Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation
ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 1, p. 22-35

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.

DOI : https://doi.org/10.1051/cocv/2010044
Classification:  35B37,  93B52,  93C05,  93C20,  93D15
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {1},
     year = {2012},
     pages = {22-35},
     doi = {10.1051/cocv/2010044},
     zbl = {1246.35120},
     mrnumber = {2887926},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 18 (2012) no. 1, pp. 22-35. doi : 10.1051/cocv/2010044. http://www.numdam.org/item/COCV_2012__18_1_22_0/

[1] R.F. Curtain, The Salamon-Weiss class of well-posed infinite dimensional linear systems : a survey. IMA J. Math. Control Inform. 14 (1997) 207-223. | MR 1470034 | Zbl 0880.93021

[2] R. Datko, Two questions concerning the boundary control of certain elastic systems. J. Diff. Equ. 92 (1991) 27-44. | MR 1113587 | Zbl 0747.93066

[3] 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. | MR 1300424 | Zbl 0812.93013

[4] R. Datko, Two examples of ill-posedness with respect to time delays revisited. IEEE Trans. Automat. Control 42 (1997) 511-515. | MR 1442585 | Zbl 0878.73046

[5] R. Datko and Y.C. You, Some second-order vibrating systems cannot tolerate small time delays in their damping. J. Optim. Theory Appl. 70 (1991) 521-537. | MR 1124776 | Zbl 0791.34045

[6] R. Datko, J. Lagnese and M.P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24 (1986) 152-156. | MR 818942 | Zbl 0592.93047

[7] 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).

[8] W.H. Fleming Ed., Future Directions in Control Theory. SIAM, Philadelphia (1988).

[9] I. Gumowski and C. Mira, Optimization in Control Theory and Practice. Cambridge University Press, Cambridge (1968). | Zbl 0242.49002

[10] B.Z. Guo and Y.H. Luo, Controllability and stability of a second order hyperbolic system with collocated sensor/actuator. Syst. Control Lett. 46 (2002) 45-65. | MR 2011071 | Zbl 0994.93021

[11] B.Z. Guo and Z.C. Shao, 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 2512487 | Zbl 1159.93026

[12] B.Z. Guo and C.Z. Xu, The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation. IEEE Trans. Automat. Contr. 52 (2007) 371-377. | MR 2295025 | Zbl 1168.93017

[13] B.Z. Guo and K.Y. Yang, Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica 45 (2009) 1468-1475. | MR 2879451 | Zbl 1166.93360

[14] B.Z. Guo, J.M. Wang and K.Y. Yang, Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation. Syst. Control Lett. 57 (2008) 740-749. | MR 2446459 | Zbl 1153.93026

[15] 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). | MR 1745476 | Zbl 0942.93001

[16] H. Logemann, R. Rebarber and G. Weiss, 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 1377713 | Zbl 0853.93081

[17] S. Nicaise and C. Pignotti, 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 2272156 | Zbl 1180.35095

[18] F. Oberhettinger and L. Badii, Tables of Laplace Transforms. Springer-Verlag, Berlin (1973). | MR 352889 | Zbl 0285.65079

[19] A. Smyshlyaev and M. Krstic, Backstepping observers for a class of parabolic PDEs. Syst. Control Lett. 54 (2005) 613-625. | MR 2142358 | Zbl 1129.93334

[20] L.N. Trefethen, Spectral Methods in Matlab. SIAM, Philadelphia (2000). | MR 1776072 | Zbl 0953.68643

[21] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhäuser, Basel (2009). | MR 2502023 | Zbl 1188.93002