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Journal of Convex Analysis 13 (2006), No. 1, 061--080
Copyright Heldermann Verlag 2006

The Bilateral Minimal Time Function

Chadi Nour
Dept. of Mathematics and Statistics, Concordia University, 1400 De Maisonneuve Boul. West, Montreal, Quebec H3G 1M8, Canada
Permanent Address: Computer Science and Mathematics Division, Lebanese American University, Byblos Campus, Lebanon

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.

Keywords: Minimal time function, Hamilton-Jacobi equations, viscosity solutions, regularity of value functions, nonsmooth analysis, proximal analysis.

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