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.

Keywords: second order evolution equations, wave equations, delay, stabilization functional

@article{COCV_2010__16_2_420_0, author = {Nicaise, Serge and Valein, Julie}, title = {Stabilization of second order evolution equations with unbounded feedback with delay}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {420--456}, publisher = {EDP-Sciences}, volume = {16}, number = {2}, year = {2010}, doi = {10.1051/cocv/2009007}, mrnumber = {2654201}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2009007/} }

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 PY - 2010 SP - 420 EP - 456 VL - 16 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2009007/ DO - 10.1051/cocv/2009007 LA - en ID - COCV_2010__16_2_420_0 ER -

%0 Journal Article %A Nicaise, Serge %A Valein, Julie %T Stabilization of second order evolution equations with unbounded feedback with delay %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 420-456 %V 16 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2009007/ %R 10.1051/cocv/2009007 %G en %F COCV_2010__16_2_420_0

Nicaise, Serge; Valein, Julie. Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 420-456. doi : 10.1051/cocv/2009007. http://archive.numdam.org/articles/10.1051/cocv/2009007/

[1] Delayed positive feedback can stabilize oscillatory systems, in ACC' 93 (American Control Conference), San Francisco (1993) 3106-3107.

, , and ,[2] Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J. Control Optim. 39 (2000) 1160-1181 (electronic). | Zbl

and ,[3] Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV 6 (2001) 361-386 (electronic). | EuDML | Numdam | Zbl

and ,[4] Feedback stabilization of a class of evolution equations with delay. J. Evol. Eq. (Submitted). | Zbl

, , and ,[5] Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 305 (1988) 837-852. | Zbl

and ,[6] Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar. 97 (2002) 55-95. | Zbl

, and ,[7] Wave propagation, observation and control in 1-d flexible multi-structures, MathÃ©matiques & Applications 50. Springer-Verlag, Berlin (2006). | Zbl

and ,[8] 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. | Zbl

,[9] Two examples of ill-posedness with respect to time delays revisited. IEEE Trans. Automat. Contr. 42 (1997) 511-515. | Zbl

,[10] An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24 (1986) 152-156. | Zbl

, and ,[11] Delay equations in biology, in Functional differential equations and approximation of fixed points, Lect. Notes Math. 730, Springer, Berlin (1979) 136-156. | Zbl

,[12] Introduction to functional differential equations, Applied Mathematical Sciences 99. Springer (1993). | Zbl

and ,[13] Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936) 367-379. | Zbl

,[14] Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235 (1999) 13-57. | Zbl

, and .[15] 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. | Zbl

, and ,[16] 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). | Zbl

and ,[17] 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).

and ,[18] Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. 44 (1983). | Zbl

,[19] Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33 (1995) 1-28. | Zbl

,[20] Robustness with respect to delays for exponential stability of distributed parameter systems. SIAM J. Control Optim. 37 (1999) 230-244. | Zbl

and ,[21] Use of time delay action in the controller design. IEEE Trans. Automat. Contr. 25 (1980) 600-603. | Zbl

and ,[22] 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. | Zbl

and ,[23] Stabilization of wave systems with input delay in the boundary control. ESAIM: COCV 12 (2006) 770-785 (electronic). | Numdam | Zbl

, and ,*Cited by Sources: *