Feedback theory extended for proving generation of contraction semigroups

M. Kurula, H.J. Zwart

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Abstract

Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated with a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, and some non-restrictive technical assumptions are fulfilled, then the original operator generates a contraction semigroup as well. Beginning with the undamped wave equation, this general idea can be applied to show that the heat equation and wave equations with damping are well-posed. In the present paper we show how this approach can benefit from feedback techniques and recent developments in well-posed systems theory, at the same time generalizing the previously known results. Among others, we show how well-posedness of degenerate parabolic equations can be proved.
Original languageEnglish
Pages (from-to)617-647
JournalJournal of Evolution Equations
Volume16
Issue number3
DOIs
Publication statusPublished - Sep 2016

Keywords

  • Existence of solutions
  • Output feedback
  • Contraction semigroup
  • Well-posed system

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