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A class of local variational problems with an arbitrary degeneration in the goal function. (English. Russian original) Zbl 0889.49013
Dokl. Math. 51, No. 2, 301-303 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 341, No. 4, 460-462 (1995).
The application of the extension procedure of optimal trajectories [A. P. Afanas’ev, V. V. Dikusar, A. A. Milyutin and S. A. Chukanov: “A necessary condition in optimal control” (1990; Zbl 0724.49014)], reduces to solving the local variational problems that arise for critical values of the extension parameter and simulate the arising switching situations. Here, I study the class of local problems that describe a switching situation. The technique I use to study these problems almost completely corresponds to that developed to study the generalized isoperimetric problem [A. P. Afanas’ev, Differ. Uravn. 29, No. 11, 1856-1867 (1993; Zbl 0815.93039)].
49K15 Optimality conditions for problems involving ordinary differential equations
49Q20 Variational problems in a geometric measure-theoretic setting