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Asymptotic Behaviour of Concrete Dissipative Systems

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Stabilization of Elastic Systems by Collocated Feedback

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2124))

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Abstract

We study the large time behavior of the solution of a homogenous string equation with a homogenous Dirichlet boundary condition at the left end and a homogenous Dirichlet or Neumann boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term.

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Bibliography

  1. K. Ammari, Stabilisation d’une classe d’équations d’évolution du deuxième ordre en temps, Ph.D. thesis, Ecole Polytechnique, 2000

    Google Scholar 

  2. K. Ammari, Dirichlet boundary stabilization of the wave equation. Asymptot. Anal. 30(2), 117–130 (2002)

    MATH  MathSciNet  Google Scholar 

  3. K. Ammari, A. Henrot, M. Tucsnak, 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)

    MATH  MathSciNet  Google Scholar 

  4. K. Ammari, M. Jellouli, Stabilization of star-shaped networks of strings. Differ. Integral Equ. 17(11–12), 1395–1410 (2004)

    MATH  MathSciNet  Google Scholar 

  5. K. Ammari, M. Jellouli, Remark on stabilization of tree-shaped networks of strings. Appl. Math. 52(4), 327–343 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. K. Ammari, M. Jellouli, M. Khenissi, Stabilization of generic trees of strings. J. Dyn. Control Syst. 11(2), 177–193 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. K. Ammari, Z. Liu, M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback. Math. Control Signals Syst. 15(3), 229–255 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. K. Ammari, D. Mercier, V. Régnier, J. Valein, Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Commun. Pure Appl. Anal. 11(2), 785–807 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. C. Bardos, L. Halpern, G. Lebeau, J. Rauch, E. Zuazua, Stabilisation de l’équation des ondes au moyen d’un feedback portant sur la condition aux limites de Dirichlet. Asymptot. Anal. 4(4), 285–291 (1991)

    MATH  MathSciNet  Google Scholar 

  12. C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30(5), 1024–1065 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. C. Castro, S.J. Cox, Achieving arbitrarily large decay in the damped wave equation. SIAM J. Control Optim. 39(6), 1748–1755 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Chen, M. Coleman, H.H. West, Pointwise stabilization in the middle of the span for second order systems, nonuniform and uniform exponential decay of solutions. SIAM J. Appl. Math. 47(4), 751–780 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Chen, M.C. Delfour, A.M. Krall, G. Payre, Modeling, stabilization and control of serially connected beams. SIAM J. Control Optim. 25(3), 526–546 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  16. G. Chen, S.G. Krantz, D.L. Russell, C.E. Wayne, H.H. West, M. P. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams. SIAM J. Appl. Math. 49(6), 1665–1693 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. F. Conrad, Stabilization of beams by pointwise feedback control. SIAM J. Control Optim. 28(2), 423–437 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Cox, E. Zuazua, The rate at which energy decays in a damped string. Commun. Partial Differ. Equ. 19(1–2), 213–243 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Dáger, Observation and control of vibrations in tree-shaped networks of strings. SIAM J. Control Optim. 43(2), 590–623 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. R. Dáger, E. Zuazua, Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci. Paris Sér. I Math. 332(12), 1087–1092 (2001)

    Article  MATH  Google Scholar 

  21. R. Dáger, E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 50 (Springer, Berlin, 2006)

    Google Scholar 

  22. B. Dekoninck, S. Nicaise, The eigenvalue problem for networks of beams. Linear Algebra Appl. 314(1–3), 165–189 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. P. Freitas, Optimizing the rate of decay of solutions of the wave equation using genetic algorithms: a counterexample to the constant damping conjecture. SIAM J. Control Optim. 37(2), 376–387 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24 (Pitman, Boston, London, Melbourne, 1985)

    Google Scholar 

  25. F.L. Huang, Characteristic conditions for exponential stability of linear dynamical system in Hilbert spaces. Ann. Differ. Equ. 1, 43–56 (1985)

    MATH  Google Scholar 

  26. A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41(1), 367–379 (1936)

    Article  MathSciNet  Google Scholar 

  27. S. Jaffard, M. Tucsnak, E. Zuazua, Singular internal stabilization of the wave equation. J. Differ. Equ. 145(1), 184–215 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. D. Jerison, C.E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. J.E. Lagnese, G. Leugering, E. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures (Birkhäuser, Boston, 1994)

    Book  MATH  Google Scholar 

  30. I. Lasiecka, J.-L. Lions, R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. (9) 65(2), 149–192 (1986)

    Google Scholar 

  31. I. Lasiecka, R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with \(L_{2}(0,\infty;L_{2}(\Gamma ))\)-feedback control in the Dirichlet boundary conditions. J. Differ. Equ. 66(3), 340–390 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  32. I. Lasiecka, R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim. 25(2), 189–224 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  33. G. Lebeau, Équation des ondes amorties, in Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993). Mathematical Physics Studies, vol. 19 (Kluwer Academic, Dordrecht, 1996), pp. 73–109

    Google Scholar 

  34. G. Lebeau, L. Robbiano, Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86(3), 465–491 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  35. G. Leugering, E. Zuazua, On exact controllability of generic trees, in Contrôle des systèmes gouvernés par des équations aux dérivées partielles (Nancy, 1999). ESAIM Proceedings, vol. 8 (Society of Industrial and Applied Mathematics, Paris, 2000), pp. 95–105

    Google Scholar 

  36. J.-L. Lions, Contrôle des systèmes distribués singuliers. Méthodes Mathématiques de l’Informatique, vol. 13. (Gauthier-Villars, Montrouge, 1983)

    Google Scholar 

  37. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8 (Masson, Paris, 1988) Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch

    Google Scholar 

  38. J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications. vol. 1. Travaux et Recherches Mathématiques, No. 17 (Dunod, Paris, 1968)

    Google Scholar 

  39. K.S. Liu, Energy decay problems in the design of a point stabilizer for coupled string vibrating systems. SIAM J. Control Optim. 26(6), 1348–1356 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  40. K.S. Liu, F.L. Huang, G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage. SIAM J. Appl. Math. 49(6), 1694–1707 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  41. F.A. Mehmeti, A characterisation of generalized c ∞ notion on nets. Integral Eq. Operator Theory 9, 753–766 (1986)

    Article  MATH  Google Scholar 

  42. F.A. Mehmeti, Nonlinear Wave in Networks. Mathematical Research, vol. 80 (Akademie Verlag, Berlin, 1994)

    Google Scholar 

  43. S. Nicaise, Diffusion sur les espaces ramifiés, Ph.D. thesis, U. Mons (Belgium), 1986

    Google Scholar 

  44. S. Nicaise, Spectre des réseaux topologiques finis. Bull. Sc. Math., 2ème série 111, 401–413 (1987)

    Google Scholar 

  45. J. Prüss, On the spectrum of C 0-semigroups. Trans. Am. Math. Soc. 284(2), 847–857 (1984)

    Article  MATH  Google Scholar 

  46. R. Rebarber, Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33(1), 1–28 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  47. L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptotic Anal. 10(2), 95–115 (1995)

    MATH  MathSciNet  Google Scholar 

  48. M. Tucsnak, Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim. 34(3), 922–930 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  49. M. Tucsnak, On the pointwise stabilization of a string, in Control and Estimation of Distributed Parameter Systems (Vorau, 1996). International Series of Numerical Mathematics, vol. 126 (Birkhäuser, Basel, 1998), pp. 287–295

    Google Scholar 

  50. J. von Below, A characteristic equation associated to an eigenvalue problem on c 2-networks. Linear Algebra Appl. 71, 309–325 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  51. J. von Below, Parabolic network equations, Habilitation thesis, 1993

    Google Scholar 

  52. G. Weiss, R.F. Curtain, Exponential stabilization of a Rayleigh beam using collocated control. IEEE Trans. Automat. Control 53(3), 643–654 (2008)

    Article  MathSciNet  Google Scholar 

  53. G. Weiss, O.J. Staffans, M. Tucsnak, Well-posed linear systems—a survey with emphasis on conservative systems. Int. J. Appl. Math. Comput. Sci. 11(1), 7–33 (2001). Mathematical theory of networks and systems (Perpignan, 2000)

    Google Scholar 

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Authors and Affiliations

  1. Dept of Mathematics Faculty of Sciences, University of Monastir, Monastir, Tunisia

    Kaïs Ammari

  2. Lab. de Mathématiques et de leurs applications de Valenciennes (LAMAV), Univ. de Valenciennes et du Hainaut Cambrésis, Valenciennes, France

    Serge Nicaise

Authors
  1. Kaïs Ammari
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  2. Serge Nicaise
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Ammari, K., Nicaise, S. (2015). Asymptotic Behaviour of Concrete Dissipative Systems. In: Stabilization of Elastic Systems by Collocated Feedback. Lecture Notes in Mathematics, vol 2124. Springer, Cham. https://doi.org/10.1007/978-3-319-10900-8_4

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