Nearly time optimal stabilizing patchy feedbacks

We consider the time optimal stabilization problem for a nonlinear control system x' = f (x, u). Let T(y) be the minimum time needed to steer the system from the state y \in R^n to the origin, and call A(tau) the set of initial states that can be steered to the origin in time T(y) <= tau. Given any epsilon > 0, in this paper we construct a patchy feedback u = U(x) such that every solution of x' = f (x, U(x)), x(0) = y \in A(tau) reaches an epsilon-neighborhood of the origin within time T(y) + epsilon.