Nearly Time Optimal Stabilizing Patchy Feedbacks

We consider the time optimal stabilization problem for a nonlinear control system $\dot x=f(x,u)$. Let $\tau(y)$ be the minimum time needed to steer the system from the state $y\in\R^n$ to the origin, and call $\A(T)$ the set of initial states that can be steered to the origin in time $\tau(y)\leq T$. Given any $\ve>0$, in this paper we construct a patchy feedback $u=U(x)$ such that every solution of $\dot x=f(x, U(x))$, $x(0)=y\in \A(T)$ reaches an $\ve$-neighborhood of the origin within time $\tau(y)+\ve$.