Properties of Measure Controls and Their Trajectories

TL;DR

This paper explores measure controls and measure vector fields within measure differential equations, establishing their properties, equivalence, and stability, thus advancing the mathematical understanding of uncertain dynamic systems.

Contribution

It introduces the existence, well-posedness, and equivalence of measure controls and measure vector fields in measure differential equations, with analysis of their stability and trajectory properties.

Findings

Proved existence and well-posedness of control systems with measure controls.

Established the equivalence between measure controls and measure vector fields.

Analyzed stability and closure properties of the trajectory set.

Abstract

This paper deals with the concepts of measure controls and of measure vector fields, within the mathematical framework of measure differential equations (MDEs), recently proposed in~\cite{piccoli_measure_2019}. Measure controls can be seen as a generalization of relaxed control. Moreover, they are particularly suitable for studying dynamics with uncertainty. The main results of this paper include establishing the existence and well-posedness of control systems with measure controls and proving the equivalence between measure controls and measure vector fields. The stability and closure properties of the trajectory set are also studied.