Optimal control and Galois theory
Mikhail Il'ich Zelikin, Denis Dmitrievich Kiselev, Lev Vyacheslavovich Lokutsievskiy · Sbornik Mathematics · 2013
An important role is played in the solution of a class of opti- mal control problems by a certain special polynomial of degree 2(n − 1) with integer coefficients. The linear independence of a family of k roots of this polynomial over the field Q implies the existence of a solution of the original problem with optimal control in the form of an irrational winding of a k-dimensional Clifford torus, which is passed in finite time. In the paper, we prove that for n 6 15 one can take an arbitrary positive inte- ger not exceeding (n/2) for k. The apparatus developed in the paper is applied to the systems of Chebyshev-Hermite polynomials and generalized Chebyshev-Laguerre polynomials. It is proved that for such polynomials of degree 2m every subsystem of ((m + 1)/2) roots with pairwise distinct squares is linearly independent over the field Q. Bibliography: 11 titles.