Under general hypotheses on the target set S and the dynamics of the system, we show that the minimal time function TS(·) is a proximal solution to the Hamilton-Jacobi equation. Uniqueness results are obtained with two different kinds of boundary conditions. A new propagation result is proven, and as an application, we give necessary and sufficient conditions for TS(·) to be Lipschitz continuous near S. A Petrov-type modulus condition is also shown to be sufficient for continuity of TS(·) near S.
- Continuity of value functions
- Hamilton-Jacobi equations
- Minimal time function
- Nonsmooth analysis
- Proximal analysis