Geodesic
Control Theory, Integral Matrices, and Orthogonal Polynomials
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 172-181.

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In control theory and approximation theory, there naturally arise matrices which are the inverses of the Gram matrices for the monomial basis in the space of square integrable functions with respect to a measure. For example, such a matrix arises in the problem of finite time feedback stabilization of a linear system, and in the Hilbert problem on the minimal $L_2$-norm of an integral polynomial. We show in a series of examples that the above inverse matrix is integral and has a large divisor. Our method is based on the arithmetic study of orthogonal polynomials naturally associated with the problem.
Keywords: control of linear systems, feedback control
Mots-clés : Hilbert matrix, orthogonal polynomials. 
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A. I. Ovseevich. Control Theory, Integral Matrices, and Orthogonal Polynomials. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 172-181. http://geodesic.mathdoc.fr/item/TM_2021_315_a11/

[1] Al-Salam W.A., Carlitz L., “Congruence properties of the classical orthogonal polynomials”, Duke Math. J., 25:1 (1958), 1–9 | DOI | MR

[2] I. M. Anan'evskii, N. V. Anokhin, and A. I. Ovseevich, “Bounded feedback controls for linear dynamic systems by using common Lyapunov functions”, Dokl. Math., 82:2 (2010), 831–834 | DOI | MR

[3] Brunovský P., “A classification of linear controllable systems”, Kibernetika, 6:3 (1970), 173–188

[4] Brylinski J.-L., “Legendre polynomials and the elliptic genus”, J. Algebra, 145:1 (1992), 83–93 | DOI | MR

[5] Carlitz L., “Congruence properties of the polynomials of Hermite, Laguerre and Legendre”, Math. Z., 59 (1953), 474–483 | DOI

[6] Choi M.-D., “Tricks or treats with the Hilbert matrix”, Amer. Math. Mon., 90:5 (1983), 301–312 | DOI

[7] Choke Rivero A.E., Korobov V.I., Skorik V.A., “Funktsiya upravlyaemosti kak vremya dvizheniya. I”, Mat. fizika, analiz, geometriya, 11:2 (2004), 208–225 | MR

[8] Choke Rivero A.E., Korobov V.I., Skorik V.A., “Funktsiya upravlyaemosti kak vremya dvizheniya. II”, Mat. fizika, analiz, geometriya, 11:3 (2004), 341–354 | MR

[9] Fedorov A., Ovseevich A., “Asymptotic control theory for a system of linear oscillators”, Moscow Math. J., 16:3 (2016), 561–598 | DOI | MR

[10] Hilbert D., “Ein Beitrag zur Theorie des Legendre'schen Polynoms”, Acta math., 18 (1894), 155–159 | DOI | MR

[11] Honda T., “Two congruence properties of Legendre polynomials”, Osaka J. Math., 13 (1976), 131–133 | MR

[12] V. I. Korobov, “A general approach to the solution of the bounded control synthesis problem in a controllability problem”, Math. USSR, Sb., 37:4 (1980), 535–557 | DOI | MR

[13] S. Lang, Algebraic Numbers, Addison-Wesley, Reading, MA, 1964

[14] Ovseevich A., “A local feedback control bringing a linear system to equilibrium”, J. Optim. Theory Appl., 165:2 (2015), 532–544 | DOI | MR

[15] Szegő G., Orthogonal polynomials, AMS Colloq. Publ., 23, Amer. Math. Soc., New York, 1939