Browse > Article
http://dx.doi.org/10.7232/iems.2013.12.1.002

Stability Analysis of Linear Uncertain Differential Equations  

Chen, Xiaowei (Department of Risk Management and Insurance, Nankai University)
Gao, Jinwu (School of Information, Renmin University of China)
Publication Information
Industrial Engineering and Management Systems / v.12, no.1, 2013 , pp. 2-8 More about this Journal
Abstract
Uncertainty theory is a branch of mathematics based on normolity, duality, subadditivity and product axioms. Uncertain process is a sequence of uncertain variables indexed by time. Canonical Liu process is an uncertain process with stationary and independent increments. And the increments follow normal uncertainty distributions. Uncertain differential equation is a type of differential equation driven by the canonical Liu process. Stability analysis on uncertain differential equation is to investigate the qualitative properties, which is significant both in theory and application for uncertain differential equations. This paper aims to study stability properties of linear uncertain differential equations. First, the stability concepts are introduced. And then, several sufficient and necessary conditions of stability for linear uncertain differential equations are proposed. Besides, some examples are discussed.
Keywords
Uncertain Process; Uncertain Differential Equation; Stability Analysis;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Gao, Y. (2012), Existence and uniqueness theorem on uncertain differential equations with local lipschitz condition, Journal of Uncertain Systems, 6(3), 223-232.
2 Ito, K. (1951), On stochastic differential equations, Memoirs of the American Mathematical Society, 4, 1-51.
3 Chen, X. and Liu, B. (2010), Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, 9(1), 69-81.   DOI   ScienceOn
4 Chen, X. (2011), American option pricing formula for uncertain financial market, International Journal of Operations Research, 8(2), 32-37.
5 Kushner, H. J. (1967), Stochastic Stability and Control, Academic Press; New York, NY.
6 Kats, I. and Krasovskii, N. (1960), On the stability of systems with random parameters, Journal of Applied Mathematics and Mechanics, 24(5), 1225-1246.   DOI   ScienceOn
7 Khas'minskii, R. Z. (1962), On the stability of the trajectory of markov processes, Journal of Applied Mathematics and Mechanics, 26(6), 1554-1565.   DOI   ScienceOn
8 Khas'minskii, R. Z. (1980), Stochastic Stability of Differential Equations, Kluwer Academic Publishers, Dordrecht.
9 Kunita, H. and Watanabe, S. (1967), On square integrable martingales, Nagoya Mathematical Journal, 30, 209-245.   DOI
10 Liu, B. (2007), Uncertainty Theory (2nd ed.), Springer- Verlag, Berlin.
11 Liu, B. (2008), Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2(1), 3-16.
12 Liu, B. (2009), Some research problems in uncertainty theory, Journal of Uncertain Systems, 3(1), 3-10.
13 Liu, B. (2010), Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer- Verlag, Berlin.
14 Liu, Y. (2012), An analytic method for solving uncertain differential equations, Journal of Uncertain Systems, 6(4), 244-249.
15 Meyer, P. A. (1970), Seminaire de ProbabilitesIV Universite de Strasbourg, Springer-Verlin, Heidelberg.
16 Peng, J. and Yao, K. (2011), A new option pricing model for stocks in uncertainty markets, International Journal of Operations Research, 8(2), 18-26.
17 Zhu, Y. (2010), Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41(7), 535-547.   DOI   ScienceOn