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Clarify the role of optimal transport in stability-preserving homotopies

Elucidate the relationship between optimal transport and the existence of stability-preserving homotopies on the level of vector fields, and determine what insights stability-preserving homotopies can in turn provide for optimal transport.

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Background

Section 5 outlines how transporting probability measures associated with Lyapunov functions suggests a potential route to constructing stability-preserving homotopies at the vector-field level.

The authors ask for a deeper investigation of this bidirectional connection to understand whether optimal transport interpolations can systematically yield such homotopies and what new perspectives homotopy ideas offer to optimal transport.

References

Open question 5}: elucidate what optimal transport can tell us about the existence of stability preserving homotopies on the level of vector fields, plus, what can we tell optimal transport?

— Asymptotic stability equals exponential stability -- while you twist your eyes (2411.03277 - Jongeneel, 5 Nov 2024) in Conclusion and future work — Open question 5