Electron. J. Differential Equations, Vol. 2018 (2018), No. 188, pp. 1-12.

Global and local versions for a Phong Vu theorem for periodic evolution families in Hilbert spaces

Constantin Buse, Lan Thanh Nguyen, Donal O'Regan

Abstract:
A theorem of Gearhart concerning strongly continuous semigroups in Hilbert spaces is extremely useful for stability analysis of concrete equations; see e.g. [20]), and for control theory [27] or [13, page 475]. Phong Vu introduced an equivalent condition in [23]. The aim of this article is to extend these results from the autonomous case to time dependent 1-periodic evolution equations in Hilbert spaces. Both cases (continuous and discrete) are analyzed and global and local versions of the Phong Vu theorem are provided.

Submitted November 7, 2017. Published November 20, 2018.}
Math Subject Classifications: 35B35 47A30, 46A30.
Key Words: Uniform exponential stability; growth bounds; Fourier series; exponentially bounded evolution families of operators.

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Constantin Buse
Politehnica University of Timisoara
Department of Mathematics, Piata Victoriei No. 2
300006-Timisoara, Romania
email: constantin.buse@upt.ro, constantin.buse@wku.edu
Lan Thanh Nguyen
Western Kentucky University
Department of Mathematics
Bowling Green, KY 42101, USA
email: lan.nguyen@wku.edu
Donal O'Regan
National University of Ireland
Galway, Ireland
email: donal.oregan@nuigalway.ie

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