# A New Stability Analysis of Uncertain Delay Differential Equations

Authors: Xiao Wang; Yufu Ning
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

This paper first provides a concept of almost sure stability for uncertain delay differential equations and analyzes this new sort of stability. In addition, this paper derives three sufficient conditions for uncertain delay differential equations being stable almost surely. Finally, the relationship between almost sure stability and stability in measure for uncertain delay differential equations is discussed.


---

## Body

## 1. Introduction

In order to deal with nondeterministic phenomenon in a dynamic system, Ito [1] proposed stochastic differential equation, which was driven by Wiener process. From then on, stochastic differential equation was employed to study dynamic systems with perturbation and applied in the fields of finance, control, and aerospace engineering. In the process of researching social system, the data used to describe the dynamic system may come from the domain experts. At this time, the expert data cannot be regard as a random variable. How to deal with these expert data in such dynamic system is an immediate problem. To tackle this problem, Liu [2] established uncertainty theory and proposed uncertain variable to describe the expert data. In addition, Liu [3] also proposed the concept of uncertain process to describe the evolution of an uncertain phenomenon. As a comparison of Wiener process, Liu process was designed by Liu [4]. Based on Liu process, uncertain calculus [4] is proposed to solve the integral and differential of an uncertain process.Driven by a Liu process, uncertain differential equation was proposed [3] to deal with dynamic systems under uncertain environment. In the aspect of theory, Chen and Liu [5] and Gao [6] proved two existence and uniqueness theorems on uncertain differential equations, respectively. Since it was difficult to obtain the analytic solutions of vast majority of uncertain differential equations, some numerical methods were proposed, such as Milne method [7], Adams-Simpson method [8], Euler method [9], and Hamming method [10]. In regard to stability analysis, stability in measure of uncertain differential equations was put forward by Liu [4] and stability in measure of linear uncertain differential equations was discussed by Yao, Gao, and Gao [11]. In addition, other types of stability were studied, such as almost sure stability [12], stability in moment [13], exponential stability [14], and stability in inverse distribution [15]. So far, some researchers employed uncertain differential equations to model the financial market. Uncertain stock model [16, 17], uncertain interest rate model [18, 19], and uncertain currency model [20, 21] have become the focus of attention for many scholars. In addition, uncertain differential equations have also been introduced into string vibration [22], differential game [23], optimal control [24], and so on.By using an uncertain differential equation, we can establish a mathematical model to describe the dynamic system under an uncertain environment, where the velocity of such dynamic system only depends on the state of the system at a given instant of time. However, the velocity of the dynamic system depends not only on the current state but also upon the previous states in some real phenomena. In such case, it is inappropriate to insist on modeling by using uncertain differential equation. An extended type called uncertain delay differential equation happens to describe the dynamic system just mentioned. Uncertain delay differential equations have been widely used in the field of engineering, especially in automatic control system; in the field of natural science, such as ecosystem, infectious diseases, and population dynamics; and in the field of social science to describe economic phenomena, commercial sales and transportation scheduling [25–28]. Back to the theoretical research on uncertain delay differential equations, Barbacioru [29] and Ge and Zhu [30] explored two existence and uniqueness theorems on uncertain delay differential equations, respectively. In the aspect of stability of uncertain delay differential equations, Wang and Ning [31] defined the stability in measure, stability in mean, and stability in moment and proved the corresponding stability theorems. In addition, Jia and Sheng [32] also discussed the stability in distribution recently.In this paper, we propose a new stability called almost sure stability for an uncertain delay differential equation and give its sufficient condition. The structure of this paper is organized as follows. Section2 introduces some basic knowledge of uncertain delay differential equations. Section 3 gives the definition of almost sure stability for an uncertain delay differential equation. Then we present three sufficient conditions for an uncertain delay differential equation and linear uncertain delay differential equations being stable almost surely in Section 4. After that, Section 5 analyzes the relationship between almost sure stability and stability in measure. The last section makes a brief conclusion.


## 2. Uncertain Delay Differential Equation

This part makes a brief introduction of uncertain delay differential equation based on uncertain variable and uncertain process, where uncertain variable and uncertain process can be seen in AppendixesA and B.Definition 1 (Barbacioru [29]).
LetCt be a Liu process, and f and g are two real-valued functions.(1)dXt=ft,Xt,Xt-τdt+gt,Xt,Xt-τdCt is called anuncertain delay differential equation, where τ>0 is calledtime delay.Theorem 2 (Ge and Zhu [30]).
Uncertain delay differential equation (1) with initial states has a unique solution if the coefficients f(t,x,y) and g(t,x,y) satisfy (2)ft,x,y+gt,x,y≤L1+x+y,∀x,y∈R,t≥0 and (3)ft,x1,y-ft,x2,y+gt,x1,y-gt,x2,y≤Lx1-x2,∀x1,x2,y∈R,t≥0 for some positive constant L.Definition 3 (Wang and Ning [31]).
Uncertain delay differential equation (1) is said to be stable in measure if for any two solutions Xt and Yt with different initial states xj and yj for any j∈[-τ,0], respectively, we have (4)limsupj∈-τ,0⁡xj-yj→0MXt-Yt<ϵ=1,∀t>0for any ϵ>0, where M is uncertain measure (see Appendix A).Definition 4 (Liu [4]).
LetXt, Ct be an uncertain process and a Liu process, respectively. For any partition of the closed interval [a,b] with a=t1<t2<⋯<tk+1=b, the mesh is written as (5)Δ=sup1≤i≤k⁡ti+1-ti. Then theuncertain integral of Xt with respect to Ct is defined by (6)∫abXtdCt=limΔ→0⁡∑i=1kXti·Cti+1-Cti provided that the limit exists almost surely and is finite.Theorem 5 (Chen and Liu [5]).
Supposing thatCt is a Liu process and Xt is an integrable uncertain process on [a,b] with respect to t, then (7)∫abXtγdCtγ≤Kγ∫abXtγdt holds, where K(γ) is the Lipschitz constant of Ct(γ).


## 3. Almost Sure Stability

Uncertain delay differential equation (1) is equivalent to the uncertain delay integral equation (8)Xt=Xt0+∫t0tfs,Xs,Xs-τds+∫t0tgs,Xs,Xs-τdCs.For the sake of simplicity, we set the initial time t0 to zero. Then, the above equation can be simplified as(9)Xt=X0+∫0tfs,Xs,Xs-τds+∫0tgs,Xs,Xs-τdCs.Now let us present a definition of almost sure stability for uncertain delay differential equation (1).Definition 6.
Supposing thatXt and Yt are two solutions of uncertain delay differential equation (1) with different initial states xj and yj for any j∈[-τ,0], respectively, uncertain delay differential equation (1) is said to be stable almost surely if (10)Mγ∈Γ∣limsupj∈-τ,0⁡xj-yj→0⁡Xtγ-Ytγ=0=1,∀t>0.Example 7.
Consider an uncertain delay differential equation(11)dXt=μXt-τdt+σdCt.
The analytical solution of uncertain delay differential equation (11) with two initial states ϕ(t) and ψ(t)(t∈[-τ,0]) is (12)Xt=ϕt,t∈-τ,0ϕ0+μ∫0tϕs-τds+σCt,t∈0,τXτ+μ∫τtXs-τds+σCt-Cτ,t∈τ,2τ⋯⋯ and (13)Yt=ψt,t∈-τ,0ψ0+μ∫0tψs-τds+σCt,t∈0,τYτ+μ∫τtYs-τds+σCt-Cτ,t∈τ,2τ⋯⋯ respectively.
Then(14)Xt-Yt=ϕt-ψt,t∈-τ,0ϕ0-ψ0+μ∫0tϕs-τ-ψs-τds,t∈0,τXτ-Yτ+μ∫τtXs-τ-Ys-τds,t∈τ,2τ⋯⋯
and, therefore, we have(15)Mγ∈Γ∣limsupw∈-τ,0⁡ϕw-ψw→0⁡Xtγ-Ytγ=0=1,∀t>0.
By using Definition6, we have that uncertain delay differential equation (11) is stable almost surely.Example 8.
Consider an uncertain delay differential equation(16)dXt=μdt+σXt-τdCt.
The analytical solution of uncertain delay differential equation (16) with two initial states ϕ(t) and ψ(t)(t∈[-τ,0]) is (17)Xt=ϕt,t∈-τ,0ϕ0+μt+σ∫0tϕs-τdCs,t∈0,τXτ+μt-τ+σ∫τtXs-τdCs,t∈τ,2τ⋯⋯ and (18)Yt=ψt,t∈-τ,0ψ0+μt+σ∫0tψs-τdCs,t∈0,τYτ+μt-τ+σ∫τtYs-τdCs,t∈τ,2τ⋯⋯ respectively.
Then(19)Xt-Yt=ϕt-ψt,t∈-τ,0ϕ0-ψ0+σ∫0tϕs-τ-ψs-τdCs,t∈0,τXτ-Yτ+σ∫τtXs-τ-Ys-τdCs,t∈τ,2τ⋯⋯
and, therefore, we have(20)Mγ∈Γ∣limsupw∈-τ,0⁡ϕw-ψw→0⁡Xtγ-Ytγ=0=1,∀t>0.
This means that uncertain delay differential equation (16) is stable almost surely by using Definition 6.


## 4. Stability Theorem

A sufficient condition for uncertain delay differential equation (1) being stable almost surely is discussed and shown by the following theorem.Theorem 9.
Supposing that uncertain delay differential equation (1) has a unique solution for each given initial state. Then uncertain delay differential equation (1) is stable almost surely if the coefficients f(t,x,y) and g(t,x,y) satisfy(21)ft,x1,y-ft,x2,y+gt,x1,y-gt,x2,y≤Ltx1-x2,∀x1,x2,y∈R,t≥0where Lt≥0 and ∫0+∞Ltdt<+∞.Proof.
LetXt and Yt denote two solutions of uncertain delay differential equation (1) with different initial states ϕ(t) and ψ(t)(t∈[-τ,0]), respectively. That is, (22)dXt=ft,Xt,Xt-τdt+gt,Xt,Xt-τdCt,t∈0,+∞Xt=ϕt,t∈-τ,0 and (23)dYt=ft,Yt,Yt-τdt+gt,Yt,Yt-τdCt,t∈0,+∞Yt=ψt,t∈-τ,0.
Then for any Lipschitz continuous sampleCt(γ), we have (24)Xtγ=X0+∫0tfs,Xsγ,Xs-τγds+∫0tgs,Xsγ,Xs-τγdCsγ,t∈0,+∞Xt=ϕt,t∈-τ,0 and (25)Ytγ=Y0+∫0tfs,Ysγ,Ys-τγds+∫0tgs,Ysγ,Ys-τγdCsγ,t∈0,+∞Yt=ψt,t∈-τ,0.
By using formula (21) and Theorem 5, the inequality(26)Xtγ-Ytγ=X0-Y0+∫0tfs,Xsγ,Xs-τγ-fs,Ysγ,Ys-τγds+∫0tgs,Xsγ,Xs-τγ-gs,Ysγ,Ys-τγdCsγ≤X0-Y0+∫0tfs,Xsγ,Xs-τγ-fs,Ysγ,Ys-τγds+∫0tgs,Xsγ,Xs-τγ-gs,Ysγ,Ys-τγdCsγ≤X0-Y0+∫0tLsXsγ-Ysγds+Kγ∫0tLsXsγ-Ysγds=X0-Y0+1+Kγ∫0tLsXsγ-Ysγds holds, where K(γ) is the Lipschitz constant of Ct(γ).
By using the Gronwall’s inequality [33], we have(27)Xtγ-Ytγ≤X0-Y0exp⁡1+Kγ∫0tLsds≤X0-Y0exp⁡1+Kγ∫0+∞Lsds≤supj∈-τ,0⁡Xj-Yjexp⁡1+Kγ∫0+∞Lsds for any t>0.
Since(28)∫0+∞Ltdt<+∞ and K(γ) is finite, we have that Xt(γ)-Yt(γ)→0 as long as supj∈[-τ,0]⁡Xj-Yj→0, which implies that (29)Mlimsupj∈-τ,0⁡Xj-Yj→0⁡Xt-Yt=0=1,∀t>0.
This means that uncertain delay differential equation (1) is almost surely stable under formula (21).Example 10.
Consider an uncertain delay differential equation(30)dXt=exp⁡-tXt+μXt-τdt+σdCt.
Takef(t,x,y)=exp(-t)x+μy and g(t,x,y)=σ. Let N denote a common upper bound of |μ|,|σ| and |exp(-t)| with t>0. The inequalities (31)ft,x,y+gt,x,y≤N1+x+y,∀x,y∈R,t≥0, and(32)ft,x1,y-ft,x2,y+gt,x1,y-gt,x2,y≤exp⁡-tx1-x2,∀x1,x2,y∈R,t≥0 hold. According to Theorem 2, we obtain that uncertain delay differential equation (30) with initial states has a unique solution. In addition, by using inequality (32) and(33)∫0+∞exp⁡-tdt<+∞, uncertain delay differential equation (30) is stable almost surely by Theorem 9.Example 11 (uncertain cell population growth model).
The initial cell population growth model was provided by the following equation [34]: (34)dNt=ρ0Nt+ρ1Nt-τdt,t≥0 where Nt is the number of cells in cell population, ρ0>0 is the instantaneous growth rate, and ρ1 is the delayed growth rate.
If these biological systems operate in uncertain environment, the populationNt is an uncertain process and its growth is described by the uncertain delay differential equation(35)dNt=ρ0Nt+ρ1Nt-τdt+σdCt,t≥0 where σ is a constant and Ct is a Liu process. Uncertain delay differential equation (35) is called uncertain cell population growth model.
Takef(t,x,y)=ρ0x+ρ1y and g(t,x,y)=σ. Let M denote a common upper bound of ρ0,|ρ1| and |σ|. The inequalities (36)ft,x,y+gt,x,y≤M1+x+y,∀x,y∈R,t≥0, and (37)ft,x1,y-ft,x2,y+gt,x1,y-gt,x2,y≤ρ0x1-x2,∀x1,x2,y∈R,t≥0 hold. According to Theorem 2, we have that uncertain delay differential equation (35) with the initial states has a unique solution.
However, takeLt=ρ0 and ∫0+∞Ltdt=+∞. According to Theorem 9, we cannot judge whether the solution of uncertain delay differential equation (35) is almost surely stable or not.
From Examples10 and 11, we can see that Theorem 9 only gives the sufficient condition but not the necessary and sufficient condition for uncertain delay differential equation being almost surely stable. On the basis of Theorem 9, we immediately present a corollary about a sufficient condition for a linear uncertain delay differential equation being stable almost surely.Corollary 12.
Supposing thatuit,vit, and wit(i=1,2) are real-valued functions, then the linear uncertain delay differential equation(38)dXt=u1tXt+v1tXt-τ+w1tdt+u2tXt+v2tXt-τ+w2tdCt is almost surely stable if uit,vit, and wit(i=1,2) are bounded, (39)∫0+∞u1tdt<+∞ and (40)∫0+∞u2tdt<+∞.Proof.
Takef(t,x,y)=u1tx+v1ty+w1t and g(t,x,y)=u2tx+v2ty+w2t. Let N denote a common upper bound of |uit|,|vit| and |wit|(i=1,2). The inequalities (41)ft,x,y+gt,x,y≤2N1+x+y,∀x,y∈R,t≥0, and(42)ft,x1,y-ft,x2,y+gt,x1,y-gt,x2,y=u1tx1-x2+u2tx1-x2≤2Mx1-x2,∀x1,x2,y∈R,t≥0 hold.
According to Theorem2, we have that linear uncertain delay differential equation (38) with initial states has a unique solution. Since(43)ft,x1,y-ft,x2,y+gt,x1,y-gt,x2,y=u1tx1-x2+u2tx1-x2≤u1t+u2tx1-x2,∀x1,x2,y∈R,t≥0, we take Lt=|u1t|+|u2t| which is integrable on [0,+∞). Hence, we have (44)∫0+∞u1tdt<+∞ and (45)∫0+∞u2tdt<+∞.
By using Theorem9, the linear uncertain delay differential equation (38) is almost surely stable.Example 13.
Consider a linear uncertain delay differential equation (30):(46)dXt=exp⁡-tXt+μXt-τdt+σdCt.
The real-valued functionsexp(-t), μ, and σ are bounded on the interval [0,+∞), and(47)∫0+∞exp⁡-tdt=1<+∞.
By using Corollary12, we also have that uncertain delay differential equation (30) is stable almost surely.
Up to now, we have discussed the almost sure stability of linear uncertain delay differential equation (38). In what follows, let us consider another type of uncertain delay differential equation.Theorem 14.
Supposing thatut,vt, and wt are real-valued functions, then the uncertain delay differential equation(48)dXt=utXt-τ+vtdt+wtdCt is almost surely stable if ut,vt, and wt are bounded, and (49)∫0+∞utdt<+∞.Proof.
According to Theorem2, we have that uncertain delay differential equation (48) has a unique solution with given initial states when ut,vt, and wt are bounded. It is supposed that Xt and Yt are the solutions of uncertain delay differential equation (48) with different initial states ϕ(t) and ψ(t)(t∈[-τ,0]), respectively. That is, (50)dXt=utXt-τ+vtdt+wtdCt,t∈0,+∞Xt=ϕt,t∈-τ,0 and (51)dYt=utYt-τ+vtdt+wtdCt,t∈0,+∞Yt=ψt,t∈-τ,0. Then, we have (52)dXt-Yt=utXt-τ-Yt-τdt, and(53)Xt-Yt=X0-Y0+∫0tusXs-τ-Ys-τds=X0-Y0+∫-τt-τup+τXp-Ypdp≤X0-Y0+∫-τt-τup+τXp-Ypdp≤X0-Y0+∫-τt-τup+τXp-Ypdp≤X0-Y0+∫-τtup+τXp-Ypdp.
By using the Gronwall’s inequality [33], we have(54)Xtγ-Ytγ≤X0-Y0exp⁡∫-τtup+τdp≤X0-Y0exp⁡∫-τ+∞up+τdp=X0-Y0exp⁡∫0+∞usds≤supj∈-τ,0⁡Xj-Yjexp⁡∫0+∞usds for any t>0 and γ∈Γ.
Hence, uncertain delay differential equation (48) is almost surely stable if (55)exp∫0+∞usds<+∞. The above inequality is equivalent to the following inequality (56)∫0+∞usds<+∞.
Therefore,(57)Mlimsupj∈-τ,0⁡Xj-Yj→0⁡Xt-Yt=0=1,∀t>0, holds, and uncertain delay differential equation (48) is almost surely stable following from Definition 6.Example 15.
Consider an uncertain delay differential equation(58)dXt=exp⁡-tXt-τ+μdt+σdCt.
It follows from Theorem2 that uncertain delay differential equation (58) has a unique solution with given initial states. In addition, the real-valued functions exp(-t), μ, and σ are bounded on the interval [0,+∞), and (59)∫0+∞exp⁡-tdt=1<+∞.
According to Theorem14, uncertain delay differential equation (58) is stable almost surely.


## 5. Comparison

The relationship between almost sure stability and stability in measure for uncertain delay differential equation (1) is shown as below.Theorem 16.
If uncertain delay differential equation (1) is almost surely stable, uncertain delay differential equation (1) is stable in measure.Proof.
Supposing thatXt and Yt are two solutions of the uncertain delay differential equation (1) with different initial states xj and yj for any j∈[-τ,0], respectively, according to Definition 6, we have (60)Mγ∈Γ∣limsupj∈-τ,0⁡xj-yj→0⁡Xtγ-Ytγ=0=1,∀t>0.
That is, there exists a setΓ0 in Γ with M{Γ0}=1 such that, for any γ∈Γ0,(61)limsupj∈-τ,0⁡xj-yj→0⁡Xtγ-Ytγ=0.According to (61), for any ϵ>0 and γ∈Γ0, we have |Xt(γ)-Yt(γ)|≤ϵ as long as supj∈-τ,0⁡xj-yj→0. It directly leads to (62)limsupj∈-τ,0⁡xj-yj→0⁡Mγ∈Γ∣Xtγ-Ytγ≤ϵ≥1.
However,(63)limsupj∈-τ,0⁡xj-yj→0⁡Mγ∈Γ∣Xtγ-Ytγ≤ϵ≤1.
Hence, for anyϵ>0, we have (64)limsupj∈-τ,0⁡xj-yj→0⁡MXt-Yt≤ϵ=1,∀t>0. Thus almost sure stability implies the stability in measure.


## 6. Conclusion

In this paper, we proposed a concept of almost sure stability of analytical solutions of uncertain delay differential equations. Meanwhile, we provided three sufficient conditions for uncertain delay differential equations being stable almost surely. At last, we analyzed the relationship between almost sure stability and stability in measure and found that almost sure stability could imply the stability in measure for uncertain delay differential equations. In future, we will focus on the application of uncertain delay differential equations in the area of biological systems and finance.



---

*Source: 1257386-2019-01-06.xml*