Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Optimal arcs and the minimum value function in problems of Lagrange

Author: R. Tyrrell Rockafellar
Journal: Trans. Amer. Math. Soc. 180 (1973), 53-83
MSC: Primary 49A10
MathSciNet review: 0320852
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Existence theorems are proved for basic problems of Lagrange in the calculus of variations and optimal control theory, in particular problems for arcs with both endpoints fixed. Emphasis is placed on deriving continuity and growth properties of the minimum value of the integral as a function of the endpoints of the arc and the interval of integration. Control regions are not required to be bounded. Some results are also obtained for problems of Bolza.

Conjugate convex functions and duality are used extensively in the development, but the problems themselves are not assumed to be especially ``convex". Constraints are incorporated by the device of allowing the Lagrangian function to be extended-real-valued. This necessitates a new approach to the question of what technical conditions of regularity should be imposed that will not only work, but will also be flexible and general enough to meet the diverse applications. One of the underlying purposes of the paper is to present an answer to this question.

References [Enhancements On Off] (What's this?)

  • [1] L. 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 34 #3392. MR 0203542 (34:3392)
  • [2] -, Closure, lower closure, and semicontinuity theorems in optimal control, SIAM J. Control 9 (1971), 287-315. MR 44 #4607. MR 0287401 (44:4607)
  • [3] A. F. Filippov, On certain questions in the theory of optimal control, Vestnik Moskov. Univ. Ser. Mat. Meh. Astronom. Fiz. Him. 1959, no. 2, 25-32; English transl., J. SIAM Control Ser. A 1 (1962), 76-84. MR 22 #13373; MR 26 #7469. MR 0149985 (26:7469)
  • [4] E. B. Lee and L. Markus, Optimal control for nonlinear processes, Arch. Rational Mech. Anal. 8 (1961), 36-58. MR 23 #B1610. MR 0128571 (23:B1610)
  • [5] E. J. McShane, Existence theorems for ordinary problems of the calculus of variation, Ann. Scuola Norm. Sup. Pisa 3 (1934), 181-211.
  • [6] M. Nagumo, Über die gleichmässige Summierbarkeit und ihre Anwendung auf ein Variations problem, Japan. J. Math. 6 (1929), 178-182.
  • [7] C. Olech, Existence theorems for optimal problems with vector-valued cost functions, Trans. Amer. Math. Soc. 136 (1969), 159-180. MR 38 #2655. MR 0234338 (38:2655)
  • [8] -, Existence theorems for optimal control problems involving multiple integrals, J. Differential Equations 6 (1969), 512-526. MR 39 #7488. MR 0246184 (39:7488)
  • [9] R. T. Rockafellar, Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc. 159 (1971), 1-40. MR 43 #7995. MR 0282283 (43:7995)
  • [10] -, State constraints in convex problems of Bolza, SIAM J. Control 10 (1972), 691-715. MR 0324505 (48:2857)
  • [11] -, Measurable dependence of convex sets and functions on parameters, J. Math. Anal. Appl. 28 (1969), 4-25. MR 40 #288. MR 0247019 (40:288)
  • [12] R. T. Rockafellar, Convex integral functionals and duality, Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, pp. 215-236. MR 0390870 (52:11693)
  • [13] -, Convex analysis, Princeton, Univ. Press, Princeton Math. Series, no 28, Princeton, N. J., 1970. MR 43 #445. MR 0274683 (43:445)
  • [14] E. O. Roxin, The existence of optimal controls, Michigan Math. J. 9 (1962), 109-119. MR 25 #305. MR 0136844 (25:305)
  • [15] L. Tonneli, Sugli integrali del calcolo delle variazioni in forma ordinaria, Ann. Scuola Norm. Sup. Pisa 3 (1934), 401-450.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 49A10

Retrieve articles in all journals with MSC: 49A10

Additional Information

Keywords: Optimal control, problems of Lagrange, problems of Bolza, existence of solutions, minimum value function, nonfinite Lagrangians, conjugate convex functions, unbounded controls
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society