ABSTRACT
Our goal is to study approximations of a vector field, or control system, which preserve structural properties. For example, if the structural property is asymptotic stability of the rest solution x = 0 for the equation ẋ = X(x) with X(0) = 0, one might attempt to write X(x) = A(x) + R(x) where A would be an approximating vector field and R a “higher order” remainder vector field. If x = 0 is an asymptotically-stable solution of ẋ = A(x) when is this also true for ẋ = Χ(χ)? We will deal with the situation where the linear approximation gives no definitive information concerning the desired property (and this aspect is coordinate free), i.e., in the above example A would be nonlinear in any choice of local coordinates.
