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Abstract

Backward uncertain differential equation is a specical type of differential equation driven by a Liu process. There are some concepts of stability such as stability in measure, stability in mean, stability in p-th moment, stability moment exponential and almost sure stability of backward uncertain differential equations have been proposed. As a supplement, this paper gives a concept of stability in uncertain distribution of backward uncertain differential equation. Some sufficient conditions for a backward uncertain differential equation being stable in uncertain distribution are provided. In addition, this paper further discusses their relationships among stability in uncertain distribution, stability in uncertain measure, stability in mean and stability in p-th moment. Last, this paper discusses some examples to illustrate the theoretical considerations.

Keywords

Uncertain distribution uncertain process backward uncertain differential equation stability in distribution

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References

Chen and

Liu , Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim DecisMaking 9(1) (2010), 69–81.

Chen and

Ralescu , Liu process and uncertain calculus, J Uncertain Anal Appl 1 (2013), Article 2.

Gao , Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition, J Uncertain Syst 6 (3) (2012), 223–232.

Ge and

Zhu , A necessary condition of optimality for uncertain optimal control problem, Fuzzy Optim Decis Making 12 (1) (2013), 41–51.

Liu , Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007.

Liu , Fuzzy process, hybrid process and uncertain process, J Uncertain Syst 2 (1) (2008), 3–16.

Liu , Some research problems in uncertainty theory, J Uncertain Syst 3 (1) (2009), 3–10.

Liu , Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.

Liu and

Yao , Uncertain integral with respect to multiple canonical processes, J Uncertain Syst 6 (4) (2012), 250–255.

10.

Liu ,

Ke and

Fei , Almost sure stability for uncertain differential equation, Fuzzy Optim Decis Making 13(4) (2014), 463–473.

11.

Sheng and

Wang , Stability in p-th moment for uncertain differential equation, J Int Fuzz Syst 26 (3) (2014), 1263–1271.

12.

Sheng and

Gao , Exponential stability for uncertain differential equation, SoftComput 20 (9) (2016), 3673–3678.

13.

Wang and

Ning , Stability analysis of backward uncertain differential equations, Technical Report.

14.

Wang and

Ning , Almost sure and pth moment exponential stability of backward uncertain differential equations, J Int Fuzz Syst 33 (2017), 1413–1422.

15.

Yao and

Chen , A numerical method for solving uncertain differential equations, J Int Fuzz Syst 25 (3) (2013), 825–832.

16.

Yao ,

Gao and

Gao , Some stability theorems of uncertain differential equation, Fuzzy Optim Decis Making 12 (1) (2013), 3–13.

17.

Yao ,

Ke and

Sheng , Stability in mean for uncertain differential equation, Fuzzy Optim Decis Making 14(3) (2015), 365–379.

18.

Yang ,

Ni and

Zhang , Stability in inverse distribution for uncertain differential equations, J Int Fuzzz Syst 32(3) (2017), 2051–2059.