The inverse problem of the calculus of variations and the stabilization of controlled Lagrangian systems

TL;DR

This paper uses the inverse calculus of variations to design feedback controls that stabilize equilibrium points in controlled mechanical systems, including pendulums, by leveraging variational principles and energy methods.

Contribution

It introduces a novel application of the inverse problem of the calculus of variations to stabilize controlled Lagrangian systems, extending to systems like inverted pendulums.

Findings

Derived feedback controls making the system variational.

Established stability criteria using controlled Lagrangian energy.

Applied methods to classical systems like inverted pendulum.

Abstract

We apply methods of the so-called `inverse problem of the calculus of variations' to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the inverted pendulum on a cart and the inertia wheel pendulum. By making use of a condition that follows from Douglas' classification, we derive feedback controls for which the control system is variational. We then use the energy of a suitable controlled Lagrangian to provide a stability criterion for the equilibrium.