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

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [ (2001) 21–40], is due to Li and Nirenberg...

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