Comptes Rendus
no. 2
Optimal Control
Description of all privileged coordinates in the homogeneous approximation problem for nonlinear control systems
[Description de toutes les coordonnées privilégiées dans le problème d'approximation homogène pour les systèmes contrôlés non linéaires]

Présenté par : Pierre-Louis Lions

Grigory Sklyar1 ; Svetlana Ignatovich2
1 University of Szczecin, Wielkopolska str., 15, 70-451, Szczecin, Poland
2 Kharkov National University, Svoboda sqr. 4, Kharkov, 61077, Ukraine
Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 109-114.

Dans le problème d'approximation homogène pour des systèmes contrôlés affines les coordonnées privilégiées sont celles dans lesquelles le système a une forme « triangulaire » qui permet de trouver un système d'approximation. Nous donnons les conditions nécessaires et suffisantes pour que des coordonnées soient privilégiées. Nous utilisons une technique algébrique basée sur la représentation par des séries de systèmes de commande affines.

In a homogeneous approximation problem for affine control systems, privileged coordinates are those in which the system takes a ‘triangular’ form allowing one to find an approximating system. We give the necessary and sufficient conditions for coordinates to be privileged. We apply an algebraic technique based on the series representation of affine control systems.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.11.016
Grigory Sklyar 1 ; Svetlana Ignatovich 2

1 University of Szczecin, Wielkopolska str., 15, 70-451, Szczecin, Poland
2 Kharkov National University, Svoboda sqr. 4, Kharkov, 61077, Ukraine 
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Grigory Sklyar; Svetlana Ignatovich. Description of all privileged coordinates in the homogeneous approximation problem for nonlinear control systems. Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 109-114. doi : 10.1016/j.crma.2006.11.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.11.016/

[1] A.A. Agrachev; R.V. Gamkrelidze; A.V. Sarychev Local invariants of smooth control systems, Acta Appl. Math., Volume 14 (1989), pp. 191-237

[2] A. Bellaïche The tangent space in sub-Riemannian geometry, Progress in Mathematics, vol. 144, 1996, pp. 4-78

[3] R.M. Bianchini; G. Stefani Graded approximation and controllability along a trajectory, SIAM J. Control Optim., Volume 28 (1990), pp. 903-924

[4] M. Fliess Fonctionnelles causales non linéaires et indéterminées non commutatives, Bull. Soc. Math. France, Volume 109 (1981), pp. 3-40

[5] H. Hermes Nilpotent and high-order approximations of vector field systems, SIAM Rev., Volume 33 (1991), pp. 238-264

[6] B. Jakubczyk Local realizations of nonlinear causal operators, SIAM J. Control Optim., Volume 24 (1986), pp. 230-242

[7] M. Kawski The combinatorics of nonlinear controllability and noncommuting flows, Abdus Salam ICTP Lecture Notes Series, vol. 8, 2002, pp. 223-312

[8] M. Kawski; H. Sussmann Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory, Operators, Systems, and Linear Algebra, Teubner, 1997, pp. 111-128

[9] G. Melançon; C. Reutenauer Lyndon words, free algebras and shuffles, Canad. J. Math., Volume XLI (1989), pp. 577-591

[10] C. Reutenauer Free Lie Algebras, Clarendon Press, Oxford, 1993

[11] R. Ree Lie elements and an algebra associated with shuffles, Ann. of Math., Volume 68 (1958), pp. 210-220

[12] G.M. Sklyar; S.Yu. Ignatovich Moment approach to nonlinear time optimality, SIAM J. Control Optim., Volume 38 (2000), pp. 1707-1728

[13] G.M. Sklyar; S.Yu. Ignatovich Approximation of time-optimal control problems via nonlinear power moment min-problems, SIAM J. Control Optim., Volume 42 (2003), pp. 1325-1346

[14] G.M. Sklyar; S.Yu. Ignatovich Representations of control systems in the Fliess algebra and in the algebra of nonlinear power moments, Systems Control Lett., Volume 47 (2002), pp. 227-235

[15] G.M. Sklyar; S.Yu. Ignatovich; P.Yu. Barkhaev Algebraic classification of nonlinear steering problems with constraints on control, Adv. Math. Res., vol. 6, 2005, pp. 37-96

[16] H. Sussmann; V. Jurdjevic Controllability of nonlinear systems, J. Differential Equations, Volume 12 (1972), pp. 95-116

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