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Remarks on stabilization of second-order evolution equations by unbounded dynamic feedbacks

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Abstract

In this paper, we consider second-order evolution equations with unbounded dynamic feedbacks. Under a regularity assumption, we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non-uniform decay properties.

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References

  1. Abbas Z., Nicaise S.: Polynomial decay rate for a wave equation with general acoustic boundary feedback laws. S \({{\vec{\rm e}}}\) MA J., 61, 19–47 (2013)

  2. Ammari K., Henrot A., Tucsnak M.: Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string. Asymptot. Anal., 28(3–4), 215–240 (2001)

    MathSciNet  MATH  Google Scholar 

  3. K. Ammari and S. Nicaise. Stabilization of elastic systems by collocated feedback, volume 2124 of Lecture Notes in Mathematics. Springer, Cham, 2015.

  4. K. Ammari, G. Tenenbaum, and M. Tucsnak. Spectral conditions for the stability of a class of second order systems. Preprint

  5. Ammari K., Tucsnak M.: Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J. Control Optim., 39(4), 1160–1181 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ammari K., Tucsnak M.: Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM Control Optim. Calc. Var., 6, 361–386 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guo B.-Z., Wang J.-M., Yang K.-Y.: Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation. Systems Control Lett., 57(9), 740–749 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mercier D., Nicaise S.: Polynomial decay rate for a wave equation with weak dynamic boundary feedback laws. J. Abstr. Differ. Equ. Appl., 2(1), 29–53 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Miller L.: Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal., 218(2), 425–444 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Muñoz Rivera J.E., Racke R.: Large solutions and smoothing properties for nonlinear thermoelastic systems. J. Differential Equations, 127(2), 454–483 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Math. Sciences. Springer-Verlag, New York, 1983.

  12. L. Toufayli. Stabilisation polynomiale et contrôlabilité exacte des équations des ondes par des contrôles indirects et dynamiques. PhD thesis, Univ. Strasbourg, 2013.

  13. H. Triebel. Interpolation theory, function spaces, differential operators, volume 18 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1978.

  14. Wehbe A.: Rational energy decay rate for a wave equation with dynamical control. Appl. Math. Lett., 16(3), 357–364 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Weiss G., Curtain R.F.: Dynamic stabilization of regular linear systems. IEEE Trans. Automat. Control, 42(1), 4–21 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. UR Analysis and Control of PDEs, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019, Monastir, Tunisia

    Kaïs Ammari

  2. Laboratoire de Mathématiques et ses Applications de Valenciennes, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313, Valenciennes Cedex 9, France

    Zainab Abbas & Denis Mercier

Authors
  1. Zainab Abbas
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  2. Kaïs Ammari
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  3. Denis Mercier
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Correspondence to Kaïs Ammari.

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Abbas, Z., Ammari, K. & Mercier, D. Remarks on stabilization of second-order evolution equations by unbounded dynamic feedbacks. J. Evol. Equ. 16, 95–130 (2016). https://doi.org/10.1007/s00028-015-0294-2

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  • DOI: https://doi.org/10.1007/s00028-015-0294-2

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