DETERMINATION OF STABILITY BY LINEAR APPROXIMATION OF A PERIODIC SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDES
DETERMINATION OF STABILITY BY LINEAR APPROXIMATION OF A PERIODIC SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDES
复制标题
右侧不连续微分方程组周期解的线性逼近确定稳定性
DOI：
10.1093/qjmam/11.4.385
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发表时间：
1958
影响因子：
0.9
通讯作者：
F. Gantmacher
中科院分区：
文献类型：
--
作者：
M. Aizerman;F. Gantmacher
Liapounoff（1）研究了微分方程组dzidt=fi（z1，...，zn，t）[fi]的周期解zi= z ∈ i（t）（周期为τ）的稳定性问题 （z1，... znt +τ）π fi（z1，. zn，t）]（0.1）在本金中减少线性逼近系统<$dxidt=∑k（<$fi <$zk）z=z <$（t）的零解xi= 0的稳定性问题的（非临界）情形xk当方程的右侧（0.1）是解析函数，或在积分曲线邻域内连续且关于z1...，zn可微的函数，本文的目的是分析系统周期解zi=z <$i（t）[z <$i（t+τ）<$z <$i（t）]的稳定性（0.1）当右边在积分曲线zi= z ∈ i（t）的某些点处不连续时。这种“不连续”的情况下的线性逼近的意义的解释，和定理，类似于Liapounoff稳定的线性逼近在连续的情况下，提出。
In the works of Liapounoff (1) the problem of stability of a periodic solution zi= z˜i(t) (with period τ) of a system of differential equations dzidt=fi(z1,…,zn,t) [fi(z1,…znt+τ)≡fi(z1,…zn,t)](0.1) was reduced in the principal (non-critical) cases to the problem of stability of the zero-solution xi= 0 of the system of linear approximation‡ dxidt=∑k(∂fi∂zk)z=z˜(t)xk This reduction is applicable when the right-hand sides of equations (0.1) are analytic functions, or functions continuous in the neighbourhood of the integral curve and differentiable with respect to z1…, zn, uniformly in regard totat the points of this curve.It is the purpose of this paper to analyse the stability of a periodic solution zi=z˜i(t)[z˜i(t+τ)≡z˜i(t)] of system (0.1) when the right-hand sides fiare discontinuous at some points of the integral curve zi= z˜i(t). An explanation of the meaning of the linear approximation for this ‘discontinuous’ case is offered, and theorems, analogous to those of Liapounoff on stability by linear approximation in the continuous case, are presented.