ABSTRACT
This chapter provides 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. The general theory regarding controllability questions for discrete time systems remained until recently much weaker than that possible in the more classical continuous time case. In principle, noninvert-ibility of transition maps in discrete time implies that semigroups appear where groups would appear in the continuous case, so less algebraic structure is available. One of the main results to be proved is a theorem that describes the orbits under such actions as sub-manifolds associated to certain vector fields. One of the main applications of the integrability theorem is in verifying the various controllability properties by checking appropriate rank conditions on vector fields. The lack of infinitesimal information is dealt with by substituting derivations with respect to control values, assuming that, as is often the case, there is a differentiable structure in the control value set.
