We consider an optimal control problem for a system of the form $\dot{x}$ = f(x,u), with a running cost L. We prove an interior sphere property for the level sets of the corresponding value function V. From such a property we obtain a semiconcavity result for V, as well as perimeter estimates for the attainable sets of a symmetric control system.

This paper studies the attainable set at time T>0 for the control system $$\dot{y}\left(t\right)=f(y\left(t\right),u\left(t\right))u\left(t\right)\in U$$ showing that, under suitable assumptions on f, such a set satisfies a uniform interior sphere condition. The interior sphere property is then applied to recover a semiconcavity result for the value function of time optimal control problems with a general target, and to deduce C1,1-regularity for boundaries of attainable sets.