×

A nonintersection property for extremals of variational problems with vector-valued functions. (English) Zbl 1208.49023

The author considers the classical problem of calculus of variations, i.e., the one of minimizing the integral of some Lagrangian function over a function whose end-point values are fixed (the function of time to be minimized has value in an Euclidean space). He studies the class of functions of time which are bounded and solution of these problems of calculus of variations for any initial and final time, with of course compatible end-point conditions. His main results are that (i) if such a function has the same value at two different times, then it is periodic, (ii) if two such functions have the same value at given times, then they coincide (perhaps up to a translation).

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49N20 Periodic optimal control problems
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Bangert, V., Mather sets for twist maps and geodesics on tori, (Dynamics Reported, vol. 1 (1988), Teubner: Teubner Stuttgart), 1-56 · Zbl 0664.53021
[2] Bangert, V., On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6, 95-138 (1989) · Zbl 0678.58014
[3] Cesari, L., Optimization - Theory and Applications (1983), Springer-Verlag: Springer-Verlag New York
[4] Gale, D., On optimal development in a multi-sector economy, Rev. Economic Studies, 34, 1-18 (1967)
[5] Giaquinta, M.; Guisti, E., On the regularity of the minima of variational integrals, Acta Math., 148, 31-46 (1982) · Zbl 0494.49031
[6] Hedlund, G. A., Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math., 33, 719-739 (1984) · Zbl 0006.32601
[7] Leizarowitz, A., Infinite horizon autonomous systems with unbounded cost, Appl. Math. Optim., 13, 19-43 (1985) · Zbl 0591.93039
[8] Leizarowitz, A.; Mizel, V. J., One dimensional infinite horizon variational problems arising in continuum mechanics, Arch. Rational Mech. Anal., 106, 161-194 (1989) · Zbl 0672.73010
[9] Marcus, M.; Zaslavski, A. J., The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16, 593-629 (1999) · Zbl 0989.49003
[10] Marcus, M.; Zaslavski, A. J., The structure and limiting behavior of locally optimal minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19, 343-370 (2002) · Zbl 1035.49001
[11] Morse, M., A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc., 26, 25-60 (1924) · JFM 50.0466.04
[12] Moser, J., Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3, 229-272 (1986) · Zbl 0609.49029
[13] Rabinowitz, P. H.; Stredulinsky, E., On some results of Moser and of Bangert, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21, 673-688 (2004) · Zbl 1149.35341
[14] Rabinowitz, P. H.; Stredulinsky, E., On some results of Moser of Bangert. II, Adv. Nonlinear Stud., 4, 377-396 (2004) · Zbl 1229.35047
[15] Zaslavski, A. J., The existence of periodic minimal energy configurations for one dimensional infinite horizon variational problems arising in continuum mechanics, J. Math. Anal. Appl., 194, 459-476 (1995) · Zbl 0869.49003
[16] Zaslavski, A. J., Dynamic properties of optimal solutions of variational problems, Nonlinear Anal., 27, 895-931 (1996) · Zbl 0860.49003
[17] Zaslavski, A. J., Existence and uniform boundedness of optimal solutions of variational problems, Abstr. Appl. Anal., 3, 265-292 (1998) · Zbl 0963.49002
[18] Zaslavski, A. J., The turnpike property for extremals of nonautonomous variational problems with vector-valued functions, Nonlinear Anal., 42, 1465-1498 (2000) · Zbl 0968.49003
[19] Zaslavski, A. J., A turnpike property for a class of variational problems, J. Convex Anal., 12, 331-349 (2005) · Zbl 1104.49003
[20] Zaslavski, A. J., Turnpike Properties in the Calculus of Variations and Optimal Control (2006), Springer: Springer New York · Zbl 1100.49003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.