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 \({\mathbb {R}}^N\). 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 \(\mathcal {S}\). Furthermore, we show that \(\mathcal {S}\) is countably \(\mathcal {H}^{N-1}\)-rectifiable with positive \(\mathcal {H}^{N-1}\)-measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set \(\mathcal {S}\) in a neighborhood of the origin in a similar way and prove the \(\mathcal {H}^1\)-rectifiability of \(\mathcal {S}\) and that \(\mathcal {H}^1(\mathcal {S})>0\). 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 \(\mathcal {S}\), 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 \(\mathcal {H}^{N-1}\)-rectifiable set. With small changes, our results are valid also in the case of multiple control input.
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Communicated by L. Ambrosio.
This work was partially supported by M.I.U.R., project “Viscosity, metric, and control theoretic methods for nonlinear partial differential equations”, by CARIPARO Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”, and University of Padova research project “Some analytic and differential geometric aspects in Nonlinear Control Theory, with applications to Mechanics”. The authors are also supported by the European Union under the 7th Framework Programme “FP7-PEOPLE-2010-IT”, Grant agreement number 264735-SADCO. In particular, the third author is a SADCO PhD fellow, position ESR4. K.T.N. did this work when he was supported by the ERC Starting Grant 2009 n.240385 ConLaws in Padova.
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Colombo, G., Nguyen, K.T. & Nguyen, L.V. Non-Lipschitz points and the \({\textit{SBV}}\) regularity of the minimum time function. Calc. Var. 51, 439–463 (2014). https://doi.org/10.1007/s00526-013-0682-9
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DOI: https://doi.org/10.1007/s00526-013-0682-9