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Дата последнего поиска статьи во внешних источниках: 30 августа 2018 г.
Аннотация: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 $ \lbrack {n}/{2} \rbrack $ 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 $ \lbrack (m+1)/2 \rbrack $ roots with pairwise distinct squares is linearly independent over the field $ \mathbb{Q}$.