Indirect stabilization of a Mindlin–Timoshenko plate

Abstract

In this note, we investigate stability issues for a Mindlin–Timoshenko plate with internal dissipation distributed everywhere in the domain under consideration. The damping occurs only in the elasticity equations describing the motion of the angles of rotation of a filament; the vertical deflection equation has no damping, but the effect of the damping is transmitted there by the coupling. A natural question is then: is the coupling robust enough to induce the exponential stability of the Mindlin–Timoshenko plate? A positive response to that question is given in this note, provided that the plate is clamped (Dirichlet boundary conditions), and the speed of propagation of the wave generated by the longitudinal component of the rotation angles and that of the wave generated by the vertical deflection are identical. This result strongly improves earlier results where only polynomial stability was established. When those speeds are distinct, an improved polynomial stability of the system is established. The proofs of those stability results are constructive, and they rely on the frequency domain method combined with the multipliers technique. In particular, the exponential stability result seems to be the first one for the Mindlin–Timoshenko system where the mechanical damping is present in the rotation angle equations only.