Networks of geometrically exact beams: well-posedness and stabilization

TL;DR

This paper analyzes the well-posedness and exponential stabilization of tree-shaped networks of geometrically exact beams, accounting for large motions, using intrinsic formulations and velocity feedback controls.

Contribution

It establishes well-posedness and exponential stabilization results for GEB networks with nonlinear equations and rigid joints, using Lyapunov functionals and control theory.

Findings

The GEB network is locally well-posed in time.

Velocity feedback stabilizes the network exponentially.

The zero steady state is stable in H^1 and H^2 norms.

Abstract

In this work, we are interested in tree-shaped networks of freely vibrating beams which are geometrically exact (GEB) -- in the sense that large motions (deflections, rotations) are accounted for in addition to shearing -- and linked by rigid joints. For the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional and the theory developed by Bastin \& Coron in \cite{BC2016} that the zero steady state of this network is exponentially stable for the $H^1$ and $H^2$ norms. The major obstacles to overcome in the intrinsic formulation of the GEB network, are that the governing equations are semilinar, containing a quadratic nonlinearity, and that linear lower order terms cannot be neglected.