Exponential stabilization of well-posed systems by colocated feedback

Ruth F. Curtain*, George Weiss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider well-posed linear systems whose state trajectories satisfy ẋ = Ax + Bu, where u is the input and A is an essentially skew-adjoint and dissipative operator on the Hilbert space X. This means that the domains of A* and A are equal and A* + A = -Q, where Q ≥ 0 is bounded on X. The control operator B is possibly unbounded, but admissible and the observation operator of the system is B*. Such a description fits many wave and beam equations with colocated sensors and actuators, and it has been shown for many particular cases that the feedback u = -ky + v, with K > 0, stabilizes the system, strongly or even exponentially. Here, y is the output of the system and v is the new input. We show, by means of a counterexample, that if B is sufficiently unbounded, then such a feedback may be unsuitable: the closed-loop semigroup may even grow exponentially. (Our counterexample is a simple regular system with feedthrough operator zero.) However, we prove that if the original system is exactly controllable and observable and if k is sufficiently small, then the closed-loop system is exponentially stable.

Original languageEnglish
Pages (from-to)273-297
Number of pages25
JournalSIAM Journal on Control and Optimization
Volume45
Issue number1
DOIs
StatePublished - 2006
Externally publishedYes

Keywords

  • Colocated sensors and actuators
  • Exact controllability and observability
  • Exponential stability
  • Output feedback
  • Positive-real transfer function
  • Regular linear system
  • Skew-adjoint operator
  • Well-posed linear system

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