On the set of points of smoothness for the value function for affine optimal control problems

TL;DR

This paper investigates the smoothness and regularity of the value function in affine optimal control problems, showing it is continuous and smooth on large open dense subsets without assuming conditions on singular minimizers.

Contribution

It proves the value function is continuous and smooth on open dense subsets in affine control problems without conditions on singular minimizers.

Findings

Value function is continuous on an open dense subset.

Value function is smooth on a possibly smaller open dense subset.

Results hold without assumptions on singular minimizers.

Abstract

We study the regularity properties of the value function associated with an affine optimal control problem with quadratic cost plus a potential, for a fixed final time and initial point. Without assuming any condition on singular minimizers, we prove that the value function is continuous on an open and dense subset of the interior of the attainable set. As a byproduct we obtain that it is actually smooth on a possibly smaller set, still open and dense.