Lyapunov theory and perturbation of stable and asymptotically stable systems
Lyapunov theory and perturbation of stable and asymptotically stable systems
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DOI：
10.1016/0022-0396(74)90082-5
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发表时间：
1974-03
影响因子：
2.4
通讯作者：
S. Chow;J. Yorke
中科院分区：
文献类型：
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作者：
S. Chow;J. Yorke
本文研究了向量常微分方程 *= f（t，x）的同零函数是其解，即f（t，0）= 0对所有时间t.我们用0简单地表示这个特殊的解。现在假设我们知道（E）的所有从0开始的解在未来的所有时间内都保持在0附近，或者甚至随着时间的增加它们接近0。如果微分方程（E）受到某些小扰动，上述关于0附近解的性质可能仍然成立，也可能不成立。这个问题的一个更精确的表述如下：如果0对于（E）是渐近稳定的，并且如果函数p（t）在某种意义下很小，给出f的条件，使得0对于扰动方程2 = f（t，x）+ p（t）是（最终）渐近稳定的。(P)In特别地，已知一个例子[1，定理C]，其中p（= Eexp（-t2））比指数更快地趋于0，并且f是实值的、一致连续的和局部Lipschitz函数，并且（E）的所有解随着t+ co而指数地和单调地接近零;然而（P）的许多解开始于附近
In this paper WC study the vector ordinary differential equation*= f (t, x) for which the identically zero function is a solution, ie, f (t, 0)= 0 for all time t. We denote this special solution simply by 0. Now suppose one knows that all the solutions of (E) which start near 0 remain near 0 for all future time, or even that they approach 0 as time increases. If the differential equation (E) is subjected to certain small perturbations, the above property concerning the solutions near 0 may or may not remain true. A more precise formulation of this problem is as follows: if 0 is asymptotically stable for (E), and if the functionp (t) is small in some sense, give conditions onf so that 0 is (eventually) asymptotically stable for the perturbed equation2= f (t, x)+ p (t).(P)In particular, an example is known [1, Theorem C] in which p (= E exp (-t2)) tends to 0 faster than exponentials and f is a real-valued, uniformly continuous, and locally Lipschitz function, and all solutions of (E) approach zero exponentially and monotonically as t+ co; yet many solutions of (P) starting near