Stabilization of second order evolution equations with unbounded feedback with delay
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 2, pp. 420-456.

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.

DOI : 10.1051/cocv/2009007
Classification : 93D15, 93C20
Mots-clés : second order evolution equations, wave equations, delay, stabilization functional 
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Nicaise, Serge; Valein, Julie. Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 2, pp. 420-456. doi : 10.1051/cocv/2009007. https://www.numdam.org/articles/10.1051/cocv/2009007/

[1] 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.

[2] 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). | Zbl

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

[4] 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). | Zbl

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

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

[7] R. Dáger and E. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures, Mathématiques & Applications 50. Springer-Verlag, Berlin (2006). | Zbl

[8] 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. | Zbl

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

[10] 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. | Zbl

[11] 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. | Zbl

[12] J. Hale and S. Verduyn Lunel, Introduction to functional differential equations, Applied Mathematical Sciences 99. Springer (1993). | Zbl

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

[14] 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. | Zbl

[15] 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. | Zbl

[16] 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). | Zbl

[17] S. Nicaise and J. Valein, 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).

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

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

[20] 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. | Zbl

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

[22] 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. | Zbl

[23] G.Q. Xu, S.P. Yung and L.K. Li, Stabilization of wave systems with input delay in the boundary control. ESAIM: COCV 12 (2006) 770-785 (electronic). | Numdam | Zbl

  • Ayadi, Habib; Jlassi, Mariem Boundary exponential stabilization of a time-delay ODE-KdV cascaded system, European Journal of Control, Volume 81 (2025), p. 101141 | DOI:10.1016/j.ejcon.2024.101141
  • Salhi, Makrem; Shel, Farhat Stabilization by unbounded and periodic feedback, International Journal of Control, Volume 97 (2024) no. 3, p. 567 | DOI:10.1080/00207179.2022.2159537
  • Boumasmoud, Soufiane; Ezzinbi, Khalil The impact of delay on second-order evolution equations, Nonautonomous Dynamical Systems, Volume 11 (2024) no. 1 | DOI:10.1515/msds-2024-0002
  • Cheng, Yi; Wang, Xin; Feng, Baowei; Regan, Donal O’ Semigroup well-posedness and exponential stability for the von Kármán beam equation under the combined boundary control of nonlinear delays and non-delays, Nonlinear Analysis: Real World Applications, Volume 80 (2024), p. 104143 | DOI:10.1016/j.nonrwa.2024.104143
  • Jin, Kun-Peng; Wang, Li Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects, Advances in Nonlinear Analysis, Volume 12 (2023) no. 1 | DOI:10.1515/anona-2022-0285
  • Ayadi, Habib; Jlassi, Mariem Global exponential stabilization of the linearized Korteweg-de Vries equation with a state delay, IMA Journal of Mathematical Control and Information, Volume 40 (2023) no. 3, p. 516 | DOI:10.1093/imamci/dnad016
  • Majumdar, Subrata Asymptotic behavior of the linearized compressible barotropic Navier‐Stokes system with a time varying delay term in the boundary or internal feedback, Mathematical Methods in the Applied Sciences, Volume 46 (2023) no. 16, p. 17288 | DOI:10.1002/mma.9500
  • Ammari, Kaïs; Chentouf, Boumediène; Smaoui, Nejib Note on the stabilization of a vibrating string via a switching time-delay boundary control: a theoretical and numerical study, SeMA Journal, Volume 80 (2023) no. 4, p. 647 | DOI:10.1007/s40324-022-00315-z
  • Chellaoua, Houria; Boukhatem, Yamna; Feng, Baowei Optimal decay of an abstract nonlinear viscoelastic equation in Hilbert spaces with delay term in the nonlinear internal damping, Asymptotic Analysis, Volume 126 (2022) no. 1-2, p. 65 | DOI:10.3233/asy-201664
  • Freitas, M.M.; Ramos, A.J.A.; Dos Santos, M.J.; Miranda, L.G.R.; Almeida, J.L.L. Asymptotic analysis and upper semicontinuity with respect to delay term of attractors to binary mixtures of solids, Asymptotic Analysis, Volume 129 (2022) no. 3-4, p. 519 | DOI:10.3233/asy-211739
  • Chellaoua, Houria; Boukhatem, Yamna General decay for second-order abstract viscoelastic equation in Hilbert spaces with time delay, Boletim da Sociedade Paranaense de Matemática, Volume 41 (2022), p. 1 | DOI:10.5269/bspm.52175
  • Ammari, Kaïs; Chentouf, Boumediène; Smaoui, Nejib Internal feedback stabilization of multi-dimensional wave equations with a boundary delay: a numerical study, Boundary Value Problems, Volume 2022 (2022) no. 1 | DOI:10.1186/s13661-022-01589-y
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  • Freitas, Mirelson M.; Ramos, Anderson J. A.; Dos Santos, Manoel J.; Almeida, Jamille L.L. Dynamics of piezoelectric beams with magnetic effects and delay term, Evolution Equations Control Theory, Volume 11 (2022) no. 2, p. 583 | DOI:10.3934/eect.2021015
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  • Valein, Julie On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback, Mathematical Control and Related Fields, Volume 12 (2022) no. 3, p. 667 | DOI:10.3934/mcrf.2021039
  • Parada, Hugo; Crépeau, Emmanuelle; Prieur, Christophe Delayed stabilization of the Korteweg–de Vries equation on a star-shaped network, Mathematics of Control, Signals, and Systems, Volume 34 (2022) no. 3, p. 559 | DOI:10.1007/s00498-022-00319-0
  • Braik, Abdelkader; Beniani, Abderrahmane; Zennir, Khaled Well-posedness and general decay for Moore–Gibson–Thompson equation in viscoelasticity with delay term, Ricerche di Matematica, Volume 71 (2022) no. 2, p. 689 | DOI:10.1007/s11587-021-00561-9
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  • Chellaoua, Houria; Boukhatem, Yamna Optimal decay for second‐order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay, Mathematical Methods in the Applied Sciences, Volume 44 (2021) no. 2, p. 2071 | DOI:10.1002/mma.6917
  • Chellaoua, Houria; Boukhatem, Yamna Stability results for second-order abstract viscoelastic equation in Hilbert spaces with time-varying delay, Zeitschrift für angewandte Mathematik und Physik, Volume 72 (2021) no. 2 | DOI:10.1007/s00033-021-01477-y
  • Ma, Yonghao; He, Xiuyu; Li, Guang; He, Wei, 2020 39th Chinese Control Conference (CCC) (2020), p. 809 | DOI:10.23919/ccc50068.2020.9189454
  • Enyi, Cyril Dennis; Mukiawa, Soh Edwin Decay estimate for a viscoelastic plate equation with strong time-varying delay, ANNALI DELL'UNIVERSITA' DI FERRARA, Volume 66 (2020) no. 2, p. 339 | DOI:10.1007/s11565-020-00346-2
  • Luong, Vu Trong; Tung, Nguyen Thanh Exponential decay for elastic systems with structural damping and infinite delay, Applicable Analysis, Volume 99 (2020) no. 1, p. 13 | DOI:10.1080/00036811.2018.1484907
  • Chentouf, Boumediène; Smaoui, Nejib Time-Delayed Feedback Control of a Hydraulic Model Governed by a Diffusive Wave System, Complexity, Volume 2020 (2020), p. 1 | DOI:10.1155/2020/4986026
  • Ramos, Anderson J. A.; Santos, Manoel J. Dos; Freitas, Mirelson M.; Almeida Júnior, Dilberto S. Existence of Attractors for a Nonlinear Timoshenko System with Delay, Journal of Dynamics and Differential Equations, Volume 32 (2020) no. 4, p. 1997 | DOI:10.1007/s10884-019-09799-2
  • Feng, Yi-Hu; Hou, Lei A Class of Shock Wave Solution to Singularly Perturbed Nonlinear Time-Delay Evolution Equations, Shock and Vibration, Volume 2020 (2020), p. 1 | DOI:10.1155/2020/8829092
  • Bayili, Gilbert; Aissa, Akram Ben; Nicaise, Serge Same decay rate of second order evolution equations with or without delay, Systems Control Letters, Volume 141 (2020), p. 104700 | DOI:10.1016/j.sysconle.2020.104700
  • Baudouin, Lucie; Crepeau, Emmanuelle; Valein, Julie Two Approaches for the Stabilization of Nonlinear KdV Equation With Boundary Time-Delay Feedback, IEEE Transactions on Automatic Control, Volume 64 (2019) no. 4, p. 1403 | DOI:10.1109/tac.2018.2849564
  • Chen, Hao; Xie, Yaru; Genqi, Xu Rapid stabilisation of multi-dimensional Schrödinger equation with the internal delay control, International Journal of Control, Volume 92 (2019) no. 11, p. 2521 | DOI:10.1080/00207179.2018.1444283
  • Feng, Baowei; Yang, Xinguang; Su, Keqin Well-posedness and stability for a viscoelastic wave equation with density and time-varying delay in Rn, Journal of Integral Equations and Applications, Volume 31 (2019) no. 4 | DOI:10.1216/jie-2019-31-4-465
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  • Feng, Baowei Global Well-Posedness and Stability for a Viscoelastic Plate Equation with a Time Delay, Mathematical Problems in Engineering, Volume 2015 (2015), p. 1 | DOI:10.1155/2015/585021
  • Dias Silva, Flávio R.; Nascimento, Flávio A. F.; Rodrigues, José H. General decay rates for the wave equation with mixed-type damping mechanisms on unbounded domain with finite measure, Zeitschrift für angewandte Mathematik und Physik, Volume 66 (2015) no. 6, p. 3123 | DOI:10.1007/s00033-015-0547-5
  • Zhang, Zaiyun; Huang, Jianhua; Liu, Zhenhai; Sun, Mingbao Boundary Stabilization of a Nonlinear Viscoelastic Equation with Interior Time-Varying Delay and Nonlinear Dissipative Boundary Feedback, Abstract and Applied Analysis, Volume 2014 (2014), p. 1 | DOI:10.1155/2014/102594
  • Zhang, Qiong; Wang, Jun-Min; Guo, Bao-Zhu Stabilization of the Euler–Bernoulli equation via boundary connection with heat equation, Mathematics of Control, Signals, and Systems, Volume 26 (2014) no. 1, p. 77 | DOI:10.1007/s00498-013-0107-5
  • Nicaise, Serge; Pignotti, Cristina Stabilization of second-order evolution equations with time delay, Mathematics of Control, Signals, and Systems, Volume 26 (2014) no. 4, p. 563 | DOI:10.1007/s00498-014-0130-1
  • Chentouf, Boumediene Stabilization of the rotating disk-beam system with a delay term in boundary feedback, Nonlinear Dynamics, Volume 78 (2014) no. 3, p. 2249 | DOI:10.1007/s11071-014-1592-x
  • Cavalcanti, Marcelo M.; Dias Silva, Flávio R.; Domingos Cavalcanti, Valéria N. Uniform Decay Rates for the Wave Equation with Nonlinear Damping Locally Distributed in Unbounded Domains with Finite Measure, SIAM Journal on Control and Optimization, Volume 52 (2014) no. 1, p. 545 | DOI:10.1137/120862545
  • Abdallah, Farah; Nicaise, Serge; Valein, Julie; Wehbe, Ali Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications, ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, p. 844 | DOI:10.1051/cocv/2012036
  • Xu, Genqi; Wang, Hongxia Stabilisation of Timoshenko beam system with delay in the boundary control, International Journal of Control, Volume 86 (2013) no. 6, p. 1165 | DOI:10.1080/00207179.2013.787494
  • Zamorano, S.; Henríquez, H. R. Feedback stabilization of abstract neutral linear control systems, Mathematics of Control, Signals, and Systems, Volume 25 (2013) no. 3, p. 345 | DOI:10.1007/s00498-012-0103-1
  • Rebiai, Salah-Eddine Exponential Stability of the System of Transmission of the Wave Equation with a Delay Term in the Boundary Feedback, System Modeling and Optimization, Volume 391 (2013), p. 276 | DOI:10.1007/978-3-642-36062-6_28
  • Abdallah, Farah; Nicaise, Serge; Valein, Julie; Wehbe, Ali Stability results for the approximation of weakly coupled wave equations, Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, p. 29 | DOI:10.1016/j.crma.2011.12.004
  • Pignotti, Cristina A note on stabilization of locally damped wave equations with time delay, Systems Control Letters, Volume 61 (2012) no. 1, p. 92 | DOI:10.1016/j.sysconle.2011.09.016
  • Henríquez, Hernán R.; Cuevas, Claudio; Rabelo, Marcos; Caicedo, Alejandro Stabilization of distributed control systems with delay, Systems Control Letters, Volume 60 (2011) no. 9, p. 675 | DOI:10.1016/j.sysconle.2011.04.012
  • Said-Houari, Belkacem; Laskri, Yamina A stability result of a Timoshenko system with a delay term in the internal feedback, Applied Mathematics and Computation, Volume 217 (2010) no. 6, p. 2857 | DOI:10.1016/j.amc.2010.08.021
  • Fridman, Emilia; Nicaise, Serge; Valein, Julie Stabilization of Second Order Evolution Equations with Unbounded Feedback with Time-Dependent Delay, SIAM Journal on Control and Optimization, Volume 48 (2010) no. 8, p. 5028 | DOI:10.1137/090762105
  • Ammari, Kaïs; Nicaise, Serge; Pignotti, Cristina Feedback boundary stabilization of wave equations with interior delay, Systems Control Letters, Volume 59 (2010) no. 10, p. 623 | DOI:10.1016/j.sysconle.2010.07.007

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