Controllability with quadratic drift

Roger Brockett

doi:10.3934/mcrf.2013.3.433

Article Contents

2013, Volume 3, Issue 4: 433-446. Doi: 10.3934/mcrf.2013.3.433

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Controllability with quadratic drift

Roger Brockett^1,

1.
School of Engineering and Applied Sciences, Harvard University, Cambridge MA 02138

Received: February 28, 2013

Revised: July 31, 2013

Published: November 2013

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Abstract

We study the controllability properties of systems of the form $ \dot{x}=Ax+Bu\;;\; \dot{w}=q(x) $ with $q$ being a vector of quadratic functions of $x$. This class of nonlinear systems is interesting because it is both remarkably tractable and because it is the second order approximation to a larger class of nonlinear systems. We not only describe the distribution generated by the vector fields associated with this system but, in important cases, we are able to give a precise description of which points are reachable from a given initial state, distinguishing between those points that are reachable immediately and those that are only reachable after a sufficient length of time.

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Mathematics Subject Classification: Primary: 93B03, 93B27; Secondary: 93C10.

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