**Title:** Control theory without controls (11 pages)

**Detail:** AMS Joint Mathematics Meeting, San Antonio, 2006/01/15

In control theory one often considers systems of the form

d*x*/d*t* = *f*_{0}(*x*) +
*u*^{a}*f*_{a}(*x*),

where *f*_{0},*f*_{1},...,*f*_{m} are
vector fields on a manifold *M*, *t* -> *u*(*t*) is the
**R**^{m}-valued control, and *t* ->
*x*(*t*) is the *M*-valued trajectory. While in applications
the appearance of the controls is natural to the problem formulation, for a
study of the *geometry* of the problem, the controls can be thought of
as playing the role of a specific set of coordinates: a possibly convenient,
but also possibly obfuscating, device.
Thus we propose a formulation of control systems of the form above, but
without making a specific choice of controls. Time will prohibit
presentation of detailed results, so the talk will focus on the formulation
of problems in this abstract setting for control theory. Particularly, we
formulate stabilisation and controllability problems, and give a few simple
results and examples to suggest that the formulation is a promising one.
No online version avaliable (but check this out).

*Andrew D. Lewis (andrew at mast.queensu.ca)*