Optimal control and Galois theory

An important role is played in the solution of a class of optimal 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≤15 one can take an arbitrary positive integer 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