Skip to main content
SpringerLink
Log in

Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions

  • Published:
Applied Mathematics and Optimization Submit manuscript

Abstract

In this paper we eliminate altogether geometrical conditions that were assumed (even) with control action on the entire boundary in prior literature: (i) strict convexity of our paper [LT4] on uniform stabilization of the wave equation in the (optimal) state spaceL 2(Ω)×H −1(Ω) withL 2(Σ) Dirichlet feedback control, as well as (ii) “star-shaped” conditions in papers [C1], [La1], and [Tr1] on uniform stabilization and [Lio1] and [LT5] on exact controllability in the energy spaceH 1(Ω)×L 2(Ω) of the wave equation withL 2(Σ)-Neumann feedback control. Key to the present improvements is a pseudodifferential analysis which permits us to express certain boundary traces of the solution in terms of other traces modulo lower-order interior terms. See Lemma 3.1 for the Dirichlet case and Lemma 7.2 for the Neumann case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. Anderson and R. Melrose, The propagation of singularities along gliding rays, Invent. Math. 41 (1977), 197–232.

    Google Scholar 

  2. C. Bardos, L. Halperin, G. Lebeau, J. Rauch, and E. Zuazua, Stabilisation de l'equation des ondes an moyen d'um feedback portant sur la condition aux limites des Dirichlet, Asymptotic Analysis (to appear).

  3. C. Bardos, G, Lebeau, and R. Rauch, Controle et stabilisation dans des problems hyperboliques, Appendix II in J. L. Lions [Lio2].

  4. C. Bardos, G. Lebeau, and J. Rauch, Un example d'utilisation des notions de propagation pour le controle et la stabilisation de problemes hyperboliques, Rend. Sem. Math. Univ. Politec. Torino, (1988). Fascicolo spéciale.

  5. C. Bardos, G. Lebeau, and J. Rauch,Microlocal Ideas in Control and Stabilization, Lectures Notes in Control and Information Sciences, Vol. 125, Springer-Verlag, Berlin, 1990, pp. 14–30.

    Google Scholar 

  6. C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation control and stabilization of waves from the boundary, SIAM J. Control. Optim.

  7. G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl. (9) 58 (1979), 249–274,

    Google Scholar 

  8. G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim. 19 (1981), 106–113.

    Google Scholar 

  9. F. Flandoli, I. Lasiecka, and R. Triggiani, Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli equations, Ann. Mat. Pura Appl. (4) CLIII (1988), 307–382.

    Google Scholar 

  10. F. L. Ho, Observabilité frontiere de l'equation des ondes, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), 443–446.

    Google Scholar 

  11. L. Hormander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1969.

    Google Scholar 

  12. L. Hormander, The Analysis of Linear Partial Differential Operators, Vols. I, III, Springer-Verlag, Berlin, 1983, 1985.

    Google Scholar 

  13. N. Iwasaki, Local decay of solutions for symmetric hyperbolic systems with dissipative and coercive boundary conditions in exterior domains, Publ. Res. Inst. Math. Sci. 5 (1969), 193–218.

    Google Scholar 

  14. B. Kellog, Properties of elliptic B.V.P., in The Mathematical Foundations of the Finite Element Method, Academic Press, New York, 1972, Chapter 3.

    Google Scholar 

  15. J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50 (1983), 163–182.

    Google Scholar 

  16. J. Lagnese, A note on the boundary stabilization of wave equations, SIAM J. Control Optim. 26 (1988), 1250–1256.

    Google Scholar 

  17. J. Lagnese, Uniform Boundary Stabilization of Homogeneous Isotropie Plates, Lecture Notes in Control and Information Sciences, Vol. 102, Springer-Verlag, Berlin, 1988, pp. 204–215.

    Google Scholar 

  18. J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1989.

    Google Scholar 

  19. J. Lagnese and J. L. Lions, Modelling, Analysis and Control of Thin Plates, Masson, Paris, 1988.

    Google Scholar 

  20. I. Lasiecka, J. L. Lions, and R. Triggiani, Non-homogeneous boundary value problems for second-order hyperbolic operators, J. Math. Pures Appl. 65 (1986), 149–192.

    Google Scholar 

  21. I. Lasiecka and D. Tataru. Uniform boundary stabilization of semilinear wave equation with nonlinear boundary conditions, 1990.

  22. I. Lasiecka and R. Triggiani, A cosine operator approach to modelingL 2(0,T; L 2(Γ))-boundary input hyperbolic equations, Appl. Math. Optim. 7 (1981), 35–83.

    Google Scholar 

  23. I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations underL 2(0,T; L 2(Γ))-boundary terms, Appl. Math. Optim. 10 (1983), 275–286.

    Google Scholar 

  24. I. Lasiecka and R. Triggiani, Riccati equations for hyperbolic partial differential equations withL 2(0,T; L 2(Γ))-Dirichlet boundary terms, SIAM J. Control Optim. 24 (1986), 884–926.

    Google Scholar 

  25. I. Lasiecka and R. Triggiani, Uniform exponential energy decay of the wave equation in a bounded region withL 2(0,T; L 2(Γ))-feedback control in the Dirichlet boundary conditions, J. Differential Equations 66 (1987), 340–390.

    Google Scholar 

  26. I. Lasiecka and R. Triggiani, Exact controllability for the wave equation with Neumann boundary control, Appl. Math. Optim. 19 (1989), 243–290. Preliminary version: Lecture Notes in Central and Information Sciences, Vol. 100, Springer-Verlag, Berlin, pp. 317–371.

    Google Scholar 

  27. I. Lasiecka and R. Triggiani, Sharp regularity results for mixed second-order hyperbolic equations of Neumann type: theL 2-boundary case, Ann. Mat. Pura Appl. (4) CLVII (1990), 285–367.

    Google Scholar 

  28. I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in Dirichlet and Neumann boundary conditions: a non-conservative case, SIAM J. Control Optim. 27 (1989), 330–373.

    Google Scholar 

  29. I. Lasiecka and R. Triggiani, Sharp trace estimates for solutions of Kirchhoff and Euler-Bernoulli equations, 1991.

  30. P. D. Lax and R. S. Phillips, Decaying models for the wave equation in the exterior of an obstacle, Comm. Pure Appl. Math. 22 (1969), 737–787.

    Google Scholar 

  31. J. Lebeau, Control de l'équation de Schrodinger, J. Analyse Math. (to appear).

  32. J. L. Lions, Control of Singular Distributed Systems, Gauthier-Villars, Paris, 1983.

    Google Scholar 

  33. J. L. Lions, Exact controllability, stabilization, and perturbations, SIAM Rev. 30 (1988), 1–68. Extended version: Collection RMA, Vol. 8, Masson, Paris, 1988.

    Google Scholar 

  34. W. Littman, Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. V (1978), 567–580.

    Google Scholar 

  35. W. Littman, Near Optimal Time Boundary Controllability for a Class of Hyperbolic Equations, Lecture Notes in Control and Information Sciences, Vol. 97, Springer-Verlag, Berlin, 1987, pp. 307–312.

    Google Scholar 

  36. W. Littman, Talk at Workshop at Saphie-Antipolis, October 1990.

  37. R. Melrose and J. Sjostrand, Singularities of boundary value problems, I, II, Comm. Pure Appl. Math. 31 (1978), 593–617, 35 (1982), 129–168.

    Google Scholar 

  38. C. S. Morawetz, Energy identities of the wave equation, Research Report No. IMM 346, NYU, Courant Institute Mathematical Sciences, New York, 1976.

    Google Scholar 

  39. C. Morawetz, J. Ralston, and W. Strauss, Decay of solutions of the wave equ non-trapping obstacle, Comm. Pure Appl. Math. 30 (1977), 447–508.

    Google Scholar 

  40. J. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math. 22 (1969), 807–823.

    Google Scholar 

  41. J. Ralston, Gaussian beams and the propagation of singularities, in Studies in Partial Differential Equations, ed. W. Littman, MAA Studies in Mathematics, Vol. 23, Mathematical Association of America, Washington, DC, 1982, pp. 206–248.

    Google Scholar 

  42. J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J. 24 (1974), 79–86.

    Google Scholar 

  43. D. L. Russell, Exact boundary controllability theorems for wave and heat processes in star complemented regions, in Differential Games and Control Theory, Roxin, Lin, Sternberg eds., Marcell Dekker, New York, 1974, pp. 291–320.

    Google Scholar 

  44. J. Simon, Compact sets in the spaceL p (0, T; B), Ann. Mat. Pura Appl. (4) CXLVI (1987), 65–96.

    Google Scholar 

  45. W. Strauss, Dispersal of waves vanishing on the boundary of an exterior domain, Comm. Pure Appl. Math. 28 (1976), 265–278.

    Google Scholar 

  46. B. M. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981.

    Google Scholar 

  47. M. Taylor, Reflection of singularities of solutions to systems of differential equations, Comm. Pure Appl. Math. 28 (1975), 457–478.

    Google Scholar 

  48. R. Triggiani, Exact boundary controllability onL 2(Ω)×H −1(Ω) of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems, Appl. Math. Optim. 18 (1988), 241–277. Preliminary version: Lecture Notes in Control and Information Sciences, Vol. 102, Springer-Verlag, Berlin, 1987, pp. 291–332; Proceedings, Workshop on Control for Distributed Parameter Systems, University of Graz, Austria, July 1986.

    Google Scholar 

  49. R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, J. Math. Anal. Appl. 137 (1989), 438–461. Preliminary version in Operator Methods for Optimal Control Problems (S. J. Lee, ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 108, Marcel Dekker, New York, 1988, pp. 283–309.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Thornton Hall, University of Virginia, Charlottesville, VA, 22903, USA

    I. Lasiecka & R. Triggiani

Authors
  1. I. Lasiecka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Triggiani
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lasiecka, I., Triggiani, R. Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl Math Optim 25, 189–224 (1992). https://doi.org/10.1007/BF01182480

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01182480

Keywords

Search

Navigation