# Stabilization of second order evolution equations with unbounded feedback with delay

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 2, page 420-456
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topNicaise, Serge, and Valein, Julie. "Stabilization of second order evolution equations with unbounded feedback with delay." ESAIM: Control, Optimisation and Calculus of Variations 16.2 (2010): 420-456. <http://eudml.org/doc/250719>.

@article{Nicaise2010,

abstract = {
We consider abstract second order evolution equations with unbounded
feedback with delay. Existence results are obtained under some
realistic 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.
},

author = {Nicaise, Serge, Valein, Julie},

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

keywords = {Second order evolution equations; wave equations; delay; stabilization functional; second order evolution equations},

language = {eng},

month = {4},

number = {2},

pages = {420-456},

publisher = {EDP Sciences},

title = {Stabilization of second order evolution equations with unbounded feedback with delay},

url = {http://eudml.org/doc/250719},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Nicaise, Serge

AU - Valein, Julie

TI - Stabilization of second order evolution equations with unbounded feedback with delay

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/4//

PB - EDP Sciences

VL - 16

IS - 2

SP - 420

EP - 456

AB -
We consider abstract second order evolution equations with unbounded
feedback with delay. Existence results are obtained under some
realistic 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.

LA - eng

KW - Second order evolution equations; wave equations; delay; stabilization functional; second order evolution equations

UR - http://eudml.org/doc/250719

ER -

## References

top- 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.
- K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J. Control Optim.39 (2000) 1160–1181 (electronic). Zbl0983.35021
- K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV6 (2001) 361–386 (electronic). Zbl0992.93039
- 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). Zbl1331.93181
- W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc.305 (1988) 837–852. Zbl0652.47022
- C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar.97 (2002) 55–95. Zbl1012.42022
- R. Dáger and E. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures, Mathématiques & Applications50. Springer-Verlag, Berlin (2006). Zbl1083.74002
- R. Datko, 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. Zbl0643.93050
- R. Datko, Two examples of ill-posedness with respect to time delays revisited. IEEE Trans. Automat. Contr.42 (1997) 511–515. Zbl0878.73046
- 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. Zbl0592.93047
- 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 Sciences99. Springer (1993). Zbl0787.34002
- A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z.41 (1936) 367–379. Zbl0014.21503
- I. Lasiecka, R. Triggiani and P.-F. Yao. Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl.235 (1999) 13–57. Zbl0931.35022
- 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. Zbl0853.93081
- 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 (electronic). Zbl1180.35095
- S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Netw. Heterog. Media2 (2007) 425–479 (electronic). Zbl1211.35050
- A. Pazy, Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci.44 (1983). Zbl0516.47023
- R. Rebarber, Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim.33 (1995) 1–28. Zbl0819.93042
- R. Rebarber and S. Townley, Robustness with respect to delays for exponential stability of distributed parameter systems. SIAM J. Control Optim.37 (1999) 230–244. Zbl0919.93041
- I.H. Suh and Z. Bien, Use of time delay action in the controller design. IEEE Trans. Automat. Contr.25 (1980) 600–603. Zbl0432.93044
- M. Tucsnak and G. Weiss, 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. Zbl1125.93383
- G.Q. Xu, S.P. Yung and L.K. Li, Stabilization of wave systems with input delay in the boundary control. ESAIM: COCV12 (2006) 770–785 (electronic). Zbl1105.35016

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.