Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Existence of optimal controls for the diffusion equation

  • 55 Accesses

  • 20 Citations

Abstract

The existence is considered of a boundary control which drives a system governed by the one-dimensional diffusion equation from the zero state to a given final state, and at the same time minimizes a given functional. The problem is first modified to one in which the minimum is sought of a functional defined on a set of Radon measures. The existence of a minimizing measure is demonstrated, and it is shown that this measure may be approximated by a piecewise constant control. Finally, conditions are given under which a minimizing measurable control exists for the unmodified problem.

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

References

  1. 1.

    Fattorini, H. O., andRussell, D. L.,Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension, Archive for Rational Mechanics and Analysis, Vol. 4, pp. 272–292, 1971.

  2. 2.

    Young, L. C.,Calculus of Variations and Optimal Control Theory, W. B. Saunders Company, Philadelphia, Pennyslvania, 1969.

  3. 3.

    Rudin, W.,Real and Complex Analysis, McGraw-Hill Book Company, New York, New York, 1966.

  4. 4.

    Ghouila-Houri, A.,Sur la Généralization de la Notion de Commande d'un Systeme Guidable, Revue Française d'Automatique, Informatique et Recherche Operationelle, Vol. 1, pp. 7–32, 1967.

  5. 5.

    Robertson, A. P., andRobertson, W. J.,Topological Vector Spaces, Cambridge University Press, Cambridge, England, 1973.

  6. 6.

    Walsh, G. R.,Methods of Optimization, John Wiley and Sons, London, England, 1975.

  7. 7.

    Ewing, G. M.,Calculus of Variations with Applications, Norton, New York, New York, 1969.

Download references

Author information

Additional information

Communicated by G. Leitmann

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Wilson, D.A., Rubio, J.E. Existence of optimal controls for the diffusion equation. J Optim Theory Appl 22, 91–101 (1977). https://doi.org/10.1007/BF00936723

Download citation

Key Words

  • Optimal control
  • Radon measures
  • existence theory
  • diffusion equation