On the structure of the value function of optimal exit time problems

TL;DR

This paper investigates the structure and regularity of the value function in optimal exit time problems for nonlinear control systems, establishing new geometric and differentiability properties without requiring Petrov's condition.

Contribution

It introduces a novel representation of supergradients, proves regularity results, and analyzes the singularities of the value function under mild controllability assumptions.

Findings

Value function is almost everywhere twice differentiable.

Hausdorff dimension of singularities is at most d-1/2.

Value function is a function of special bounded variation.

Abstract

In this paper, we study an optimal exit time problem with general running and terminal costs and a target $\mathcal{S}\subset\mathbb{R}^d$ having an inner ball property for a nonlinear control system that satisfies mild controllability assumptions. In particular, Petrov's condition at the boundary of $\mathcal{S}$ is not required and the value function $V$ may fail to be locally Lipschitz. In such a weakened set-up, we first establish a representation formula for proximal (horizontal) supergradients of $V$ by using transported proximal normal vectors. 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 $d-1/2$. Furthermore, besides optimality conditions for trajectories of the optimal control problem, we extend the analysis to propagation of singularities and differentiability properties of the value function. An upper bound for the Hausdorff measure of the singular set is also studied, which implies that $V$ is a function of special bounded variation.