Geodesic
Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 355-366
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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 ${\cal X}$, $B(t)$ is a variable bounded operator in ${\cal X}$. 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 $\|S {\rm e}^{At}\|$ 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.
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 ${\cal X}$, $B(t)$ is a variable bounded operator in ${\cal X}$. 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 $\|S {\rm e}^{At}\|$ 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.
DOI : 10.21136/CMJ.2023.0188-21
Classification : 34G10, 35B35, 35K51, 47D06
Keywords: Banach space; differential equation; linear nonautonomous equation; exponential stability; commutator; parabolic equation 
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     title = {Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a {Banach} space},
     journal = {Czechoslovak Mathematical Journal},
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Gil', Michael. Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 355-366. doi: 10.21136/CMJ.2023.0188-21

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