Long-time dynamics of a nonlinear Timoshenko beam with discrete delay term and nonlinear damping

The purpose of this manuscript is to establish well posedness as well as the existence of global and exponential attractor for a nonlinear Timoshenko system subject to control terms in the two equations of the system. Since the control terms act on both equations, we will not use the nonphysical relationship known as equal speeds of propagation of waves. A combination involving friction-delay and friction controls will act on the angle of rotation equation, while a nonlinear friction control will act on the transverse motion equation. The result will be established by showing that the system is quasi-stable and by using a relationship involving the size of the friction type controls inserted in the rotation angle equation.