Strong bi-metric regularity in affine optimal control problems

TL;DR

This paper establishes new sufficient conditions for strong bi-metric regularity in affine optimal control problems, crucial for analysis and numerical methods, without requiring convexity of the objective functional.

Contribution

It introduces novel sufficient conditions for strong bi-metric regularity in affine problems that do not depend on convexity assumptions.

Findings

Conditions ensure regularity without convexity.

Proves uniform convergence of Euler discretization.

Applicable to a broad class of affine control problems.

Abstract

The paper presents new sufficient conditions for the property of strong bi-metric regularity of the optimality map associated with an optimal control problem which is affine with respect to the control variable ({\em affine problem}). The optimality map represents the system of first order optimality conditions (Pontryagin maximum principle), and its regularity is of key importance for the qualitative and numerical analysis of optimal control problems. The case of affine problems is especially challenging due to the typical discontinuity of the optimal control functions. A remarkable feature of the obtained sufficient conditions is that they do not require convexity of the objective functional. As an application, the result is used for proving uniform convergence of the Euler discretization method for a family of affine optimal control problems.