Feedback equivalence for general control systems

We discuss the solution to the problem of local equivalence of control systems with n states and p controls in a neighbourhood of a generic point, under the Lie pseudo-group of local time independent feedback transformations. We have shown earlier that this problem is identical with the problem of simple equivalence of the time optimal variational problem. Here we indicate the way in which this identification may be used to obtain closed loop time critical controls for general systems. We show that the classification of general nonlinear systems depends on the classification of all n−p dimensional affine subspaces of the space of symmetric forms in p variables and that the case of control linear systems depends on the classification of all n−p dimensional affine subspaces of the space of skew forms in p variables. We show that in the latter case the G-structure is the prolongation of one determined by a state space transformation group. We give a complete list of normal forms for control linear systems in the case p=n−1.