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

This article is cited in 4 scientific papers (total in 4 papers)

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

Full text: PDF file (580 kB)
References: PDF file   HTML file

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
Received: 17.01.2013 and 09.04.2013

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref
    2. D. D. Kiselev, “On a dense winding of the 2-dimensional torus”, Sb. Math., 207:4 (2016), 581–589  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref  crossref  adsnasa  isi  elib
    4. D. D. Kiselev, “Optimal control, everywhere dense torus winding, and Wolstenholme primes”, Moscow University Mathematics Bulletin, 73:4 (2018), 162–163  mathnet  crossref  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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