Some Backgrounds

  • Kaïs Ammari
  • Serge Nicaise
Part of the Lecture Notes in Mathematics book series (LNM, volume 2124)


In the whole book (except in Chaps.  4 and  5), X is a complex and separable Hilbert space with norm and inner product denoted respectively by \(\|\cdot \|_{X}\) and (⋅ , ⋅ ) X .


  1. 25.
    S.A. Avdonin, S.A. Ivanov, Families of Exponentials (Cambridge University Press, Cambridge, 1995). The method of moments in controllability problems for distributed parameter systems, Translated from the Russian and revised by the authorsGoogle Scholar
  2. 26.
    C. Baiocchi, V. Komornik, P. Loreti, Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hung. 97, 55–95 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 37.
    J.W. Cassals, An Introduction to Diophantine Approximation (Cambridge University Press, Cambridge, 1966)Google Scholar
  4. 60.
    I.C. Gohberg, M.G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, vol. 18 (American Mathematical Society, Providence, 1969)Google Scholar
  5. 67.
    G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 6th edn. (Oxford University Press, Oxford, 2008). Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew WilesGoogle Scholar
  6. 69.
    A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41(1), 367–379 (1936)CrossRefMathSciNetGoogle Scholar
  7. 71.
    S. Jaffard, M. Tucsnak, E. Zuazua, Singular internal stabilization of the wave equation. J. Differ. Equ. 145(1), 184–215 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 75.
    V. Komornik, P. Loreti, Fourier Series in Control Theory. Springer Monographs in Mathematics (Springer, New York, 2005)Google Scholar
  9. 80.
    S. Lang, Introduction to Diophantine Approximations, 2nd edn. (Springer, New York, 1995)CrossRefzbMATHGoogle Scholar
  10. 107.
    S. Nicaise, C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45(5), 1561–1585 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 108.
    S. Nicaise, J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2(3), 425–479 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 111.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. (Springer, New York, 1983)Google Scholar
  13. 114.
    R. Rebarber, Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33(1), 1–28 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 123.
    M. Tucsnak, Regularity and exact controllability for a beam with piezoelectric actuator. SIAM J. Control Optim. 34(3), 922–930 (1996)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Kaïs Ammari
    • 1
  • Serge Nicaise
    • 2
  1. 1.Dept of Mathematics Faculty of SciencesUniversity of MonastirMonastirTunisia
  2. 2.Lab. de Mathématiques et de leurs applications de Valenciennes (LAMAV)Univ. de Valenciennes et du Hainaut CambrésisValenciennesFrance

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