基于约束区间运算的区间线性微分方程组零解的稳定性
Stability of Zero Solutions of Interval Linear Differential Equations Based on Constrained Interval Algorithm
摘要: 基于约束区间运算，本文研究了区间线性微分方程组的基本解矩阵，利用约束区间矩阵特征值，给出了区间线性微分方程组零解渐近稳定、稳定、不稳定的充要条件。具体的例子表明，区间线性微分方程零解的稳定性，不仅随着约束区间表达式中参数的变化而变化，个别问题也随着方程组维数的变化而变化。
Abstract: Based on the constrained interval algorithm, this paper studies the fundamental solution matrix of interval linear differential equations, and gives the necessary and sufficient conditions for the asymptotic stability, stability and instability of the zero solution of interval linear differential equations by using the eigenvalues of the constrained interval matrix. Specific examples show that the stability of the zero solution of interval linear differential equations changes not only with the change of the parameters in the constrained interval expression, but also with the change of the dimension of some equations.
文章引用：智照丹, 陶娟. 基于约束区间运算的区间线性微分方程组零解的稳定性[J]. 运筹与模糊学, 2022, 12(3): 1016-1028. https://doi.org/10.12677/ORF.2022.123108

参考文献

[1] Moore, R.E. (1966) Interval Analysis. Prentice-Hall, London.
[2] Moore, R.E., Kearfott, R.B. and Cloud, M.J. (2009) Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia. [Google Scholar] [CrossRef] 
[3] Lodwick, W.A. (1999) Constrained Interval Arithmetic. CCM Report 138.
[4] Lodwick, W.A. and Jenkins, O.A. (2013) Constrained Intervals and Interval Spaces. Soft Computing, 17, 1393-1402. [Google Scholar] [CrossRef] 
[5] Michel, A.N. and Oppenheimer, E.P. (1987) Application of Interval Analysis Techniques to Linear Systems. IFAC Proceedings, 20, 251-256. [Google Scholar] [CrossRef] 
[6] Michel, A.N. and Oppenheimer, E.P. (1988) Application of Interval Analysis Techniques to Linear Systems: Part I. Fundamental Results. IEEE Transactions on Circuits and Systems, 35, 1129-1138. [Google Scholar] [CrossRef] 
[7] Oppenheimer, E.P. and Michel, A.N. (1988) Application of Interval Analysis Techniques to Linear Systems: Part II. The Interval Matrix Exponential Function. IEEE Transactions on Circuits and Systems, 35, 1230-1242. [Google Scholar] [CrossRef] 
[8] Michel, A.N., Pai, M.A., Sun, H.F. and Kulig, C. (1993) Interval-Analysis Techniques in Linear Systems: An Application in Power Systems. Circuits Systems and Signal Processing, 12, 51-60. [Google Scholar] [CrossRef] 
[9] Goldsztejn, A. and Neumaier, A. (2014) On the Exponentiation of Inter-val Matrices. Reliable Computing Electronic Edition, 20, 53-72.
[10] Keyanpour, M., Tabar, M.M. and Lodwick, W.A. (2018) A Solution Algorithm for a System of Interval Linear Equations Based on the Constraint Interval Point of View. Soft Computing, 26, 26-42.
[11] Lodwick, W.A. and Jamison, K.D. (2017) A Constraint Fuzzy Interval Analysis Approach to Fuzzy Optimization. Information Sciences, 426, 38-49. [Google Scholar] [CrossRef] 
[12] Maqui-Huamán, G.G., Silva, G. and Leal, U. (2018) Necessary Op-timality Conditions for Interval Optimization Problems with Inequality Constraints Using Constrained Interval Arithmetic. Fuzzy Information Processing, 831, 439-449. [Google Scholar] [CrossRef] 
[13] Mizukoshi, M.T. and Lodwick, W.A. (2020) The Interval Eigenvalue Problem Using Constraint Interval Analysis with an Application to Linear Differential Equations. Fuzzy Sets and Systems, 419, 141-157. [Google Scholar] [CrossRef] 
[14] Mizukoshi, M.T., Jacquemard, A. and Lodwick, W.A. (2020) Classification of Hyperbolic Singularities in Interval 3-Dimensional Linear Differential Systems. Nature Public Health Emergency Collection, 1238, 13-27. [Google Scholar] [CrossRef] 
[15] Cecconello, M.S., Mizukoshi, M.T. and Lodwick, W.A. (2021) Interval Nonlinear Initial-Valued Problem Using Constraint Intervals: Theory and an Application to the Sars-Cov-2 Outbreak. Information Sciences, 577, 871-882. [Google Scholar] [CrossRef] [PubMed]
[16] Chalco-Cano, Y., Lodwick, W.A. and Bede, B. (2014) Single Level Constraint Interval Arithmetic. Fuzzy Sets and Systems, 257, 146-168. [Google Scholar] [CrossRef] 
[17] Campos, J.R., Assunção, E., Silva, G.N., Lodwick, W.A. and Leal, U. (2022) A Necessary and Sufficient Condition for the Stability of Interval Difference Equation via Interval Lyapunov Equation. Soft Computing, 26, 5043-5056. [Google Scholar] [CrossRef] 
[18] Campos, J.R., Assunção, E., Silva, G.N. and Lodwick, W.A. (2018) Biological Control of Sugarcane Caterpillar (Diatraea saccharalis) Using Interval Mathematical Models. International Journal on Mathematical Methods and Models in Biosciences, 5, Article ID: 1604232. [Google Scholar] [CrossRef] 
[19] Costa, T.M., Bouwmeester, H., Lodwick, W.A. and Lavor, C. (2017) Calculating the Possible Conformations Arising from Uncertainty in the Molecular Distance Geometry Problem Using Constraint Interval Analysis. Information Sciences, 415-416, 41-52. [Google Scholar] [CrossRef] 
[20] 马知恩, 周义仓. 常微分方程定性与稳定性方法[M]. 北京: 科学出版社, 2001.
[21] Stefanini, L. and Bede, B. (2009) Generalized Hukuhara Differentiability of Interval-Valued Functions and Interval Differential Equations. Nonlinear Analysis, 71, 1311-1328. [Google Scholar] [CrossRef] 

为你推荐