Perimeter Estimates for Reachable Sets of Control Systems

The reachable set in time $T>0$, ${\cal R}(T)$, is here investigated for the symmetric control system $\dot x (t)=f(x(t))u(t)$, $u(t)\in\overline{B}$. It turns out that, for $f(x)$ smooth and nondegenerate, ${\cal R}(T)$ has finite perimeter, and a sharp estimate for the time-dependence of the perimeter and volume of such a set can be obtained.