Bounds on the response of a drilling pipe model

The drill pipe model described by the wave equation with boundary conditions is reduced through the d’Alembert transformation to a difference equation model. Assuming that the boundary condition at the bottom is perturbed by bounded additive noise, an ultimate bound for the velocity at the bottom of the pipe is obtained. The proposal of a Lyapunov functional for the distributed model allows to provide an ultimate bound for a measure of the distributed variables describing the system in terms of linear matrix inequality conditions. The two approaches are compared through an illustrative example.