TY - GEN

T1 - Strong stabilization of almost passive linear systems

AU - Weiss, George

AU - Curtain, Ruth F.

PY - 2006

Y1 - 2006

N2 - The plant to be stabilized is a system node E with generating triple (A, B, C) and transfer function G, where A generates a contraction semigroup on the Hilbert space X. The control and observation operators B and C may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator E such that, if we replace G(s) by G(s) + E, the new system ΣE becomes impedance passive. A trivial case is when G is already impedance passive and a special case is when Σ has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback u = -ky + v, where u is the input of the plant and k > 0, stabilizes Σ, strongly or even exponentially. Here, y is the output of Σ and v is the new input. Our main result is that if for some E ∈ ℒ(U), ΣE is impedance passive, and Σ is approximately observable or approximately controllable in infinite time, then for sufficiently small k the closed-loop system is weakly stable. If, moreover, σ (A) ∩iℝ. is countable, then the closed-loop semigroup and its dual are both strongly stable. We illustrate our results with three classes of second order systems, only one of which has colocated actuators and sensors.

AB - The plant to be stabilized is a system node E with generating triple (A, B, C) and transfer function G, where A generates a contraction semigroup on the Hilbert space X. The control and observation operators B and C may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator E such that, if we replace G(s) by G(s) + E, the new system ΣE becomes impedance passive. A trivial case is when G is already impedance passive and a special case is when Σ has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback u = -ky + v, where u is the input of the plant and k > 0, stabilizes Σ, strongly or even exponentially. Here, y is the output of Σ and v is the new input. Our main result is that if for some E ∈ ℒ(U), ΣE is impedance passive, and Σ is approximately observable or approximately controllable in infinite time, then for sufficiently small k the closed-loop system is weakly stable. If, moreover, σ (A) ∩iℝ. is countable, then the closed-loop semigroup and its dual are both strongly stable. We illustrate our results with three classes of second order systems, only one of which has colocated actuators and sensors.

UR - http://www.scopus.com/inward/record.url?scp=39649096051&partnerID=8YFLogxK

U2 - 10.1109/cdc.2006.377191

DO - 10.1109/cdc.2006.377191

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:39649096051

SN - 1424401712

SN - 9781424401710

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 4688

EP - 4693

BT - Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 45th IEEE Conference on Decision and Control 2006, CDC

Y2 - 13 December 2006 through 15 December 2006

ER -