In this paper, we study an optimal exit time problem with general running and terminal costs and a target S having an inner ball property for a nonlinear control system that satisfies mild controllability assumptions. In particular, the Petrov's condition at the boundary of S is not required and the value function V fails to be locally Lipschitz. In such a weakened set-up, we establish a representation formula of proximal (horizontal) supergradients of V at every point x in S. This allows us to obtain an external sphere condition for the hypograph of V which yields several regularity properties. In particular, V is almost everywhere twice differentiable and the Hausdorff dimension of its singularities is not greater than (d-1/2).
On the structure of the value function of optimal exit time problems
Khai T. Nguyen, North Carolina State University
Authors: Piermarco Cannarsa, Marco Mazzola, Khai T. Nguyen
2023 AWM Research Symposium
Recent Developments in Control, Optimization, and the Analysis of Partial Differential Equations [Organized by Lorena Bociu and Pelin Guven Geredeli]