Sweeping by a tame process

TL;DR

This paper proves that semi-algebraic sweeping processes have piecewise absolutely continuous solutions with finite-length trajectories, extending results to o-minimal structures and surpassing previous work on gradient dynamical systems.

Contribution

It establishes finite-length and piecewise absolute continuity of solutions for semi-algebraic and o-minimal sweeping processes, broadening the scope beyond monotone sets.

Findings

Solutions are piecewise absolutely continuous.

Bounded trajectories have finite length.

Results extend to o-minimal structures.

Abstract

We show that any semi-algebraic sweeping process admits piecewise absolutely continuous solutions, and any such bounded trajectory must have finite length. Analogous results hold more generally for sweeping processes definable in o-minimal structures. This extends previous work on (sub)gradient dynamical systems beyond monotone sweeping sets.