  In this article we consider diffeomorphism groups of manifolds with smooth
boundary. We show that the diffeomorphism groups of the manifold and its
boundary fit into a short exact sequence which admits local sections. In other
words, they form an infinite-dimensional fibre bundle. Manifolds with boundary
are of interest in numerical analysis and with a view towards applications in
machine learning we establish controllability results for families of vector
fields. This generalises older results due to Agrachev and Caponigro in the
boundary-less case. Our results show in particular that the diffeomorphism
group of a manifold with smooth boundary is generated by the image of the
exponential map.
