Title : Non - Lipschitz points and the SBV regularity of the mini - mum time function Luong

We study the Hausdorff dimension of the set of non-Lipschitz points S of the minimum time function T under controllability conditions which imply the continuity of T . We consider first the case of normal linear control systems with constant coefficients in RN . We characterize S as the set of points which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of S. Furthermore, we show that S is HN−1-rectifiable and has positive HN−1-measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set S in a neighborhood of the origin in a similar way and prove its H1-rectifiability and that H1(S) > 0. In both (linear and nonlinear) cases, T is known to have epigraph with positive reach, hence to be a locally BV function. Since the Cantor part of DT must be concentrated in S, our analysis yields that T is SBV , i.e., the Cantor part of DT vanishes. This talk is based on a joint work with Giovanni Colombo and Khai T. Nguyen.