no. 4
Ensemble controllability by Lie algebraic methods
Agrachev, Andrei1, 2 ; Baryshnikov, Yuliy3 ; Sarychev, Andrey4
1 International School for Advanced Studies (SISSA), v. Bonomea, 265, 34136 Trieste, Italy
2 Steklov Mathematical Institute, Russian Acad. Sciences, Moscow, Russia
3 University of Illinois at Urbana-Champaign 1409 W. Green Str., Urbana IL 61801, USA
4 University of Florence, DiMaI, v. delle Pandette 9, 50127 Firenze, Italy
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 921-938.

We study possibilities to control an ensemble (a parameterized family) of nonlinear control systems by a single parameter-independent control. Proceeding by Lie algebraic methods we establish genericity of exact controllability property for finite ensembles, prove sufficient approximate controllability condition for a model problem in R 3 , and provide a variant of Rashevsky−Chow theorem for approximate controllability of control-linear ensembles.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2016029
Classification : 93C15, 93B05, 93B27
Mots clés : Infinite-dimensional control systems, ensemble controllability, Lie algebraic methods
Agrachev, Andrei 1, 2 ; Baryshnikov, Yuliy 3 ; Sarychev, Andrey 4

1 International School for Advanced Studies (SISSA), v. Bonomea, 265, 34136 Trieste, Italy
2 Steklov Mathematical Institute, Russian Acad. Sciences, Moscow, Russia
3 University of Illinois at Urbana-Champaign 1409 W. Green Str., Urbana IL 61801, USA
4 University of Florence, DiMaI, v. delle Pandette 9, 50127 Firenze, Italy 
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     title = {Ensemble controllability by {Lie} algebraic methods},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {921--938},
     publisher = {EDP-Sciences},
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Agrachev, Andrei; Baryshnikov, Yuliy; Sarychev, Andrey. Ensemble controllability by Lie algebraic methods. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 921-938. doi : 10.1051/cocv/2016029. http://archive.numdam.org/articles/10.1051/cocv/2016029/

A. Agrachev and M. Caponigro, Controllability on the group of diffeomorphisms. Ann. Inst. Henri Poincaré, Anal. Non Lin. 26 (2009) 2503–2509. | DOI | Zbl

A. Agrachev and Yu. Sachkov, Control Theory from the Geometric Viewpoint. Springer (2004). | Zbl

A. Agrachev and A. Sarychev, The control of rotation for asymmetric rigid body. Probl. Control Inform. Theory 12 (1983) 335–347. | Zbl

A. Agrachev and A. Sarychev, Solid Controllability in Fluid Dynamics, in Instabilities in Models Connected with Fluid Flows I, edited by C. Bardos and A. Sarychev. Springer (2008) 1–35. | Zbl

J.M. Ball, J.E. Marsden and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim. 20 (1982) 575–597. | DOI | Zbl

K. Beauchard, J.-M. Coron and P. Rouchon, Controllability issues for continuous-spetrum systems and ensemble controllability of Bloch equations. Comm. Math. Phys. 296 (2010) 525–557. | DOI | Zbl

B. Bonnard, Contrôllabilité des systèmes non linéaires. C.R. Acad. Sci. 292 (1981) 535–537. | Zbl

B. Bonnard, Contrôle de l’attitude d’un satellite rigide, in Outils et modèles mathématiques pour l’automatique, l’analyse de systèmes et le traitement du signal, 3 CNRS (1983) 649–658. | Zbl

P.M. Dudnikov and S.N. Samborski, Criterion of controllability for systems in Banach space (generalization of Chow’s theorem). Ukrain. Matem. Zhurnal 32 (1980) 649–653 (in Russian). | Zbl

Yu. Ledyaev, On Infinite-Dimensional Variant of Rashevsky−Chow Theorem. Dokl. Akad. Nauk 398 (2004) 735–737.

J.S. Li and N. Khaneja, Noncommuting vector fields, polynomial approximations and control of inhomogeneous quantum ensembles, Preprint [quant-ph] (2005). | arXiv

J.S. Li and N. Khaneja, Control of inhomogeneous quantum ensembles. Phys.Rev. A 73 (2006) 030302. | DOI

J.S. Li and N. Khaneja, Ensemble Control of Bloch Equations. IEEE Trans. Automatic Control 54 (2009) 528–536. | DOI | Zbl

C. Lobry, Une propriete generique des couples de champs de vecteurs. Czechoslovak Mathem. J. (1972) 230–237. | Zbl

C. Lobry, Controllability of nonlinear systems on compact manifolds. SIAM J. Control 12 (1974) 1–4. | DOI | Zbl

M.K. Salehani and I. Markina, Controllability on Infinite-Dimensional Manifolds: a Chow-Rashevsky theorem. Acta Appl. Math. 134 (2014) 229–246. | DOI | Zbl

H.J. Sussmann, Some properties of vector fields, which are not altered by small perturbations. J. Differ. Eq. 20 (1976) 292–315. | DOI | Zbl

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