On the exponential stabilization of a flexible structure with dynamic delayed boundary conditions via one boundary control only

This study deals with the stability analysis of a flexible structure with one and only one boundary control. The system is composed of three parts: a cart (motorized platform), a flexible cable, and a load mass attached to the lower part of the cable. This situation leads to a hybrid system as a mathematical model for the cable dynamics: one partial differential equation coupled to two ordinary differential equations. Despite the presence of a time-delay in the top-end of the cable, we are able to prove that the hybrid system is well-posed in the sense of semigroups theory and more importantly, only one boundary control can guarantee the exponentially decay of the energy of the system under reasonable conditions on the parameters of the system. This outcome considerably improves the result recently established in [17], where two more controls are required: one interior (Kelvin–Voigt) damping which acts over the entire cable and another boundary control which is exerted on the lower-end of the cable. Furthermore, we provide an estimate of the exponential decay of the system by means an appropriate Lyapunov functional. Lastly, numerical examples are presented in order to ascertain and highlight our theoretical outcomes.