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On the hamiltonian structure in the computation of singular values for a class of Hankel operators

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1496)

Abstract

The paper deals with the problem of computing the singular values of a Hankel operator with a symbol of the type m * w with m, w ε H + , m inner and w rational. In this case the problem reduces to the computation of zeroes of some functional determinant. This has been shown in Foias, Tannenbaum and Zames [1988], Lypchuk, Smith and Tannenbaum [1988], Smith [1989], Ozbay [1990]. The emphasis in this paper is to give a more illuminating and geometrical proof of the conjecture in Zhou and Khargonekar [1987]. This proof is based on the theory of polynomial models developed by the author, and it leans heavily on Smith [1989].

Keywords

  • Half Plane
  • Invariant Subspace
  • Polynomial Model
  • Singular Vector
  • Hankel Operator

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

  • [1968a] V. M. Adamjan, D. Z. Arov and M. G. Krein, “Infinite Hankel matrices and generalized problems of Caratheodory-Fejer and F. Riesz”, Funct. Anal. Appl. 2, 1–18.

    CrossRef  Google Scholar 

  • [1968b] V. M. Adamjan, D. Z. Arov and M. G. Krein, “Infinite Hankel matrices and generalized problems of Caratheodory-Fejer and I. Schur”, Funct. Anal. Appl. 2, 269–281.

    CrossRef  Google Scholar 

  • [1971] V. M. Adamjan, D. Z. Arov and M. G. Krein, “Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem”, Math. USSR Sbornik 15 (1971), 31–73.

    CrossRef  Google Scholar 

  • [1978] V. M. Adamjan, D. Z. Arov and M. G. Krein, “Infinite Hankel block matrices and related extension problems”, Amer. Math. Soc. Transl., series 2, Vol. 111, 133–156.

    Google Scholar 

  • [1971] R. G. Douglas, H. S. Shapiro and A. L. Shields, “Cyclic vectors and invariant subspaces for the backward shift”, Ann. Inst. Fourier, Grenoble, 20, 37–76.

    CrossRef  MathSciNet  Google Scholar 

  • [1957] N. Dunford and J. T. Schwartz, Linear Operators, vol. 1, Interscience, New York.

    Google Scholar 

  • [1987] H. Bercovici, C. Foias & A. Tannenbaum, “On skew Toeplitz operators, I”, Operator Theory: Advances and Applications, 29, 21–43.

    MathSciNet  Google Scholar 

  • [1987] C. Foias & A. Tannenbaum, “On the Nehari problem for a certain class of L -functions appearing in control, II”, Journal of Functional Analysis, 74, 146–159.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [1986] C. Foias, A. Tannenbaum & G. Zames, “Some explicit formulae for the singular values of certain Hankel operators with factorizable symbol”, SIAM J. Mathematical Analysis, 19, 1081–1091.

    CrossRef  MathSciNet  Google Scholar 

  • [1986] C. Foias, A. Tannenbaum & G. Zames, IEEE Trans. Aut. Contr. 31, 763–767.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [1976] P. A. Fuhrmann, “On series and parallel coupling of a class of discrete time infinite dimensional linear systems”, SIAM J. Contr. Optim. 14, 339–358.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [1981] P. A. Fuhrmann, Linear Systems and Operators in Hilbert Space, McGraw-Hill, New York.

    MATH  Google Scholar 

  • [1981] P. A. Fuhrmann, “Polynomial models and algebraic stability criteria”, Proceedings of Joint Workshop on Synthesis of Linear and Nonlinear Systems, Bielefeld June 1981, 78–90.

    Google Scholar 

  • [1983] P. A. Fuhrmann, “On symmetric rational transfer functions”, Linear Algebra and Appl., 50, 167–250.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [1984] P. A. Fuhrmann, “On Hamiltonian transfer functions”, Lin. Alg. Appl., 84, 1–93.

    CrossRef  MathSciNet  Google Scholar 

  • [1990] P. A. Fuhrmann, “A polynomial approach to Hankel norm and balanced approximations”, to appear, Lin. Alg. Appl., 101 pp.

    Google Scholar 

  • [1991] P. A. Fuhrmann and R. Ober, “A functional approach to Riccati balancing”, to appear.

    Google Scholar 

  • [1984] K. Glover, “All optimal Hankel-norm approximations and their L -error bounds”, Int. J. Contr. 39, 1115–1193.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [1989] U. Helmke and P. A. Fuhrmann, “Bezoutians”, Lin. Alg. Appl., vols. 122–124, 1039–1097.

    CrossRef  MathSciNet  Google Scholar 

  • [1988] T. A. Lypchuk, M. C. Smith &, A. Tannenbaum, “Weighted sensitivity minimization: general plants in H and rational weights,”, Lin. Alg. Appl. vol. 109, 71–90.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • [1985] N. K. Nikolskii, Treatise on the Shift Operator, Springer Verlag, Berlin.

    Google Scholar 

  • [1990] H. Ozbay “A simpler formula for the singular values of a certain Hankel operator”, Sys. and Contr. Lett..

    Google Scholar 

  • [1989] M. C. Smith, “Singular values and vectors of a class of Hankel operators”, Sys. and Contr. Lett. 12, 301–308.

    CrossRef  MATH  Google Scholar 

  • [1949] B. L. Van der Waerden, Modern Algebra, F. Ungar, New York.

    Google Scholar 

  • [1981] G. Zames, IEEE Trans. Aut. Contr. 16, 301, 320.

    CrossRef  MathSciNet  Google Scholar 

  • [1987] K. Zhou and P. P. Khargonekar, “On the weighted sensitivity minimization problem for delay systems”, Sys. and Contr. Lett. 8, 307–312.

    CrossRef  MATH  MathSciNet  Google Scholar 

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© 1991 Springer-Verlag Berlin Heidelberg

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Fuhrmann, P.A. (1991). On the hamiltonian structure in the computation of singular values for a class of Hankel operators. In: Mosca, E., Pandolfi, L. (eds) H-Control Theory. Lecture Notes in Mathematics, vol 1496. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084471

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  • DOI: https://doi.org/10.1007/BFb0084471

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