Abstract
We give existence theorems of solutions for Lagrange and Bolza problems of optimal control. These results are obtained without convexity assumptions on the cost function with respect to the control variable. We extend a Cesari's theorem to cost functions which are nonlinear with respect to the space variable and to problems which are governed by a differential inclusion. Moreover, we consider the case where the control variable belongs to a space of measurable functions and the case where this variable belongs to a Lebesgue space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aubert, G., andTahraoui, R.,Théorèmes d'Existence pour des Problèmes du Calcul des Variations du Type Inf ∝ LO f(x, u′(x))dx et Inf ∝ LO f(x, u(x),u′(x))dx, Journal of Differential Equations, Vol. 33, pp. 1–15, 1979.
Ekeland, I.,Discontinuités de Champs Hamiltoniens et Existence de Solutions Optimales en Calcul des Variations, Publications Mathématiques de l'Institut des Hautes Etudes Scientifiques, No. 47, pp. 5–32, 1977.
Marcellini, P., Alcune Osservazioni sull' Esistenza del Minimo di Integrali del Calcolo delle Variazioni senza Ipotesi di Convessità, Rendiconti di Matematica, Vol. 13, pp. 271–281, 1980.
Raymond, J. P., Conditions Nécessaires et Suffisantes d'Existence de Solutions en Calcul des Variations, Annales de l'Institut Henri Poincaré, Analyse Nonlinéaire, Vol. 4, pp. 169–202, 1987.
Cellina, A., andColombo, G.,On a Classical Problem of the Calculus of Variations without Convexity Assumptions, Conference on Nonsmooth Optimization and Related Topics, Erice, Italy, 1988.
Cesari, L.,Optimization: Theory and Applications, Springer-Verlag, New York, New York, 1983.
Klok, F.,On Nonexistence of Minimizing Solutions of Variational Problems, Journal of Mathematical Analysis and Applications, Vol. 114, pp. 129–155, 1986.
Raymond, J. P.,Champs Hamiltoniens, Relaxation et Existence de Solutions en Calcul des Variations, Journal of Differential Equations, Vol. 70, pp. 226–274, 1987.
Cesari, L.,An Existence Theorem without Convexity Conditions, SIAM Journal on Control and Optimization, Vol. 12, pp. 319–331, 1974.
Olech, C.,Integrals of Set-Valued Functions and Linear Optimal Control Problems, Colloque sur la Théorie Mathématique du Contrôle Optimal, Centre Belge de Recherches Mathématiques, Vander Louvain, Belgium, pp. 109–125, 1970.
Cesari, L.,Convexity of the Range of Certain Integrals, SIAM Journal on Control and Optimization, Vol. 13, pp. 666–676, 1975.
Ioffe, A. D., andTihomirov, V. M.,Theory of Extremal Problems, North-Holland, Amsterdam, Holland, 1979.
Ciarlet, P. G.,Introduction à l'Analyse Numérique Matricielle et à l'Optimisation, Masson, Paris, France, 1982.
Ekeland, I., andTemam, R.,Analyse Convexe et Problèmes Variationnels, Dunod, Paris, France, 1974.
Buttazzo, G.,Some Relaxation Problems in Optimal Control Theory, Journal of Mathematical Analysis and Its Applications, Vol. 125, pp. 272–287, 1987.
Krasnosel'ski, M. A., andRutickii, Y. B.,Convex Functions and Orlicz Spaces, Noordhoff, Gröningen, Holland, 1961.
Rockafellar, R. T.,Integrals, Functionals, Normal Integrands, and Measurable Selections, Nonlinear Operators and the Calculus of Variations, Edited by L. Waelbroeck, Springer-Verlag, Berlin, Germany, pp. 157–207, 1976.
Rockafellar, R. T.,Existence Theorems for General Control Problems of Bolza and Lagrange, Advances in Mathematics, Vol. 15, pp. 312–333, 1975.
Author information
Authors and Affiliations
Additional information
Communicated by L. Cesari
Rights and permissions
About this article
Cite this article
Raymond, J.P. Existence theorems in optimal control problems without convexity assumptions. J Optim Theory Appl 67, 109–132 (1990). https://doi.org/10.1007/BF00939738
Issue Date:
DOI: https://doi.org/10.1007/BF00939738