【作者】 白洁；

【导师】 范猛；

【作者基本信息】 东北师范大学 ， 应用数学， 2006， 硕士

【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。更多还原

【Abstract】 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.更多还原