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

Abstract

We study a problem of boundary stabilization of the vibrations of elastic structure governed by the nonlinear integro-differential equation u = (a 2 + b Ω |∇u| 2 dx)Δu+ f , in a bounded domain Ω in R n 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.