Optimal arcs and the minimum value function in problems of Lagrange

Rockafellar, R. Tyrrell

doi:10.1090/S0002-9947-1973-0320852-1

Optimal arcs and the minimum value function in problems of Lagrange
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by R. Tyrrell Rockafellar

Trans. Amer. Math. Soc. 180 (1973), 53-83

DOI: https://doi.org/10.1090/S0002-9947-1973-0320852-1

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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

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
Lamberto Cesari, Closure, lower closure, and semicontinuity theorems in optimal control. , SIAM J. Control 9 (1971), 287–315. MR 0287401, DOI 10.1137/0309022
A. F. Filippov, On certain questions in the theory of optimal control, J. SIAM Control Ser. A 1 (1962), 76–84. MR 0149985
E. B. Lee and L. Markus, Optimal control for nonlinear processes, Arch. Rational Mech. Anal. 8 (1961), 36–58. MR 128571, DOI 10.1007/BF00277429

Existence theorems for ordinary problems of the calculus of variation

3

Über die gleichmässige Summierbarkeit und ihre Anwendung auf ein Variations problem

6

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
Czesław Olech, Existence theorems for optimal control problems involving multiple integrals, J. Differential Equations 6 (1969), 512–526. MR 246184, DOI 10.1016/0022-0396(69)90007-2
R. T. Rockafellar, Existence and duality theorems for convex problems of Bolza, Trans. Amer. Math. Soc. 159 (1971), 1–40. MR 282283, DOI 10.1090/S0002-9947-1971-0282283-0
R. Tyrrell Rockafellar, State constraints in convex control problems of Bolza, SIAM J. Control 10 (1972), 691–715. MR 0324505, DOI 10.1137/0310051
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. Tyrrell Rockafellar, Convex integral functionals and duality, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Publ. Math. Res. Center Univ. Wisconsin, No. 27, Academic Press, New York, 1971, pp. 215–236. MR 0390870
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683, DOI 10.1515/9781400873173
Emilio Roxin, The existence of optimal controls, Michigan Math. J. 9 (1962), 109–119. MR 136844

Sugli integrali del calcolo delle variazioni in forma ordinaria

3