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Global and local versions for a Phóng Vũ theorem for periodic evolution families in Hilbert spaces. (English) Zbl 06983567
Summary: A theorem of Gearhart concerning strongly continuous semigroups in Hilbert spaces is extremely useful for stability analysis of concrete equations; see e.g. [F. Huang, Ann. Differ. Equations 1, 43–56 (1985; Zbl 0593.34048)]), and for control theory [G. Weiss, J. Differ. Equations 76, No. 2, 269–285 (1988; Zbl 0675.47031)] or [K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations. Berlin: Springer (2000; Zbl 0952.47036), p. 475]. Phong Vu introduced an equivalent condition in [Vu Quoc Phong, Proc. Am. Math. Soc. 129, No. 10, 2871–2879 (2001; Zbl 0998.47022)]. 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.
35B35 Stability in context of PDEs
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
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