Stabilization of wave systems with input delay in the boundary control

Gen Qi Xu; Siu Pang Yung; Leong Kwan Li

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 4, page 770-785
  • ISSN: 1292-8119

Abstract

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In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight ( 1 - μ ) is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop system generates a C0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors and generalized eigenvectors that forms a Riesz basis for the state Hilbert space. Furthermore, we show that when the weight μ > 1 2 , for any time delay, we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When μ = 1 2 , we show that the system is at most asymptotically stable. When μ < 1 2 , the system is always unstable.

How to cite

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Xu, Gen Qi, Yung, Siu Pang, and Li, Leong Kwan. "Stabilization of wave systems with input delay in the boundary control." ESAIM: Control, Optimisation and Calculus of Variations 12.4 (2006): 770-785. <http://eudml.org/doc/249665>.

@article{Xu2006,
abstract = { In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight $(1-\mu)$ is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop system generates a C0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors and generalized eigenvectors that forms a Riesz basis for the state Hilbert space. Furthermore, we show that when the weight $\mu>\frac\{1\}\{2\}$, for any time delay, we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When $\mu=\frac\{1\}\{2\}$, we show that the system is at most asymptotically stable. When $\mu<\frac\{1\}\{2\}$, the system is always unstable. },
author = {Xu, Gen Qi, Yung, Siu Pang, Li, Leong Kwan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Wave equation; time delay; stabilization; Riesz basis.; Riesz basis; boundary velocity feedback law; closed loop system},
language = {eng},
month = {10},
number = {4},
pages = {770-785},
publisher = {EDP Sciences},
title = {Stabilization of wave systems with input delay in the boundary control},
url = {http://eudml.org/doc/249665},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Xu, Gen Qi
AU - Yung, Siu Pang
AU - Li, Leong Kwan
TI - Stabilization of wave systems with input delay in the boundary control
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/10//
PB - EDP Sciences
VL - 12
IS - 4
SP - 770
EP - 785
AB - In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight $(1-\mu)$ is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop system generates a C0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors and generalized eigenvectors that forms a Riesz basis for the state Hilbert space. Furthermore, we show that when the weight $\mu>\frac{1}{2}$, for any time delay, we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When $\mu=\frac{1}{2}$, we show that the system is at most asymptotically stable. When $\mu<\frac{1}{2}$, the system is always unstable.
LA - eng
KW - Wave equation; time delay; stabilization; Riesz basis.; Riesz basis; boundary velocity feedback law; closed loop system
UR - http://eudml.org/doc/249665
ER -

References

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Citations in EuDML Documents

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  1. Abbes Benaissa, Mostefa Miloudi, Mokhtar Mokhtari, Global existence and energy decay of solutions to a Bresse system with delay terms
  2. Zhong-Jie Han, Gen-Qi Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks
  3. Zhong-Jie Han, Gen-Qi Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks
  4. Serge Nicaise, Julie Valein, Stabilization of second order evolution equations with unbounded feedback with delay

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