Stabilisation of a viscoelastic flexible marine riser under unknown spatiotemporally varying disturbance

In this work, we study a viscoelastic flexible marine riser problem with vessel dynamics subject to a distributed disturbance. The dynamic of the problem is modelled as a viscoelastic Euler–Bernoulli beam structure. Based on the multiplier method and some ideas introduced by N.-E. Tatar (J. Contemp. Math. Anal., 48(6):285–296, 2013), we shall suppress the riser's vibration in a certain manner that we will determine. In fact, we prove uniform stability of the system for a large class of relaxation functions. Moreover, the relationship between the behaviour of the relaxation function and the decay rate of the energy is established. This improves earlier work where a control on the top of the structure has been imposed and the ocean disturbance was ignored.