December  2013, 3(4): 433-446. doi: 10.3934/mcrf.2013.3.433

Controllability with quadratic drift

1. 

School of Engineering and Applied Sciences, Harvard University, Cambridge MA 02138, United States

Received  March 2013 Revised  August 2013 Published  September 2013

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.
Citation: Roger Brockett. Controllability with quadratic drift. Mathematical Control & Related Fields, 2013, 3 (4) : 433-446. doi: 10.3934/mcrf.2013.3.433
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Cesar O. Aguilar, Local controllability of control-affine systems with quadratic drift and constant control-input vector fields,, in, (2012), 1877. doi: 10.1109/CDC.2012.6425807. Google Scholar

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P. Brunovský, A classification of controllable linear systems,, Kybernetika (Prague), 6 (1970), 173. Google Scholar

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R. E. Kalman, When is a linear control system optimal?,, J. Basic Eng., 86 (1964), 51. doi: 10.1115/1.3653115. Google Scholar

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J. W. Melody, T. Basar and F. Bullo, On nonlinear controllability of homogeneous systems linear in the control,, IEEE Transactions on Automation and Control, 48 (2000), 139. Google Scholar

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H. Sussmann, A general theorem on local controllability,, SIAM J. on Control and Optimization, 25 (1987), 158. doi: 10.1137/0325011. Google Scholar

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R. W. Brockett, Feedback invariants for nonlinear systems,, in, (1978), 1115. Google Scholar

show all references

References:
[1]

Cesar O. Aguilar, Local controllability of control-affine systems with quadratic drift and constant control-input vector fields,, in, (2012), 1877. doi: 10.1109/CDC.2012.6425807. Google Scholar

[2]

R. W. Brockett, Nonholonomic regulators,, in, (2012), 1865. doi: 10.1109/CDC.2012.6426321. Google Scholar

[3]

P. Brunovský, A classification of controllable linear systems,, Kybernetika (Prague), 6 (1970), 173. Google Scholar

[4]

R. E. Kalman, When is a linear control system optimal?,, J. Basic Eng., 86 (1964), 51. doi: 10.1115/1.3653115. Google Scholar

[5]

J. W. Melody, T. Basar and F. Bullo, On nonlinear controllability of homogeneous systems linear in the control,, IEEE Transactions on Automation and Control, 48 (2000), 139. Google Scholar

[6]

H. Sussmann, A general theorem on local controllability,, SIAM J. on Control and Optimization, 25 (1987), 158. doi: 10.1137/0325011. Google Scholar

[7]

R. W. Brockett, Feedback invariants for nonlinear systems,, in, (1978), 1115. Google Scholar

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