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Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
47
Issue
2
Identifiers
DOI: 10.1016/0022-0396(83)90036-0

Abstract

Abstract General second-order parabolic and hyperbolic equations on a bounded domain are considered. The input is applied in the Neumann or mixed boundary condition and is expressed as a finite-dimensional feedback. In the parabolic case, the feedback acts, in particular, on the Dirichlet trace of the solution: here it is shown that the resulting closed loop system defines a (feedback) C 0-semigroup on L 2(Ω) (in fact, on H 3 2 − 2ϱ (Ω), ρ > 0 ), that is both analytic and compact for positive times, and whose generator has compact resolvent. In the hyperbolic case, the feedback acts on the position vector only, or on its Dirichlet trace in a special case: here a similar result is established regarding the existence of a feedback C 0-cosine operator. Moreover, an example is given, which hints that the class of prescribed feedbacks acting on the Dirichlet trace cannot be substantially enlarged. Functional analytic techniques are employed, in particular perturbation theory. However, perturbation theory for the original variable fails on L 2(Ω) , the space in which the final result is sought. Therefore, our approach employs perturbation theory, after a suitable continuous extension, on the larger space [H 1 2 + 2ϱ (Ω)]′ .

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