  This paper deals with the exponential stability of a drilling pipe controlled
by a PI controller. The model used leads to a coupled ODE / PDE and is
consequently of infinite dimension. Using recent advances in time-delay
systems, we derive a new Lyapunov functional based on an state extension made
up of projections of the Riemann coordinates. Two cases will be considered.
First, we will provide an exponential stability result expressed using the LMI
framework. This result is dedicated to a linear version of the torsional
dynamic. On a second hand, the nonlinear terms in the initial model, that
generates the well-known stick-slip phenomenon is captured through a new
stability theorem. Numerical simulations show the effectiveness of the method
and that the stick-slip oscillations cannot be weaken using a PI controller.
