Asymptotic Stabilization of Oilwell Drillstring Torsional and Axial Vibrations

This paper takes as its starting point the distributed parameter models for both torsional and axial vibrations of the oilwell drillstring. While integrating several accepted features, the considered models are deduced following the Hamilton variational principle in the distributed parameter case. Then, these models are completed in order to take into account the elastic strain in driving signal transmission to the drillstring motions—rotational and axial (vertical). Stability and stabilization are tackled within the framework of the energy type Lyapunov functionals. From such “weak” Lyapunov functionals, only non-asymptotic Lyapunov stability can be obtained; therefore, asymptotic stability follows from the application of the Barbashin–Krasovskii–LaSalle invariance principle. This use of the invariance principle is carried out by associating a system of coupled delay differential and difference equations, recognized to be of neutral type. For this system of neutral type, the corresponding difference operator is strongly stable; hence, the Barbashin–Krasovskii–LaSalle principle can be applied. Note that this strong stability of the difference operator has been ensured by the aforementioned model completion with the elastic strain induced by the driving signals.