Local asymptotic feedback stabilization to a submanifold: Topological conditions

We consider the problem of local asymptotic feedback stabilization of a control system x˙=f(x,u) defined in RnRn to a compact, connected, oriented, embedded codimension one submanifold P   of RnRn using a continuous feedback law. This generalizes the problem of local asymptotic feedback stabilization to a point which has been previously considered in the control theory literature. It is natural to expect the topology of P to play a role in deciding whether or not local asymptotic stabilization to P   of the system x˙=f(x,u) is feasible via continuous feedback, and our aim in this paper is precisely to outline necessary conditions on the topology of P for stabilization via continuous feedback to be achievable.