We study the controlled dynamics of the ensembles of points of a Riemannian manifold $M$. Parameterized ensemble of points of $M$ is the image of a continuous map $gamma:Theta o M$, where $Theta$ is a compact set of parameters. The dynamics of ensembles is defined by the action $gamma( heta) mapsto P_t(gamma( heta))$ of the semigroup of diffeomorphisms $P_t:M o M, t in mathbb{R}$, generated by the controlled equation $dot{x}=f(x,u(t))$ on $M$. Therefore, any control system on $M$ defines a control system on (generally infinite-dimensional) space $mathcal{E}_Theta(M)$ of the ensembles of points. We wish to establish criteria of controllability for such control systems. As in our previous work [A. Agrachev, Y. Baryshnikov, and A. Sarychev, ESAIM Control Optim. Calc. Var., 22 (2016), pp. 921--938], we seek to adapt the Lie-algebraic approach of geometric control theory to the infinite-dimensional setting. We study the case of finite ensembles and prove the genericity of the exact controllability property for them. We also find a sufficient approximate controllability criterion for continual ensembles and prove a result on motion planning in the space of flows on $M$. We discuss the relation of the obtained controllability criteria to various versions of the Rashevsky--Chow theorem for finite- and infinite-dimensional manifolds.

Control in the spaces of ensembles of points / Agrachev, A; Sarychev, A.. - In: SIAM JOURNAL ON CONTROL AND OPTIMIZATION. - ISSN 0363-0129. - 58:3(2020), pp. 1579-1596. [10.1137/19M1273049]

Control in the spaces of ensembles of points

Agrachev, A;Sarychev, A.
2020

Abstract

We study the controlled dynamics of the ensembles of points of a Riemannian manifold $M$. Parameterized ensemble of points of $M$ is the image of a continuous map $gamma:Theta o M$, where $Theta$ is a compact set of parameters. The dynamics of ensembles is defined by the action $gamma( heta) mapsto P_t(gamma( heta))$ of the semigroup of diffeomorphisms $P_t:M o M, t in mathbb{R}$, generated by the controlled equation $dot{x}=f(x,u(t))$ on $M$. Therefore, any control system on $M$ defines a control system on (generally infinite-dimensional) space $mathcal{E}_Theta(M)$ of the ensembles of points. We wish to establish criteria of controllability for such control systems. As in our previous work [A. Agrachev, Y. Baryshnikov, and A. Sarychev, ESAIM Control Optim. Calc. Var., 22 (2016), pp. 921--938], we seek to adapt the Lie-algebraic approach of geometric control theory to the infinite-dimensional setting. We study the case of finite ensembles and prove the genericity of the exact controllability property for them. We also find a sufficient approximate controllability criterion for continual ensembles and prove a result on motion planning in the space of flows on $M$. We discuss the relation of the obtained controllability criteria to various versions of the Rashevsky--Chow theorem for finite- and infinite-dimensional manifolds.
58
3
1579
1596
https://epubs.siam.org/doi/abs/10.1137/19M1273049
http://arxiv.org/abs/1907.00905v2
Agrachev, A; Sarychev, A.
File in questo prodotto:
File Dimensione Formato  
19m1273049.pdf

non disponibili

Tipologia: Versione Editoriale (PDF)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 408.95 kB
Formato Adobe PDF
408.95 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
M127304-gg.pdf

accesso aperto

Descrizione: Postprint
Tipologia: Documento in Post-print
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 404.93 kB
Formato Adobe PDF
404.93 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/108833
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 6
social impact