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 $y\in {\mathbb{R}}^n$ to the origin, and call $\text{A}(T)$ the set of initial states that can be steered to the origin in time $\tau (y)\le T$. Given any $\text{ve}>0$, in this paper we construct a patchy feedback $u=U(x)$ such that every solution of $\dot{x}=f(x,U(x))$, $x(0)=y\in \text{A}(T)$ reaches an $\text{ve}$-neighborhood of the origin within time $\tau (y)+\text{ve}$.