Boundary stabilization of a flexible structure with dynamic boundary conditions via one time-dependent delayed boundary control

<p style='text-indent:20px;'>This article deals with the dynamic stability of a flexible cable attached at its top end to cart and load mass bottom end. The model is governed by system one partial differential equation coupled two ordinary equations. Assuming that time-dependent delay occurs in boundary, main concern this paper stabilize dynamics as well dynamical terms related mass. To do so, we first prove problem well-posed sense semigroups theory provided some conditions on are satisfied. Thereafter, an appropriate Lyapunov function put forward, which leads exponential decay energy estimate rate.</p>