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EXPONENTIAL STABILIZATION OF WELL-POSED SYSTEMS BY COLOCATED FEEDBACK.
- Source :
-
SIAM Journal on Control & Optimization . 2006, Vol. 45 Issue 1, p273-297. 25p. 1 Diagram. - Publication Year :
- 2006
-
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 = -κy+v, with κ > 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 κ is sufficiently small, then the closed-loop system is exponentially stable. [ABSTRACT FROM AUTHOR]
- Subjects :
- *LINEAR systems
*TRAJECTORY optimization
*HILBERT space
*LOOP spaces
*WAVE equation
Subjects
Details
- Language :
- English
- ISSN :
- 03630129
- Volume :
- 45
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- SIAM Journal on Control & Optimization
- Publication Type :
- Academic Journal
- Accession number :
- 21876604
- Full Text :
- https://doi.org/10.1137/040610489