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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.
MSC:
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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