On a Variational Problem Connected with Phase Transitions of Means in Controllable Dynamical Systems

We study the optimization of the integral of a given smooth function along the distribution determined by a density bounded by given functions from above and from below. Phase transitions are nonsmooth dependence of optimal means, as well as optimal strategies, on parameters. In this variational problem and, in particular, in the case of functions of even number of variables, they necessarily appear by topological reasons, which leads to logarithmic singularities. Two-dimensional variational problems in hydrodynamics and in magnetohydrodynamics are also considered. In these problems, as in the case studied in this paper, singularities are caused by topological reasons which I discussed with great pleasure with O. A. Ladyzhenskaya in Dilizhan in 1973.