# On singular control problems, the time-stretching method, and the weak-M1 topology

## TL;DR

This paper demonstrates that the weak-M1 topology provides a more effective framework than the J1-topology for establishing the existence of solutions in singular control problems with state constraints, simplifying the analysis by avoiding time-transformations.

## Contribution

The paper shows that the weak-M1 topology is better suited than the J1-topology for singular control problems, embedding the time-stretching method's scheme and facilitating existence proofs.

## Key findings

- Weak-M1 topology simplifies existence proofs for singular control problems.
- Time-scaling in weak-M1 embeds the time-stretching method.
- Potential applications in queueing control under heavy traffic.

## Abstract

We consider a general class of singular control problems with state constraints. Budhiraja and Ross (2006) established the existence of optimal controls for a relaxed version of this class of problems by using the so-called `time-stretching' method and the J1-topology. We show that the weak-M1 topology is better suited for establishing existence, since it allows to bypass the time-transformations, without any additional effort. Furthermore, we reveal how the time-scaling feature in the definition of the weak-M1 distance embeds the time-stretching method's scheme. This case study suggests that one can benefit from working with the weak-M1 topology in other singular control frameworks, such as queueing control problems under heavy traffic.