A Hadamard-type open map theorem for submersions and applications to   completeness results in Control Theory
  We prove a quantitative openness theorem for $C^1$ submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carath\'eodory spaces and we improve a classical completeness result by Palais. 
Andrea Bonfiglioli, Annamaria Montanari, Daniele Morbidelli
math.CA math.DG