Controllability theory for control systems deals with the description of the set of states reachable, using the controls, from an initial state. This appears to be a difficult description to obtain, although significant progress has been made.

After understanding what a ``solution'' of the problem might look like, the status the current state of the art will be reviewed. The first-order conditions for controllability are currently understood. A geometric approach to the second-order conditions will be proposed, and preliminary results will be given.

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