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Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation∗

Published online by Cambridge University Press:  02 December 2010

Bao-Zhu Guo ,
Cheng-Zhong Xu  and
Hassan Hammouri
Bao-Zhu Guo
Affiliation:
Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, P.R. China. Bao-Zhu.Guo@wits.ac.za School of Mathematical Sciences, Shanxi University, Taiyuan 030006, P.R. China School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
Cheng-Zhong Xu
Affiliation:
Université de Lyon, LAGEP, Bâtiment CPE, Université Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
Hassan Hammouri
Affiliation:
Université de Lyon, LAGEP, Bâtiment CPE, Université Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
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Abstract

The stabilization with time delay in observation or control represents difficult mathematical challenges in the control of distributed parameter systems. It is well-known that the stability of closed-loop system achieved by some stabilizing output feedback laws may be destroyed by whatever small time delay there exists in observation. In this paper, we are concerned with a particularly interesting case: Boundary output feedback stabilization of a one-dimensional wave equation system for which the boundary observation suffers from an arbitrary long time delay. We use the observer and predictor to solve the problem: The state is estimated in the time span where the observation is available; and the state is predicted in the time interval where the observation is not available. It is shown that the estimator/predictor based state feedback law stabilizes the delay system asymptotically or exponentially, respectively, relying on the initial data being non-smooth or smooth. Numerical simulations are presented to illustrate the effect of the stabilizing controller.

Keywords

Wave equationtime delayobserverpredictorfeedback controlstability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 22 - 35
Copyright
© EDP Sciences, SMAI, 2010

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References

Curtain, R.F., The Salamon-Weiss class of well-posed infinite dimensional linear systems : a survey. IMA J. Math. Control Inform. 14 (1997) 207–223. Google Scholar
Datko, R., Two questions concerning the boundary control of certain elastic systems. J. Diff. Equ. 92 (1991) 27–44. Google Scholar
R. Datko, Is boundary control a realistic approach to the stabilization of vibrating elastic systems?, in Evolution Equations, Baton Rouge (1992), Lecture Notes in Pure and Appl. Math. 168, Dekker, New York (1995) 133–140.
Datko, R., Two examples of ill-posedness with respect to time delays revisited. IEEE Trans. Automat. Control 42 (1997) 511–515. Google Scholar
Datko, R. and You, Y.C., Some second-order vibrating systems cannot tolerate small time delays in their damping. J. Optim. Theory Appl. 70 (1991) 521–537. Google Scholar
Datko, R., Lagnese, J. and Polis, M.P., An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24 (1986) 152–156. Google Scholar
A.J. Deguenon, G. Sallet and C.Z. Xu, A Kalman observer for infinite-dimensional skew-symmetric systems with application to an elastic beam, Proc. of the Second International Symposium on Communications, Control and Signal Processing, Marrakech, Morocco (2006).
W.H. Fleming Ed., Future Directions in Control Theory. SIAM, Philadelphia (1988).
I. Gumowski and C. Mira, Optimization in Control Theory and Practice. Cambridge University Press, Cambridge (1968).
Guo, B.Z. and Luo, Y.H., Controllability and stability of a second order hyperbolic system with collocated sensor/actuator. Syst. Control Lett. 46 (2002) 45–65. Google Scholar
Guo, B.Z. and Shao, Z.C., Stabilization of an abstract second order system with application to wave equations under non-collocated control and observations. Syst. Control Lett. 58 (2009) 334–341. Google Scholar
Guo, B.Z. and Xu, C.Z., The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation. IEEE Trans. Automat. Contr. 52 (2007) 371–377. Google Scholar
Guo, B.Z. and Yang, K.Y., Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica 45 (2009) 1468–1475. Google Scholar
Guo, B.Z., Wang, J.M. and Yang, K.Y., Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation. Syst. Control Lett. 57 (2008) 740–749. Google Scholar
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations : Continuous and Approxiamation Theories – II : Abstract Hyperbolic-Like Systems over a Finite Time Horizon. Cambridge University Press, Cambridge (2000).
Logemann, H., Rebarber, R. and Weiss, G., Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop. SIAM J. Control Optim. 34 (1996) 572–600. Google Scholar
Nicaise, S. and Pignotti, C., Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45 (2006) 1561–1585. Google Scholar
F. Oberhettinger and L. Badii, Tables of Laplace Transforms. Springer-Verlag, Berlin (1973).
Smyshlyaev, A. and Krstic, M., Backstepping observers for a class of parabolic PDEs. Syst. Control Lett. 54 (2005) 613–625. Google Scholar
L.N. Trefethen, Spectral Methods in Matlab. SIAM, Philadelphia (2000).
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhäuser, Basel (2009).