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.
Mots-clés : wave equation, time delay, observer, predictor, feedback control, stability
@article{COCV_2012__18_1_22_0,
author = {Guo, Bao-Zhu and Xu, Cheng-Zhong and Hammouri, Hassan},
title = {Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {22--35},
publisher = {EDP-Sciences},
volume = {18},
number = {1},
year = {2012},
doi = {10.1051/cocv/2010044},
mrnumber = {2887926},
zbl = {1246.35120},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2010044/}
}
TY - JOUR AU - Guo, Bao-Zhu AU - Xu, Cheng-Zhong AU - Hammouri, Hassan TI - Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 22 EP - 35 VL - 18 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2010044/ DO - 10.1051/cocv/2010044 LA - en ID - COCV_2012__18_1_22_0 ER -
%0 Journal Article %A Guo, Bao-Zhu %A Xu, Cheng-Zhong %A Hammouri, Hassan %T Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 22-35 %V 18 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2010044/ %R 10.1051/cocv/2010044 %G en %F COCV_2012__18_1_22_0
Guo, Bao-Zhu; Xu, Cheng-Zhong; Hammouri, Hassan. Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 22-35. doi : 10.1051/cocv/2010044. http://archive.numdam.org/articles/10.1051/cocv/2010044/
[1] , The Salamon-Weiss class of well-posed infinite dimensional linear systems : a survey. IMA J. Math. Control Inform. 14 (1997) 207-223. | MR | Zbl
[2] , Two questions concerning the boundary control of certain elastic systems. J. Diff. Equ. 92 (1991) 27-44. | MR | Zbl
[3] , 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. | MR | Zbl
[4] , Two examples of ill-posedness with respect to time delays revisited. IEEE Trans. Automat. Control 42 (1997) 511-515. | MR | Zbl
[5] and , Some second-order vibrating systems cannot tolerate small time delays in their damping. J. Optim. Theory Appl. 70 (1991) 521-537. | MR | Zbl
[6] , and , An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24 (1986) 152-156. | MR | Zbl
[7] , and , 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).
[8] W.H. Fleming Ed., Future Directions in Control Theory. SIAM, Philadelphia (1988).
[9] and , Optimization in Control Theory and Practice. Cambridge University Press, Cambridge (1968). | Zbl
[10] and , Controllability and stability of a second order hyperbolic system with collocated sensor/actuator. Syst. Control Lett. 46 (2002) 45-65. | MR | Zbl
[11] and , 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. | MR | Zbl
[12] and , The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation. IEEE Trans. Automat. Contr. 52 (2007) 371-377. | MR | Zbl
[13] and , Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica 45 (2009) 1468-1475. | MR | Zbl
[14] , and , Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation. Syst. Control Lett. 57 (2008) 740-749. | MR | Zbl
[15] and , 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). | MR | Zbl
[16] , and , 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. | MR | Zbl
[17] and , 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. | MR | Zbl
[18] and , Tables of Laplace Transforms. Springer-Verlag, Berlin (1973). | MR | Zbl
[19] and , Backstepping observers for a class of parabolic PDEs. Syst. Control Lett. 54 (2005) 613-625. | MR | Zbl
[20] , Spectral Methods in Matlab. SIAM, Philadelphia (2000). | MR | Zbl
[21] and , Observation and Control for Operator Semigroups. Birkhäuser, Basel (2009). | MR | Zbl
Cité par Sources :
