Skip to main content
SpringerLink
Log in

Feedback exact null controllability for unbounded control problems in Hilbert space

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

Abstract

Terminal constraint optimal control problems with unbounded control operators are considered. It is shown that the optimal solutions can be represented in a feedback form via a solution of an appropriate Riccati equation. In particular, it is proved that, for systems described by partial differential equations with infinite speed of propagation, boundary exact null controllability can be realized in feedback form.

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. Brunovsky, P., andKomornik, J.,The Riccati Equation Solution of the Linear Quadratic Problem with Terminal State, IEEE Transactions on Automatic Control, Vol. AC-26, pp. 388–402, 1981.

    Google Scholar 

  2. Chen, S.,Optimal Control Problem with Terminal Constraints in Hilbert Spaces, Mathematical Report, Hangzhou University, 1989.

  3. Da Prato, G., andIchikawa, A.,Riccati Equations with Unbounded Coefficients, Annali di Matematica Pura e Applicata, Vol. 140, pp. 209–221, 1985.

    Google Scholar 

  4. Flandoli, F.,Riccati Equation Arising in a Boundary Control Problem with Distributed Parameters, SIAM Journal on Control and Optimization, Vol. 22, pp. 76–86, 1984.

    Google Scholar 

  5. Flandoli, F., Lasiecka, I., andTriggiani, R.,Algebraic Riccati Equations with Nonsmoothing Observation Arising in Hyperbolic and Euler-Bernouli Equations, Annali di Matematica Pura e Applicata, Vol. 153, pp. 307–382, 1988.

    Google Scholar 

  6. Da Prato, G., Lasiecka, I., andTriggiani, R.,A Direct Study of Riccati Equations Arising in Boundary Control Problems for Hyperbolic Equations, Journal of Differential Equations, Vol. 64, pp. 26–47, 1986.

    Google Scholar 

  7. Lasiecka, I., andTriggiani, R.,Dirichlet Boundary Control Problem for Parabolic Equations with Quadratic Cost: Analyticity and Riccati's Feedback Synthesis, SIAM Journal on Control and Optimization, Vol. 21, pp. 41–68, 1983.

    Google Scholar 

  8. Lasiecka, I., andTriggiani, R.,Riccati Equations for Hyperbolic Partial Differential Equations with L 2(0,T:L 2(Γ))-Dirichlet Boundary Terms, SIAM Journal on Control and Optimization, Vol. 24, pp. 884–924, 1986.

    Google Scholar 

  9. Siedman, T.,Exact Boundary Controllability for Some Evolution Equations, SIAM Journal on Control and Optimization, Vol. 16, pp. 979–999, 1978.

    Google Scholar 

  10. Lions, J. L.,Exact Controllability, Stabilization, and Perturbations, SIAM Review, Vol. 30, pp. 1–68, 1988.

    Google Scholar 

  11. Lasiecka, I., andTriggiani, R.,Exact Controllability of the Euler-Bernoulli Equation with Controls in Dirichlet and Neumann Boundary Conditions: A Non-conservative Case, SIAM Journal on Control and Optimization, Vol. 27, pp. 330–373, 1989.

    Google Scholar 

  12. Lasiecka, I., andTriggiani, R.,Exact Controllability of the Euler-Bernoulli Equation with Boundary Controls for Displacement and Moment, Journal of Mathematical Analysis and Applications, Vol. 146, pp. 1–33, 1990.

    Google Scholar 

  13. Lasiecka, I., andTriggiani, R.,Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Information Sciences, Springer-Verlag, New York, Vol. 164, 1991.

    Google Scholar 

  14. Flandoli, F.,A New Approach to the LQR Problem for Hyperbolic Dynamics with Boundary Control, Lecture Notes in Control and Information Sciences, Springer-Verlag, New York, New York, Vol. 102, pp. 89–111, 1987.

    Google Scholar 

  15. Lasiecka, I., andTriggiani, R.,Riccati Differential Equations with Unbounded Coefficients and Nonsmooth Terminal Condition: The Case of Analytic Semigroups, SIAM Journal on Mathematical Analysis (to appear).

  16. Balakrishnan, A. V.,Applied Functional Analysis, 2nd Edition, Springer-Verlag, New York, New York, 1981.

    Google Scholar 

  17. Groetsch, C. W.,Generalized Inverses of Linear Operators: Representation and Approximation, Pure and Applied Mathematics, Vol. 37, Marcel Dekker, New York, New York, 1977.

    Google Scholar 

  18. Chen, S., andLasiecka, I.,Feedback Realization of Terminal Constraint Unbounded Action Control Problems in Hilbert Spaces with Applications to Feedback Exact Null Controllability, Report No. 91-02, Department of Applied Mathematics, University of Virginia, 1989.

  19. Kato, T.,Perturbation Theory for Linear Operator, Springer-Verlag, New York, New York, 1976.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Zhejiang University, Hangzhou, China

    S. Chen (Professor)

  2. Department of Applied Mathematics, University of Virginia, Charlottesville, Virginia

    I. Lasiecka (Professor)

Authors
  1. S. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Lasiecka
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by R. Conti

This work was partially supported by the National Science Foundation, Grant No. DMS-89-02811, and by the Air Force Office of Scientific Research, Grant No. AFOSR-89-0511 DEF.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, S., Lasiecka, I. Feedback exact null controllability for unbounded control problems in Hilbert space. J Optim Theory Appl 74, 191–219 (1992). https://doi.org/10.1007/BF00940891

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00940891

Key Words

Search

Navigation