Skip to main content Accessibility help
×
Home
Hostname: page-component-55b6f6c457-qgndx Total loading time: 0.171 Render date: 2021-09-28T11:42:53.465Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

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
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
Get access

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Curtain, R.F., The Salamon-Weiss class of well-posed infinite dimensional linear systems : a survey. IMA J. Math. Control Inform. 14 (1997) 207223. Google Scholar
Datko, R., Two questions concerning the boundary control of certain elastic systems. J. Diff. Equ. 92 (1991) 2744. 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) 511515. 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) 521537. 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) 152156. 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) 4565. 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) 334341. 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) 371377. 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) 14681475. 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) 740749. 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) 572600. 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) 15611585. 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) 613625. 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).

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *