Global and local versions for a Phóng Vũ theorem for periodic evolution families in Hilbert spaces.

*(English)*Zbl 06983567Summary: 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) |

##### Keywords:

uniform exponential stability; growth bounds; Fourier series; exponentially bounded evolution families of operators
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\textit{C. Buse} et al., Electron. J. Differ. Equ. 2018, Paper No. 188, 12 p. (2018; Zbl 06983567)

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[18] | J. Zabczyk; Mathematical control theory: An introduction, Birkh¨auser, Systems and Control, 1992. Constantin Bus¸e Politehnica University of Timisoara, Department of Mathematics, Piata Victoriei No. 2, 300006-Timisoara, Romania. Western Kentucky University, Department of Mathematics, Bowling Green, KY 42101, USA E-mail address: constantin.buse@upt.ro, constantin.buse@wku.edu Lan Thanh Nguyen Western Kentucky University, Department of Mathematics, Bowling Green, KY 42101, USA E-mail address: lan.nguyen@wku.edu 1. Introduction2. Main results3. Discrete caseAcknowledgementsReferences |

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