Non-Lipschitz points and the SBV regularity of the minimum time function

作者：Colombo Giovanni^*; Nguyen Khai T; Nguyen Luong V

来源：Calculus of Variations and Partial Differential Equations, 2014, 51(1-2): 439-463.

DOI：10.1007/s00526-013-0682-9

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function under controllability conditions which do not imply the Lipschitz continuity of . We consider first the case of normal linear control systems with constant coefficients in . We characterize points around which 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 . Furthermore, we show that is countably -rectifiable with positive -measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set in a neighborhood of the origin in a similar way and prove the -rectifiability of and that . In both cases, is known to have epigraph with positive reach, hence to be a locally 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 must be concentrated in , our analysis yields that is locally , i.e., the Cantor part of vanishes. Our results imply also that is differentiable outside a -rectifiable set. With small changes, our results are valid also in the case of multiple control input.

• 出版日期2014-9