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Coupled points in the calculus of variations and applications to periodic problems. (English) Zbl 0677.49020
Summary: The aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions, and to develop second order necessary conditions for optimality. When one of the end points is fixed, our necessary conditions reduce to the known ones involving conjugate points. We also apply our results to the periodic problems of the calculus of variations.

MSC:
49K15 Optimality conditions for problems involving ordinary differential equations
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[1] J. M. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal. 90 (1985), no. 4, 325 – 388. · Zbl 0585.49002
[2] Gilbert A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill., 1946. · Zbl 0036.34401
[3] Lamberto Cesari, Optimization — theory and applications, Applications of Mathematics (New York), vol. 17, Springer-Verlag, New York, 1983. Problems with ordinary differential equations. · Zbl 0506.49001
[4] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc. 289 (1985), no. 1, 73 – 98. · Zbl 0563.49009
[5] M. R. Hestenes, Calculus of variations, Wiley, New York, 1969. · Zbl 0173.35703
[6] M. Morse, Variational analysis, Wiley, New York, 1973. · Zbl 0255.49002
[7] William T. Reid, Riccati differential equations, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 86. · Zbl 0254.34003
[8] H. Sagan, Introduction to the calculus of variations, McGraw-Hill, New York, 1969.
[9] Jason L. Speyer and Richard T. Evans, A second variational theory for optimal periodic processes, IEEE Trans. Automat. Control 29 (1984), no. 2, 138 – 148. · Zbl 0534.49021
[10] V. Zeidan and P. Zezza, An extension of the conjugate point conditions to the case of variable endpoints, Proc. 27th IEEE Conf. on Decision and Control, 1988, pp. 1187-1191.
[11] Vera Zeidan and Pier Luigi Zezza, Variable end points problems in the calculus of variations: coupled points, Analysis and optimization of systems (Antibes, 1988) Lect. Notes Control Inf. Sci., vol. 111, Springer, Berlin, 1988, pp. 372 – 380. · Zbl 0789.49012
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