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Examples from the calculus of variations. IV: Concluding review. (English) Zbl 1008.49015

Summary: Variational integrals containing several functions of one independent variable subjected moreover to an underdetermined system of ordinary differential equations (the Lagrange problem) are investigated within a survey of examples. More systematical discussion of two crucial examples from Part I [Math. Bohem. 125, No. 1, 55–76 (2000; Zbl 0968.49001)] with help of the methods of Parts II and III [Math. Bohem. 125, No. 2, 187–197 (2000; Zbl 0968.49002) and ibid. 126, No. 1, 93-111 (2001; Zbl 0980.49024)] is performed not excluding certain instructive subcases to manifest the significant role of generalized Poincaré-Cartan forms without undetermined multipliers. The classical Weierstrass-Hilbert theory is simulated to obtain sufficient extremality conditions. Unlike the previous parts, this article is adapted to the category of continuous objects and mappings without any substantial references to the general principles, which makes the exposition self-contained.

MSC:

49K27 Optimality conditions for problems in abstract spaces
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
58E30 Variational principles in infinite-dimensional spaces
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