  This article deals with the stability analysis of a drilling system which is
modelled as a coupled ordinary differential equation / string equation. The
string is damped at the two boundaries but leading to a stable open-loop
system. The aim is to derive a linear matrix inequality ensuring the
exponential stability with a guaranteed decay-rate of this interconnected
system. A strictly proper dynamic controller based on boundary measurements is
proposed to accelerate the system dynamics and its effects are investigated
through the stability theorem and simulations. It results in an efficient
finite dimension controller which subsequently improves the system
performances.
