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

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Abstract

We consider the equation dy(t)/dt = (A + B(t))y(t) (t ≽ 0), where A is the generator of an analytic semigroup (eAt)t≽0 on a Banach space χ, B(t) is a variable bounded operator in χ. 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 ∥SeAt∥ is integrable and the operator \(K\left( t \right)S_l^{ - 1}\) is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations.

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Acknowledgements

I am very grateful to the reviewer of this paper for really deep and helpful remarks.

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  1. Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva, 84105, Israel

    Michael Gil’

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  1. Michael Gil’
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Correspondence to Michael Gil’.

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Gil’, M. Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space. Czech Math J 73, 355–366 (2023). https://doi.org/10.21136/CMJ.2023.0188-21

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  • DOI: https://doi.org/10.21136/CMJ.2023.0188-21

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