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Cited article:

On the existence of solutions to a class of minimum time control problems and applications to Fermat's Principle and to the Brachystocrone

A. Cellina, A. Ferriero and E.M. Marchini
Systems & Control Letters 55 (2) 119 (2006)
https://doi.org/10.1016/j.sysconle.2005.06.002

Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Vanishing Lagrangians

Michael Malisoff
SIAM Journal on Control and Optimization 40 (5) 1358 (2002)
https://doi.org/10.1137/S0363012900368594