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Exact controllability for wave equation with Neumann boundary control

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Boundary Control and Boundary Variations

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 100))

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Abstract

the author presented a summary of very recent results on exact controllability for the wave equation under boundary control exercised either in the Dirichlet or else in the Neumann boundary conditions. For lack of space, the present paper deals exclusively with the Neumann case, while for the Dirichlet case reference is made to [T.1].

The results presented in this paper were obtained jointly with I. Lasiecka and J. L. Lions during the period February – July 1987 and are part of a more complete and more comprehensive joint work by I. Lasiecka, J. L. Lions, and R. Triggiani presently in progress.

The work of the first two authors was sponsored by AFOSR under Grant 84-0365A and by NSF under Grant DMS-8301668 whose financial support is gratefully acknowledged.

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References

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

    Google Scholar 

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

    Google Scholar 

  3. K. Graham and D. L. Russell, Boundary value control of the wave equation, in a spherical region, SIAM J. Control 13 (1975), 174–196.

    Google Scholar 

  4. L. F. Ho, Obserbilite frontiere de l'equation des CRAS, 302, Paris, 1986.

    Google Scholar 

  5. F. John, On linear partial differential equations with analytic coefficients — Unique continuation of data. Comm. Pure Appl. Math 2 (1949) 209–253.

    Google Scholar 

  6. J. Lagnese, De cay of solutions of wave equations in a bounded region with boundary dissipation J. Diff. Equats. 50 (1983), 163–182.

    Google Scholar 

  7. J. L. Lions, Controle des systemes distribues singuliers, Gauthier Villars, 1983.

    Google Scholar 

  8. J. L. Lions, Controlabilite exacte de systemes distribues C.R.A.S. 302, Paris (1986), 471–475.

    Google Scholar 

  9. J. L. Lions, Controlabilite exacte de systemes distribues: remarques sur le theorie generale et les applications, Proceedings of 7th International Conference on Analysis & Optimization of Systems, Antibes, France, June 25–27, 1986; Lecture Notes in CIS, 1–13.

    Google Scholar 

  10. J. L. Lions, Exact controllability of distributed systems. An introduction Proceedings of 25th Conference on Decision and Control, Athens, Greece, December 1986.

    Google Scholar 

  11. J. L. Lions, Exact controllability, stabilization and perturbations, J. Von Neumann Lecture, SIAM July 1986.

    Google Scholar 

  12. P. L. Lions, private communication.

    Google Scholar 

  13. W. Littman, Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients, Anneli Scuole Normale Superiore de Pisa, Classe Suarze, Serie IV, Vol. 3 (1978), 567–580.

    Google Scholar 

  14. W. Littman, Near optimal time boundary controllability for a class of hyperbolic equations Proceedings of the IFIP WG7.2 Working conference on ‘Control Systems governed by partial differential equations’ held at the University of Florida, Gainesville, February 3–6, 1986, to appear in Springer Verlag series.

    Google Scholar 

  15. W. Littman, private communication.

    Google Scholar 

  16. I. Lasiecka, J. L. Lions, and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, Journ. de Mathematiques Pures et Appliques, 65, (1986), 149–192.

    Google Scholar 

  17. I. Lasiecka, and R. Triggiani, A cosine opoerator approach to modeling L2(0,T; L2(Γ)) — boundary input hyperbolic equations, Applied Math & Optimization 7(1981), 35–83.

    Google Scholar 

  18. I. Lasiecka, and R. Triggiani, Uniform exponential energy decay of the wave equation in a bounded region with L2(0,∞; L2(Γ)) — feedback control in the Dirichlet boundary conditions, J. Diff. Eqts. 66 (1987), 340–390.

    Google Scholar 

  19. I. Lasiecka, and R. Triggiani, Sharp regularity results for second order hyperbolic equations of Neumann type, preprint 1986, to appear.

    Google Scholar 

  20. D. L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions.

    Google Scholar 

  21. R. Triggiani, Exact boundary controllability on L2(Q)xH−1(Q) for the wave equation with Dirichlet control acting on a portion of the boundary, and related problems, preprint 1986, submitted.

    Google Scholar 

  22. R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, to appear in “Operator Methods for Optimal Control Problems” Proceedings of special session at Annual Meeting of the A.M.S. held at New Orleans, January 1986, Marcel Dekker.

    Google Scholar 

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J. P. Zolésio

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© 1988 Springer-Verlag

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Triggiani, R. (1988). Exact controllability for wave equation with Neumann boundary control. In: Zolésio, J.P. (eds) Boundary Control and Boundary Variations. Lecture Notes in Control and Information Sciences, vol 100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0041924

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  • DOI: https://doi.org/10.1007/BFb0041924

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