Nonsmooth analysis and control theory
Nonsmooth analysis and control theory
Approximating Subdifferentials by Random Sampling of Gradients
Mathematics of Operations Research
Some New Regularity Properties for the Minimal Time Function
SIAM Journal on Control and Optimization
On the Structure of the Minimum Time Function
SIAM Journal on Control and Optimization
Exterior Sphere Condition and Time Optimal Control for Differential Inclusions
SIAM Journal on Control and Optimization
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We consider the class of continuous functions that map an open set Ω ⊆ Rn to R with an epigraph having locally positive reach with an additional property. This class contains all finite convex and C1, 1 functions, but also ones that are not necessarily Lipschitz continuous. We provide a representation formula for the Clarke generalized gradient of such functions using convex combinations and limits of gradients at differentiability points, thus offering an alternative to the well-known proximal normal formula by replacing a pointedness assumption by one of positive reach. Our proof consists of a detailed analysis of singularities using methods taken from both nonsmooth analysis and geometric measure theory, and is based on an induction argument. As an application, we prove for a particular class of Hamilton-Jacobi equations that an a.e. solution whose hypograph has positive reach and satisfies an additional property is indeed the unique viscosity solution.