当前位置: X-MOL 学术Math. Nachr. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Filippov lemma for measure differential inclusion
Mathematische Nachrichten ( IF 0.8 ) Pub Date : 2021-02-02 , DOI: 10.1002/mana.201800457
Andrzej Fryszkowski 1 , Jacek Sadowski 1
Affiliation  

In this work we propose a Filippov‐type lemma for the differential inclusion
d d μ x ( t ) F ( t , x ( t ) ) , x ( 0 ) = x 0 , (0.1)
where F : [ 0 , T ] × R d R d is a given multifunction and μ is a finite Borel signed measure on [0, T] (possibly atomic). By a solution of (0.1) we mean a function x : [ 0 , T ] R d such that x ( 0 ) = x 0 and
x ( t ) = x 0 + S ( t ) v ( s ) d μ ( s ) for t > 0 ,
where v ( · ) is a μ‐integrable function such that v ( t ) F ( t , x ( t ) ) for μ‐almost every t [ 0 , T ] and S ( t ) stands for either (0, t] for each t J or [0, t). Such setting leads to at least two nonequivalent notions of a solution to (0.1) and therefore we formulate two different Filippov‐type inequalities (Theorems 2.1 and 2.2). These two concepts coincide in case of the Lebesgue measure. The purpose of our considerations is to cover a class of impulsive control systems, a class of stochastic systems and differential systems on time scales.


中文翻译:

Filippov引理用于度量微分包含

在这项工作中,我们为差分包含提出了Filippov型引理
d d μ X Ť F Ť X Ť X 0 = X 0 (0.1)
在哪里 F [ 0 Ť ] × [R d [R d 是给定的多功能且μ是对[0,T ](可能是原子)的有限Borel有符号度量。通过(0.1)的解,我们的意思是一个函数 X [ 0 Ť ] [R d 这样 X 0 = X 0
X Ť = X 0 + 小号 Ť v s d μ s 为了 Ť > 0
在哪里 v · 是一个μ可积函数,使得 v Ť F Ť X Ť 几乎每个 Ť [ 0 Ť ] 小号 Ť 分别代表(0,t ] Ť Ĵ 或[0,t)。这样的设置至少导致(0.1)解的两个非等价概念,因此我们公式化了两个不同的Filippov型不等式(定理2.1和2.2)。如果采用勒贝格测度,这两个概念是一致的。我们考虑的目的是在时标上涵盖一类脉冲控制系统,一类随机系统和差分系统。
更新日期:2021-03-09
down
wechat
bug