Regularity and subgradient formulas for the minimal time function

Peter Wolenski (Louisiana State University)

Abstract: The regularity of the minimum time function is an old subject, both for linear and nonlinear control systems. Typically conditions are imposed on the target and vector field to imply properties such as differentiability, Lipschitz continuity, Hölder continuity, semiconvexity or semiconcavity of this function outside the target. The most fully understood and studied example is the Euclidean distance function to a closed set, which is the minimum time function for a control system whose dynamics is constantly the unit ball. Recent advances in nonsmooth analysis have characterized those sets in which the distance function is differentiable in a neighborhood of the target and have provided detailed formulas for the derivative. The present paper extends these results to more complicated dynamics, and provides conditions for semiconvexity and formulas for the subgradient of the minimal time function for more complicated linear dynamics.