  Brockett's necessary condition yields a test to determine whether a system
can be made to stabilize about some operating point via continuous, purely
state-dependent feedback. For many real-world systems, however, one wants to
stabilize sets which are more general than a single point. One also wants to
control such systems to operate safely by making obstacles and other
"dangerous" sets repelling.
  We generalize Brockett's necessary condition to the case of stabilizing
general compact subsets having a nonzero Euler characteristic in general
ambient state spaces (smooth manifolds). Using this generalization, we also
formulate a necessary condition for the existence of "safe" control laws. We
illustrate the theory in concrete examples and for some general classes of
systems including a broad class of nonholonomically constrained Lagrangian
systems. We also show that, for the special case of stabilizing a point, the
specialization of our general stabilizability test is stronger than Brockett's.
