Controllability types of a circular membrane with rotationally symmetric data

Various controllability types are demonstrated for a circular membrane with rotationally symmetric initial data and boundary control depending on time only. We prove that the set of initial states, which can be steered to res t in the critical time interval (equal to the diameter of the membrane) by means of L-2-controls is dense in the energy space but contains no eigenmod e. We also show that any initial data from a Sobolev space can be transferr ed to a stationary state. The proof is based on study of exponential famili es arising in the approach using the method of moments.