  An analysis of necessary conditions for the existence of controlled dynamics
with an attractor of a specified topological type is given. It uses the Hopf
classification by degree for Gauss maps of manifolds to spheres of the same
dimension, in this case noting that by a linear homotopy, the degree is the
same for the negative of the gradient flow of a Lyapunov function and for the
stabilising vector field. The manifolds are Lyapunov level sets. More general
contexts are given by considering maps in homology. Some account is taken of
cases where the degree is zero and the surjectivity fails. A version of this
work was submitted for publication in 2009, and the paper is now in the process
of revision and expansion.
