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 Mat. Sb., 2013, Volume 204, Number 11, Pages 83–98 (Mi msb8211)

Optimal control and Galois theory

M. I. Zelikin, D. D. Kiselev, L. V. Lokutsievskii

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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, Galois group, orthogonal polynomials.

DOI: https://doi.org/10.4213/sm8211

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English version:
Sbornik: Mathematics, 2013, 204:11, 1624–1638

Bibliographic databases:

Document Type: Article
UDC: 512.623.3+517.587+517.977.57
MSC: Primary 49J21; Secondary 49J15, 49K21

Citation: M. I. Zelikin, D. D. Kiselev, L. V. Lokutsievskii, “Optimal control and Galois theory”, Mat. Sb., 204:11 (2013), 83–98; Sb. Math., 204:11 (2013), 1624–1638

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8211
• https://doi.org/10.4213/sm8211
• http://mi.mathnet.ru/eng/msb/v204/i11/p83

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This publication is cited in the following articles:
1. M. I. Zelikin, L. V. Lokutsievskii, R. Hildebrand, “Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side”, Journal of Mathematical Sciences, 221:1 (2017), 1–136
2. D. D. Kiselev, “On a dense winding of the 2-dimensional torus”, Sb. Math., 207:4 (2016), 581–589
3. D. D. Kiselev, “Galois theory, the classification of finite simple groups and a dense winding of a torus”, Sb. Math., 209:6 (2018), 840–849
4. D. D. Kiselev, “Optimal control, everywhere dense torus winding, and Wolstenholme primes”, Moscow University Mathematics Bulletin, 73:4 (2018), 162–163
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