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

On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations1 Tian Zhang†Huabin Chen†2Chenggui Yuan‡3and Tom´as Caraballo♯4 †Department of Mathematics, Nanchang University, Nanchang 330031, China ‡Department of Mathematics, Swansea University, Swansea SA2 8PP, UK ♯The Depto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain Abstract. In this paper, under a local Lipschitz condition and a monotonicity condition, the problems on the existence and uniqueness theorem as well as the almost surely asymptotic behavior for the global solution of highly nonlinear stochastic differential equations with time-varying delay and Markovian switching are discussed by using the Lyapunov function and some stochastic analysis techniques. Two integral lemmas are firstly established to overcome the difficulty stemming from the coexistence of the stochastic perturbation and the time-varying delay. Then, without any redundant restrictive condition on the time-varying delay, by utilizing the integral inequality, the exponential stability in pth(p≥1)-moment for such equations is investigated. By employing the nonnegative semi-martingale convergence theorem, the almost sure exponential stability is analyzed. Finally, two examples are given to show the usefulness of the results obtained. AMS Subject Classification: 60H15 Key words and phrases: Stochastic differential equations, time-varying delay, asymptotic behavior, stability, Markov switching. 1The research of Huabin Chen is supported by the National Natural Science Foundation of China (61364005, 11401292, 61773401), the Natural Science Foundation of Jiangxi Province of China (20171BAB201007, 20171BCB23001), and the Foundation of Jiangxi Provincial Educations of China (GJJ160061, GJJ14155). The research of Tom´as Caraballo was partially supported by the projects MTM2015-63723-P (MINECO/ FEDER, EU) and P12-FQM-1492 (Junta de Andaluc´ıa). 2E-mail address: chb [email protected] (H. Chen) 3E-mail address: [email protected](C. Yuan). 4E-mail address: [email protected] (T. Caraballo). 1 1 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 in [2, 3, 4]. When DDEs are subject to the environmental disturbances, it can be characterized by stochastic delay differential equations (SDDEs), see [5, 6, 7, 8, 9, 10, 11]. One of the important issues in the study of SDDEs is automatic control, with consequent emphasis being placed on the stability analysis. Some excellent works on the stochastic stability analysis have been presented in [5, 12, 13, 14, 15, 16, 17] and the references therein. For instance, in [12], the dynamical behavior for stochastic delay Lotka-Volterra model as a particularly important application of SDDEs was analyzed. In [15], 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 [16], 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 [18, 19, 20], etc. Many excellent works are seen in [21, 22] and the references therein. 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 in [18, 20, 23, 24, 25, 26, 27, 28, 29, 30] and the references therein. For instance, in [23], the comparison principle was used to study the stability for SDDEs with Markovian switching. In [25], 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 [28], by utilizing a linear matrix inequality approach, the delay-dependent exponential stability of stochastic systems with time-varying delays, Markovian switching and nonlinearities has been discussed. In [18], by using the Lyapunov functional approach, the delay feedback control was designed to achieve the stabilization of hybrid SDDEs. In [21], in order to reduce the control cost, the feedback control based on discrete-time state observations was designed to guarantee the stabilization of hybrid SDEs. Note that there are some results on the stability analysis of SDDEs with Markovian switching, see [18, 20, 23, 24, 25, 27, 28] and the references therein, 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 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 representive 2 works on the stability analysis for highly nonlinear SDDEs with Markovian switching, see [31, 32, 33, 34, 35]. For example, In [31], the delay-dependent stability criteria for highly nonlinear SDDEs with Markovian switching have been discussed 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 [32, 33]. However, the obtained results in [31, 32, 33, 34, 35] are 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 [36]. Thus, we will analyze the existence and uniqueness of solutions as well as their stability properties when the restrictive conditions imposed on the time-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 highly nonlinear SDDEs with Markovian switching is primarily considered under a local Lipschitz condition and a monotonicity condition. Without any redundant restrictive condition on the time-varying delay, 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 results obtained. Notations: Throughout this paper, unless otherwise specified, we use the following notation. Let |·| denote the Euclidean norm in Rn. If Ais a vector or matrix, its transpose is denoted by AT. If Ais a matrix, its trace norm is denoted by |A|=√trace(ATA). Let (Ω,F,{Ft}t≥0,P) represents a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is increasing and right-continuous while F0contains all P-null sets). Let B(t) = col[B1(t), B2(t), . . . , Bm(t)] be an m-dimensional Brownian motion on (Ω,F,{Ft}t≥0,P). For τ > 0, let C([−τ, 0]; Rn) represent the family of all continuous Rn-valued functions on [−τ, 0] with norm ∥φ∥C= sup{|φ(θ)|:−τ≤θ≤0} for any φ∈ C([−τ, 0]; Rn). CFt([−τ, 0]; Rn) denotes the family of all Ft-measurable and C([−τ, 0]; Rn)-valued random variables ξ={ξ(θ) : −τ≤θ≤0}. Let E{·} stand for the expectation operator. For any two numbers a,b,a∨band a∧bdenote the maximum value and the minimum value between aand b, respectively. H(a−) denotes the left-hand limit of the function H(·) at a,i. e. H(a−) = limu→0−H(a+u). 3 2 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 = (γij)N×Ngiven by P{r(t+△) = j|r(t) = i}={γij△+o(△),if i=j, 1 + γij△+o(△),if i=j, where lim△↓0o(△) △= 0. Here, γij ≥0 is the transition rate from ito j, if i=jwhile γii =−∑j=iγij. 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 dr(t) = ∫R ¯ h(r(t−), y)ν(dt, dy), t ≥0 with the initial value r(0) = i0∈ S, where ν(dt, dy) is a Poisson random measure with intensity dt×m(dy) in which mis the Lebesgue measure on R, while the explicit definition of ¯ h:S × R→Rcan be founded in [12]. Consider the following highly nonlinear hybrid stochastic delay differential equations: dx(t) = f(t, x(t), x(t−τ(t)), r(t))dt +g(t, x(t), x(t−τ(t)), r(t))dB(t), t ≥0,(2.1) with the initial value {x(θ) : −τ≤θ≤0}=φ∈ CF0([−τ, 0]; Rn) and r(0) = i0∈ S, where x(t) = col[x1(t), x2(t), . . . , xn(t)] ∈Rnis the state vector. The time-varying delay τ(·) : [0,∞)→[0, τ] is a bounded measurable function. f(·,·,·,·) : [0,∞)×Rn×Rn×S → Rn is the drift coefficient vector, and g(·,·,·,·) : [0,∞)×Rn×Rn× S → Rn×mis the diffusion coefficient matrix. In this paper, it is also assumed that the Markov chain r(·) is independent of the Brownian motion B(·). Let x(t, 0, φ, i0) be the solution of Eq. (2.1). For simplicity, x(t) = x(t, 0, φ, r(0)). In this paper, the existence-uniqueness theorem, and the asymptotic behavior of Eq. (2.1) will be checked. In general, the following assumptions are given for the existence and uniqueness of the solution to Eq. (2.1), see [18]. Hypothesis I (Local Lipschitz condition): For each k= 1,2, . . ., there exists a positive constant cksuch that |f(t, x, y, i)−f(t, ¯x, ¯y, i)| ∨ |g(t, x, y, i)−g(t, ¯x, ¯y, i)| ≤ ck(|x−¯x|+|y−¯y|) for any (t, i)∈[0, T]× S (T > 0), x, y, ¯x, ¯y∈Rnwith |x|∨|y|∨|¯x|∨|¯y| ≤ k.In addition, f(t, 0,0, i) = 0 and g(t, 0,0, i) = 0. Hypothesis II (Linear growth condition): There is a positive constant Lsuch that |f(t, x, y, i)| ∨ |g(t, x, y, i)| ≤ L(1 + |x|+|y|) for any (t, x, y, i)∈[0, T]×Rn×Rn× S. 4 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−2x3+ 0.4y,f(t, x, y, 2) = −2x−0.5xy4+0.82y,g(t, x, y, 1) = 2x2, and g(t, x, y, 2) = xy2, 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 C1,2= C1,2([0,∞)×Rn× S; [0,∞)) denote the family of all continuous nonnegative functions V(t, x, i) defined on [0,∞)×Rn× S, such that for each i∈ S, they are continuously once differentiable in tand twice in x. Given V∈ C1,2, then we define the Itˆo operator LV : [0,∞)×Rn×Rn× S −→ Rby LV (t, x, y, i) =Vt(t, x, i) + Vx(t, x, i)f(t, x, y, i) + 1 2trace[gT(t, x, y, i)Vxx(t, x, i)g(t, x, y, i)] + N ∑ j=1 γijV(t, x, j). where Vt(t, x, i) = ∂V (t, x, i) ∂t , Vx(t, x, i) = (∂V (t, x, i) ∂x1 ,∂V (t, x, i) ∂x2 , . . . , ∂V (t, x, i) ∂xn), and Vxx(t, x, i) = (∂2V(t, x, i) ∂xl∂xm)n×n . To obtain the main results, one more general condition is presented as follows: Hypothesis III (Monotonicity condition): There exists one Lyapunov function V∈ C1,2, one function U∈ C(Rn; [0,∞)) and some positive constants c1,c2,λ1and λ2such that for any x, y ∈Rn,t≥0, and i∈ S, c1U(x)≤V(t, x, i)≤c2U(x),(2.2) and LV (t, x, y, i)≤ −λ1U(x) + λ2U(y),(2.3) and lim |x|→∞ U(x) = ∞.(2.4) when U(x) = |x|p,Hypothesis III can be written as the following form: Hypothesis IV : There exists one Lyapunov function V∈ C1,2, and some positive constants p,c1,c2,λ1and λ2such that for any x, y ∈Rn,t≥0, and i∈ S, c1|x|p≤V(t, x, i)≤c2|x|p,(2.5) 5 and LV (t, x, y, i)≤ −λ1|x|p+λ2|y|p,(2.6) where p≥1 and λ2c2< λ1c1. Remark 2.1 In [15, 17, 18, 25], Hypothesis IV has been imposed with τ(t)≡τor 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 [15, 17, 18, 25] can not be used here. Even if the asymptotic behavior for high nonlinear SDDEs with Markovian switching has been been considered under the general monotonicity condition [31, 32, 33, 34], but this restrictive condition is also added. Definition 2.2 Let x(t):−τ≤t<σ∞be a continuous Ft-adapted Rn-valued local process, where σ∞is a stopping time and we set Ft=F0for t∈[−τ, 0]. It is called a local solution of Eq. (2.1) with initial data φ∈ CF0([−τ, 0]; Rn).If x0=φ={x(θ) : −τ≤θ≤0}and for all t≥0 x(t∧σk) =φ(0) + ∫t∧σk 0 f(s, x(s), x(s−τ(s)), r(s))ds +∫t∧σk 0 g(s, x(s), x(s−τ(s)), r(s))dB(s) holds for any k≥1, where {σk}k≥1is a nondecreasing sequence of finite stopping times such that σk↑σ∞a.s. Furthermore, if lim supk→∞ |x(σk)|=∞is satisfied whenever σ∞<∞, it is called a maximal solution and σ∞is called the explosion time. A maximal local solution x(t) : −τ≤t<σ∞, is said to be unique if for any other maximal local solution ˆx(t) : −τ≤t < ˆσ∞, we have σ∞= ˆσ∞a.s. and x(t) = ˆx(t) for all −τ≤t < σ∞ a.s. Definition 2.3 The solution of Eq. (2.1) is said to be exponentially stable in pth(p≥1) moment with decay etof order γ, if there exists a positive constant γsuch that lim sup t→∞ log(E|x(t)|p) t≤ −γ holds for any φ∈ CF0([−τ, 0]; Rn). Furthermore, the solution of Eq. (2.1) is said to be almost surely exponentially stable with exponential decay etof order γ, if lim sup t→∞ log(|x(t)|) t≤ −γ a.s. holds for any φ∈ CF0([−τ, 0]; Rn). Lemma 2.4 ([37]) For γ > 0, there exist two positive constants: λ,λ′with λ′< γ, and a function y: [−τ, ∞)→[0,∞). If the inequality y(t)≤{λe−γt +λ′∫t 0e−γ(t−s)supθ∈[−τ,0] y(s+θ)ds, for t ≥0, λe−γt, for t ∈[−τ, 0],(2.7) 6 holds, then we have y(t)≤˜ Me−µt, for any t∈[−τ, ∞), where µis a unique positive root of the algebra equation: λ′eµτ γ−µ= 1 and ˜ M= max{λ(γ−µ) λ′eµτ , λ}>0. 3 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 λ2c2< λ1c1, holds, then we have ∆(ε) = ∫∞ 0 eεt sup θ∈[−τ,0] EU(x(t+θ))dt < ∞,(3.1) where ε∈(0, ε0),ε0is a unique positive solution of the algebraic equation: λ2c2eετ λ1c1−c1c2ε= 1. Proof : Define the function: H(ε) = λ2c2eετ λ1c1−c1c2ε−1. It can be proved that H(0) <0, H((λ1 c2)−) = ∞, and H(ε) is a nondecreasing function on (0,λ1 c2). Therefore, there exists a scalar ε0∈(0,λ1 c2) satisfying H(ε0) = 0. That is, for any ε∈(0, ε0), we have Λ(ε)≡λ2c2eετ λ1c1−c1c2ε<1.(3.2) Using the Itˆo formula, for any t≥0, it follows e λ1 c2tV(t, x(t), r(t)) ≤V(0, x(0), r(0)) + ∫t 0 e λ1 c2s[λ1 c2 V(s, x(s), r(s)) + LV (s, x(s), x(s−τ(s)), r(s))]ds +∫t 0 e λ1 c2sVx(s, x(s), r(s))g(s, x(s), x(s−τ(s)), r(s))dB(s) +∫t 0∫R e λ1 c2s[V(s, x(s), i0+¯ h(r(s−), l)−V(s, x(s), r(s))]µ(ds, dl), (3.3) 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. From conditions (2.2) and (2.3), we obtain λ1 c2 V(s, x(s), r(s)) + LV (s, x(s), x(s−τ(s)), r(s)) ≤λ2U(x(s−τ(s)),(3.4) 7 Substituting (3.4) into (3.3), and then taking the expectation, it yields e λ1 c2tEV(t, x(t), r(t)) ≤EV(0, x(0), r(0)) + λ2∫t 0 e λ1 c2sEU(x(s−τ(s))ds. By using condition (2.2), it concludes that for any t≥0, EU(x(t)) ≤EV(0, x(0), r(0)) c1 e−λ1 c2t+λ2 c1∫t 0 e−λ1 c2(t−s)EU(x(s−τ(s))ds ≤M′e−λ1 c2t+λ2 c1∫t 0 e−λ1 c2(t−s)EU(x(s−τ(s))ds, (3.5) where M′=EV(0,x(0),r(0)) c1>0. For any t≥τand θ∈[−τ, 0], from (3.5), we have EU(x(t+θ)) ≤M′e−λ1 c2(t+θ)+λ2 c1∫t+θ 0 e−λ1 c2(t+θ−s)EU(x(s−τ(s))ds ≤M′e−λ1 c2(t+θ)+λ2 c1∫t+θ 0 e−λ1 c2(t+θ−s)sup u∈[−τ,0] EU(x(s+u))ds. Multiplying by eεt(ε∈(0, ε0)) on both sides of inequality above in turn, and then integrating with τto T(T > τ), it follows ∫T τ eεtEU(x(t+θ))dt ≤M′∫T τ eεt−λ1 c2(t+θ)dt +λ2 c1∫T τ∫t+θ 0 eεt−λ1 c2(t+θ−s)sup u∈[−τ,0] EU(x(s+u))dsdt. (3.6) Note that for any θ∈[−τ, 0] and t≥τ, the formula of integration by parts implies ∫T τ∫t+θ 0 eεt−λ1 c2(t+θ−s)sup u∈[−τ,0] EU(x(s+u))dsdt ≤eετ ∫T τ e−(λ1 c2−ε)(t+θ)∫t+θ 0 e λ1 c2ssup u∈[−τ,0] EU(x(s+u))dsdt ≤e λ1 c2τ λ1 c2−ε∫τ 0 e λ1 c2ssup u∈[−τ,0] EU(x(s+u))ds +eετ λ1 c2−ε∫T 0 eεs sup u∈[−τ,0] EU(x(s+u))ds. (3.7) 8 Substituting (3.7) to (3.6) implies ∫T τ eεtEU(x(t+θ))dt ≤M′e λ1 c2τ∫T τ eεt−λ1 c2tdt +λ2c2e λ1 c2τ λ1c1−c1c2ε∫τ 0 e λ1 c2ssup u∈[−τ,0] EU(x(s+u))ds +λ2c2eετ λ1c1−c1c2ε∫T 0 eεs sup u∈[−τ,0] EU(x(s+u))ds. (3.8) From (3.8), we have ∫T 0 eεtEU(x(t+θ))dt =∫τ 0 eεtEU(x(t+θ))dt +∫T τ eεtEU(x(t+θ))dt ≤¯ M+ Λ(ε)∫T 0 eεs sup u∈[−τ,0] EU(x(s+u))ds, (3.9) where ¯ M=∫τ 0eεtEU(x(t+θ))dt +c2M′eετ λ1−c2ε+λ2c2e λ1 c2τ λ1c1−c1c2ε∫τ 0e λ1 c2ssupu∈[−τ,0] EU(x(s+u))ds. Combing (3.2) and (3.9), it gives ∫T 0 eεt sup θ∈[−τ,0] EU(x(t+θ))dt ≤¯ M 1−Λ(ε)<∞. Let T→ ∞, the desired result (3.1) is obtained. 2 Remark 3.2 From (3.1), it follows that ∆ = ∫∞ 0 sup θ∈[−τ,0] EU(x(t+θ))dt < ∞,(p≥1).(3.10) Theorem 3.3 Suppose that the conditions of Lemma 3.1 hold, for any initial data φ∈ CF0([−τ, 0]; Rn), there is a unique solution x(t)to Eq. (2.1) on t∈[−τ, ∞)with probability one. Proof : By Hypothesis I, for any initial data φ∈ CF0([−τ, 0]; Rn), by using Theorem 7.12 (see, pp. 278 [18]), it is shown that there exist a unique maximal local strong solution x(t) on [−τ, σe], where σeis the explosion time. To show that this solution is global, we only need to prove σe=∞,a.s. Note that φ∈ CF0([−τ, 0]; Rn), consequently, there must exist a positive number k0such that ||φ||C≤k0. For each integer k > k0, define the stopping time τk= inf{t∈[0, σe) : |x(t)| ≥ k}. 9 From (3.27), it yields P({α2i−1<∞, βh=∞} ∩ { sup 0≤t≤T |U(x(α2i−1+t)) −U(y(α2i−1))|< ε} ≥P({α2i−1<∞, βh=∞} ∩ { sup 0≤t≤T |x(α2i−1+t)−x(α2i−1)|< δ}) > ε. (3.29) Set ˆ Ωi={sup 0≤t≤T |U(x(α2i−1+t)) −U(y(α2i−1))|< ε}, and note that α2i(ω)−α2i−1(ω)≥T, if ω∈ {α2i−1<∞, βh=∞} ∩ ˆ Ωi. Using (3.25) and (3.29), we have ∞ ≥ ε ∞ ∑ i=1 E{I{α2i<∞,βh=∞}[α2i−α2i−1]} ≥ε ∞ ∑ i=1 E{I{α2i<∞,βh=∞}∩ˆ Ωi[α2i−α2i−1]} ≥εT ∞ ∑ i=1 P({α2i<∞, βh=∞} ∩ ˆ Ωi) > εT ∞ ∑ i=1 ε=∞, which is a contradiction. Hence, (3.18) holds (i.e. limt→∞ U(x(t)) = 0). Step 4: Now, it is necessary to show that Ker(U)=∅. From (3.18), it is seen that there exists an Ω0⊂Ω with P(Ω0) = 1 such that lim t→∞ U(x(t)) = 0 and sup 0≤t<∞ |x(t)|<∞,for any ω∈Ω0.(3.30) Choose any ω∈Ω0, then {x(t)}t≥0is bounded in Rn. Then, there must be an increasing sequence {tk}k≥1such that tk→ ∞ and {x(tk)}k≥1converges to some ¯x∈Rn. Thus, U(¯x) = lim k→∞ U(x(tk)) = 0, which implies that ¯x∈Ker(U). That is, Ker(U)=∅. Step 5: It is necessary to show that for any ω∈Ω0, lim t→∞ d(x(t), Ker(U)) = 0.(3.31) 16 If this is false, then there exists some ¯ω∈Ω0such that lim sup t→∞ d(x(t, ¯ω), Ker(U)) >0. Thus, there exists a subsequence {x(tk,¯ω)}k≥0of {x(t, ¯ω)}t≥0satisfying lim sup k→∞ d(x(tk,¯ω), Ker(U)) >¯ε, for some ¯ε > 0. Since {x(tk,¯ω)}k≥0is bounded, we can find a subsequence converging to some ˜x∈Rn. Clearly, ˜x /∈Ker(U) and U(˜x)>0. However, from (3.30), U(˜x) = lim k→∞ U(x(tk,¯ω)) = 0. This is a contradiction. Therefore, (3.31) must be satisfied. In addition, if U(x) = 0 ⇔ x= 0, then Ker(U) = 0. Consequently, from (3.31), we deduce that lim t→∞ x(t) = 0. a.s. The proof is therefore complete. 2 Corollary 3.8 Let x(t;φ)be a solution to Eq. (2.1) with the initial condition φ. Suppose that Hypotheses I and IV are satisfied, for any initial data φ∈ CF0([−τ, 0]; Rn), the pth(p≥1)-moment Lyapunov exponent of the solution of the Eq. (2.1) obeys lim t→∞ sup 1 tlog(E|x(t;φ)|p)≤ −¯µ, where ¯µ∈(0,λ1 c2)is a root of the algebra equation : λ2c2eµτ λ1c1−c1c2µ= 1. That is, the solution of the Eq. (2.1) is exponentially stable in pth(p≥1) mean. Corollary 3.9 Let x(t)be a solution to Eq. (2.1) with the initial condition φ. Suppose that Hypotheses I and IV hold, for any initial data φ∈ CF0([−τ, 0]; Rn), the sample Lyapunov exponent of the solution of the Eq. (2.1) obeys lim t→∞ sup 1 tlog(|x(t;φ)|)≤ −ε p,a.s. where p≥1, and ε∈(0, ε0), where ε0is given in Lemma 3.1. That is, the solution of the Eq. (2.1) is almost surely exponentially stable. 4 Two Examples In order to illustrate the advantages of the main results, two examples are provided. 17 Example 4.1: Let B(t)be a scalar Brownian motion on (Ω,F,{Ft}t≥0,P). Consider one dimensional stochastic differential equations with time-varying delay and Markovian switching: dx(t) = f(t, x(t), x(t−τ(t)), r(t))dt +g(t, x(t), x(t−τ(t)), r(t))dB(t), t ≥0,(4.1) with the initial value {x(θ) : −τ≤θ≤0}=φ∈ CF0([−τ, 0]; Rn) and r(0) = i0∈ S and r(0) = 1 ∈ S ={1,2}, where x(t) and x(t−τ(t)) are the state scalar and the delayed state scalar, respectively. τ(t) is a bounded measurable function with 0 ≤τ(t)≤τ(t≥0, τ > 0), and r(t) is a right-continuous Markov chain taking values in Swith the generator Γ = (γij)2×2=[−2 2 1−1]. In (4.1), we assume that f, g : [0,∞)×R×R× S → Rwith f(t, x, y, i) = {−0.15x−2x3+ 0.4y, if i= 1, −2x−0.5xy4+ 0.82y, if i= 2, and g(t, x, y, i) = {2x2,if i= 1, xy2,if i= 2. Define a Lyapunov function V(t, x, i) = {x2,if i= 1, 0.5x2,if i= 2, then, it is computed for the Itˆo operator to Eq. (4.1) that LV (t, x, y, 1) = 2x[−0.15x−2x3+ 0.4y]+4x4+ 2 ∑ j=1 γ1jV(t, x, j) =−1.3x2+ 0.8xy ≤ − 0.9x2+ 0.4y2, and LV (t, x, y, 2) = x[−2x−0.5xy4+ 0.82y]+0.5x2y4+ 2 ∑ j=1 γ2jV(t, x, j) =−1.5x2+ 0.82xy ≤ − 1.09x2+ 0.41y2. Hence, we have LV (t, x, y, i)≤ −0.9x2+ 0.41y2, with λ1= 0.9, λ2= 0.41, c1= 0.5 and c2= 1. Then, λ2c2< λ1c1holds, which implies that the existence and uniqueness, the exponential stability in mean square, the almost sure 18 0 5 10 15 −0.5 0 0.5 1 1.5 2 Time t (a) E|x(t)|2 x(t) Figure 1: Asymptotic behavior in mean square of the global solution for Eq. (4.1) 0 5 10 15 −1.5 −1 −0.5 0 0.5 1 1.5 2 Time t (a) x(t) x(t) Figure 2: Asymptotic behavior in almost sure sense of the global solution for Eq. (4.1) exponential stability and the almost sure asymptotical stability of the global solution for Eq. (4.1) are guaranteed. When the initial condition x(t) = −1 (t∈[−2.3,0]), r(0) = 1, and τ(t) = 1.1|sin(t)|+1.2 are fixed, Fig. 1 and Fig. 2 illustrate the asymptotic behavior in mean square and in almost sure sense of the global solution for Eq. (1), respectively. Example 4.2: One coupled system consists of a mass-spring-damper (MSD) model [39]. An actuator is taken to a transfer system. The mathematical expression of the system is DDEs, which are written as M¨y(t) + C˙y(t) + Ky(t) = 0 (4.2) on t≥0, where M,C,Kare the mass, stiffness and damping of a mass-spring-damper model, and y(t), ˙y(t), ¨y(t) denote the position, velocity and acceleration of MSD at time t. If this physical model is affected by the external force, then Eq. (4.2) is further described as M¨y(t) + C˙y(t) + Ky(t) + F(t) = 0 (4.3) on t≥0, where F(t) denotes the external force, M= 10, C= 25, and K= 15. Assume that this external force is subject to the environmental noise and abrupt changes in the 19 parameters, which is characterized by F(t) = F1( ˙y(t),˙y(t−τ(t)), r(t)) + F2( ˙y(t), y(t−τ(t)),˙y(t−τ(t)), r(t)) ˙ B(t) where ˙ B(t)is a scalar white noise ( i.e. ˙ B(t)is a scalar Brownian motion), τ(t) is the time-varying delay, r(t) is a Markovian switching taking values in S={1,2}with its generator Γ = [−2 2 3−3], F1( ˙y(t),˙y(t−τ(t)), r(t)) = {5.4 ˙y(t) ˙y2(t−τ(t)),if i= 1, 15 ˙y3(t) ˙y2(t−τ(t)),if i= 2, and F2( ˙y(t), y(t−τ(t)),˙y(t−τ(t)), r(t)) ={6 ˙y(t) ˙y(t−τ(t)) + 3y(t−τ(t)) + 3 ˙y(t−τ(t)),if i= 1, 10 ˙y2(t) ˙y(t−τ(t)) + 2y(t−τ(t)) + 2 ˙y(t−τ(t)),if i= 2. Let x1(t) = y(t) and x2(t) = ˙y(t), Eq. (4.3) can be written as highly nonlinear SDDEs with Markovian switching: dx(t) = f(t, x(t), x(t−τ(t)), r(t))dt +g(t, x(t), x(t−τ(t)), r(t))dB(t)(4.4) where x(t) = col[x1(t), x2(t)], f(t, x(t), x(t−τ(t)),1) = [x2(t) −1.5x1(t)−2.5x2(t)−0.54x2(t)x2 2(t−τ(t)) ], f(t, x(t), x(t−τ(t)),2) = [x2(t) −1.5x1(t)−2.5x2(t)−1.5x3 2(t)x2 2(t−τ(t)) ], g(t, x(t), x(t−τ(t)),1) = [0 0.6x1(t−τ(t)) + 0.3x2(t−τ(t)) + 0.3x2(t)x2(t−τ(t)) ], and g(t, x(t), x(t−τ(t)),2) = [0 x1(t−τ(t)) + 0.2x2(t−τ(t)) + 0.2x2(t)x2(t−τ(t)) ]. For Eq. (4.4), consider a Lyapunov function V(t, x, i) = {|x|2,if i= 1, 0.8|x|2,if i= 2, with |x|2=x2 1+x2 2. Then, for Eq. (4.4), the Itˆo operator is computed as LV (t, x(t), x(t−τ(t)), i) = 2qixT(t)f(t, x(t), x(t−τ(t)), r(t)) + qitrace[gT(t, x(t), x(t−τ(t)), i) ×g(t, x(t), x(t−τ(t)), i)] + 2 ∑ j=1 γijV(t, x(t), j), 20 012345 −0.5 0 0.5 1 1.5 2 Time t (a) E|x(t)|2 x1(t) x2(t) Figure 3: Asymptotic behavior in mean square of the global solution for Eq. (4.4) 0 1 2 3 4 5 6 7 8 −1 −0.5 0 0.5 1 1.5 2 Time t (a) x(t) x1(t) x2(t) Figure 4: Asymptotic behavior in almost sure sense of the global solution for Eq. (4.4) where q1= 1, q2= 0.8. Consequently, when i= 1, we have LV (t, x(t), x(t−τ(t),1) ≤ − 2[x2 1(t) + x2 2(t)] −1.08x2 2(t)x2 2(t−τ(t)) + [0.6x2(t)x2(t−τ(t)) + 0.3x1(t−τ(t)) + 0.3x2(t−τ(t))]2 −0.4[x2 1(t) + x2 2(t)] ≤ − 2.4|x(t)|2+ 0.27|x(t−τ(t))|2, and when i= 2, LV (t, x(t), x(t−τ(t),2) ≤ − 1.6[x2 1(t) + x2 2(t)] −2.4x4 2(t)x2 2(t−τ(t)) + 0.8[x2 2(t)x2(t−τ(t)) + 0.2x1(t−τ(t)) + 0.2x2(t−τ(t))]2 + 0.6[x2 1(t) + x2 2(t)] ≤ − |x(t)|2+ 0.096|x(t−τ(t))|2. Thus, for any i∈ S. LV (t, x(t), x(t−τ(t)), i)≤ −|x(t)|2+ 0.27|x(t−τ(t))|2. 21 with λ1= 1, λ2= 0.27, c1= 0.8 and c2= 1. Thus, λ2c2< λ1c1is satisfied. Consequently, the existence and uniqueness, the exponential stability in mean square, the almost sure exponential stability and the almost sure asymptotical stability of the global solution for Eq. (4.4) are guaranteed. When taking the initial condition x(t) = col[−sin(t),0.5 cos(t)] (t∈ [−2.3,0]), r(0) = 1, and τ(t) = 1.1|cos(t)|+ 1.2, Fig. 3 and Fig. 4 show the asymptotic behavior in mean square and in almost sure sense of the global solution for Eq. (4.4), respectively. 5 Conclusion The method of Lyapunov function has been widely used in the study of the stability of SDDEs 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. To remove this restrictive condition, firstly, two integral lemmas have been proposed. Then, by using the integral inequality, some stochastic analysis technique and the nonnegative semi-martingale convergence theorem, the existence-uniqueness theorem and the stability analysis for the global solution of highly nonlinear hybrid SDDEs have been discussed. 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