Energy cost of dynamical stabilization: stored versus dissipated energy

TL;DR

This paper systematically analyzes the energy requirements for dynamical stabilization using an autonomous Kapitza's pendulum model, revealing conditions where stored energy stabilizes states without ongoing dissipation.

Contribution

It generalizes the classical Kapitza's pendulum model to include autonomous dynamics and studies the energetic costs of stabilization, including scenarios with no constant dissipation.

Findings

Stored energy can stabilize the upper state without ongoing energy dissipation.

Asymptotic stability does not guarantee stability against multiple perturbations.

Certain regimes require constant energy dissipation for stabilization.

Abstract

Dynamical stabilization processes (homeostasis) are ubiquitous in nature, but energetic resources needed for their existence were not studied systematically. Here we undertake such a study using the famous model of Kapitza's pendulum, which attracted attention in the context of classical and quantum control. This model is generalized and made autonomous. We show that friction and stored energy stabilize the upper (normally unstable) state of the pendulum. The upper state can be made asymptotically stable and yet it does not cost any constant dissipation of energy, only a transient energy dissipation is needed. The asymptotic stability under a single perturbation does not imply stability with respect to multiple perturbations. For a range of pendulum-controller interactions, there is also a regime where constant energy dissipation is needed for stabilization. Several mechanisms are studied for the decay of dynamically stabilized states.