Reachable subspaces, control regions and heat equations with memory.

We study the controlled heat equations with analytic memory from two perspectives: reachable subspaces and control regions. Due to the hybrid parabolic-hyperbolic phenomenon of the equations, the support of a control needs to move in time to efficiently control the dynamics.We show that under a sharp sufficient geometric condition imposed to the control regions, the difference between reachable subspaces of the controlled heat equations, with and without memory, is precisely given by a Sobolev space. The appearance of this Sobolev space is attributed to the memory which makes the equation having the wave-like nature. The main ingredients in the proofs of our main results are as: first, the decomposition of the flow (generated by the equation with the null control) given in [31], second, an observability inequality built up in this paper.