Existence and duality theorems for convex problems of Bolza
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- by R. T. Rockafellar PDF
- Trans. Amer. Math. Soc. 159 (1971), 1-40 Request permission
Abstract:
The theory of conjugate convex functions is applied to a fundamental class of “convex” problems in the calculus of variations and optimal control. This class has many special properties which have not previously been exploited and for which the standard methods of approach are inadequate. Duality theorems are established which yield new results on the existence of optimal arcs, as well as necessary and sufficient conditions for optimality. These results have some relevance also to the study of “nonconvex” problems.References
- E. Asplund and R. T. Rockafellar, Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1969), 443–467. MR 240621, DOI 10.1090/S0002-9947-1969-0240621-X
- Lamberto Cesari, Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I, Trans. Amer. Math. Soc. 124 (1966), 369–412. MR 203542, DOI 10.1090/S0002-9947-1966-0203542-1 J. J. Moreau, Fonctionelles convexes, mimeographed lecture notes, Collège de France, 1967.
- Jean-Jacques Moreau, Sur la fonction polaire d’une fonction semi-continue supérieurement, C. R. Acad. Sci. Paris 258 (1964), 1128–1130 (French). MR 160093
- Czesław Olech, Existence theorems for optimal problems with vector-valued cost function, Trans. Amer. Math. Soc. 136 (1969), 159–180. MR 234338, DOI 10.1090/S0002-9947-1969-0234338-5
- R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original; Princeton Paperbacks. MR 1451876
- R. T. Rockafellar, Integrals which are convex functionals, Pacific J. Math. 24 (1968), 525–539. MR 236689, DOI 10.2140/pjm.1968.24.525
- R. T. Rockafellar, Measurable dependence of convex sets and functions on parameters, J. Math. Anal. Appl. 28 (1969), 4–25. MR 247019, DOI 10.1016/0022-247X(69)90104-8
- R. T. Rockafellar, Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl. 32 (1970), 174–222. MR 266020, DOI 10.1016/0022-247X(70)90324-0
- R. Tyrrell Rockafellar, Generalized Hamiltonian equations for convex problems of Lagrange, Pacific J. Math. 33 (1970), 411–427. MR 276853, DOI 10.2140/pjm.1970.33.411
- R. T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966), 46–63. MR 192318, DOI 10.1090/S0002-9947-1966-0192318-X —, Convex Bolza functionals in control problems with state constraints (to appear). J. L. Joly, Une famille de topologies et de convergences sur l’ensemble des fonctionelles convexes, Thèse, Grenoble, 1970.
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 159 (1971), 1-40
- MSC: Primary 49.10
- DOI: https://doi.org/10.1090/S0002-9947-1971-0282283-0
- MathSciNet review: 0282283