NDLI: On Local and Global Controllability

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Hermes, H.
Copyright Year	1974
Abstract	Computable sufficient conditions to determine local controllability along a reference trajectory are developed both by considering controllability of the linear variational equation and by direct use of differential geometric techniques with special types of control perturbations. The equivalence of the results obtained by the two methods is shown.A collection, $\mathcal{S}$, of smooth vector fields on a manifold M is said to be controllable on M if every pair of points of M can be connected by a solution of $\mathcal{S}$. If the set of points attainable by solutions of $\mathcal{S}$ from every point $x \in M$ has nonempty interior, is said to have the accessibility property. Jurdjevic has posed the problem of whether every family of analytic vector fields on a connected analytic manifold M which has the accessibility property is controllable on M. We give a counter-example on the two-torus. It is next shown that every commuting two-field on the two-torus is controllable. We also show that any n-manifold admits a collection $\mathcal{S}$ of $2n$ smooth $(C^\infty )$ vector fields with the accessibility property, but such that $\mathcal{S}$ is not controllable on M.
Starting Page	252
Ending Page	261
Page Count	10
File Format	PDF
ISSN	00361402
DOI	10.1137/0312019
Issue Number	2
Volume Number	12
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-18
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article
Subject	Engineering

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