We prove that the higher integrability of the data $f,f_0$ improves on the integrability of minimizers $u$ of functionals $ℱ$, whose model is $${\int }_{\Omega }\left[{\left|Du\right|}^p+{(det\left(Du\right))}^2-\langle f,Du\rangle +\langle f_0,u\rangle \right]dx,$$ where $u:\Omega \subset {ℝ}^n\to {ℝ}^n$ and $p\ge 2$.

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.