# The Boundedness and Exponential Stability Criterions for Nonlinear Hybrid Neutral Stochastic Functional Differential Equations

Authors: Xiaofeng Zong; Fuke Wu; Chengming Huang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Neutral differential equations have been used to describe the systems that not only depend on the present and past states but also involve derivatives with delays. This paper considers hybrid nonlinear neutral stochastic functional differential equations (HNSFDEs) without the linear growth condition and examines the boundedness and exponential stability. Two illustrative examples are given to show the effectiveness of our theoretical results.


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## Body

## 1. Introduction

Many dynamic systems not only depend on the present and past states but also involve derivatives with delays. Neutral differential equations have been used to model such systems. Deterministic neutral differential equations were introduced by Hale and Meyer [1] and discussed in Hale and Lunel  (see [2]) and Kolmanovskii et al. (for details see also [3, 4]), among others. Such equations were used to study two or more simple oscillatory systems with some interconnections between them, such as Brayton [5], Rubanik [6], and Driver [7].Generally speaking, many practical systems commonly encounter stochastic perturbations and may experience abrupt changes in their structure and parameters caused by phenomena such as component failures or repairs and abrupt environmental disturbances. Of course, there is no exception to neutral systems, mentioned previous. Taking these stochastic factors into account, Mao and Yuan developed hybrid systems driven by Brownian motion and continuous-time Markovian chain to cope with such a situation (see [8]). Hu et al. [9] investigated the stability and boundedness of stochastic differential delay equations with Markovian switching. Kolmanovskii et al. [10] discussed the neutral stochastic delay differential equations with Markovian switching, also known as hybrid neutral stochastic delay differential equations (HNSDDEs).The boundedness and stability analysis of the neutral stochastic systems without switching has attracted much attention; see [11–18] to mention a few. For hybrid neutral systems, studying boundedness and stability of the solutions is also a challenging and interesting work. Kolmanovskii et al. [10] established a fundamental theorem of HNSDDEs and discussed the boundedness and exponential stability of the solutions. They also gave an example to show that Markovian can average the subsystems; that is, when some subsystems are stable and others are not stable, the overall system formed by the Markovian switching may be stable. Bao et al. [19] discussed stability in distribution of the HNSDDEs. Hu and Wang [20] studied the stability in distribution for the general HNSFDEs. The stability of HNSDDEs with interval uncertainty was investigated in [21]. Mao et al. [22] gave a criterion related to almost surely asymptotic stability of HNSDDEs. These results are undoubtedly remarkable.However, there are few publications on the boundedness and exponential stability of the general HNSFDEs with highly nonlinear terms. To fill in this gap, this work gives the boundedness and exponential stability criterions for such HNSFDEs. Moreover, when this HNSFDE degenerates to the HNSDDE, our stability criterions improve the related results in [10]. Further, these stability criterions can also be used to investigate the exponential stability of NSFDEs or NSDDEs with more accurate Lyapunov exponent bound than that obtained in [23, 24].The rest of the paper is arranged as follows. The next section provides necessary notations and definitions for the use of this paper. Section3 establishes the boundedness and exponential stability criterions of the solutions to HNSFDEs. Section 4 further gives the generalized results for the HNSDDEs with variable time delay. Finally, two illustrative examples are provided to show the effectiveness of our theoretical results.


## 2. Notations and Definitions

Throughout this paper, unless otherwise specified, we use the following notations.|·| denotes both the Euclidean vector norm in ℝn and Frobenius matrix norm in ℝn×d. If A is a vector or matrix, its transpose is denoted by AT. If A is a matrix, its trace norm is denoted by |A|=trace(ATA). Let (Ω,𝔉,P) be a complete probability space with a filtration {𝔉𝔱}t≥0 satisfying the usual conditions; that is, it is right continuous and increasing while 𝔉0 contains all P-null sets. Let w(t)=(w1(t),…,wd(t)) be a d-dimensional Brownian motion defined on this probability space. Let ℝ+=[0,∞) and τ>0. Denote by C([-τ,0],ℝn) the family of continuous functions from [-τ,0] to ℝn with the norm ∥φ∥=sup-τ≤θ≤0|φ(θ)|. Let C𝔉0b([-τ,0],ℝn) be the family of all 𝔉0-measurable bounded C([-τ,0],ℝn) valued random variables ξ={ξ(θ):-τ≤θ≤0}. a∨b represents max{a,b}, and a∧b denotes min{a,b}.Letr(t) be a Markov chain (independent of w(t)) taking values in a finite state space 𝕊={1,2,…,m}. Assume the generator of r(t) is denoted by Γ=(γij)m×m, so that
(1)ℙ{r(t+δ)=j∣r(t)=i}={γijδ+o(δ),ifi≠j,1+γijδ+o(δ),ifi=j,
where δ>0. Here γij is the transition rate from i to j and γij>0 if i≠j while γii=-∑j≠i‍γij. Let us consider the following n-dimensional nonlinear HNSFDE:
(2)d[x(t)-u(xt,r(t))]=f(xt,r(t))dt+g(xt,r(t))dw(t),t≥0,
with initial data x0=ξ∈C𝔉0b([-τ,0];ℝn) and r(0)=r0∈𝕊, where
(3)xt={x(t+θ);-τ≤θ≤0}∈C([-τ,0];ℝn),u:C([-τ,0],ℝn)×𝕊→ℝn, f:C([-τ,0],ℝn)×𝕊→ℝn, and g:C([-τ,0],ℝn)×𝕊→ℝn×d. In order to guarantee the existence and uniqueness of the solution to (2), we give the following assumptions for the functionals u, f, and g.Assumption 1 (local Lipschitz condition).
fand g satisfy the local Lipschitz condition; that is, for each j>0 there exists a positive constant Cj such that for any maps ϕ,  φ∈C([-τ,0];ℝn) with ∥ϕ∥∨∥φ∥≤j(4)|f(ϕ,i)-f(φ,i)|∨|g(ϕ,i)-g(φ,i)|≤Cj∥ϕ-φ∥,∀i∈𝕊,
where ∥ϕ-φ∥≤sup-τ≤θ≤0|ϕ(θ)-φ(θ)|.Assumption 2 (contractive mapping).
There exists a positive constantκ∈(0,1) such that for all ϕ,  φ∈C([-τ,0];ℝn) and i∈𝕊(5)|u(ϕ,i)-u(φ,i)|≤κ∥ϕ-φ∥
and u(0,i)=0.Note that the previous assumptions are standard for the existence and uniqueness of the local solutions (see [19, 22]). Additional conditions should be imposed for the local solution to be global. In view of this, we need a few more notations. Let C2(ℝn×𝕊;ℝ+) denote the family of all nonnegative functions V(x,i) on ℝn×𝕊 which are continuously twice differentiable in x. For each V(x,i)∈C2(ℝn×𝕊;ℝ+), define an operator ℒV from C([-τ,0];ℝn)×𝕊 to ℝ:
(6)ℒV(φ,i)=Vx(φ(0)-u(φ,i),i)f(φ,i)+∑j∈𝕊‍γijV(φ(0)-u(φ,i),j)+12trace[gT(φ,i)Vxx(φ(0)-u(φ,i),i)g(φ,i)],
where
(7)Vx(x,i)=(∂V(x,i)∂x1,…,∂V(x,i)∂xn),Vxx(x,i)=(∂2V(x,i)∂xj∂xl)n×n.In the following sections, we will impose the some conditions on the diffusion operatorℒV for the global solution and its asymptotic behavior.


## 3. The Boundedness and Exponential Stability of HNSFDEs

The following theorem gives the boundedness and exponential stability criterions of the solution to (2).Theorem 3.
Let Assumptions1 and 2 hold. Assume that there are two functions V∈C2(ℝn×𝕊;ℝ+), U∈C(ℝn;ℝ+), three probability measures η, μ, μ¯ on [-τ,0], and a number of positive constants κ∈(0,1), c, c1, c2, p≥1, λ1, λ2, λ3, λ4 such that for any x∈ℝn and (φ,i)∈C([-τ,0],ℝ+)×𝕊(8)|u(φ,i)|≤κ∫-τ0‍φ(θ)η(dθ),(9)c1|x|p≤V(x,i)≤c2|x|p,(10)ℒV(φ,i)≤c-λ1|φ(0)|p+λ2∫-τ0‍|φ(θ)|pμ(dθ)-λ3U(φ(0))+λ4∫-τ0‍U(φ(θ))μ¯(dθ).
If λ1>λ2 and λ3>λ4, then we have the following assertions:(i)
for any given initial dataξ∈C𝔉0b([-τ,0],ℝn), there is a unique global solution x(t)=x(t;ξ) to the hybrid system (2) on t∈[-τ,∞);(ii)
the solutionx(t) obeys
(11)limsupt→∞𝔼|x(t)-u(xt,r(t))|p≤cΛ,(12)limsupt→∞1t∫0t‍𝔼U(x(s))ds≤cλ3-λ4,
where Λ:=γ¯∧(1/τ)log(λ3/λ4)∧r with γ¯ and r defined by
(13)γ¯=max{[c2q(1+ɛ1/(p-1))p-1κpɛ+λ2]eqτq>0;c2q(1+ɛ1/(p-1))p-1-λ1======+[c2q(1+ɛ1/(p-1))p-1κpɛ+λ2]eqτ=0,ɛ>0}
and r:=(p/τ)log(1/κ)-ℓ for sufficiently small ℓ>0.(iii)
If, in addition,c=0, then the solution of (2) has properties that
(14)limsupt→∞log(𝔼|x(t)|p)t≤-Λ,(15)∫0∞‍𝔼U(x(s))ds≤1λ3-λ4[𝔼V(y0,r(0))+λ2∫-τ0‍𝔼|x(s)|pds≤1λ3-λ4+λ4𝔼∫-τ0‍U(x(s))ds],
where y0=x(0)-u(x0,r(0)).Proof.
We prove these three assertions, separately. For any given initial dataξ∈C𝔉0b([-τ,0],ℝn), by Assumptions 1 and 2, there exists a unique maximal local strong solution x(t) to (2) on t∈[-τ,ρe), where ρe is the explosion time. To show that this solution is global, we only need to prove that ρe=∞ a.s. Define yt=x(t)-u(xt,r(t)), then by Assumption 2, we have
(16)|y0|≤|x(0)|+|u(x0,r(0))|≤(1+κ)∥ξ∥.
Let k0 be sufficiently lager positive number, such that ∥ξ∥<k0. For each k>(1+κ)k0, define the stopping time ρk=inf{t∈[0,ρe):|yt|≥k}. Clearly, ρk is increasing as k→∞ and ρk→ρ∞≤ρe a.s. If we can show ρ∞=∞ a.s., then ρe=∞, which implies that the solution x(t) is actually global. By the generalized Itô formula (see [20]) and condition (10), we can obtain that, for any k>k0 and t≥0,
(17)𝔼V(y(t∧ρk),r(t∧ρk))=𝔼V(y0,r(0))+𝔼∫0t∧ρk‍ℒV(xs,r(s))ds≤ct+𝔼V(y0,r(0))-λ1𝔼∫0t∧ρk‍|x(s)|pds+λ2𝔼∫0t∧ρk‍∫-τ0‍|x(s+θ)|pμ(dθ)ds-λ3𝔼∫0t∧ρk‍U(x(s))ds+λ4𝔼∫0t∧ρk‍∫-τ0‍U(x(s+θ))μ¯(dθ)ds.
By the Fubini theorem, we compute
(18)𝔼∫0t∧ρk‍∫-τ0‍|x(s+θ)|pdμ(θ)ds=𝔼∫-τ0‍∫0t∧ρk‍|x(s+θ)|pdsdμ(θ)≤∫-τ0‍𝔼|x(s)|pds+𝔼∫0t∧ρk‍|x(s)|pds.
Similarly,
(19)𝔼∫0t∧ρk‍∫-τ0‍U(x(s+θ))μ¯(dθ)ds≤∫-τ0‍𝔼U(x(s))ds+𝔼∫0t∧ρk‍U(x(s))ds.
Substituting (18) and (19) into (17) yields
(20)𝔼V(y(t∧ρk),r(t∧ρk))≤𝔼V(y0,r(0))+λ2∫-τ0‍𝔼|x(s)|pds+λ4𝔼∫-τ0‍U(x(s))ds+ct-(λ1-λ2)𝔼×∫0t∧ρk‍|x(s)|pds-(λ3-λ4)𝔼∫0t∧ρk‍U(x(s))ds≤H0+ct,
where
(21)H0=𝔼V(y0,r(0))+λ2∫-τ0‍𝔼|x(s)|pds+λ4𝔼∫-τ0‍U(x(s))ds.
Note that
(22)𝔼V(y(t∧ρk),r(t∧ρk))≥𝔼[V(y(t∧ρk),r(t∧ρk))1{ρk≤t}]≥c1kpℙ{ρk≤t}.
Hence,(23)c1kpℙ{ρk≤t}≤H0+ct.
Then, for anyt>0,
(24)limk→∞ℙ{ρk≤t}=0,
which together with the arbitrariness of t implies that ρ∞=∞ a.s. Therefore, the solution x(t) is global, and assertion (i) follows.
Then by Itô’s formula and condition (10), we have, for any γ>0,
(25)𝔼eγ(t∧ρk)V(y(t∧ρk)),r(t∧ρk))=𝔼V(y0,r(0))+𝔼∫0t∧ρk‍eγs[γV(ys,r(s))+ℒV(xs,r(s))]ds≤𝔼V(y0,r(0))+γ𝔼∫0t∧ρk‍eγsV(ys,r(s))ds-λ1𝔼∫0t∧ρk‍eγs|x(s)|pds+λ2𝔼∫0t∧ρk‍∫-τ0‍eγs|x(s+θ)|pμ(dθ)ds+c∫0t‍eγsds-λ3𝔼∫0t∧ρk‍eγsU(x(s))ds+λ4𝔼∫0t∧ρk‍eγs∫-τ0‍U(x(s+θ))μ¯(dθ)ds.
For p≥1 and any ɛ>0, we have
(26)V(ys,r(s))≤c2|x(s)-u(xs,r(s))|p≤c2[1+ɛ1/(p-1)]p-1(|x(s)|p+1ɛ|u(xs,r(s))|p)≤c2[1+ɛ1/(p-1)]p-1[|x(s)|p+κpɛ∫-τ0‍|x(s+θ)|pdη(θ)],
where we used the Hölder inequality and condition (8). Substituting (26) into (25), we obtain
(27)eγ(t∧ρk)𝔼V(y(t∧ρk),r(t∧ρk))≤𝔼V(y0)+[c2γ(1+ɛ1/(p-1))p-1-λ1]×𝔼∫0t∧ρk‍eγsV(x(s))ds+c2γ(1+ɛ1/(p-1))p-1κpɛ×𝔼∫0t∧ρk‍∫-τ0‍eγs|x(s+θ)|pη(dθ)ds+λ2𝔼∫0t∧ρk‍∫-τ0‍eγs|x(s+θ)|pμ(dθ)ds+c∫0t‍eγsds-λ3𝔼∫0t∧ρk‍eγsU(x(s))ds+λ4𝔼∫0t∧ρk‍eγs∫-τ0‍U(x(s+θ))μ¯(dθ)ds.
Define a probability measureν on [-τ,0](28)dν(θ)=c2γ(1+ɛ1/(p-1))p-1(κp/ɛ)η(θ)+λ2μ(θ)c2γ(1+ɛ1/(p-1))p-1(κp/ɛ)+λ2;
then from (27), we have
(29)𝔼eγ(t∧ρk)V(y(t∧ρk),r(t∧ρk))≤𝔼V(y0)+c∫0t‍eγsds+[c2γ(1+ɛ1/(p-1))p-1-λ1]×𝔼∫0t∧ρk‍eγsV(x(s))ds+[c2γ(1+ɛ1/(p-1))p-1κpɛ+λ2]×𝔼∫0t∧ρk‍eγs∫-τ0‍|x(s+θ)|pν(dθ)ds-λ3𝔼∫0t∧ρk‍eγsU(x(s))ds+λ4𝔼∫0t∧ρk‍eγs∫-τ0‍U(x(s+θ))μ¯(dθ)ds.
By the Fubini theorem
(30)∫0t‍eγs∫-τ0‍|x(s+θ)|pdν(θ)ds=∫-τ0‍∫0t‍eγ(s+θ)|x(s+θ)|pdsdν(θ)≤eγτ∫-τ0‍eγs|x(s)|pds+eγτ∫0t‍eγs|x(s)|pds,
we have from (29)
(31)𝔼eγ(t∧ρk)V(y(t∧ρk),r(t∧ρk))≤𝔼V(y0)+[c2γ(1+ɛ1/(p-1))p-1κpɛ+λ2]×eγτ∫-τ0‍eγs𝔼V(x(s))ds+c∫0t‍eγsds+([c2γ(1+ɛ1/p-1)p-1κpɛ+λ2]c2γ(1+ɛ1/(p-1))p-1-λ1+[c2γ(1+ɛ1/(p-1))p-1κpɛ+λ2]eγτ)×𝔼∫0t∧ρk‍eγsV(x(s))ds+λ4eγτ∫-τ0‍eγs𝔼U(x(s))ds-[λ3-λ4eγτ]×𝔼∫0t∧ρk‍eγsU(x(s))ds.
Denote
(32)l(γ)=λ3-λ4eγτ,h(γ,ɛ)=c2γ(1+ɛ1/(p-1))p-1-λ1+[c2γ(1+ɛ1/(p-1))p-1κpɛ+λ2]eγτ.
Let ɛ be fixed; then it is easy to obtain hγ′(γ,ɛ)>0 and h(0,ɛ)=-λ1+λ2<0, which implies that for any fixed ɛ>0, function h(·,ɛ) has a unique positive root, denoted by q. Choose a ɛ=ɛ*>0 such that
(33)γ¯=supɛ>0,h(q,ɛ)=0q=suph(q,ɛ*)=0q.
Noting that for any γ∈(0,Λ], h(γ,ɛ*)≤0 and l(γ)≥0, we therefore have
(34)𝔼eγ(t∧ρk)V(y(t∧ρk),r(t∧ρk))≤𝔼V(y0)+λ4eγτ∫-τ0‍eγs𝔼U(x(s))ds+c∫0t‍eγsds+[c2γ(1+ɛ*(1/(p-1)))p-1κpɛ*+λ2]eγτ×∫-τ0‍eγs𝔼V(ξ(s))ds≤c1C0sup-τ≤θ≤0𝔼|ξ(θ)|p+ceγtγ
for some positive constant C0>1. Letting k→∞, we have
(35)eγt𝔼V(yt,r(t))≤c1C0sup-τ≤θ≤0𝔼|ξ(θ)|p+ceγtγ,
which implies the desired assertion (11). Assertion (12) can be obtained from (20) by letting k→∞. Hence assertion (ii) follows.
Letc=0. For any ϵ>0, we have that
(36)|x(s)|p≤[1+ϵ1/(p-1)]p-1|x(s)|p≤×[|ys|p+|u(xs,r(s))|pϵ]|x(s)|p≤[1+ϵ1/(p-1)]p-1|x(s)|p≤×[|ys|p+κpϵ∫-τ0‍|x(s+θ)|pdη(θ)].
By (34) and (36), we have for t>s>0(37)eγs𝔼|x(s)|p≤[1+ϵ1/(p-1)]p-1×[eγs𝔼|ys|p+κpϵeγs∫-τ0‍𝔼|x(s+θ)|pdη(θ)]≤[1+ϵ1/(p-1)]p-1×[eγs𝔼|ys|p+κpϵeγτsups-τ≤θ≤s[eγθ𝔼|x(θ)|p]]≤[1+ϵ1/(p-1)]p-1×[C0sup-τ≤θ≤0𝔼|ξ(θ)|p+κpϵeγτsup-τ≤θ≤t[eγθ𝔼|x(θ)|p].
This inequality also holds for all -τ≤s≤0. In view of γ≤r<(p/τ)log(1/κ), there exists a positive number ϵ0>0 such that
(38)a¯(ϵ0):=[1+ϵ01/(p-1)]p-1κpϵ0eγτ=[1+1ϵ01/(p-1)]p-1κpeγτ<1.
Therefore,
(39)sup-τ≤s≤teγs𝔼|x(s)|p≤[1+ϵ01/(p-1)]p-1C0sup-τ≤θ≤0𝔼|ξ(θ)|p+a¯(ϵ0)sup-τ≤s≤teγs𝔼|x(s)|p,
which implies(40)eγt𝔼|x(t)|p≤[1+ϵ01/(p-1)]p-1C01-a¯(ϵ0)sup-τ≤t≤0𝔼|ξ(t)|p.
Finally, the required inequality (14) follows by taking logarithm and limitation. Inequality (15) can be also obtained from (20) by letting k→∞. Hence assertion (iii) follows.Remark 4.
If the Markovian switching vanishes, Theorem3 is also true and gives the pth moment exponential stability with the decay rate bigger than that in [24, Theorem 2]. Since the decay rate in [24] is the special case of Theorem 3 with ɛ=1 in (13).Ifλ3=λ4=0, then we directly obtain the following corollary.Corollary 5.
Let Assumptions1 and 2 hold. Assume that there exist a function V∈C2(ℝn×𝕊;ℝ+)×𝕊, two probability measures η, μ on [-τ,0], and a number of positive constants κ∈(0,1), c1, c2, p≥1, λ1, λ2 such that for any x∈ℝn and (φ,i)∈C([-τ,0],ℝ+)×𝕊(41)|u(φ,i)|≤κ∫-τ0‍φ(θ)η(dθ),c1|x|p≤V(x,i)≤c2|x|p,ℒV(φ,i)≤-λ1|φ(0)|p+λ2∫-τ0‍|φ(θ)|pμ(dθ).
If λ1>λ2, then for any given initial data ξ∈C𝔉0b([-τ,0],ℝn), the solution of (2), denoted by x(t)=x(t;ξ), has property that
(42)limsupt→∞log(𝔼|x(t)|p)t≤-(γ¯∧r),
where γ¯ and r satisfy
(43)γ¯=max{[c2q(1+ɛ1/p-1)p-1κpɛ+λ2]eqτ=0,ɛ>0q>0;c2q(1+ɛ1/(p-1))p-1-λ1+[c2q(1+ɛ1/(p-1))p-1κpɛ+λ2]eqτ=0,ɛ>0}
and r:=(p/τ)log(1/κ)-ℓ for sufficiently small ℓ>0.Although thepth moment exponential stability and almost sure exponential stability of the exact solution do not imply each other in general, under a restrictive condition the pth moment exponential stability implies almost sure exponential stability (cf. [11]). Here, we give the following theorem about the almost sure exponential stability of the exact solution to (2).Theorem 6.
Letp≥1. Assume that there exists a constant K>0 such that
(44)|f(φ,i)|+|g(φ,i)|≤K∥φ∥,∀(φ,i)∈C([-τ,0],ℝ+)×𝕊.
Then (42) implies
(45)limsupt→∞1tlog|x(t)|≤-γ¯∧rpa.s.
In other words, the pth moment exponential stability implies almost sure exponential stability.Remark 7.
One may question that whether the semimartingale technique can be used to obtain the almost sure exponential stability directly. In fact, semimartingale technique may fail, since it may not be true to transfer the almost sure exponential stability fromx(t)-u(xt,r(t)) to x(t).


## 4. The Boundedness and Exponential Stability of HNSDDEs

In this section, we investigate the exponential stability of the hybrid NSDDE with varying delay(46)d[x(t)-N(x(t-τ(t)),r(t))]=F(x(t),x(t-τ(t)),r(t))dt+G(x(t),x(t-τ(t)),r(t))dw(t),
where τ(t):ℝ+→[0,τ] is a continuously differentiable function such that
(47)dτ(t)dt≤τ¯
for some constant τ¯<1, while
(48)N:ℝn×𝕊→ℝn,F:ℝn×ℝn×𝕊→ℝn,G:ℝn×ℝn×𝕊→ℝn×d.For (46), we impose the following assumptions.Assumption 8 (local Lipschitz condition).
F and G satisfy the local Lipschitz condition; that is, for each j>0 there exists a positive constant Cj such that
(49)|F(x,y,i)-F(x¯,y¯,i)|∨|G(x,y,i)-G(x¯,y¯,i)|≤Cj(|x-x¯|+|y-y¯|)
for all i∈𝕊 and x,y,x¯,y¯∈ℝn with |x|∨|y|∨|x¯|∨|y¯|≤j.Assumption 9 (contractive mapping).
N is a contractive mapping; that is, there exists a positive constant κ∈(0,1) such that for all x,  y∈ℝn and i∈𝕊(50)|N(x,i)-N(y,i)|≤κ|x-y|.Under the previous two assumptions, HNSDDE (46) admits a unique local solution. We also need more conditions to guarantee that the local solution is actually global. So we introduce an operator LV from ℝn×ℝn×𝕊 to ℝ by
(51)LV(x,y,i)=Vx(x-N(y,i),i)F(x,y,i)+∑j∈𝕊‍γijV(x-N(y,i),j)+12trace[GT(x,y,i)Vxx(x-N(y,i),i)G(x,y,i)]
for each V(x,i)∈C2(ℝn×𝕊;ℝ+), and we will impose the same conditions on the diffusion operator LV for the global solution and its asymptotic behavior.Although HNSDDE can be regarded as the special case of HNSFDEs, we still establish the boundedness and exponential stability criterions of the solution for (46) so as to obtain more accurate results.Theorem 10.
Let Assumptions8 and 9 hold. Assume that there are functions V∈C2(ℝn×𝕊;ℝ+), U∈C(ℝn;ℝ+) as well as a number of positive constants c, c1, c2, p≥1, λ1, λ2, λ3, λ4 such that for any x,  y∈ℝn and i∈𝕊,
(52)c1|x|p≤V(x,i)≤c2|x|p,(53)LV(x,y,i)≤c-λ1|x|p+λ2|y|p-λ3U(x)+λ4U(y).
If λ1>λ2/(1-τ¯) and λ3>λ4/(1-τ¯), then for any given initial data ξ∈C𝔉0b([-τ,0],ℝn), (46) admits a unique global solution x(t)=x(t;ξ). Moreover, we have the following assertions:(i)
the solutionx(t) obeys
(54)limsupt→∞𝔼|x(t)-N(x(t-τ(t)),r(t))|p≤cΛ,(55)limsupt→∞1t∫0t‍𝔼U(x(s))ds≤cλ3-λ4/(1-τ¯),
where Λ:=γ¯∧(1/τ)log(λ3/λ4)∧r with γ¯ and r defined by
(56)γ¯=max{[c2q(1+ɛ1/p-1)p-1κpɛ+λ2]eqτ=0,ɛ>0q>0;c2q(1+ɛ1/(p-1))p-1-λ1+[c2q(1+ɛ1/(p-1))p-1κpɛ+λ2]eqτ=0,ɛ>0}
and r:=(p/τ)log(1/κ)-ℓ for sufficiently small ℓ>0.(ii)
If, in addition,c=0, then the solution to (46) has properties that
(57)limsupt→∞log(𝔼|x(t)|p)t≤-Λ,(58)∫0∞‍𝔼U(x(s))ds≤1λ3-λ4/(1-τ¯)×[λ21-τ¯∫-τ0‍𝔼V(x(0)-N(x(-τ(0)),r(0)),r(0))+λ21-τ¯∫-τ0‍𝔼|x(s)|pds+λ41-τ¯𝔼∫-τ0‍U(x(s))ds].Proof.
The proof is similar to that of Theorem3, so we only give an outlined one. Denote yt=x(t)-N(x(t-τ(t)),r(t)). Let ρk be the stopping time defined similarly in the proof of Theorem 3. By the generalized Itô formula (see [10]) and condition (53), we can obtain that, for any k>k0 and t≥0,
(59)𝔼V(y(t∧ρk),r(t∧ρk))≤ct+𝔼V(y0,r(0))-λ1𝔼∫0t∧ρk‍|x(s)|pds+λ2𝔼∫0t∧ρk‍|x(s-τ(s))|pds-λ3𝔼∫0t∧ρk‍U(x(s))ds+λ4𝔼∫0t∧ρk‍U(x(s-τ(s)))ds.
Noting that(60)∫0t‍|x(s-τ(s))|pds≤11-τ¯∫-τ0‍|x(s)|pds+11-τ¯∫0t‍|x(s)|pds,
then we have
(61)𝔼V(y(t∧ρk),r(t∧ρk))≤ct+𝔼V(y0,r(0))+λ21-τ¯𝔼∫-τ0‍|x(s)|pds+λ41-τ¯∫-τ0‍𝔼U(x(s))ds-(λ1-λ21-τ¯)𝔼∫0t∧ρk‍|x(s)|pds-(λ3-λ41-τ¯)𝔼∫0t∧ρk‍U(x(s))ds.
Then by the similar arguments used in the proof of Theorem 3, we easily obtain ρk→∞ as k→∞; that is, the solution x(t) is global. The desired assertions (55) and (58) follow from (61) by letting k→∞.
Applying Ito^’s formula to eγtV(yt,r(t)) and using condition (53), we have for any γ∈(0,Λ](62)𝔼eγ(t∧ρk)V(yt,r(t))=𝔼V(y0,r(0))+𝔼∫0t∧ρk‍eγs[γV(ys,r(s))+𝔼∫0t∧ρk‍eγs+LV(x(s),x(s-τ(s)),r(s))]ds≤𝔼V(y0,r(0))+γ𝔼∫0t∧ρk‍eγsV(ys,r(s))ds-λ1𝔼∫0t∧ρk‍eγs|x(s)|pds+λ2𝔼∫0t∧ρk‍eγs|x(s-τ(s))|pds-λ3𝔼∫0t∧ρk‍eγsU(x(s))ds+λ4𝔼∫0t∧ρk‍eγsU(x(s-τ(s)))ds.
For p≥1 and any ɛ>0, we have
(63)V(ys,r(s))≤c2|x(s)-N(x(s-τ(s)),r(s))|p≤c2[1+ɛ1/(p-1)]p-1×(|x(s)|p+1ɛ|N(x(s-τ(s)),r(s))|p)≤c2[1+ɛ1/(p-1)]p-1×[|x(s)|p+κpɛ|x(s-τ(s))|p],
where we used the Ho¨lder inequality and Assumption 9. Substituting (63) into (62), we therefore obtain
(64)𝔼eγ(t∧ρk)V(y(t∧ρk),r(t∧ρk))≤𝔼V(y0)+[c2γ(1+ɛ1/(p-1))p-1-λ1]×𝔼∫0t∧ρk‍eγsV(x(s))ds+c2γ(1+ɛ1/(p-1))p-1κpɛ𝔼×∫0t∧ρk‍eγs|x(s-τ(s))|pds+λ2𝔼∫0t∧ρk‍eγs|x(s-τ(s))|pds-λ3𝔼∫0t∧ρk‍eγsU(x(s))ds+λ4𝔼∫0t∧ρk‍eγsU(x(s-τ(s)))ds.
Noting that
(65)∫0t‍eγs|x(s-τ(s))|pds≤eγτ∫0t‍eγ(s-τ(s))|x(s-τ(s))|pds≤eγτ1-τ¯∫-τ0‍eγs|x(s)|pds+eγτ1-τ¯∫0t‍eγs|x(s)|pds,
then we have from (64)
(66)𝔼eγ(t∧ρk)V(y(t∧ρk),r(t∧ρk))≤𝔼V(y0)+[c2γ(1+ɛ1/(p-1))p-1κpɛ+λ2]×eγτ∫-τ0‍eγs𝔼V(x(s))ds+h(γ,ɛ)𝔼∫0t∧ρk‍eγsV(x(s))ds-l(γ)𝔼∫0t∧ρk‍eγsU(x(s))ds+λ4eγτ1-τ¯∫-τ0‍eγs𝔼U(x(s))ds,
where
(67)l(γ)=λ3-λ4eγτ1-τ¯,h(γ,ɛ)=c2γ(1+ɛ1/(p-1))p-1-λ1+[c2γ(1+ɛ1/(p-1))p-1κpɛ+λ2]eγτ.
Let ɛ be fixed; then it is easy to obtain hγ′(γ,ɛ)>0 and h(0,ɛ)=-λ1+λ2<0, which implies that for any fixed ɛ>0 function h(·,ɛ) has a unique positive root denoted by q. Choose a ɛ=ɛ*>0 such that
(68)γ¯=supɛ>0,h(q,ɛ)=0q=suph(q,ɛ*)=0q.
Noting that for any γ∈(0,Λ], h(γ,ɛ*)≤0 and l(γ)≥0. We therefore have
(69)𝔼eγ(t∧ρk)V(y(t∧ρk),r(t∧ρk))≤𝔼V(y0)+λ4eγτ1-τ¯∫-τ0‍eγs𝔼|x(s)|qds+[c2γ(1+ɛ*(1/(p-1)))p-1κpɛ*+λ2]×eγτ∫-τ0‍eγs𝔼V(ξ(s))ds≤c1C0sup-τ≤θ≤0𝔼|ξ(θ)|p
for some positive constant C0>1. By the similar skills used in the proof of Theorem 3, we can easily obtain the desired assertions (54) and (57).If the delayτ(t)=τ is a fixed constant, then τ¯=0. Hybrid system (46) becomes the following HNSDDE:
(70)d[x(t)-N(x(t-τ),r(t))]=F(x(t),x(t-τ),r(t))dt+G(x(t),x(t-τ),r(t))dw(t).
Resorting to Theorem 10, we have the following corollary.Corollary 11.
Let Assumptions8 and 9 hold. Assume that there are functions V∈C2(ℝn×𝕊;ℝ+), U∈C(ℝn;ℝ+) as well as a number of positive constants c, c1, c2, p≥1, λ1, λ2, λ3, λ4 such that for any x,y∈ℝn and i∈𝕊(71)c1|x|p≤V(x,i)≤c2|x|p,LV(x,y,i)≤c-λ1|x|pLV(x,y,i)=+λ2|y|p-λ3U(x)+λ4U(y).
If λ1>λ2 and λ3>λ4, then for any given initial data ξ∈C𝔉0b([-τ,0],ℝn), (70) admits a unique global solution x(t). Moreover, we have the following assertions: (i)
the solutionx(t) obeys
(72)limsupt→∞𝔼|x(t)-N(x(t-τ),r(t))|p≤cΛ,(73)limsupt→∞1t∫0t‍𝔼U(x(s))ds≤cλ3-λ4,
where Λ:=γ¯∧(1/τ)log(λ3/λ4)∧r with γ¯ and r defined by
(74)γ¯=max{[c2q(1+ɛ1/p-1)p-1κpɛ+λ2]eqτ=0,ɛ>0q>0;c2q(1+ɛ1/(p-1))p-1-λ1======+[c2q(1+ɛ1/(p-1))p-1κpɛ+λ2]eqτ=0,ɛ>0}
and r:=(p/τ)log(1/κ)-ℓ for sufficiently small ℓ>0.(ii)
If, in addition,c=0, then the solution of (70) has properties that
(75)limsupt→∞log(𝔼|x(t)|p)t≤-Λ,∫0∞‍𝔼U(x(s))ds≤1λ3-λ4×[𝔼V(y0,r(0))+λ2∫-τ0‍𝔼|x(s)|pds×+λ4𝔼∫-τ0‍U(x(s))ds].Further, ifU(x)≡0, Corollary 11 implies.Corollary 12.
Let Assumptions8 and 9 hold. Assume that there is a function V∈C2(ℝn×𝕊;ℝ+) and a number of positive constants c1, c2, p≥1, λ1, λ2 such that for any x,y∈ℝn and i∈𝕊(76)c1|x|p≤V(x,i)≤c2|x|p,LV(x,y,i)≤-λ1|x|p+λ2|y|p.
If λ1>λ2, then for any given initial data ξ∈C𝔉0b([-τ,0],ℝn), (70) admits a unique global solution x(t). Moreover, the solution x(t) obeys
(77)limsupt→∞log(𝔼|x(t)|p)t≤-Λ,
where Λ:=γ¯∧r with γ¯ and r defined by
(78)γ¯=max{[c2q(1+ɛ1/p-1)p-1κpɛ+λ2]eqτ=0,ɛ>0q>0;c2q(1+ɛ1/(p-1))p-1-λ1======+[c2q(1+ɛ1/(p-1))p-1κpɛ+λ2]eqτ=0,ɛ>0}
and r:=(p/τ)log(1/κ)-ℓ for sufficiently small ℓ>0.Remark 13.
Corollary12 improves Theorem  5.2 in [10, Chap. 5, pp. 838]. In [10], Theorem 5.2 states that if the assumptions and conditions in Corollary 12 hold, then
(79)limsupt→∞1tlog(𝔼|x(t)|p)≤-Λ*,
where
(80)Λ*=γ¯*∧12τlog(1κ)
with γ¯*>0 being the unique root to equation
(81)γ¯*c2(1+κ)p-1+eγ¯*τ[λ2+γ¯*c2κ(1+κ)p-1]=λ1.
It is easy to see γ¯≥γ¯*. Moreover, (p/τ)log(1/κ)>(1/2τ)log(1/κ). That means Λ*≤Λ for sufficiently small ℓ>0, where Λ is defined in Corollary 12.


## 5. Examples

In this section, we give an example to illustrate the usefulness and flexibility of the theorems developed previously. Letw(t) be a scalar Brownian motion. Let r(t) be a right-continuous Markov chain value in 𝕊={1,2} with generator
(82)Γ=(γij)2×2=(-221-1).
Assume that w(t) and r(t) are independent.Example 1.
Consider the one-dimensional linear HNSDDEs(83)d[x(t)-κ(r(s))∫-τ0‍x(s+θ)dθ]=[μ(r(s))x(t)-2x(t)3+∫-τ0‍x(t+θ)dθ]dt+[σ(r(s))∫-τ0‍|x(t+θ)|2dθ+c]dw(t),
where κ(1)=1/8,  κ(2)=1/4,  μ1=-3,  μ2=-4,  σ(1)=1/8,  σ(2)=1/2, and τ=1. To find out whether (83) is mean-square exponential stability, we use the Lyapunov function
(84)V(x,i)=qi|x|2,
where q1=1 and q2=0.5. One can show that
(85)ℒV(φ,i)=2qi[φ(0)+κ(i)∫-10‍φ(θ)dθ]×[μ(i)φ(0)-3φ(0)3+∫-10‍φ(θ)dθ]+∑j=1,2‍γijqj|φ(0)+18∫-10‍φ(θ)dθ|2+2qiσ(i)2|∫-10‍|φ(θ)|2dθ|2+2c2.
By the elementary inequalities aαbβ≤(α/(α+β))aα+β+(β/(α+β))bα+β, we have
(86)ℒV(φ,1)≤2[38(34|φ(0)|4+14∫-10‍|φ(θ)|4dθ)-3|φ(0)|2-3|φ(0)|4+12(|φ(0)|2+∫-10‍|φ(θ)|2dθ)+3812(|φ(0)|2+∫-10‍|φ(θ)|2dθ)+38(34|φ(0)|4+14∫-10|φ(θ)|4dθ)]-φ(0)+18∫-10‍|φ(θ)dθ|2+14∫-10‍|φ(θ)|4dθ+2c2≤2c2-458|φ(0)|2+9764∫-10‍|φ(θ)|2dθ-8716|φ(0)|4+716∫-10‍|φ(θ)|4dθ.
Similarly,
(87)ℒV(φ,2)≤2c2-298|φ(0)|2+1332∫-10‍|φ(θ)|2dθ-3916|φ(0)|4+2316∫-10‍|φ(θ)|4dθ.
Let λ1=29/8, λ2=97/64, λ3=39/16, and λ4=23/16. Then we have from (86), and (87) for each i∈𝕊,
(88)ℒV(φ,i)≤2c2-λ1|φ(0)|2+λ2∫-10‍|φ(θ)|2dθ-λ3|φ(0)|4+λ4∫-10‍|φ(θ)|4dθ.
Then one can compute Λ=0.5281 by Theorem 3. If c≠0, then
(89)limsupt→∞𝔼|x(t)-u(xt,r(t))|2≤2c20.5281,limsupt→∞1t∫0t‍𝔼|x(s)|4ds≤2c2λ3-λ4.
If c=0, the solution to HNSDDE (83) has the property
(90)limsupt→∞log(𝔼|x(t)|2)t≤-0.5281.Example 2.
Consider the HNSDDE:(91)d[x(t)-N(x(t-τ),r(t))]=F(x(t),x(t-τ),r(t))dt+G(x(t),x(t-τ),r(t))dw(t),
where the functions N(y,1)=1/3y, N(y,2)=1/4y, F(x,y,1)=-2x-3x3+y/4, F(x,y,2)=-3x-4x3+y/2, G(x,y,1)=1/2y2, and G(x,y,2)=y/2. Let V(x,i)=qi|x|2 for q1=1,  q2=0.5. Then one can compute
(92)LV(x,y,1)≤-412x2+2324y2-214x4+16y4,LV(x,y,2)≤-2916x2+3164y2-134x4+14y4.
Then we have from (92) that for i∈𝕊(93)LV(x,y,i)≤-2916x2+2324y2-134x4+14y4.
One can compute Λ=0.2767 by Corollary 11. Then the solution to HNSDDE (91) has the property
(94)limsupt→∞log(𝔼|x(t)|2)t≤-0.2767.



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*Source: 138031-2013-08-05.xml*