Back to Search
Start Over
Control in the Spaces of Ensembles of Points
- Source :
- SIAM Journal on Control and Optimization. 58:1579-1596
- Publication Year :
- 2020
- Publisher :
- Society for Industrial & Applied Mathematics (SIAM), 2020.
-
Abstract
- We study the controlled dynamics of the {\it ensembles of points} of a Riemannian manifold $M$. Parameterized ensemble of points of $M$ is the image of a continuous map $\gamma:\Theta \to M$, where $\Theta$ is a compact set of parameters. The dynamics of ensembles is defined by the action $\gamma(\theta) \mapsto P_t(\gamma(\theta))$ of the semigroup of diffeomorphisms $P_t:M \to M, \ t \in \mathbb{R}$, 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 ([1]) 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 genericity of exact controllability property for them. We also find 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 Rashevsky-Chow theorem for finite- and infinite-dimensional manifolds.<br />Comment: 24 pages
- Subjects :
- 0209 industrial biotechnology
Pure mathematics
Control and Optimization
Continuous map
Parameterized complexity
Infinite-dimensional control systems , Nonlinear control , Controllability , Lie-algebraic methods
02 engineering and technology
Nonlinear control
01 natural sciences
020901 industrial engineering & automation
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
Mathematics - Optimization and Control
Mathematics - Classical Analysis and ODEs
93B05, 93C25, 58E25
0101 mathematics
Control (linguistics)
Mathematics
Applied Mathematics
Image (category theory)
010102 general mathematics
93C25
Riemannian manifold
93B05
58E25
Controllability
Optimization and Control (math.OC)
Mathematics::Differential Geometry
Subjects
Details
- ISSN :
- 10957138 and 03630129
- Volume :
- 58
- Database :
- OpenAIRE
- Journal :
- SIAM Journal on Control and Optimization
- Accession number :
- edsair.doi.dedup.....000fa85d5e3b24f4701aa322511fd29e