Integrability of certain distributions associated with actions on manifolds and applications to control problems

The first objective of this paper is to provide a tutorial introduction to integrability results for distributions-or “singular vector bundles”-on manifolds. These distributions arise from actions of smoothly parametrized families of diffeomorphisms. Such results generalize Frobenious’ Theorem in two ways: they deal with diffeomorphisms not necessarily associated to flows, and they do not require the distribution to be nonsingular. Results along these lines are important in various areas of control theory, and they originated in the work of Hermann ([7]) and subsequent research by Sussmann ([19]) and Stefan ([14]) in the early 70‘s, who removed the non-singularity assumption and showed that a form of the theorem still holds in the singular case. We shall present an abstract version which summarizes all that is needed for various applications. Our result is more abstract in that it deals with rather general classes of diffeomorphisms, not just those arising from flows as in [19] and [14], but the main ideas of the proof are 82very similar to the ones in the former reference. Following [19], we also show how more special results due to Nagano, Lobry, and others can be obtained as consequences of the general theorem.