Abstract

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function T under controllability conditions which do not imply the Lipschitz continuity of T. We consider first the case of normal linear control systems with constant coefficients in RN. We characterize points around which T is not Lipschitz as those 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 such set, that we call S. Furthermore, we show that S is countably HN−1-rectifiable with 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 the H1-rectifiability of S and that H1(S)>0. In both cases, T is known to have epigraph with positive reach, hence to be a locally BV function (see Colombo et al.: SIAM J Control Optim 44:2285–2299, 2006; Colombo and Nguyen.: Math Control Relat 3: 51–82, 2013). Since the Cantor part of DT must be concentrated in S, our analysis yields that T is locally SBV, i.e., the Cantor part of DT vanishes. Our results imply also that T is differentiable outside a HN−1-rectifiable set. With small changes, our results are valid also in the case of multiple control input.