On the Local Structure of Time-Optimal Bang-Bang Trajectories in $\mathbb{R}^3$

Abstract

We consider the problem of time-optimal control for systems of the form $\dot x = f(x) + ug(x)$ where f and g are smooth vector fields and admissible controls are measurable functions u with values in $ - 1 \leqq u \leqq 1$. Under the assumption that f, g and $[f,g]$ are independent, we prove that generically every point has a neighborhood $\mathcal{U}$ such that bang-bang trajectories that lie in $\mathcal{U}$ and have more than 7 switchings are not time-optimal.