Control in the spaces of ensembles of points

2019/07/01 by Agrachev, Andrei, Sarychev, Andrey · 4 citations
#58E25 #93B05 #93C25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1907.00905

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 γ:Θ→ M, where Θ is a compact set of parameters. The dynamics of ensembles is defined by the action γ(θ) ↦ P_t(γ(θ)) of the semigroup of diffeomorphisms P_t:M → M, t ∈ ℝ, generated by the controlled equation x=f(x,u(t)) on M. Therefore any control system on M defines a control system on (generally infinite-dimensional) space E_Θ(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.

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