Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in Rn

We study a problem of boundary stabilization of the vibrations of elastic structure governed by the nonlinear integro-differential equation , in a bounded domain Ω in Rn with a smooth boundary Γ, under mixed boundary conditions. To stabilize this system, we apply a velocity feedback control only on a part of the boundary. We prove that the solution of such system is stable subject to some restriction on the uncertain disturbing force f. We also estimate the total energy of the system over any time interval [0,T], with a tolerance level of the disturbances. Finally, we establish the uniform decay of solution by a direct method, with an explicit form of exponential energy decay estimate, when this disturbing force f is insignificant.