Boundary feedback stabilization for the intrinsic geometrically exact beam model

TL;DR

This paper develops boundary feedback stabilization techniques for a geometrically exact shearable beam model, proving local exponential stability of the zero steady state using Lyapunov functions and nonlinear transformations.

Contribution

It introduces a novel approach to stabilize large deflections in shearable beams by leveraging intrinsic variables and a nonlinear transformation, extending stability analysis to complex geometries.

Findings

Zero steady state is locally exponentially stable in IGEB model.

Lyapunov function construction proves stability despite nonlinearities.

Existence and asymptotic properties of solutions are established.

Abstract

In this work we address the problem of boundary feedback stabilization for a geometrically exact shearable beam, allowing for large deflections and rotations and small strains. The corresponding mathematical model may be written in terms of displacements and rotations (GEB), or intrinsic variables (IGEB). A nonlinear transformation relates both models, allowing to take advantage of the fact that the latter model is a one-dimensional first-order semilinear hyperbolic system, and deduce stability properties for both models. By applying boundary feedback controls at one end of the beam, while the other end is clamped, we show that the zero steady state of IGEB is locally exponentially stable for the $H^1$ and $H^2$ norms. The proof rests on the construction of a Lyapunov function, where the theory of Coron \& Bastin '16 plays a crucial role. The major difficulty in applying this theory stems from the complicated nature of the nonlinearity and lower order term where no smallness arguments apply. Using the relationship between both models, we deduce the existence of a unique solution to the GEB model, and properties of this solution as time goes to $+\infty$.