Filippov Lemma for matrix fourth order differential inclusions

• Autorzy: Grzegorz Bartuzel , Andrzej Fryszkowski
• Języki publikacji: EN

Abstrakty

EN

In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions
𝒟y = y'''' - (A² + B²)y'' + A²B²y ∈ F(t,y), (*)
with the initial conditions
y(0) = y'(0) = y''(0) = y'''(0) = 0, (**)
where the matrices $A,B ∈ ℝ^{d×d}$ are commutative and the multifunction $F: [0,1] × ℝ^{d} ⇝ cl(ℝ^{d})$ is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||².
Main theorem. Assume that $F: [0,1] × ℝ^{d} ⇝ cl(ℝ^{d}) is measurable in t and integrably bounded. Let $y₀ ∈ W^{4,1}$ be an arbitrary function satisfying (**) and such that
$d_{H}(𝒟y₀(t),F(t,y₀(t))) ≤ p₀(t)$ a.e. in [0,1],
where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W^{4,1} of (*) with (**) such that
|𝒟y(t)-𝒟y₀(t)| ≤ p₀(t) + l(Y₄(⋅,α,β)∗p₀)(t)
|y(t)-y₀(t)| ≤ (Y₄(⋅,α,β)∗p₀)(t) a.e. in [0,1],
where
$Y₄(x,α,β) = (α^{-1}sinh(αx) - β^{-1}sinh(βx))/(α²-β²)$
and α,β depend on ||A||, ||B|| and l.

Kategorie tematyczne

• 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems
• 46G05: Derivatives
• 34A60: Differential inclusions
• 49J21: Optimal control problems involving relations other than differential equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2014
• Tom: 101
• Numer: 1
• Strony: 9-18

Daty

wydano

2014

Twórcy

autor

Grzegorz Bartuzel

• Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland

autor

Andrzej Fryszkowski

• Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland

Identyfikatory

DOI

10.4064/bc101-0-1