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Stabilization of second order evolution equationswith unbounded feedback with delay

Published online by Cambridge University Press:  21 April 2009

Serge Nicaise  and
Julie Valein
Serge Nicaise
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France. [email protected]; [email protected]
Julie Valein
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France. [email protected]; [email protected]
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Abstract

We consider abstract second order evolution equations with unboundedfeedback with delay. Existence results are obtained under somerealistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented.

Keywords

Second order evolution equationswave equationsdelaystabilization functional
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 2 , April 2010 , pp. 420 - 456
Copyright
© EDP Sciences, SMAI, 2009

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References

C. Abdallah, P. Dorato, J. Benitez-Read and R. Byrne, Delayed positive feedback can stabilize oscillatory systems, in ACC' 93 (American Control Conference), San Francisco (1993) 3106–3107.
Ammari, K. and Tucsnak, M., Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J. Control Optim. 39 (2000) 1160–1181 (electronic). CrossRef
Ammari, K. and Tucsnak, M., Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV 6 (2001) 361–386 (electronic). CrossRef
K. Ammari, E.M. Ait Ben Hassi, S. Boulite and L. Maniar, Feedback stabilization of a class of evolution equations with delay. J. Evol. Eq. (Submitted).
Arendt, W. and Batty, C.J.K., Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 305 (1988) 837–852. CrossRef
Baiocchi, C., Komornik, V. and Loreti, P., Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar. 97 (2002) 55–95. CrossRef
R. Dáger and E. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures, Mathématiques & Applications 50. Springer-Verlag, Berlin (2006).
Datko, R., Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26 (1988) 697–713. CrossRef
Datko, R., Two examples of ill-posedness with respect to time delays revisited. IEEE Trans. Automat. Contr. 42 (1997) 511–515. CrossRef
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. CrossRef
K.P. Hadeler, Delay equations in biology, in Functional differential equations and approximation of fixed points, Lect. Notes Math. 730, Springer, Berlin (1979) 136–156.
J. Hale and S. Verduyn Lunel, Introduction to functional differential equations, Applied Mathematical Sciences 99. Springer (1993).
Ingham, A.E., Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936) 367–379. CrossRef
Lasiecka, I., Triggiani, R. and Yao, P.-F.. Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235 (1999) 13–57. CrossRef
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. CrossRef
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 (electronic). CrossRef
Nicaise, S. and Valein, J., Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2 (2007) 425–479 (electronic). CrossRef
A. Pazy, Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. 44 (1983).
Rebarber, R., Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33 (1995) 1–28. CrossRef
Rebarber, R. and Townley, S., Robustness with respect to delays for exponential stability of distributed parameter systems. SIAM J. Control Optim. 37 (1999) 230–244. CrossRef
Suh, I.H. and Bien, Z., Use of time delay action in the controller design. IEEE Trans. Automat. Contr. 25 (1980) 600–603. CrossRef
Tucsnak, M. and Weiss, G., How to get a conservative well-posed linear system out of thin air. II. Controllability and stability. SIAM J. Control Optim. 42 (2003) 907–935. CrossRef
Xu, G.Q., Yung, S.P. and Stabilization, L.K. Li of wave systems with input delay in the boundary control. ESAIM: COCV 12 (2006) 770–785 (electronic). CrossRef