This paper deals with the stability analysis of a drilling pipe controlled by a PI controller. The model is 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 a state extension made up of projections of the Riemann coordinates. 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 influence of the nonlinear friction force, which may generate the well-known stick-slip phenomenon, is analyzed 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.