On conditions that prevent steady-state controllability of certain linear partial differential equations

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 $n\geq 2$, 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 a 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