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 $\mathbb{Q}$ 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 $n\le15$ 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 $\mathbb{Q}$.
Bibliography: 11 titles.