In this paper, we investigate the connections between controllability
properties of distributed systems and existence of non zero entire
functions subject to restrictions on their growth and on their sets of
zeros. Exploiting these connections, we first show that, for generic
bounded open domains in dimension at leat 2, the steady--state
controllability for the heat equation, with boundary controls dependent
only on time, does not hold.
In a second step, we study a model of water tank whose dynamics is
given by a wave equation on a two-dimensional bounded open domain. We
provide a condition which prevents steady--state controllability of
such a system, where the control acts on the boundary and is only
dependent on time. Using that condition, we prove that the
steady--state controllability does not hold for generic tank shapes. Full paper.