Local Regularity of the Minimum Time Function



We consider the minimum time problem of optimal control theory. It is well known that under appropriate controllability type conditions the minimum time function has an open domain of definition and is locally Lipschitz on it. Thus, it is differentiable almost everywhere on its domain. Furthermore, in general, it fails to be differentiable at points where there are multiple time optimal trajectories and its differentiability at a point does not guarantee continuous differentiability around this point. In this paper, however, we show that, under some regularity assumptions, the nonemptiness of proximal subdifferential of the minimum time function at a point implies its continuous differentiability on a neighborhood of this point.


Hamiltonian systems Conjugate times Maximum principle Minimum time function 

Mathematics Subject Classification

49N60 49J15 49K15 



Partial support by the European Commission (FP7-PEOPLE-2010-ITN, Grant Agreement no. 264735-SADCO) is gratefully acknowledged. The second author expresses his thanks to the Institut de Mathématiques de Jussieu at Université Pierre and Marie Curie, Paris for the hospitality during his secondment.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.CNRS, IMJ-PRG, UMR 7586, Sorbonne Universités, UPMC Univ Paris 06, Univ Paris Diderot Sorbonne Paris CitéParisFrance
  2. 2.Dipartimento di MatematicaUniversità di PadovaPadovaItaly

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