On global asymptotic stability of solutions of differential equations.
On global asymptotic stability of solutions of differential equations.
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DOI：
10.1090/s0002-9947-1962-0145152-7
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发表时间：
1962
影响因子：
1.3
通讯作者：
P. Hartman;C. Olech
中科院分区：
文献类型：
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作者：
P. Hartman;C. Olech
（1.3） H(x)是负定的（对于固定的x # 0），则x=0是（1.1）的全局渐近稳定解；也就是说，对于大t和x(t)->0, x=x(t)或（1.1）的每个解都存在。在处理n =2的情况下，[4]的结果如下：如果（1.4）x=0是（1.1）的局部渐近稳定解，（1.3）被（1.5）tr J(x) = tr H(x) 0替换为lxI > const。> 0，则x =0是（1.1）的全局渐近稳定解。本文第一部分的主要结果之一是将后一个定理推广到任意n_2的情况。在这种情况下，H(x)的轨迹将被函数（1.7）a(x) = max(Xi（x) + Xj(x)）代替，其中Xj(x), * * *, Xn(x)是H(x)的特征值，条件（1.5）by (1.8) a(x) _ 0，
and (1.3) H(x) is negative definite (for fixed x # 0), then x=0 is a globally asymptotically stable solution of (1.1); i.e., every solution x=x(t) or (1.1) exists for large t and x(t)->0 as too. Among the results of [4], which deals with the case n =2, is the following: if (1.4) x=0 is a locally asymptotically stable solution of (1.1) and (1.3) is replaced by the conditions (1.5) tr J(x) = tr H(x) 0 for lxI > const. > 0, then again x =0 is a globally asymptotically stable solution of (1.1). One of the main results of the first part of this paper will be a generalization of the latter theorem to the case of arbitrary n _ 2. In this situation, the trace of H(x) will be replaced by the function (1.7) a(x) = max(Xi(x) + Xj(x)) for 1 < i < j < n, where Xj(x), * * *, Xn(x) are the eigenvalues of H(x), the condition (1.5) by (1.8) a (x) _ 0,