Extremal values of a class of functionals with moment constraints (Q1898536)

The author investigates the following optimization problem. Let the functions \(u_1 (t), \dots, u_n (t)\), \(w_1 (t), \dots, w_n(t)\) be continuous on \(T = [a,b]\) (or \(T = [a, \infty))\) and linear independent and \(\varphi\) be a continuous function. Let \(\vec u(t)\) and \(\vec w(t)\) denote curves in \(R^{n + 1}\). The problem considered is to find an extremum of the functional \[ J (\sigma) = \varphi \Bigl( \int_T \vec w(t) d \sigma (t) \Bigr) \] in the class of non-decreasing functions, continuous from the right and of bounded variation satisfying the equation \[ c = \int_T \vec u(t) d \sigma (t). \] Sufficient conditions ensuring the existence of the solution are presented and the upper bound for a number of discontinuity points of optimal distribution are found. Three examples are given to illustrate applications of the developed theory.