How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance

George Weiss, Marius Tucsnak

Research output: Contribution to journalArticlepeer-review

Abstract

Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from D(A1/20) to another Hilbert space U. We prove that the system of equations (Formula Presented) determines a well-posed linear system with input u and output y. The state of this system is (Formula Presented) where X is the state space. Moreover, we have the energy identity (Formula Presented) We show that the system described above is isomorphic to its dual, so that a similar energy identity holds also for the dual system and hence, the system is conservative. We derive various other properties of such systems and we give a relevant example: a wave equation on a bounded n-dimensional domain with boundary control and boundary observation on part of the boundary.

Original languageEnglish
Pages (from-to)247-273
Number of pages27
JournalESAIM - Control, Optimisation and Calculus of Variations
Volume9
DOIs
StatePublished - Feb 2003
Externally publishedYes

Keywords

  • Conservative system
  • Dual system
  • Energy balance equation
  • Operator semigroup
  • Wave equation
  • Well-posed linear system

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