Simultaneous temperature and flux controllability for heat equations with memory

It is known that, in the case of heat equation with memory, temperature can be controlled to an arbitrary square integrable target provided that the system evolves for a sufficiently long time. The control is the temperature on the boundary. In this paper we consider heat equations with memory (one dimensional space variable) and we first show that when the control is square integrable then the flux is square integrable too. Then we prove that both temperature and flux can be simultaneously controlled to a pair of independent targets, both square integrable. This solves a problem first raised by Renardy. The method of proof relies on moment theory and one of the contributions of this paper is the identification of $\mathcal{L} $-bases and Riesz bases especially suited to heat equations with memory, which so appear to be endowed with a very rich bases structure.