Abstract
Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Rubio, J. E.,An Existence Theorem for Control Problems in Hilbert Spaces, Bulletin of the London Mathematical Society, Vol. 9, pp. 70–78, 1977.
Rubio, J. E.,On Optimal Control Problems in Hilbert Spaces: The Case of the Unbounded Controls, Proceedings of the IMA Conference on Recent Theoretical Developments in Control, Leicester, England, 1976; Edited by M. Gregson, pp. 241–254, Academic Press, London, England, 1978.
Rubio, J. E.,Existence and Approximation in Control Problems in Hilbert Spaces, Journal of Mathematical Analysis and Applications, Vol. 69, pp. 419–427, 1979.
Kretschmer, K. S.,Programmes in Paired Spaces, Canadian Journal of Mathematics, Vol. 13, pp. 221–238, 1961.
Temelit, V.,Duality of Infinite-Dimensional Linear Programming Problems and Some Related Matters, Kibernetika, Vol. 5, pp. 447–455, 1969.
Duffin, R. J.,Infinite Programs, Linear Inequalities and Related Systems, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, 1956.
Rosenbloom, P. C., Quelques Classes de Problèmes Extremaux, Bulletin de la Société Mathématique de France, Vol. 80, pp. 183–216, 1952.
Hewitt, E.,Linear Functionals on Spaces of Continuous Functions, Fundamenta Mathematicae, Vol. 37, pp. 161–189, 1950.
Ghouila-Houri, A., Sur la Generalization de la Notion de Commande d'un Système Guidable, Revue Française d'Automatique, Informatique, et Recherche Operationelle, Vol. 48, pp. 7–32, 1965.
Rubio, J. E.,Generalized Curves and Extremal Points, SIAM Journal on Control, Vol. 13, pp. 28–47, 1975.
Rubio, J. E.,Extremal Points and Optimal Control Theory, Annali di Matematica, Vol. 109, pp. 165–176, 1976.
Leray, J., Étude de Diverses Equations Intégrales Non Lineaires et de Quelques Problèmes que Pose l'Hydrodynamique, Journal de Mathématiques Pures et Appliquées, Vol. 12, pp. 1–82, 1933.
Young, L. C.,On Approximation by Polygons in the Calculus of Variations, Proceedings of the Royal Society, Series A, Vol. 141, pp. 325–341, 1933.
Temam, R.,Navier-Stokes Equations, North-Holland Publishing Company, Amsterdam, Holland, 1977.
Author information
Authors and Affiliations
Additional information
Communicated by L. Cesari
Rights and permissions
About this article
Cite this article
Rubio, J.E. Solution of nonlinear optimal control problems in Hilbert spaces by means of linear programming techniques. J Optim Theory Appl 30, 643–661 (1980). https://doi.org/10.1007/BF01686727
Issue Date:
DOI: https://doi.org/10.1007/BF01686727