The $\Lambda$-Set and Its Role in Local Controllability and Necessary Conditions for Free-Time Optimal Control

TL;DR

This paper introduces the $\Lambda$-set as a key concept linking controllability, optimality conditions, and attainability in free-time control problems, providing new insights and methods for non-convex optimization.

Contribution

It establishes a unified framework connecting the $\Lambda$-set with controllability and optimality, offering new necessary conditions and constructive approximation procedures.

Findings

Emptiness of the $\Lambda$-set implies local controllability.

Provides explicit methods for approximating generalized controls.

Extends classical control theory to non-convex and higher-order problems.

Abstract

This paper establishes a unified framework connecting local controllability, necessary conditions for optimality, and attainability in free-time optimal control problems. The central object of our investigation is the $\Lambda$-set, which governs the relationship between original control systems and their convexifications. Our main results demonstrate that emptiness of the $\Lambda$-set implies local controllability and guarantees the existence of minimizing sequences achieving the target in reduced time. We derive strengthened necessary conditions for time-optimal control and provide explicit constructive procedures for approximating generalized controls by ordinary trajectories. These results resolve longstanding questions about relaxation phenomena while extending classical theory to address modern challenges in non-convex optimization, establishing foundations for higher-order analysis in free-time problems.