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.