On the asymptotic behavior of highly nonlinear hybrid stochastic delay diﬀerential equations 1

. In this paper, the existence and uniqueness, the stability analysis for the global solution of highly nonlinear stochastic diﬀerential equations with time-varying delay and Markovian switching are analyzed under a locally Lipschitz condition and a monotonicity condition. In order to overcome a diﬃculty stemming from the existence of the time-varying delay, one integral lemma is established. It should be mentioned that the time-varying delay is a bounded measurable function. By utilizing the integral inequality, the Lyapunov function and some stochastic analysis techniques, some suﬃcient conditions are proposed to guarantee the stability in both moment and almost sure senses for such equations, which can be also used to yield the almost surely asymptotic behavior. As a by-product, the exponential stability in p th( p ≥ 1)-moment and the almost sure exponential stability are analyzed. Finally, two examples are given to show the usefulness of the results obtained.


Introduction
Many dynamical systems not only depend on the present state but also the past ones, which are described by differential delay equations (DDEs) [1]. Since DDEs have been used in many fields, such as the population ecology, steam or water pipes, heat exchangers, lossless transmission lines, and the mass-spring-damper model, etc, the dynamical behavior for DDEs has been widely investigated. When DDEs are subject to the environmental disturbances, it can be characterized by stochastic delay differential equations (SDDEs)(see [2]- [7] and the references therein). One of the important issues in the study of SDDEs is automatic control, with consequent emphasis being placed on the stability analysis. Many papers on the stochastic stability analysis have been published. For instance, in [8], the dynamical behavior for stochastic delay Lotka-Volterra model as a particularly important application of SDDEs was analyzed. In [9], the exponential stability analysis for linear stochastic delay differential equation has been investigated by one useful and advanced method such as the comparison principle. In [10], by establishing the LaSalle theorem, the stability analysis for SDDEs has been investigated.
Hybrid systems driven by continuous-time Markov chains have been used to describe many practical systems, in which they may experience abrupt changes in their structure and parameters, for example, electric power systems, manufacturing systems, financial systems, etc. The hybrid systems comprise two parts: one is that the state takes values continuously, and the other is that the state takes discrete values. Recently, the stability analysis for SDDEs with Markovian switching has been extensively studied. For example, in [13], the comparison principle was used to study the stability for SDDEs with Markovian switching. In [15], by using the Lyapunov functional approach, the exponential stability in pth(p ≥ 1)-moment and the almost sure exponential stability for SDDEs with Markovian switching have been investigated under one monotonicity condition, which likes (2.6) (see Hypothesis IV ). In [18], by utilizing a linear matrix inequality approach, the delaydependent exponential stability of stochastic systems with time-varying delays, Markovian switching and nonlinearities has been discussed. In [14], by using the Lyapunov functional approach, the delay feedback control was designed to achieve the stabilization of hybrid SDDEs. In [16,17], in order to reduce the control cost, the feedback control based on discrete-time state observations was designed to guarantee the stabilization of SDDEs with Markovian switching.
Note that there are some results on the stability analysis of SDDEs with Markovian switching, in which the diffusion term and the drift term of the SDDEs obey the local Lipschitz condition and the linear growth condition. Usually, for many nonlinear SDDEs, these two terms often do not satisfy the linear growth condition, but satisfy the local Lipschitz condition. When the linear growth condition is replaced with the monotonicity condition, one of the most powerful technique used in the study of stability of SDDEs with Markovian switching is based on a stochastic version of the Lyapunov direct method, and there are some representative works on the stability analysis for highly nonlinear SDDEs with Markovian switching. For example, In [19], the delay-dependent stability criteria for highly nonlinear SDDEs with Markovian switching have been derived by using the Lyapunov function approach. Without the linear growth condition, the existence and uniqueness, the stability analysis and boundedness for the global solution of highly nonlinear SDDEs with Markovian switching were considered in [20,21].
However, the obtained results in the literature are only suitable for the constant delay or the time-varying delay with its derivative value being less than one. It is well known that in most industrial process involving transportation of materials, delay variation is one among the well-known structural time variations in the process plants. Since the transportation time varies frequently according to varying flow rates, time-varying delay is an inherent characteristics of these processes, which varies around a constant value and depends on the frequency of the external excitation [23]. Thus, we will analyze the existence and uniqueness of the global solution as well as its stability properties when this restrictive condition imposed on the time-varying delay is removed, the local Lipschitz condition is satisfied for the drift term and the diffusion term, and the linear growth condition is replaced by the monotonicity condition.
In this paper, the existence and uniqueness theorem for the global solution of highly nonlinear SDDEs with Markovian switching is primarily considered under a local Lipschitz condition and a monotonicity condition. Without the derivative value of the time-varying delay being less than one, the exponential stability in pth(p ≥ 1)-moment for such equations is discussed by using the integral inequality, and the almost sure exponential stability is analyzed by employing the nonnegative semi-martingale convergence theorem. The almost sure asymptotical stability for the global solution of highly nonlinear SDDEs with Markovian switching is also investigated by virtue of some stochastic analysis technique. Finally, two examples including one coupled systems consisting of a mass-spring-damper with the nonlinear external random forces are provided to validate the effectiveness of the theoretical findings obtained.
N otations: Throughout this paper, unless otherwise specified, we use the following notation. Let |·| denote the Euclidean norm in 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). Let (Ω, F, {F t } t≥0 , P) represent 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 B(t) = col[B 1 (t), B 2 (t), . . . , B m (t)] be an m-dimensional Brownian motion on (Ω, F, {F t } t≥0 , P).

Problem statement and preliminaries
Let r(t)(t ≥ 0) be a right-continuous Markov chain on the probability space taking values in a finite state space S = {1, 2, ..., N } with generator Γ = (γ ij ) N ×N given by For a continuous-time Markov chain r(t) with its generator Γ, it can be given as one stochastic integral with respect to a Poisson random measure with the initial value r(0) = i 0 ∈ S, where ν(dt, dy) is a Poisson random measure with intensity dt×m(dy) in which m is the Lebesgue measure on R, while the explicit definition ofh : S × R → R can be founded in [12].
Note that Hypothesis II is a conservative condition to check the existence of the global solution. For example, when S = {1, 2}, f (t, x, y, 1) = −0.15x − 2x 3 + 0.4y, f (t, x, y, 2) = −2x−0.5xy 4 +0.82y, g(t, x, y, 1) = 2x 2 , and g(t, x, y, 2) = xy 2 , for any t ≥ 0, Hypothesis II does not hold for f (·, ·, ·, ·) and g(·, ·, ·, ·). Here, we shall persist Hypothesis I but replace Hypothesis II by a more general condition to guarantee the existence of the unique global solution to Eq. (2.1). To state a general condition, we need a few notations. Let such that for each i ∈ S, they are continuously once differentiable in t and twice in x. Given V ∈ C 1,2 , then we define the Itô operator x, y, i)] To obtain the main results, one more general condition is presented as follows: Hypothesis III (Monotonicity condition): There exist one Lyapunov function V ∈ C 1,2 , one function U ∈ C(R n ; [0, ∞)) and some positive constants c 1 , c 2 , λ 1 and λ 2 such that for any x, y ∈ R n , t ≥ 0, and i ∈ S, and When U (x) = |x| p , Hypothesis III can be written as the following form: Hypothesis IV : There exist one Lyapunov function V ∈ C 1,2 , and some positive constants p, c 1 , c 2 , λ 1 and λ 2 with λ 2 c 2 < λ 1 c 1 such that for any x, y ∈ R n , t ≥ 0, and i ∈ S, and where p ≥ 1.
Remark 2.1 In [9,11,12,15], Hypothesis IV has been imposed with τ (t) ≡ τ or dτ (t) dt ∈ (0, 1). It should be mentioned that the restrictive condition that the derivative value of time-varying delay is less than one is not required in this paper. Thus, the proposed methods in [9,11,12,15] can not be used here. The asymptotic behavior for high nonlinear SDDEs with Markovian switching has been considered under the general monotonicity condition [19,20,21,22], but this restrictive condition is also imposed.

Definition 2.3
The global solution x(t) of Eq. (2.1) is said to be exponentially stable in pth(p ≥ 1) moment with decay e t of order γ, if there exists a positive constant γ such that is said to be almost surely exponentially stable with exponential decay e t of order γ , if holds for any ϕ ∈ C F 0 ([−τ, 0]; R n ).

Main results
Lemma 3.1 Let x(t) be a solution to Eq. (2.1) with the initial condition ϕ. Suppose that Hypotheses I and III hold. Assume that the inequality holds, then we have where ε ∈ (0, ε 0 ), ε 0 is a unique positive solution of the algebraic equation: Proof : Define a function: is a nondecreasing function on (0, λ 1 c 2 ). Therefore, there exists a scalar ε 0 ∈ (0, λ 1 c 2 ) satisfying H(ε 0 ) = 0. That is, for any ε ∈ (0, ε 0 ), we have Using the Itô formula, for any t ≥ 0, it follows where µ(ds, dl) = ν(ds, dl)−m(dl) is a martingale measure, which is related to the Markov chain but not the Brownian motion.
Let T → ∞, the desired result (3.1) is obtained. EU (x(t + θ))dt < ∞. Proof : From Hypothesis I, for any initial data ϕ ∈ C F 0 ([−τ, 0]; R n ), by using Theorem 7.12 (see, pp. 278 [12]), it is shown that there exist a unique maximal local strong solution where σ e is the explosion time. To show that this solution is global, we only need to prove σ e = ∞, a.s. Note that ϕ ∈ C F 0 ([−τ, 0]; R n ), consequently, there must exist a positive number k 0 such that ||ϕ|| C ≤ k 0 . For each integer k > k 0 , define the stopping time with the traditional convention inf ∅ = ∞, where ∅ denotes the empty set. Clearly, τ k is increasing and τ k → τ ∞ ≤ σ e a.s. (as k → ∞). If we can show τ ∞ = ∞ a.s., then σ e = ∞ a.s., which implies that x(t) is actually global. This is equivalent to proving that for any t > 0, P(τ k ≤ t) → 0, as k → ∞.
According to the definition of the function ψ(·), we have From (3.20) and (3.21), it yields where |x(β h )| = h by the definition of stopping time β h .
(3.24) From Remark 3.5, we obtain (3.25) From Hypothesis I, we have for any |x| ∨ |y| ≤ h, where C h is positive constant.
By the Hölder inequality and the Doob's martingale inequality, it is derived that for any T > 0, (3.26) Since U (x) is continuous in R n , it must be uniformly continuous. That is, when |x| ∨ |y| ≤ h, we can therefore choose δ = δ(ε) satisfying Choose T sufficiently small such that From (3.26) and (3.28), it follows that Accordingly, we have Using (3.23) and (3.24), we have

Two Examples
In order to illustrate the advantages of the main results, two examples are provided.
Thus, for any i ∈ S.

Conclusion
The method of Lyapunov function has been widely used in the study for the stability of highly nonlinear stochastic differential delay equations with Markovian switching. However, so far, most of the existing results in this area usually require that the delay is a constant or the time-varying delay with its derivative value being less than one, which limits their applications to some extent. When the involved delay is time-varying with it being a bounded measurable function, one integral lemma has first been given. Then, under a locally Lipschitz condition and a monotonicity condition, the existence and uniqueness for the global solution of stochastic differential delay equations with Markovian switching has been proved; by using the integral inequality, some stochastic analysis technique and the nonnegative semi-martingale convergence theorem, the stability analysis for the global solution of highly nonlinear stochastic differential delay equations with Markovian switching have been discussed. Finally, two examples have been provided to illustrate the effectiveness of the theoretical results obtained.