Topological Obstructions to Submanifold Stabilization

We consider the problem of local asymptotic feedback stabilization-via a continuously differentiable feedback law-of a control system ẋ = f(x,u) defined in Euclidean space Rⁿ (with f being continuously differentiable) to a compact, connected, oriented m-dimensional submanifold M of Rⁿ with codimension strictly larger than one. We obtain necessary conditions on the topology of M for such a stabilizing feedback law to exist. This extends the work done in, where only the codimension one case was treated. We also briefly discuss the case where the control is only assumed continuous.