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

Bao-Zhu Guo; Cheng-Zhong Xu; Hassan Hammouri

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 1, page 22-35
- ISSN: 1292-8119

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topGuo, Bao-Zhu, Xu, Cheng-Zhong, and 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 18.1 (2012): 22-35. <http://eudml.org/doc/277818>.

@article{Guo2012,

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. },

author = {Guo, Bao-Zhu, Xu, Cheng-Zhong, Hammouri, Hassan},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Wave equation; time delay; observer; predictor; feedback control; stability},

language = {eng},

month = {2},

number = {1},

pages = {22-35},

publisher = {EDP Sciences},

title = {Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation∗},

url = {http://eudml.org/doc/277818},

volume = {18},

year = {2012},

}

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

DA - 2012/2//

PB - EDP Sciences

VL - 18

IS - 1

SP - 22

EP - 35

AB - 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.

LA - eng

KW - Wave equation; time delay; observer; predictor; feedback control; stability

UR - http://eudml.org/doc/277818

ER -

## References

top- 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.
- R. Datko, Two questions concerning the boundary control of certain elastic systems. J. Diff. Equ.92 (1991) 27–44.
- 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.
- R. Datko, Two examples of ill-posedness with respect to time delays revisited. IEEE Trans. Automat. Control42 (1997) 511–515.
- 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.
- 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.
- 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).
- 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.
- 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.
- 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.
- B.Z. Guo and K.Y. Yang, Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica45 (2009) 1468–1475.
- 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.
- 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).
- 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.
- 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.
- F. Oberhettinger and L. Badii, Tables of Laplace Transforms. Springer-Verlag, Berlin (1973).
- A. Smyshlyaev and M. Krstic, Backstepping observers for a class of parabolic PDEs. Syst. Control Lett.54 (2005) 613–625.
- L.N. Trefethen, Spectral Methods in Matlab. SIAM, Philadelphia (2000).
- M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhäuser, Basel (2009).

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