Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space

Gil', Michael. "Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space." Czechoslovak Mathematical Journal 73.2 (2023): 355-366. <http://eudml.org/doc/299363>.

@article{Gil2023,
abstract = {We consider the equation $\{\rm d\}y(t)/\{\rm d\}t=(A+B(t))y(t)$$(t\ge 0)$, where $A$ is the generator of an analytic semigroup $(\{\rm e\}^\{At\})_\{t\ge 0\}$ on a Banach space $\{\mathcal \{X\}\}$, $B(t)$ is a variable bounded operator in $\{\mathcal \{X\}\}$. It is assumed that the commutator $K(t)=AB(t)-B(t)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^\{-1\}$ such that $\Vert S \{\rm e\}^\{At\}\Vert $ is integrable and the operator $K(t)S_l^\{-1\}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations.},
author = {Gil', Michael},
journal = {Czechoslovak Mathematical Journal},
keywords = {Banach space; differential equation; linear nonautonomous equation; exponential stability; commutator; parabolic equation},
language = {eng},
number = {2},
pages = {355-366},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space},
url = {http://eudml.org/doc/299363},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Gil', Michael
TI - Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 355
EP - 366
AB - We consider the equation ${\rm d}y(t)/{\rm d}t=(A+B(t))y(t)$$(t\ge 0)$, where $A$ is the generator of an analytic semigroup $({\rm e}^{At})_{t\ge 0}$ on a Banach space ${\mathcal {X}}$, $B(t)$ is a variable bounded operator in ${\mathcal {X}}$. It is assumed that the commutator $K(t)=AB(t)-B(t)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^{-1}$ such that $\Vert S {\rm e}^{At}\Vert $ is integrable and the operator $K(t)S_l^{-1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations.
LA - eng
KW - Banach space; differential equation; linear nonautonomous equation; exponential stability; commutator; parabolic equation
UR - http://eudml.org/doc/299363
ER -