A Baire Category Approach to the Bang-Bang Property

TL;DR

This paper introduces a novel Baire category-based technique to analyze control systems, establishing the closure of reachable sets and the bang-bang property without convexity assumptions, using Lyapunov's theorem and linear selections.

Contribution

It develops a new approach leveraging Baire category theory to prove the bang-bang property for a broader class of control systems without convexity.

Findings

Closure of reachable sets established without convexity

Bang-bang property proved for 'concave' multifunctions

New technique applicable to broader control systems

Abstract

Aim of this paper is to develop a new technique, based on the Baire category theorem, in order to establish the closure of reachable sets and the existence of optimal trajectories for control systems, without the usual convexity assumptions. The bang-bang property is proved for a new class of ``concave" multifunctions, characterized by the existence of suitable linear selections. The proofs rely on Lyapunov's theorem in connection with a Baire category argument.