 
Summary: METHODS OF CONTROL THEORY
IN NONHOLONOMIC GEOMETRY
Andrei A. Agrachev
1.Introduction. Let M be a C
manifold and TM the total space of the tangent
bundle. A control system is a subset V TM. Fix an initial point q0 M and a
segment [0, t] R. Admissible trajectories are Lipschitzian curves q(), 0
t, q(0) = q0, satisfying a differential equation of the form
(1) q = v (q),
where v (q) V TqM, q M, v (q) is smooth in q, bounded and measurable
in . The mapping q(·) q(t) which maps admissible trajectories in their end
points is called an endpoint mapping.
Control Theory is in a sense a theory of endpoint mappings. This point
of view is rather restrictive but sufficient for our purposes. For instance, attain
able sets are just images of endpoint mappings. Geometric Control Theory tends
to characterize properties of these mappings in terms of iterated Lie brackets of
smooth vector fields on M with values in V. A number of researchers have shown
a remarkable ingenuity in this regard leading to encouraging results. See, for in
stance, books [1],[2],[3] to get an idea of various periods in the development of this
domain and for other references. A complete list of references would probably run
