The Bilateral Minimal Time Function

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.