Moving constraints as stabilizing controls in classical mechanics

TL;DR

This paper investigates how moving constraints can serve as stabilizing controls in classical mechanics, analyzing the equations of motion, the role of quadratic terms, and stabilization via oscillating controls with applications to mechanical systems.

Contribution

It introduces a novel approach to stabilization in classical mechanics using moving constraints and analyzes the effects of quadratic terms in the equations of motion.

Findings

Quadratic terms relate to geodesics orthogonal to foliations.

Oscillating controls can stabilize systems to a point.

Lyapunov methods effectively analyze stability.

Abstract

The paper analyzes a Lagrangian system which is controlled by directly assigning some of the coordinates as functions of time, by means of frictionless constraints. In a natural system of coordinates, the equations of motions contain terms which are linear or quadratic w.r.t.time derivatives of the control functions. After reviewing the basic equations, we explain the significance of the quadratic terms, related to geodesics orthogonal to a given foliation. We then study the problem of stabilization of the system to a given point, by means of oscillating controls. This problem is first reduced to the weak stability for a related convex-valued differential inclusion, then studied by Lyapunov functions methods. In the last sections, we illustrate the results by means of various mechanical examples.