Topology of rearrangements of non-smoothness sets of minimum functions in variational problems
The minimal value of a smooth functional does not necessarily depend on the final point in a smooth manner. The non-smoothness set of this minimum function, called in optimal control the Bellman function, describes boundary conditions with a non-unique solution. In physics it corresponds to a shock wave. In time, this set may rearrange itself. \textit{I. A. Bogaevskij} [Leningr. Math. J. 1, No.4, 807-823 (1990); translation from Algebra Anal. 1, No.4, 1-16 (1989)] described typical rearrangements of the non-smoothness set of the minimum function in dimensions 3 and 4. However, it was observed [see \textit{S. N. Gurbatov} and \textit{A. I. Saichev}, Radiophys. Quant. Electron. 27, No.4, 303-313 (1984); translation from Izv. Vyssh. Uchebn. Zaved Radiofiz. 27, No.4, 456-458 (1984); and the joint work with \textit{S. F. Chandarin}, Mon. Not. R. Astron. Soc. 236, No.2, 385-402 (1989; Zbl 0662.76062)] that not all these rearrangements are realized as rearrangements of minimum functions of typical variational problems. For example, a spherical shock wave cannot be generated ``out of nothing''. Bogaevskij exhibited a list of rearrangements of non-smoothness sets of minimum functions of typical variational problems in dimensions \(\leq 4\). From this list it follows that the realizable rearrangements are distinguished by the following condition on the topology of the non- smoothness set: after a rearrangement, the non-smoothness set is locally contractible. In the present paper we describe another restriction on the type of rearrangement of the non-smoothness set of a minimum function of a variational problem, which consists of local coincidence of homotopy types of the complements of the non-smoothness sets at the moment of rearrangement and immediately following it. This restriction holds in arbitrary dimension, and not only for typical variational problems; it holds for all other ones as well, with the exception of a set of infinite codimension in a functional space. In low dimensions, like the Bogaevskij restriction, it is sharp, that is, it singles out from the list of rearrangements of non-smoothness sets of minimum functions of typical families precisely the rearrangements realized in typical variational problems.
non-smoothness sets of minimum functions
scientific article; zbMATH DE number 4190531