Exponential stability of laminated Timoshenko beams with boundary/internal controls

Abstract

For a laminated Timoshenko system with slip along the interface, we study existence and boundary/interior stabilization of solutions. Our first main result, using resolvent estimates in the frequency domain, is to show that interior damping created by the interfacial slip in companion with boundary feedback controls acting on the complementary displacements (traverse and rotation angle) drives the solution to zero with exponential decay rate. We impose neither conditions on the physical parameters of the model nor on the damping coefficients. Our second main result is to investigate the asymptotic behavior of solutions for partially dissipative problems. Assuming the condition that the waves travel at the same speed, we prove the decay of solutions with exponential rate. Finally, the same condition allows us to show that dissipative effect created by the interfacial slip is strong enough to exponentially stabilizes the solutions to zero.