Geodesic
Optimal control and Galois theory
Sbornik. Mathematics, Tome 204 (2013) no. 11, pp. 1624-1638 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Pontryagin's maximum principle, Lie algebra, dense winding
Mots-clés : Galois group, orthogonal polynomials. 
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M. I. Zelikin; D. D. Kiselev; L. V. Lokutsievskii. Optimal control and Galois theory. Sbornik. Mathematics, Tome 204 (2013) no. 11, pp. 1624-1638. http://geodesic.mathdoc.fr/item/SM_2013_204_11_a4/

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