Abstract

We consider the class of linear parametric differential systems \(\dot {x}=\mu A(t)x \) defined on the half-line \(t\geq 0 \), where \(\mu \in \mathbb {R} \) is a parameter, \(x\in \mathbb {R}^n \), and the matrix \(A(\cdot )\colon [0,+\infty )\to \mathbb {R}^{n\times n}\) is piecewise continuous and bounded. It is proved that for each \(n\geq 2\) some set is the irregularity set of one of such systems (i.e., the set of \(\mu \) for which this system is Lyapunov irregular) if and only if it is a \( G_{\delta \sigma }\)-set of the real line not containing zero. The necessity of the above conditions has been established earlier. This assertion solves the problem posed by N.A. Izobov in the early 1990s.

中文翻译：

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带线性参数的线性微分系统不规则集结构的描述

摘要

我们考虑在半线\（t \ geq 0 \）上定义的线性参数微分系统\（\ dot {x} = \ mu A（t）x \），其中\（\ mu \ in \ mathbb {R} \）是参数\（x \ in \ mathbb {R} ^ n \）和矩阵\（A（\ cdot）\冒号[0，+ \ infty} \到\ mathbb {R} ^ {n \ times n} \）是分段连续且有界的。证明了对于每个\（n \ geq 2 \），当且仅当它是一个系统的不规则集（即，该系统是Lyapunov不规则的\（\ mu \）的集合）时，是\（G _ {\ delta \ sigma} \）-不包含零的实线的集合。上述条件的必要性已在较早的时间确定。这一主张解决了纳·伊佐波夫（NA Izobov）在1990年代初提出的问题。