Generic uniqueness and conjugate points for optimal control problems

TL;DR

This paper investigates the structure of conjugate points and optimal trajectories in linear control problems, showing that conjugate points form a closed set with bounded measure and that the value function is smooth on a large subset.

Contribution

It establishes the closedness and measure bounds of conjugate points and characterizes the set of initial points with multiple optimal solutions for a broad class of terminal costs.

Findings

Conjugate points form a closed set with bounded Hausdorff measure.

The set of initial points with multiple optimal trajectories is contained in finitely many manifolds.

The value function is continuously differentiable on a dense open subset.

Abstract

The paper is concerned with an optimal control problem on $\mathbb{R}^n$, where the dynamics is linear w.r.t.~the control functions. For a terminal cost $\psi$ in a $mathcal{G}_\delta$ set of $\mathcal{C}^4(\mathbb{R}^n)$ (i.e., in a countable intersection of open dense subsets), two main results are proved.Namely: the set $\Gamma_\psi\subset\mathbb{R}^n$ of conjugate points is closed, with locally bounded $(n-2)$-dimensional Hausdorff measure. Moreover, the set of initial points $y\in \mathbb{R}^n\setminus\Gamma_\psi$, which admit two or more globally optimal trajectories, is contained in the union of a locally finite family of embedded manifolds. In particular, the value function is continuously differentiable on an open, dense subset of $\mathbb{R}^n$.