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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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.
Young, L. C.,Calculus of Variations and Optimal Control Theory, W. B. Saunders Company, Philadelphia, Pennyslvania, 1969.
Rudin, W.,Real and Complex Analysis, McGraw-Hill Book Company, New York, New York, 1966.
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.
Robertson, A. P., andRobertson, W. J.,Topological Vector Spaces, Cambridge University Press, Cambridge, England, 1973.
Walsh, G. R.,Methods of Optimization, John Wiley and Sons, London, England, 1975.
Ewing, G. M.,Calculus of Variations with Applications, Norton, New York, New York, 1969.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
Rights 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
Issue Date:
DOI: https://doi.org/10.1007/BF00936723