Geodesic
The Bilateral Minimal Time Function
Journal of convex analysis, Tome 13 (2006) no. 1, pp. 61-8.

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We study the minimal time function as a function of two variables (the initial and the terminal points). This function, called the "bilateral minimal time function", plays a central role in the study of the Hamilton-Jacobi equation of minimal control in a domain which contains the target set, as shown in a recent article of F. H. Clarke and the author [J. Convex Analysis 11 (2004) 413--436]. We study the regularity of the function, and characterize it as the unique (viscosity) solution of partial Hamilton-Jacobi equations with certain boundary conditions.
Mots-clés : Minimal time function, Hamilton-Jacobi equations, viscosity solutions, regularity of value functions, nonsmooth analysis, proximal analysis 
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C. Nour. The Bilateral Minimal Time Function. Journal of convex analysis, Tome 13 (2006) no. 1, pp. 61-8. http://geodesic.mathdoc.fr/item/JCA_2006_13_1_JCA_2006_13_1_a4/