Lack of controllability of thermal systems with memory

Heat equations with memory of Gurtin-Pipkin type (i.e. Eq.~(\ref{eq:CGM}) with $ \alpha=0 $) have controllability properties which strongly resemble those of the wave equation. Instead, recent counterexamples show that when $ \alpha>0 $ the control properties do not parallel those of the (memoryless) heat equation, in the sense that there are square integrable initial conditions which cannot be controlled to zero. The proof of this fact, in previous papers, consists in the construction of two quite special examples of systems with memory which cannot be controlled to zero. Here we prove that lack of controllability holds in general, for every smooth memory kernel $ M(t) $.