# 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

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topXu, 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

top- I. Gumowski and C. Mira, Optimization in Control Theory and Practice. Cambridge University Press, Cambridge (1968).
- R. Datko, J. Lagness and M.P. Poilis, An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24 (1986) 152–156.
- R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delay in their feedbacks. SIAM J. Control Optim. 26 (1988) 697–713.
- I.H. Suh and Z. Bien, Use of time delay action in the controller design. IEEE Trans. Automat. Control25 (1980) 600–603.
- W.H. Kwon, G.W. Lee and S.W. Kim, Performance improvement, using time delays in multi-variable controller design. INT J. Control52 (1990) 1455–1473.
- G. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory systems, in ACC' 93 (American control conference), San Francisco (1993) 3106–3107.
- N. Jalili and N. Olgac, Optimum delayed feedback vibration absorber for MDOF mechanical structure, in 37th IEEE CDC'98 (Conference on decision and control), Tampa, FL, December (1998) 4734–4739.
- W. Aernouts, D. Roose and R. Sepulchre, Delayed control of a Moore-Greitzer axial compressor model. Intern. J. Bifurcation Chaos10 (2000) 115–1164.
- J.K. Hale and S.M. Verduyn-Lunel, Strong stabilization of neutral functional differential equations. IMA J. Math. Control Inform.19 (2002) 5–24.
- J.K. Hale and S.M. Verduyn-Lunel, Introduction to functional differential equations, in Applied Mathematical Sciences, New York, Springer 99 (1993).
- S.I. Niculescu and R. Lozano, On the passivity of linear delay systems. IEEE Trans. Automat. Control46 (2001) 460–464.
- P. Borne, M. Dambrine, W. Perruquetti and J.P. Richard, Vector Lyapunov functions: nonlinear, time-varying, ordinary and functional differential equations. Stability and control: theory, methods and applications13, Taylor and Francis, London (2003) 49–73.
- Ö. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equation. IEEE Trans. Automat. Control40 (1995) 1626–1630.
- Ö. Mörgul, Stabilization and disturbance rejection for the wave equation. IEEE Trans. Automat. Control43 (1998) 89–95.
- J.-L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system. SIAM Rev.30 (1988) 1–68.
- G.Q. Xu and B.Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation. SIAM J. Control Optim.42 (2003) 966–984.
- M.A. Shubov, The Riesz basis property of the system of root vectors for the equation of a nonhomogeneous damped string: transformation operators method. Methods Appl. Anal.6 (1999) 571–591.
- G.Q. Xu and S.P. Yung, The expansion of semigroup and a criterion of Riesz basis. J. Differ. Equ.210 (2005) 1–24.
- I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators. AMS Transl. Math. Monographs 18 (1969).
- Lars V. Ahlfors, Complex Analysis. McGraw-Hill.

## Citations in EuDML Documents

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