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Filippov lemma for measure differential inclusion
Author(s) -
Fryszkowski Andrzej,
Sadowski Jacek
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800457
Subject(s) - mathematics , differential inclusion , lemma (botany) , measure (data warehouse) , lebesgue integration , pure mathematics , cover (algebra) , class (philosophy) , integrable system , lebesgue measure , type (biology) , function (biology) , differential (mechanical device) , borel measure , discrete mathematics , mathematical analysis , probability measure , mechanical engineering , ecology , poaceae , database , evolutionary biology , computer science , engineering , biology , artificial intelligence , aerospace engineering
In this work we propose a Filippov‐type lemma for the differential inclusion 0.1d d μ x ( t ) ∈ F ( t , x ( t ) ) , x ( 0 ) = x 0 , where F : [ 0 , T ] × R d ⇝ R dis 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 dsuch that x ( 0 ) = x 0andx ( t ) = x 0 + ∫ S ( t ) v ( s )d μ ( s )fort > 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.