Lyapunov's theorem via Baire category

TL;DR

This paper presents a new quantitative approach to Lyapunov's theorem using Baire category methods, extending its applications in optimal control theory for countable families of functions.

Contribution

It introduces a dual Baire category approach to provide a quantitative version of Lyapunov's theorem for countable integrable function families.

Findings

Provides a quantitative version of Lyapunov's theorem.

Extends Lyapunov's results to countable families of functions.

Uses dual Baire category method for analysis.

Abstract

Lyapunov's theorem is a classical result in convex analysis, concerning the convexity of the range of nonatomic measures. Given a family of integrable vector functions on a compact set, this theorem allows to prove the equivalence between the range of integral values obtained considering all possible set decompositions and all possible convex combinations of the elements of the family. Lyapunov type results have several applications in optimal control theory: they are used to prove bang-bang properties and existence results without convexity assumptions. Here, we use the dual approach to the Baire category method in order to provide a "quantitative" version of such kind of results applied to a countable family of integrable functions.