时标动力学方程的稳定性
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摘要
本文主要研究非自治时标动力学方程x~△=f(t,x),(t,x)∈T×R~n的平凡解稳定、一致稳定、渐近稳定,与不稳定的充分必要条件。本文首先介绍时标的基础知识、稳定性的概念和几个引理。在定理的证明中充分性主要利用K类函数严格单调递增和Lyapunov函数的连续性;必要性的证明中,分别构造Lyapunov函数,满足定理的条件,使得定理的结论成立。最后举例说明,利用定理构造适当的Lyapunov函数,来判断非自治时标动力学方程x~△=f(t,x)的稳定性。另一方面,已知方程解的稳定性,构造适当的Lyapunov函数,使得相应的结论成立。
We study the sufficient and necessary conditions of stability,asymptotical stabil-ity,uniform stability and instability of the equilibrium solution x = 0 to dynamic equation on time scale x~△ = f(t,x), (t,x) ∈ T x R~n.This paper firstly introduce the based knowledge of Time Scale,stability and several lemmas.In the proof,sufficient conditions are proved mainly by properties of K function and continuity of Lyapunov function:in necessary proof,we separately structure Lyapunov functions which satisfy the conditions of the theorems, such that the results establish.Finally we give an example to utilize the theorems to ensure stability of the equilibrium solution x = 0 to dynamic equation on time scale x~△ = f(t, x), (t, i) ∈ T ×R~n. On the other hand, we have known the stability of the equilibrium solution,and structure appropriate Lyapunov function V, such that relevant results establish.
We study the sufficient and necessary conditions of stability,asymptotical stabil-ity,uniform stability and instability of the equilibrium solution x = 0 to dynamic equation on time scale x~△ = f(t,x), (t,x) ∈ T x R~n.This paper firstly introduce the based knowledge of Time Scale,stability and several lemmas.In the proof,sufficient conditions are proved mainly by properties of K function and continuity of Lyapunov function:in necessary proof,we separately structure Lyapunov functions which satisfy the conditions of the theorems, such that the results establish.Finally we give an example to utilize the theorems to ensure stability of the equilibrium solution x = 0 to dynamic equation on time scale x~△ = f(t, x), (t, i) ∈ T ×R~n. On the other hand, we have known the stability of the equilibrium solution,and structure appropriate Lyapunov function V, such that relevant results establish.
引文
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[5] Jeffrey J. Dacunha, Stability for Time Varying Linear Dynamic Systems on Time Scales[J]. Journal of Computational and Applied Mathematics, 2005, 176(2): 381-410.
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[15] 沃松林,邹云.离散广义大系统的Lyapunov稳定性分析[J].控制理论与应用,2004,21(2):291-294.
[16] 郭晓丽,方建印.Lyapunov稳定性逆定理的另一种证明[J].郑州大学学报(理学版),2004,36(2):22-24.
[17] 蹇继贵,罗海庚,廖晓昕.非线性控制大系统的部分镇定[J].华中科技大学学报(自然科学版),2005,33(7):34-37.
[18] T. Gard and J. Hoffacker, Asymptotic Behavior of Natural Growth on Time Scales[J]. Dynamic Systems and Applications 12(1-2), 2003.
[19] J. Hoffacker and C.C.Tisdell, Stability and Instability for Dynamic Systems on the Time Scales[J], Computers and Mathematics with Applications49, 2005, 1327-1334.
[20] A. M. Lyapunov, The General Problem of the Stability of Motion. Internat.J.Control[J], 1992(55): 521-790.
[21] B. Kaymakcalan, Lyapunov stability theary for dynamic systems on time scales[J], J.Appl.Math.Stochastic Anal. 1992, 5(3): 275-281.
[22] A.S. Vatsala, Strict stability criteria for dynamic systems on time scales[J], J.Differ. Equations Appl. 1998, 3(3-4): 267-276.
[23] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems[M], Translated from the 1985 Dutch original, Second Edition, Universitext, Springer-Verlag, Berlin, 1996.
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[25] Paulo Ferreira da Silva Porto Jinior, Givberto Franciso Loibel. Relative finite determinacy and relative stability of function-germs[J]. BOL SOC BRAS MAT, 1978,9(2):1-18.
[26] vPaulo Ferreira da Silva Porto Junior. On relative stability of function-germs[J]. BLO SOC BRAS MAT, 1983, 14(2): 99-108.
[27] 秦华峰,丁博,谭军伟:构造Lyapunov函数判定常微分方程的稳定性[J].重庆工学院学报,2005,19(3):139-141.
[28] 王联,王慕秋.常差分方程[M].新疆大学出版社,1991.
[29] 田秀恭。部分稳定的李雅普诺夫函数与大系统的部分稳定性[J].应用数学学报,1988,11(2):238-248.
[30] 张毅,王慕秋.离散大系统关于部分变元的关联稳定性[J].应用数学学报,1989,12(4):430-439.
[31] 陈明晖,邓明立.常微分方程定性理论与稳定性理论的哲学思考[J].自然科学史研究,2005,24(1):45-52.
[32] 陈雪如,杨成梧.2-D奇异线性离散系统渐进稳定性的一类Lyapunov方法[J].控制与决策,2001,16(4):497-499.
[33] 张建华,王旬.利用Lyapunov函数判别连续广义系统的渐进稳定性[J].锦州师范学院学报,2001,22(1):55-57.
[34] 李连忠.二阶差分系统Lyapunov函数的构造[J].泰安师专学报,1999,21(6):1-5.
[35] 张庆灵。樊治平.广义系统的Lyapunov方程[J].东北大学学报(自然科学版),1997,18(1):21-25。
[36] 陈菊芳。杨霞.生态系统中Lyapunov函数的构造[J].生物数学学报,1996,11(3):105-110.
[37] 徐道义。关于稳定性的几个定理[J].数学季刊,1992,7(2):61-67.
[38] 阿.阿.玛尔德纽克,孙振绮.实用稳定性及应用[M].北京:科学出版社,2004.
[39] 蹇继贵,廖晓昕.非线性非自治系统零解的稳定性及部分稳定性研究[J].数学杂志,2005,25(6):641-644.
[40] 徐道义.稳定性理论中几个基本定理的推广[J].应用数学,1992,5(2):76-79.
[2] S.Keller.Asymptotisches Verhalten invarianter Faserbundel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen.PhD thesis,Universitat Augsburg,1999.
[3] C.P Aotzsche, S. Siegmund, and F.Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales[J], Discrete Contin. Dyn. Syst.2003(9), 1223-1241.
[4] T. Gard and J. Hoacker, Asymptotic behavior of natural growth on time scales[J], Dynam. Systems Appl. 2003(12), 131-147.
[5] Jeffrey J. Dacunha, Stability for Time Varying Linear Dynamic Systems on Time Scales[J]. Journal of Computational and Applied Mathematics, 2005, 176(2): 381-410.
[6] S.Sivasundaram, Stability of dynamic systems on the time scales[J], Nonlinear Dyn.Syst. 2002, Theary2(2): 185-202.
[7] M. Bohner and A. Peterson, Dynamic Equations on Time Scales[M]. An Introduction with Applications, Birkhauser, Boston, MA, 2001.
[8] 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,2001,41-92.
[9] 廖晓昕.稳定性的数学理论及应用(第二版)[M].武汉:华中师范大学出版社,2001,81-101.
[10] 阮炯.差分方程和常微分方程[M].上海:复旦大学出版社,2002。
[11] 李君湘.离散系统部分稳定Lyapunov函数与大系统部分稳定性[J].天津大学学报,1994,27(2):211-217.
[12] 李敬群,陆启韶.动力系统的Lyapunov稳定性和吸引性[J].北京航空航天大学学报,1996,22(5):618-622.
[13] 张庆灵,戴冠中,徐心和,等.离散广义系统稳定性分析与控制的Lyapunov方法[J).自动化学报,1998,24(5):622-629.
[14] 廖晓昕.动力系统的稳定性理论和应用[M].北京:国防工业出版社,2001.
[15] 沃松林,邹云.离散广义大系统的Lyapunov稳定性分析[J].控制理论与应用,2004,21(2):291-294.
[16] 郭晓丽,方建印.Lyapunov稳定性逆定理的另一种证明[J].郑州大学学报(理学版),2004,36(2):22-24.
[17] 蹇继贵,罗海庚,廖晓昕.非线性控制大系统的部分镇定[J].华中科技大学学报(自然科学版),2005,33(7):34-37.
[18] T. Gard and J. Hoffacker, Asymptotic Behavior of Natural Growth on Time Scales[J]. Dynamic Systems and Applications 12(1-2), 2003.
[19] J. Hoffacker and C.C.Tisdell, Stability and Instability for Dynamic Systems on the Time Scales[J], Computers and Mathematics with Applications49, 2005, 1327-1334.
[20] A. M. Lyapunov, The General Problem of the Stability of Motion. Internat.J.Control[J], 1992(55): 521-790.
[21] B. Kaymakcalan, Lyapunov stability theary for dynamic systems on time scales[J], J.Appl.Math.Stochastic Anal. 1992, 5(3): 275-281.
[22] A.S. Vatsala, Strict stability criteria for dynamic systems on time scales[J], J.Differ. Equations Appl. 1998, 3(3-4): 267-276.
[23] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems[M], Translated from the 1985 Dutch original, Second Edition, Universitext, Springer-Verlag, Berlin, 1996.
[24] A. C. Peterson and C.C. Tisdell, Boundedness and uniqueness of solutions to dynamic equations on time scales[J], J. Difference Equ. Appl. 2004, 10(13-15): 1295-1306.
[25] Paulo Ferreira da Silva Porto Jinior, Givberto Franciso Loibel. Relative finite determinacy and relative stability of function-germs[J]. BOL SOC BRAS MAT, 1978,9(2):1-18.
[26] vPaulo Ferreira da Silva Porto Junior. On relative stability of function-germs[J]. BLO SOC BRAS MAT, 1983, 14(2): 99-108.
[27] 秦华峰,丁博,谭军伟:构造Lyapunov函数判定常微分方程的稳定性[J].重庆工学院学报,2005,19(3):139-141.
[28] 王联,王慕秋.常差分方程[M].新疆大学出版社,1991.
[29] 田秀恭。部分稳定的李雅普诺夫函数与大系统的部分稳定性[J].应用数学学报,1988,11(2):238-248.
[30] 张毅,王慕秋.离散大系统关于部分变元的关联稳定性[J].应用数学学报,1989,12(4):430-439.
[31] 陈明晖,邓明立.常微分方程定性理论与稳定性理论的哲学思考[J].自然科学史研究,2005,24(1):45-52.
[32] 陈雪如,杨成梧.2-D奇异线性离散系统渐进稳定性的一类Lyapunov方法[J].控制与决策,2001,16(4):497-499.
[33] 张建华,王旬.利用Lyapunov函数判别连续广义系统的渐进稳定性[J].锦州师范学院学报,2001,22(1):55-57.
[34] 李连忠.二阶差分系统Lyapunov函数的构造[J].泰安师专学报,1999,21(6):1-5.
[35] 张庆灵。樊治平.广义系统的Lyapunov方程[J].东北大学学报(自然科学版),1997,18(1):21-25。
[36] 陈菊芳。杨霞.生态系统中Lyapunov函数的构造[J].生物数学学报,1996,11(3):105-110.
[37] 徐道义。关于稳定性的几个定理[J].数学季刊,1992,7(2):61-67.
[38] 阿.阿.玛尔德纽克,孙振绮.实用稳定性及应用[M].北京:科学出版社,2004.
[39] 蹇继贵,廖晓昕.非线性非自治系统零解的稳定性及部分稳定性研究[J].数学杂志,2005,25(6):641-644.
[40] 徐道义.稳定性理论中几个基本定理的推广[J].应用数学,1992,5(2):76-79.