Abstract The problem of damping out the vibrations of a thick plate is solved using the optimal control theory of distributed parameter systems. The plate is modelled as a Mindlin- Timoshenko plate to include shear effects and may exhibit viscous damping. The dynamic response of the structure comprises the displacement and velocity components which are combined with the amount of force expended in controlling the motion in a multiobjective cost functional. This functional is minimized with respect to distributed controls. A maximum principle is formulated to relate the control forces to adjoint variables, the use of which leads to the explicit solution of the title problem. The control over the plate is exercised by distributed moment and transverse forces which are in practice applied by means of torque and force actuators.