Stability and Long-Time Behavior of a Pendulum with an Interior Cavity Filled with a Viscous Liquid

TL;DR

This paper proves exponential stability and long-term decay of a pendulum with an interior cavity filled with viscous liquid, showing solutions become smooth and decay exponentially under certain initial conditions.

Contribution

It introduces a generalized linearization principle for evolution equations with a slow center manifold and applies it to a viscous liquid-filled pendulum.

Findings

Exponential stability of the pendulum's equilibrium configuration.

Weak solutions with finite energy become smooth and decay exponentially.

Development of a generalized linearization principle for evolution equations.

Abstract

We show asymptotic, exponential stability of the equilibrium configuration, $\smallL$, of a hollow physical pendulum with its inner part entirely filled with a viscous liquid, corresponding to the center of mass being in the lowest position. Moreover, we prove that every weak solution with initial data possessing finite total initial energy and belonging to a "large" open set, becomes eventually smooth and decays exponentially fast to the equilibrium $\smallL$. These results are obtained also as byproduct of a "generalized linearization principle" that we show for evolution equations with non-empty "slow" center manifold.