  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 $+\infty$.
