Methods of Control Theory in Nonholonomic Geometry

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(τ), 0 ≤ τ ≤ t, q(0) = q0, satisfying a differential equation of the form (1) q̇ = vτ (q), where vτ (q) ∈ V ∩ TqM, ∀q ∈ M, vτ (q) is smooth in q, bounded and measurable in τ. The mapping q(·) 7→ q(t) which maps admissible trajectories in their end points is called an end-point mapping. Control Theory is in a sense a theory of end-point mappings. This point of view is rather restrictive but sufficient for our purposes. For instance, attainable sets are just images of end-point 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 instance, 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 to thousands of items. A great part of the theory is devoted to the case of nonsmooth V such that V ∩ TqM are polytopes or worse. There is a wide-spread view that such a nonsmoothness is the essence of Control Theory. This is not my opinion, and I am making the following radical assumption. Let us assume that V forms a smooth locally trivial bundle over M with fibers Vq — smooth closed convex submanifolds in TqM of positive dimension, symmetric with respect to the origin. So we consider a very special class of control systems.