Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 1257386, 8 pages
https://doi.org/10.1155/2019/1257386

Research Article

A New Stability Analysis of Uncertain Delay Differential Equations

Xiao Wang1,2 and Yufu Ning³

¹School of Economics and Management, Beijing Institute of Petrochemical Technology, Beijing 102617, China
²Beijing Academy of Safety Engineering and Technology, Beijing 102617, China
³School of Information Engineering, Shandong Youth University of Political Science, Jinan 250103, China

Correspondence should be addressed to Xiao Wang; wangxiao19871125@163.com

Received 19 October 2018; Revised 1 December 2018; Accepted 25 December 2018; Published 6 January 2019

Academic Editor: Zhiwei Gao

Copyright © 2019 Xiao Wang and Yufu Ning. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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. Section 2 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 Appendixes A and B.

Definition 1 (Barbacioru [29]). Let be a Liu process, and and are two real-valued functions.

is called an uncertain delay differential equation, where is called time delay.

Theorem 2 (Ge and Zhu [30]). Uncertain delay differential equation (1) with initial states has a unique solution if the coefficients and satisfy

and

for some positive constant .

Definition 3 (Wang and Ning [31]). Uncertain delay differential equation (1) is said to be stable in measure if for any two solutions and with different initial states and for any , respectively, we have

for any , where is uncertain measure (see Appendix A).

Definition 4 (Liu [4]). Let , be an uncertain process and a Liu process, respectively. For any partition of the closed interval with , the mesh is written as

Then the uncertain integral of with respect to is defined by

provided that the limit exists almost surely and is finite.

Theorem 5 (Chen and Liu [5]). Supposing that is a Liu process and is an integrable uncertain process on with respect to , then

holds, where is the Lipschitz constant of .

3. Almost Sure Stability

Uncertain delay differential equation (1) is equivalent to the uncertain delay integral equation

For the sake of simplicity, we set the initial time to zero. Then, the above equation can be simplified as

Now let us present a definition of almost sure stability for uncertain delay differential equation (1).

Definition 6. Supposing that and are two solutions of uncertain delay differential equation (1) with different initial states and for any , respectively, uncertain delay differential equation (1) is said to be stable almost surely if

Example 7. Consider an uncertain delay differential equation

The analytical solution of uncertain delay differential equation (11) with two initial states and is

and

respectively.
Then

and, therefore, we have

By using Definition 6, we have that uncertain delay differential equation (11) is stable almost surely.

Example 8. Consider an uncertain delay differential equation

The analytical solution of uncertain delay differential equation (16) with two initial states and is

and

respectively.
Then

and, therefore, we have

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 and satisfy

where and

Proof. Let and denote two solutions of uncertain delay differential equation (1) with different initial states and , respectively. That is,

and

Then for any Lipschitz continuous sample , we have

and

By using formula (21) and Theorem 5, the inequality

holds, where is the Lipschitz constant of .
By using the Gronwall’s inequality [33], we have

for any .
Since

and is finite, we have that as long as , which implies that

This means that uncertain delay differential equation (1) is almost surely stable under formula (21).

Example 10. Consider an uncertain delay differential equation

Take and . Let denote a common upper bound of and with . The inequalities

and

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

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]:

where is the number of cells in cell population, is the instantaneous growth rate, and is the delayed growth rate.
If these biological systems operate in uncertain environment, the population is an uncertain process and its growth is described by the uncertain delay differential equation

where is a constant and is a Liu process. Uncertain delay differential equation (35) is called uncertain cell population growth model.
Take and . Let denote a common upper bound of and . The inequalities

and

hold. According to Theorem 2, we have that uncertain delay differential equation (35) with the initial states has a unique solution.
However, take and According to Theorem 9, we cannot judge whether the solution of uncertain delay differential equation (35) is almost surely stable or not.
From Examples 10 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 that , and are real-valued functions, then the linear uncertain delay differential equation

is almost surely stable if , and are bounded,

and

Proof. Take and . Let denote a common upper bound of and . The inequalities

and

hold.
According to Theorem 2, we have that linear uncertain delay differential equation (38) with initial states has a unique solution. Since

we take which is integrable on . Hence, we have

and

By using Theorem 9, the linear uncertain delay differential equation (38) is almost surely stable.

Example 13. Consider a linear uncertain delay differential equation (30):

The real-valued functions , , and are bounded on the interval , and

By using Corollary 12, 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 that , and are real-valued functions, then the uncertain delay differential equation

is almost surely stable if , and are bounded, and

Proof. According to Theorem 2, we have that uncertain delay differential equation (48) has a unique solution with given initial states when , and are bounded. It is supposed that and are the solutions of uncertain delay differential equation (48) with different initial states and , respectively. That is,

and

Then, we have

and

By using the Gronwall’s inequality [33], we have

for any and .
Hence, uncertain delay differential equation (48) is almost surely stable if

The above inequality is equivalent to the following inequality

Therefore,

holds, and uncertain delay differential equation (48) is almost surely stable following from Definition 6.

Example 15. Consider an uncertain delay differential equation

It follows from Theorem 2 that uncertain delay differential equation (58) has a unique solution with given initial states. In addition, the real-valued functions , , and are bounded on the interval , and

According to Theorem 14, 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 that and are two solutions of the uncertain delay differential equation (1) with different initial states and for any , respectively, according to Definition 6, we have

That is, there exists a set in with such that, for any ,

According to (61), for any and , we have as long as . It directly leads to

However,

Hence, for any , we have

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.

Appendix

A. Uncertain Variable

Definition A.1 (Liu [2, 4]). Let be a -algebra on a nonempty set . Uncertain measure is a set function from to satisfying the following axioms:

Axiom 1. .

Axiom 2. for any .

Axiom 3. For every countable sequence of , we have

Axiom 4. Let be uncertainty spaces for . The product uncertain measure is an uncertain measure satisfying

where for , respectively.

Definition A.2 (Liu [2]). An uncertain variable is a measurable function from an uncertainty space to the set of real numbers.

B. Uncertain Process

Definition B.1 (Liu [3]). Let be an uncertainty space and be a totally ordered set. An uncertain process is a function from to the set of real numbers such that

is an event for any Borel set at each .

Definition B.2 (Liu [4]). An uncertain process is called a Liu process if

(1) and almost all sample paths are Lipschitz continuous;(2) is a stationary independent increment process;(3)every increment is a normal uncertain variable with expected value 0 and variance .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was funded by National Natural Science Foundation of China (11701338) and a Project of Shandong Province Higher Educational Science and Technology Program (J17KB124).

References

K. Ito, “Stochastic integral,” in Proceedings of the Japan Academy, pp. 519–524, Tokyo, Japan, 1944.

B. Liu, Uncertainty Theory, Springer-Verlag, Berlin, Germany, 2nd edition, 2007.

B. Liu, “Fuzzy process, hybrid process and uncertain process,” Journal of Uncertain Systems, vol. 2, no. 1, pp. 3–16, 2008.

B. Liu, “Some research problems in uncertainty theory,” Journal of Uncertain Systems, vol. 1, pp. 3–10, 2009.

X. W. Chen and B. Liu, “Existence and uniqueness theorem for uncertain differential equations,” Fuzzy Optimization and Decision Making, vol. 9, no. 1, pp. 69–81, 2010.

Y. Gao, “Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition,” Journal of Uncertain Systems, vol. 6, no. 3, pp. 223–232, 2012.

R. Gao, “Milne method for solving uncertain differential equations,” Applied Mathematics and Computation, vol. 274, pp. 774–785, 2016.

X. Wang, Y. Ning, T. A. Moughal, and X. Chen, “Adams-Simpson method for solving uncertain differential equation,” Applied Mathematics and Computation, vol. 271, pp. 209–219, 2015.

K. Yao and X. W. Chen, “A numerical method for solving uncertain differential equations,” Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, vol. 25, no. 3, pp. 825–832, 2013.

Y. Zhang, J. Gao, and Z. Huang, “Hamming method for solving uncertain differential equations,” Applied Mathematics and Computation, vol. 313, pp. 331–341, 2017.

K. Yao, J. Gao, and Y. Gao, “Some stability theorems of uncertain differential equation,” Fuzzy Optimization and Decision Making, vol. 12, no. 1, pp. 3–13, 2013.

H. Liu, H. Ke, and W. Fei, “Almost sure stability for uncertain differential equation,” Fuzzy Optimization and Decision Making. A Journal of Modeling and Computation Under Uncertainty, vol. 13, no. 4, pp. 463–473, 2014.

Y. Sheng and C. Wang, “Stability in -th moment for uncertain differential equation,” Journal of Intelligent & Fuzzy Systems. Applications in Engineering and Technology, vol. 26, no. 3, pp. 1263–1271, 2014.

Y. Sheng and J. Gao, “Exponential stability of uncertain differential equation,” Soft Computing, vol. 20, no. 9, pp. 3673–3678, 2016.

X. Yang, Y. Ni, and Y. Zhang, “Stability in inverse distribution for uncertain differential equations,” Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, vol. 32, no. 3, pp. 2051–2059, 2017.

Y. Sun and T. Su, “Mean-reverting stock model with floating interest rate in uncertain environment,” Fuzzy Optimization and Decision Making. A Journal of Modeling and Computation Under Uncertainty, vol. 16, no. 2, pp. 235–255, 2017.

X. Yu, “A stock model with jumps for uncertain markets,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 20, no. 3, pp. 421–432, 2012.

X. W. Chen and J. Gao, “Uncertain term structure model of interest rate,” Soft Computing, vol. 17, no. 4, pp. 597–604, 2013.

Y. Sun, K. Yao, and Z. Fu, “Interest rate model in uncertain environment based on exponential Ornstein–Uhlenbeck equation,” Soft Computing, vol. 22, no. 2, pp. 465–475, 2018.

Y. Shen and K. Yao, “A mean-reverting currency model in an uncertain environment,” Soft Computing, vol. 20, no. 10, pp. 4131–4138, 2016.

X. Wang and Y. Ning, “An uncertain currency model with floating interest rates,” Soft Computing, vol. 21, no. 22, pp. 6739–6754, 2017.

R. Gao, “Uncertain wave equation with infinite half-boundary,” Applied Mathematics and Computation, vol. 304, pp. 28–40, 2017.

X. Yang and J. Gao, “Linear-Quadratic Uncertain Differential Game with Application to Resource Extraction Problem,” IEEE Transactions on Fuzzy Systems, vol. 24, no. 4, pp. 819–826, 2016.

Y. Zhu, “Uncertain optimal control with application to a portfolio selection model,” Cybernetics and Systems, vol. 41, no. 7, pp. 535–547, 2010.

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, Mass, USA, pp. 117–350, 1993.

M. Li and J. Wang, “Finite time stability of fractional delay differential equations,” Applied Mathematics Letters, vol. 64, pp. 170–176, 2017.

M. Z. Liu and D. Li, “Properties of analytic solution and numerical solution of multi-pantograph equation,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 853–871, 2004.

J. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667–1694, 2003.

I. C. Barbacioru, “Uncertainty functional differential equations for finance,” Surveys in Mathematics and its Applications, vol. 5, pp. 275–284, 2010.

X. Ge and Y. Zhu, “Existence and uniqueness theorem for uncertain delay differential equations,” Journal of Computational Information Systems, vol. 8, no. 20, pp. 8341–8347, 2012.

X. Wang and Y. Ning, “Stability of uncertain delay differential equations,” Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, vol. 32, no. 3, pp. 2655–2664, 2017.

L. Jia and Y. Sheng, “Stability in distribution for uncertain delay differential equation,” Applied Mathematics and Computation, vol. 343, pp. 49–56, 2019.

T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919.

E. Buckwar, “Introduction to the numerical analysis of stochastic delay differential equations,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 297–307, 2000.