Stabilization of a coupled bridge system with past history and gurtin-pipkin’s heat conduction

The asymptotic stability of the vertical vibrations of a suspension bridge and the main cables from which the bridge is suspended by the cables is investigated. The history of the materials of the bridge and the effect of heat governed by Gurtin-Pipkin’s thermal law on main cables are taking into consideration. We show that the damping induced by the infinite memory of the bridge and the heat on the main cables are strong enough to stabilize the system.