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Existence theorems for usual and approximate solutions of optimal control problems

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Abstract

The theory of measurable set-valued mappings allows us to study some problems of optimal control in the framework of minimization of convex functionals and thus to obtain existence theorems. When the functionals are nonconvex, we obtain the existence theorems for control problems which are weakly perturbed from the initial one. In this regard, we specify some theorems of nonconvex optimization.

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References

  1. Ekeland, I.,Relaxation de Problèmes de Contrôle Optimal pour des Systèmes Gouvernés par des Équations aux dérivées Partielles de Type Elliptique, University of Paris, Paris, France, Thesis, 1970.

    Google Scholar 

  2. Cesari, L.,Existence Theorems for Weak and Usual Optimal Solutions in Lagrange Problems with Unilateral Constraints, I, Transactions of the American Mathematical Society, Vol. 124, pp. 369–412, 1966.

    Google Scholar 

  3. Cesari, L.,Existence Theorems for Weak and Usual Optimal Solutions in Lagrange Problems with Unilateral Constraints, II. Transactions of the American Mathematical Society, Vol. 124, pp. 413–430, 1966.

    Google Scholar 

  4. Cesari, L.,Existence Theorems for Abstract Multidimensional Control Problems, Journal of Optimization Theory and Applications, Vol. 6, No. 3, 1970.

  5. Lions, J. L.,Ouelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, France, 1969.

    Google Scholar 

  6. Brezis, H.,Problèmes Unilatéraux, University of Paris, Paris, France, Thesis, 1970.

    Google Scholar 

  7. Castaing, C.,Sur les Multiapplications Mesurables, University of Caen, Caen, France, Thesis, 1967.

    Google Scholar 

  8. Castaing, C.,Une Nouvelle Extension du Théorème de Dragoni-Scorza, Note de Compte-Rendu à l'Académie des Sciences de Paris, Vol. 271, pp. 396–398, 1970.

    Google Scholar 

  9. Bidaut, M. F.,Théorèmes d'Existence et d'Existence “en Général” d'un Contrôle Optimal pour des Systèmes Régis par des Équations aux Dérivées Partielles non Linéaires, University of Paris, Paris, France, Thesis, 1973.

    Google Scholar 

  10. Asplund, E.,Frechet Differentiability of Convex Functions, Acta Mathematica, Vol. 121, pp. 31–47, 1968.

    Google Scholar 

  11. Brezis, H.,Perturbations non Linéaires d'Opérateurs Maximaux Monotones, Note de Compte-Rendu à l'Académie des Sciences de Paris, Vol. 269, pp. 566–569, 1969.

    Google Scholar 

  12. Stetchkine, S. B.,Caractérisation de l'Approximation par des Semi-Ensembles d'Espaces Vectoriels Normés (in Russian), Revue de Mathématique Pures et Appliquées, Vol. 8, pp. 5–18, 1963.

    Google Scholar 

  13. Edelstein, M.,Farthest Points of Sets in Uniformly Convex Banach Spaces, Israel Journal of Mathematics, Vol. 4, pp. 171–176, 1966.

    Google Scholar 

  14. Edelstein, M.,On Nearest Points of Sets in Uniformly Convex Banach Spaces, Journal of London Mathematical Society, Vol. 43, pp. 375–377, 1968.

    Google Scholar 

  15. Baranger, J.,Théorème d'Existence en Densité et Application au Contrôle, University of Grenoble, Grenoble, France, Thesis, 1973.

    Google Scholar 

  16. Robert, R.,Points de Continuité Multiapplications Semi-continues supérieurement, Note de Compte-Rendu à l'Academie des Sciences de Paris (to appear).

  17. Asplund, E., andRockafellar, R. T.,Gradients of Convex Functions, Transactions of the American Mathematical Society, Vol. 139, pp. 443–467, 1969.

    Google Scholar 

  18. Baranger, J., andTemam, R.,Un Théorème d'Existence en Densité pour le Problème supx∈X(ω∥x−y∥−f(x)), SIAM Journal on Control (to appear).

  19. Troyanski, S.,On Local Uniform Convexity and Differentiable Norms in Certain Banach Spaces, Studia Mathematica, Vol. 37, pp. 173–179, 1971.

    Google Scholar 

  20. Edelstein, M.,A Note on Nearest Points, Quarterly Journal of Mathematics of Oxford, Vol. 2, pp. 403–405, 1970.

    Google Scholar 

  21. Bidaut, M. F.,Quelques Résultats d'Existence pour des Problèmes de Contrôle Optimal, Note de Compte-Rendu à l'Académie des Sciences de Paris, Vol. 274, pp. 62–65, 1972.

    Google Scholar 

  22. Bidaut, M. F.,Quelques Résultats d'Existence “en général” pour des problèmes de Contrôle Optimal, Note de Compte-Rendu à l'Académie des Sciences de Paris, Vol. 277, pp. 97–100, 1973.

    Google Scholar 

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Authors and Affiliations

  1. Attachée de Recherche au CNRS, Paris, France

    M. F. Bidaut

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  1. M. F. Bidaut
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Translated into English by Dr. M. B. Suryanarayana, Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan.

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Bidaut, M.F. Existence theorems for usual and approximate solutions of optimal control problems. J Optim Theory Appl 15, 393–411 (1975). https://doi.org/10.1007/BF00933205

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