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Applied Mathematics and Optimization

, Volume 15, Issue 1, pp 223–250 | Cite as

Exact boundary controllability of an integrodifferential equation

  • Günter Leugering
Article

Abstract

An integrodifferential equation of the Volterra type is considered under the action of anL2(0, T, L2(Γ))-boundary control. By harmonic analysis arguments it is shown that the controllability results obtained in [17] for the underlying reference model associated with a trivial convolution kernel, carry over to the model under consideration without any smallness assumption concerning the memory kernel.

Keywords

System Theory Harmonic Analysis Mathematical Method Reference Model Boundary Control 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Balakrishnan AV (1976) Applied Functional Analysis. Springer-Verlag, New YorkGoogle Scholar
  2. 2.
    Baumeister J (1983) Boundary control of an integrodifferential equation. J Math Anal Appl 93:550–570Google Scholar
  3. 3.
    Becker E, Bürger W (1975) Kontinuumsmechanik. Teubner-Verlag, StuttgartGoogle Scholar
  4. 4.
    Chadwick RS, Inselberg A (1976) Mathematical model of the Cochlea. I: Formulation and solution. SIAM J Appl Math 30:149–163Google Scholar
  5. 5.
    Christensen RM (1971) The Theory of Linear Viscoelasticity: An Introduction. Academic Press, New YorkGoogle Scholar
  6. 6.
    Curtain RF (1984) Linear-quadratic control problems with fixed endpoints in infinite dimensions. J Optim Theory Appl 44:55–74Google Scholar
  7. 7.
    Cushing JM (1977) Integrodifferential Equations and Delay Models in Population Dynamics. Lecture Notes in Biomathematics, vol 20. Springer-Verlag, New YorkGoogle Scholar
  8. 8.
    Dafermos CM (1970) An abstract Volterra equation with applications to linear viscoelasticity. J Differential Equations 7:554–569Google Scholar
  9. 9.
    Desch, GW, Grimmer RC (1985) Initial boundary value problems for integrodifferential equations.J Integral Equations 10:73–97Google Scholar
  10. 10.
    Graham KD, Russell DL (1975) Boundary value control of the wave equation in a spherical region. SIAM J Control Optim 12:174–196Google Scholar
  11. 11.
    Grimmer RC (1982) Resolvent operators for integral equations in a Banach space. Trans Amer Math Soc 273:333–349Google Scholar
  12. 12.
    Grimmer RC, Schappacher W (1984) Weak solutions of integrodifferential equations and resolvent operators. J Integral Equations 6:205–229Google Scholar
  13. 13.
    Grimmer RC, Zeman M (1984) Wave propagation for linear integrodifferential equations in Banach space. J Differential Equations 54:274–282Google Scholar
  14. 14.
    Ingham AE (1936) Some trigonometrical inequalities with applications to the theory of series. Math Z 41:367–379Google Scholar
  15. 15.
    Kato T (1976) Perturbation Theory for Linear Operators. Springer-Verlag, New YorkGoogle Scholar
  16. 16.
    Komkov V (1972) Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems. Lecture Notes in Mathematics, vol 253. Springer-Verlag, New YorkGoogle Scholar
  17. 17.
    Krabs W, Leugering G, Seidman TI (1985) Boundary controllability of a vibrating plate. Appl Math Optim 13:205–229Google Scholar
  18. 18.
    Lasiecka I, Triggiani R (1981) A cosine operator approach to modelingL 2(0,T, L 2(Γ))-boundary hyperbolic equations. Appl Math Optim 7:35–93Google Scholar
  19. 19.
    Levinson N (1940) Gap and Density Theorems. Colloquium Publications, vol XXVI. American Mathematical Society, Providence, RIGoogle Scholar
  20. 20.
    Miller RK, Wheeler RL (1978) Well posedness and stability of linear Volterra integrodifferential equation in abstract spaces. Funkcial Ekvac 21:279–305Google Scholar
  21. 21.
    Narukawa K (1983) Boundary value control of thermoelastic systems. Hiroshima Math J 13:227–272Google Scholar
  22. 22.
    Narukawa K, Exact and admissible controllability of viscoelastic systems with boundary controls (preprint)Google Scholar
  23. 23.
    Nečas J (1967) Les méthodes directes en théorie des équation elliptique. Masson et Cie, ParisGoogle Scholar
  24. 24.
    Nerain A, Joseph DD (1982) Linearized dynamics for step jumps of velocity and displacement of shearing flows of a simple fluid. Rheol Acta 21:228–250Google Scholar
  25. 25.
    Nevanlinna R (1953) Eindeutige analytische Funktionen, 2nd edn. Grundlehren der mathematischen Wissenschaften, Band XLVI. Springer-Verlag, BerlinGoogle Scholar
  26. 26.
    Paley REAC, Wiener N (1934) Fourier Transforms in the Complex Domain. Colloquium Publications, vol XIX. American Mathematical Society, Providence, RIGoogle Scholar
  27. 27.
    Renardy M (1982) Some remarks on the propagation and nonpropagation of discontinuities in linearly viscoelastic liquids. Rheol Acta 21:251–254Google Scholar
  28. 28.
    Tsuruta K (1984) Bounded linear operators satisfying second-order integrodifferential equations in a Banach space. J Integral Equations 6:231–268Google Scholar
  29. 29.
    Velte W (1976) Direkte Methoden der Variationsrechnung. Teubner-Verlag, StuttgartGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Günter Leugering
    • 1
  1. 1.Technische Hochschule DarmstadtDarmstadtWest Germany

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