Skip to main content
SpringerLink
Log in

The Nehari problem for nonexponentially stable systems

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

The Nehari problem and its suboptimal extension are solved under the assumption that the system Σ(A, B, C) has bounded controllability and observability maps, an L2-impulse response and a transfer matrix that is bounded and holomorphic on the right half-plane. Exponential stability of the semigroup is not assumed and the Hankel operator is not compact. The new contribution is an explicit parameterization of all solutions given in terms of the system parametersA, B, C.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V.M. Adamjan, D.Z. Arov, and M.G. Krein. Infinite Hankel block matrices and related extension problems.American Mathematical Society Translations, 111:133–156, 1978.

    Google Scholar 

  2. A.V. Balakrishnan. Compensator design for stability enhancement with co-located controllers.IEEE Transactions on Automatic Control, 36:994–1007, 1991.

    Google Scholar 

  3. A.V. Balakrishnan. Shape control of plates with piezo actuators and collocated position/rate sensors.Applied Mathematics and Computation, 63:213–234, 1994.

    Google Scholar 

  4. J.A. Ball and J.W. Helton. A Beurling-Lax theorem for the Lie groupU(m, n) which contains most classical interpolation theory.Journal of Operator Theory, 9:107–142, 1983.

    Google Scholar 

  5. R.F. Curtain and A. Ichikawa. The Nehari problem for infinite-dimensional systems of parabolic type.Integral Equations and Operator Theory, 26:29–45, 1996.

    Google Scholar 

  6. R.F. Curtain and J.C. Oostveen. Riccati equations for strongly stabilizable bounded linear systems. 1997. (submitted).

  7. R.F. Curtain and J.C. Oostveen. Robustly stabilizing controllers for dissipative infinite-dimensional systems with colocated actuators and sensors. 1997. (submitted).

  8. R.F. Curtain and A. Ran. Explicit formulas for Hankel norm approximations of infinite-dimensional systems.Integral Equations and Operator Theory, 13:455–469, 1989.

    Google Scholar 

  9. R.F. Curtain and H.J. Zwart. The Nehari problem for the Pritchard-Salomon class of infinite-dimensional linear systems: a direct approach.Integral Equations and Operator Theory, 18:130–153, 1994.

    Google Scholar 

  10. R.F. Curtain and H.J. Zwart.An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York, 1995.

    Google Scholar 

  11. R.F. Curtain and H.J. Zwart. Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems.Systems and Control Letters, 28:11–22, 1996.

    Google Scholar 

  12. P.L. Duren.Theory of H p Spaces. Academic Press, New York, 1970.

    Google Scholar 

  13. C. Folas and A. Tannenbaum. On the Nehari problem for a certain class ofL ∞-functions appearing in control theory.Journal of Functional Analysis, 74:146–159, 1987.

    Google Scholar 

  14. K. Glover, R.F. Curtain, and J.R. Partington. Realization and approximation of linear infinite-dimensional systems with error bounds.SIAM Journal on Control and Optimization, 26:863–898, 1988.

    Google Scholar 

  15. P. Grabowski. On the spectral-Lyapunov approach to parametric optimization of distributed parameter systems.IMA Journal of Math. Control and Information, 7:317–338, 1990.

    Google Scholar 

  16. M. Green, K. Glover, D. Limebeer, and J. Doyle. A J-spectral factorization approach toH ∞ control.SIAM Journal on Control and Optimization, 28:1350–1371, 1990.

    Google Scholar 

  17. S. Hansen and G. Weiss. New results on the operator Carleson measure criterion.IMA Journal of Mathematical Control and Information, 14:3–32, 1997.

    Google Scholar 

  18. J.R. Partington,An Introduction to Hankel Operators. London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1988.

    Google Scholar 

  19. T.A. Posbergh.Modeling and Control of Mixed and Flexible Structures. PhD thesis, University of Maryland, 1988.

  20. A. Ran. Hankel norm approximation for infinite-dimensional systems and Wiener-Hopf factorization. In R.F. Curtain, editor,Modelling Robustness and Sensitivity Reduction in Control Systems, NATO ASI Series, pages 57–70. Springer-Verlag, 1986.

  21. M. Slemrod. Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control.Mathematics of Control, Signals and Systems 2:265–285, 1989.

    Google Scholar 

  22. S.R. Treil. The theorem of Adamjan-Arov-Krein: A vector variant.Zap. Nauchn. Semin. Leningrad. Otdel. Math. Inst. Steklov (LOMI), 141:56–71, 1985. (in Russian).

    Google Scholar 

  23. B. van Keulen. Hankel operators for non-exponentially stabilizable infinite-dimensional systems.Systems and Control Letters, 15:221–226, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV, Groningen, The Netherlands

    Ruth F. Curtain & Job C. Oostveen

Authors
  1. Ruth F. Curtain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Job C. Oostveen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Curtain, R.F., Oostveen, J.C. The Nehari problem for nonexponentially stable systems. Integr equ oper theory 31, 307–320 (1998). https://doi.org/10.1007/BF01195122

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01195122

Mathematics Subject Classification

Search

Navigation