∨ |y| ∨ |x| ∨ |ȳ| ≤ h and all (t, i) ∈ R + × S.
However, we do not state the linear growth condition as we here is to study the structured robust stability and boundedness for highly nonlinear PSDEs which do not satisfy this condition.

Boundedness and stability
Set and where ρ is a free positive parameter. By the definitions of λ i and ζ i , we see that all λ i and ζ i are positive.
Assumption 3.1: Choose ρ > 0 sufficiently small such that where λ i and ζ i have been defined by (4) and (5). Assume also that and Remark 3.2: Letb be the maximum of the row sums of Then we have for all i ∈ S.
Proof: By the definiton of V(x, i), we can see that By the generalized Itô formula, we have that By inequality |x T g(x, y, t, i)| 2 ≤ |x| 2 |g(x, y, t, i)| 2 , we have By Assumption 2.3, we can get By (4) and (5), we have Consequently, By the Young inequality, we have We hence obtain from (13) that, for i ∈ S 1 , Similarly, for i ∈ S 2 , we can show that But, by condition (6), we have Hence By condition (7) and the Young inequality, we then obtain that, for i ∈ S 2 , Combining (7), (14) and (17), we see that, for all i ∈ S, By conditions (8) and (9), we haveρ < ρq/((q − 2)θ + q − p + 2). By the definitions of ρ 1 and ρ 2 , we have ρ 1 > 0 and ρ 2 > 0. Noting that we obtain from (18) that, for all i ∈ S, Thus the proof is complete. and where H 1 and H 2 are positive constants.
Proof: Since the coefficients of the hybrid PSDE (1) are locally Lischitz continuous, for any given initial date , whereσ ∞ is the explosion time. Let k 0 > 0 be a sufficiently large integer such that |x 0 | < k 0 . For each integer k ≥ k 0 , define the stopping time where throughout this paper we set inf ∅ = ∞. Clearly, τ k is increasing as k → ∞ and τ ∞ = lim k→∞ τ k ≤σ ∞ a.s. If we can show that τ ∞ = ∞, thenσ ∞ = ∞ a.s. and the assertion (i) follows.
We now show assertion (19). It follows from (24) that Letting k → ∞ and then using the Fubini theorem, we get Dividing both sides by ρ 1 t and then letting t → ∞, we see lim sup which is the desired assertion (19). Choose a positive constant sufficiently small for < ρ 1 c 2 and ≤ 1.
By the generalized Itô formula again, we have that for any t ≥ 0, By (12) and (22), we then have By 0 < θ < 1 and ≤ 1, we can get ( − 1 + θ)/θ ≤ , so that and similarly, thus,we can get where D : R + → R is defined by By (26), we can obtain that Letting k → ∞, we have which yields The proof is complete.

Theorem 3.5: Let all the conditions in Lemma 3.3 hold and, moreover, β i1 = 0 for all i ∈ S. Then the unique global solution x(t) of the PSDE (1) has the property that
Proof: Noting that c 3 = 0 in (11) given that β i1 = 0 for all i ∈ S. Hence, (11) becomes It is then easy to show by the generalized Itô formula that Letting t → ∞ yields assertion (30). (7) is replaced by

Theorem 3.6: Let all the conditions in Lemma 3.3 hold except condition
and, moreover, β i1 = 0 for all i ∈ S. Then there is a positive number δ such that for any initial data x(0) = x 0 , the unique global solution x(t) of the PSDE (1) satisfies lim sup and lim sup Proof: In the same way that (11) was proved, we can show from (13) and (16) that This implies Let δ > 0 be sufficiently small for and 2ρ 1 ≥ δc 2 .
Applying the generalized Itô formula, we have that

Two special cases and an example
We will also assume that all coefficients of PSDEs in this section will satisfy the local Lipschitz condition and, moreover, q > p ≥ 2. To make our cases more understandable, we assume that the given hybrid system is described by a hybrid differential equation Assume that this given hybrid differential equation is either asymptotically stable or bounded. Its structured differences and various stochastic perturbations will be discussed in the following two cases.

Case 1
Assume that for each i ∈ S 1 , there is a number b i1 < 0 such that while for each i ∈ S 2 , there ia a pair of numbers b i1 ∈ R and b i2 > 0 such that for (x, t) ∈ R n × R + . This means that the differential equation in mode i ∈ S 1 is stable but may not in mode i ∈ S 2 . In order for the hybrid Equation (37) to be stable, we assume moreover that is a nonsingular M-matrix. It is then known (see, e.g. Hu, Mao, & Zhang, 2013) that Equation (37) is exponentially stable in pth moment. Suppose that Equation (37) is subject to a stochastic perturbation and the perturbed system is described by and the perturbation has its structured difference in the sense that for (y, t) ∈ R n × R + , where b i3 > 0 for all i ∈ S. We wish to obtain upper bounds on b i3 's for the perturbed system (39) to remain stable. Noting that for i ∈ S 1 we see that Assumption 2.3 is satisfied with Hence the matrix A defined by (2) is the same as the matrix A defined by (38) and hence A is a nonsingular M-matrix. Moreover, the matrix B defined by (3) becomes which is a nonsingular M-matrix too by Lemma 2.1. We choose ρ by (10), so condition (6) is satisfied by Remark 3.2. Compute λ i 's and by ζ i 's by (4) and (5), respectively. Conditions (7)-(9) yield the following bounds for i ∈ S 2 .
(42) By Theorems 3.5 and 3.6, we can therefore conclude that if the perturbed parameters b i3 satisfy (41) and (42), then the PSDE (39) is both mean square and almost surely exponentially stable.

Case 2
In this case we will discuss the robust boundedness.
Assume that and Suppose that the perturbed system is described by and the perturbation coefficients satisfy where b i3 and b i5 are all nonnegative numbers. We aim to obtain upper bounds on them so that the perturbed system (45) remains asymptotically bounded. It follows from these conditions that for i ∈ S 1 As a result, Assumption 2.3 is satisfied with  (4) and (5), respectively. Conditions (7)-(9) then By Theorems 3.4, we can therefore conclude that if the perturbed parameters b i3 satisfy (48) and (49), then the solution x(t) of the PSDE (45) has properties (19) and (20).
We can therefore conclude that if the perturbed parameters σ i satisfy (51) where H 1 and H 2 are positive constants.
To perform a computer simulation for the solution, we set σ 1 = σ 2 = 0.15, σ 3 = σ 4 = 0.07,σ 5 = 0.7, σ 6 = 0.7, x(0) = 1 and r(0) = 1. The computer simulations in Figure 1 show a single sample path of the Markov chain and that of the solution, from which we can see how the Markov chain jumps from one mode to another and also the solution evolves in a bounded domain.

Conclusion
In this paper, we have discussed robust stability and boundedness for highly nonlinear hybrid PSDEs with different structures. We have also discussed two special cases and an example to illustrate our theory.

Disclosure statement
No potential conflict of interest was reported by the authors.