High order conditions for local controllability and controlled stability

High order conditions are given to determine whether or not a specified reference trajectory of a control system is, at each time t, in the interior of the set of points attainable at time t. The conditions are derived via differential geometric methods; the test can be viewed as a high dimensional analog of the two dimensional Green´s theorem approach. A real valued function is constructed; it´s sign can be used to determine the desired property of the reference trajectory. Lie theory is then used to expand this function in a Taylor series making the test algebraic (computational).