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An introduction to the hilbert uniqueness method

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Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 185))

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Abstract

What relates to P.D.E. and what relates to general principles?

Uniqueness property requires deep results from P.D.E. theory (Holmgren’s theorem).

Exact controllability (estimate (ΓT h, h)≥c 0∥h∥ 2(D(A))′ ). A lot of developments can be made based on the properties of the pair (A, B) (system theoretic approach), more precisely the spectral properties of A play an instrumental role.

Interesting question: microlocal analysis leads to optimum estimates (unlike the method of multipliers). Can we combine this method with the system theoretic approach?

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References

  1. C. BARDOS, G. LEBEAU, J. RAUCH: Controôle et stabilisation dans des problèmes hyperboliques, Appendix 2 in J.L. LIONS [8]

    Google Scholar 

  2. A. BENSOUSSAN, G. DA PRATO, M. DELFOUR, S.K. MITTER: Infinite Dimensional System Theory, Birkhaüser, 1992.

    Google Scholar 

  3. J.E. LAGNESE: The Hilbert Uniqueness Method: A retrospective, Preprint.

    Google Scholar 

  4. J.E. LAGNESE, J.L. LIONS: Modelling, Analysis and Control of Thin Plates, RMA, Vol. 6, Masson, 1988.

    Google Scholar 

  5. I. LASIECKA, J.L. LIONS, R. TRIGGIANI: Nonhomogeneous boundary value problem for 2nd order hyperbolic operators, JMPA, 65 (1986), 149–190.

    MATH  MathSciNet  Google Scholar 

  6. I. LASIECKA, R. TRIGGIANI: Exact controllability of the wave equation with Neumann boundary control, AMO, 19 (1989), 243–290.

    MATH  MathSciNet  Google Scholar 

  7. J.L. LIONS: Exact controllability, stabilization and perturbations for distributed parameter systems, SIAM Review, 30 (1988).

    Google Scholar 

  8. J.L. LIONS: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1,2, RMA, Vol. 8,9, 1988.

    Google Scholar 

  9. D.L. RUSSEL: Controllability and stabilization theory for linear partial differential equations, SIAM Review 20 (1978), 639–739.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Université Paris Dauphine et INRIA, France

    A. Bensoussan

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  1. A. Bensoussan
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R. F. Curtain A. Bensoussan J. L. Lions

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© 1993 Springer-Verlag

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Bensoussan, A. (1993). An introduction to the hilbert uniqueness method. In: Curtain, R.F., Bensoussan, A., Lions, J.L. (eds) Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems. Lecture Notes in Control and Information Sciences, vol 185. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0115025

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  • DOI: https://doi.org/10.1007/BFb0115025

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56155-2

  • Online ISBN: 978-3-540-47480-7

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