This paper proves a corona theorem for the algebra of Radon measures compactly supported in and this result is applied to provide a necessary and sufficient Hautus–type frequency criterion for the exact controllability of linear controlled delayed difference equations (LCDDE). Hereby, it solves an open question raised in [5].
Ce papier prouve un théorème de couronne dans l’algèbre des mesures de Radon à support compact dans et ce résultat est appliqué pour fournir un critère fréquentiel de type Hautus, nécessaire et suffisant, pour la contrôlabilité exacte des équations linéaires aux différences retardées. Ainsi, il résout une question ouverte soulevée dans [5].
Revised:
Accepted:
Published online:
Keywords: Corona theorem, Bézout’s identity, difference delay equations, exact controllability
Mots-clés : Théorème de couronne, identité de Bézout, équations aux différences, contrôlabilité exacte
Sébastien Fueyo 1; Yacine Chitour 2
CC-BY 4.0
@article{CRMATH_2024__362_G8_851_0,
author = {S\'ebastien Fueyo and Yacine Chitour},
title = {A corona theorem for an algebra of {Radon} measures with an application to exact controllability for linear controlled delayed difference equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {851--861},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.604},
language = {en},
}
TY - JOUR AU - Sébastien Fueyo AU - Yacine Chitour TI - A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations JO - Comptes Rendus. Mathématique PY - 2024 SP - 851 EP - 861 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.604 LA - en ID - CRMATH_2024__362_G8_851_0 ER -
%0 Journal Article %A Sébastien Fueyo %A Yacine Chitour %T A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations %J Comptes Rendus. Mathématique %D 2024 %P 851-861 %V 362 %I Académie des sciences, Paris %R 10.5802/crmath.604 %G en %F CRMATH_2024__362_G8_851_0
Sébastien Fueyo; Yacine Chitour. A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 851-861. doi : 10.5802/crmath.604. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.604/
[1] Sufficient stability conditions for time-varying networks of telegrapher’s equations or difference-delay equations, SIAM J. Math. Anal., Volume 53 (2021) no. 2, pp. 1831-1856 | DOI | MR | Zbl
[2] Matrix-valued corona theorem for multiply connected domains, Indiana Univ. Math. J., Volume 49 (2000) no. 4, pp. 1405-1410 | DOI | MR | Zbl
[3] On the corona theorem for almost periodic functions, Integral Equations Oper. Theory, Volume 33 (1999) no. 3, pp. 253-272 | DOI | MR | Zbl
[4] Interpolations by bounded analytic functions and the corona problem, Ann. Math., Volume 76 (1962), pp. 547-559 | DOI | MR | Zbl
[5] Hautus–Yamamoto criteria for approximate and exact controllability of linear difference delay equations, Discrete Contin. Dyn. Syst., Volume 43 (2023) no. 9, pp. 3306-3337 | DOI | MR | Zbl
[6] Stability of non-autonomous difference equations with applications to transport and wave propagation on networks, Netw. Heterog. Media, Volume 11 (2016) no. 4, pp. 563-601 | DOI | MR | Zbl
[7] Approximate and exact controllability of linear difference equations, J. Éc. Polytech., Math., Volume 7 (2020), pp. 93-142 | DOI | Numdam | MR | Zbl
[8] Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems, SIAM J. Math. Anal., Volume 47 (2015) no. 3, pp. 2220-2240 | DOI | MR | Zbl
[9] A course in functional analysis, Graduate Texts in Mathematics, 96, Springer, 2019 | MR
[10] A subalgebra of the Hardy algebra relevant in control theory and its algebraic-analytic properties, The Corona Problem: Connections Between Operator Theory, Function Theory, and Geometry (Fields Institute Communications), Volume 72, Springer, 2014, pp. 85-106 | DOI | MR | Zbl
[11] Geometric and analytic number theory, Springer, 2012 | MR
[12] Introduction to functional-differential equations, Applied Mathematical Sciences, 99, Springer, 1993, x+447 pages | DOI | MR | Zbl
[13] A course in commutative Banach algebras, Graduate Texts in Mathematics, 246, Springer, 2009, xii+353 pages | DOI | MR | Zbl
[14] Functional equations and inequalities with applications, Springer Monographs in Mathematics, Springer, 2009, xxiv+810 pages | DOI | MR | Zbl
[15] Corona-type theorems and division in some function algebras on planar domains, The Corona problem. Connections between operator theory, function theory, and geometry (Fields Institute Communications), Volume 72, Springer, 2014, pp. 127-151 | DOI | MR | Zbl
[16] Generators for rings of compactly supported distributions, Integral Equations Oper. Theory, Volume 69 (2011) no. 1, pp. 63-71 | DOI | MR | Zbl
[17] In search of the invisible spectrum, Ann. Inst. Fourier, Volume 49 (1999) no. 6, pp. 1925-1998 | DOI | Numdam | MR | Zbl
[18] Functional analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., 1991, xviii+424 pages | MR | Zbl
[19] Elementary functional analysis, World Scientific, 2009, x+181 pages | DOI | MR | Zbl
[20] The matrix-valued Corona problem in the disk and polydisk, J. Funct. Anal., Volume 226 (2005) no. 1, pp. 138-172 | DOI | MR | Zbl
[21] Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems, SIAM J. Control Optim., Volume 27 (1989) no. 1, pp. 217-234 | DOI | MR | Zbl
[22] Behavioral controllability and coprimeness for a class of infinite-dimensional systems, Proceeding of the 47th IEEE Conference on Decision and Control, IEEE (2008), pp. 1513-1518 | DOI
Cited by Sources:
Comments - Policy