On motions without falling of an inverted pendulum with dry friction

TL;DR

This paper analyzes the motion of an inverted pendulum with dry friction and a moving pivot, providing conditions for solutions where the pendulum never falls below horizontal, with proofs of solution uniqueness and stability.

Contribution

It introduces sufficient conditions for non-falling solutions of an inverted pendulum with dry friction and proves solution uniqueness and continuous dependence on initial conditions.

Findings

Sufficient conditions for non-falling solutions are established.

Solutions of the differential inclusion are right-unique.

Solutions depend continuously on initial conditions.

Abstract

An inverted planar pendulum with horizontally moving pivot point is considered. It is assumed that the law of motion of the pivot point is given and the pendulum is moving in the presence of dry friction. Sufficient conditions for the existence of solutions along which the pendulum never falls below the horizontal positions are presented. The proof is based on the fact that solutions of the corresponding differential inclusion are right-unique and continuously depend on initial conditions, which is also shown in the paper.