Spectral value sets of infinite-dimensional systems

  • E. Gallestey
  • D. Hinrichsen
  • A. J. Pritchard
Part of the Communications and Control Engineering book series (CCE)


We assume that X, X, U, Y are complex or real separable Banach spaces, A with dense domain D(A) ⊂ X is a closed linear operator on X, D(A) ⊂ XX with continuous dense injections, B ∈ ℒ(U,X) and C ∈ ℒ (X,Y). Let K denote the field of scalars. Our subject is the variation of the spectrum, σ(A) under structured perturbations of the form
$$ A \rightsquigarrow {A_\Delta } = A + B\Delta C,\,\,\Delta \in \mathcal{L}\left( {Y,U} \right) $$
where D(A Δ) = D(A). The operators B, C are fixed and describe both the structure and unboundedness of the perturbations, whilst Δ ∈ ℒ (Y, U) is arbitrary. If U = X = X = Y, B = I x = C, i.e. A Δ = A + Δ, Δ ∈ ℒ (X) the perturbations are bounded and are said to be unstructured.


Stability Radius Structure Perturbation Closed Linear Operator Dimensional Linear Subspace Finite Dimensional System 
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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  1. [1]
    A. Böttcher. Pseudospectra and singular values of large convolution operators. J. Integral Equations and Applications 6: 267–301, 1994.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    F. Chatelin. Spectral Approximation of Linear Operators. Academic Press, London, 1983.MATHGoogle Scholar
  3. [3]
    E. Gallestey. Computing spectral value sets using the subharmonicity of the norm of rational matrices. BIT 38: 22–33, 1998.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    E. Gallestey, D. Hinrichsen, and A. J. Pritchard. On the spectral value set problem in infinite dimensions. In Proc. 4rd International Symposium on Methods and Models in Automation and Robotics, Miedzyzdroje (Poland), pp 109–114, 1997.Google Scholar
  5. [5]
    S. K. Godunov. Spectral portraits of matrices and criteria of spectrum dichotomy. In Proc. of Internat. Conf. on Computer Arithmetic, Scientific Computation and Mathematical Modelling. SCAN-91, Oldenburg, 1991.Google Scholar
  6. [6]
    T. Herbert. On perturbation methods in nonlinear stability theory. J. Fluid Mech., 126: 167–186, 1983.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    D. Hinrichsen and B. Kelb. Spectral value sets: a graphical tool for robustness analysis. Systems & Control Letters, 21: 127–136, 1993.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    D. Hinrichsen and B. Kelb. Stability radii and spectral value sets for real matrix perturbations. In Systems and Networks: Mathematical Theory and Applications, Vol.11 Invited and Contributed Papers, Akademie-Verlag, Berlin, 217–220, 1994.Google Scholar
  9. [9]
    D. Hinrichsen and A. J. Pritchard. On spectral variations under bounded real matrix perturbations. Numerische Mathematik 60: 509–524, 1992.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    K. Ito and F. Kappel A uniformly differentiate approximation scheme for delay systems using splines. Applied Mathematics and Optimization 23: 217–262, 1991.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    T. Kato. Perturbation Theory for Linear Operators. Springer-Verlag, Berlin-Heidelberg-New York, 1976.MATHGoogle Scholar
  12. [12]
    S. C. Reddy, P. Schmidt, and D. Henningson. Pseudospectra of the Orr-Sommerfeld operator. SIAM J. Appl. Math., 53: 15–47, 1993.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    D. Salamon. Structure and stability of finite dimensional approxima-tions for functional differential equations. SIAM J. Contr. & Opt. 23: 928–951, 1985.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    L. N. Trefethen. Pseudospectra of matrices. Report 91/10, Oxford University Computer Laboratory, 1991.Google Scholar
  15. [15]
    L. N. Trefethen. Pseudospectra of linear operators. SIAM Rev. 39: 383–406, 1997.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 1999

Authors and Affiliations

  • E. Gallestey
    • 1
  • D. Hinrichsen
    • 1
  • A. J. Pritchard
    • 2
  1. 1.Institut für Dynamische SystemeUniversität BremenBremenGermany
  2. 2.Mathematics DepartmentUniversity of WarwickCoventryUK

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