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. Lamine Bouzettouta, Sabah Baibeche, Manel Abdelli, Amar Guesmia, Stability result for a thermoelastic Bresse system with delay term in the internal feedback
  5. Kassimu Mpungu, Tijani A. Apalara, Mukhiddin Muminov, On the stabilization of laminated beams with delay
  6. Serge Nicaise, Julie Valein, Stabilization of second order evolution equations with unbounded feedback with delay

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