Skip to main content
SpringerLink
Log in

Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We present some necessary and sufficient conditions for null controllability for a class of general linear evolution equations on a Banach space with constraints on the control space. We also present a result on the existence of time-optimal controls and some partial results on the maximum principle. Some interesting insights that can be obtained from these results are discussed, and the paper is concluded with an application to a boundary control problem.

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. Russell, D. L.,Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions, SIAM Review, Vol. 20, pp. 639–739, 1978.

    Google Scholar 

  2. Fattorini, H. O.,The Time-Optimal Control Problem in Banach Spaces, Applied Mathematics and Optimization, Vol. 1, pp. 163–187, 1974.

    Google Scholar 

  3. Triggiani, R.,A Note on the Lack of Exact Controllability and Optimization, SIAM Journal on Control and Optimization, Vol. 15, pp. 407–411, 1977.

    Google Scholar 

  4. Knowles, G.,Time-Optimal Control of Infinite-Dimensional Systems, SIAM Journal on Control and Optimization, Vol. 14, pp. 919–933, 1976.

    Google Scholar 

  5. Chen, G.,Control and Stabilization for the Wave Equation in a Bounded Domain, Part 2, SIAM Journal on Control and Optimization, Vol. 19, pp. 114–122, 1981.

    Google Scholar 

  6. Washburn, D.,A Bound on the Boundary Input Map for Parabolic Equations with Applications to Time-Optimal Control, SIAM Journal on Control and Optimization, Vol. 17, pp. 652–671, 1979.

    Google Scholar 

  7. Ahmed, N. U., andTeo, K. L.,Optimal Control of Distributed-Parameter Systems, North-Holland, New York, New York, 1981.

    Google Scholar 

  8. Clarke, B. M. N., andWilliamson, D.,Control Canonical Forms and Eigenvalue Assignment by Feedback for a Class of Linear Hyperbolic Systems, SIAM Journal on Control and Optimization, Vol. 19, pp. 711–729, 1981.

    Google Scholar 

  9. Georg Schmidt, E. J. P.,The Bang-Bang Principle for the Time-Optimal Problem in Boundary Control of the Heat Equation, SIAM Journal on Control and Optimization, Vol. 18, pp. 101–107, 1980.

    Google Scholar 

  10. Schmitendorf, W. E., andBarmish, B. R.,Null Controllability of Linear Systems with Constrainted Controls, SIAM Journal on Control and Optimization, Vol. 18, pp. 327–345, 1980.

    Google Scholar 

  11. Conti, R.,Teoria del Controllo e del Controllo Ottimo, UTET, Torino, Italy, 1974.

    Google Scholar 

  12. Pandolfi, L.,Linear Control Systems: Controllability with Constrained Controls, Journal of Optimization Theory and Applications, Vol. 19, pp. 577–585, 1976.

    Google Scholar 

  13. Butzer, P. L., andBerens, H.,Semigroups of Operators and Approximation, Springer-Verlag, Berlin, Germany, 1967.

    Google Scholar 

  14. Kato, T.,Linear Evolution Equations of Hyperbolic Type, Part 2, Journal of the Mathematical Society of Japan, Vol. 25, pp. 648–666, 1973.

    Google Scholar 

  15. Dunford, N., andSchwartz, J. T.,Linear Operators, Part 1, Interscience Publishers, New York, New York, 1964.

    Google Scholar 

  16. Himmelberg, C. J., Jacobs, M. Q., andVan Vleck, V. S.,Measurable Multifunctions, Selectors, and Filippov's Implicit Functions Lemma, Journal of Mathematical Analysis and Applications, Vol. 25, pp. 276–285, 1969.

    Google Scholar 

  17. Holmes, R. B.,Geometric Functional Analysis and Its Applications, Springer-Verlag, New York, New York, 1975.

    Google Scholar 

  18. Barbu, V.,Boundary Control Problems with Convex Cost Criterion, SIAM Journal on Control and Optimization, Vol. 18, pp. 227–243, 1980.

    Google Scholar 

  19. Narukawa, K.,Admissible Controllability of Vibrating Systems with Constrained Controls, SIAM Journal on Control and Optimization, Vol. 20, pp. 770–782, 1982.

    Google Scholar 

  20. Narukawa, K.,Boundary-Value Control of Isotropic Elastodynamic Systems with Constrained Controls, Journal of Mathematical Analysis and Applications, Vol. 93, pp. 250–272, 1983.

    Google Scholar 

  21. Narukawa, K.,Admissible Null Controllability and Time-Optimal Control, Hiroshima Mathematical Journal, Vol. 11, pp. 533–551, 1981.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, University of Ottawa, Ottawa, Ontario, Canada

    N. U. Ahmed (Professor)

Authors
  1. N. U. Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by L. Cesari

This work was supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.

The author is thankful to Professor L. Cesari for many helpful suggestions and also for calling his attention to the recent papers of Professor K. Narukawa.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ahmed, N.U. Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints. J Optim Theory Appl 47, 129–158 (1985). https://doi.org/10.1007/BF00940766

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00940766

Key Words

Search

Navigation