We consider the boundary controllability problem for the wave equation and the Schrodinger equation over surfaces of revolution. Under these hypotheses, the geometric condition of Bardos, Lebeau and Rauch does does not hold, and therefore, we know that some initial data cannot be controlled within finite time. We give quantitative results about the analyticity of the space of controllable data for these problems. In the case of a barrel shaped surface, we prove that beyond an explicit time that is determined by the geometry of the surface, this space becomes constant.
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