Identification of a source factor in a control problem for the heat equation with a boundary memory term

We consider a material with thermal memory occupying a bounded region $\Omega$ with boundary $\Gamma$. The evolution of the temperature $u(t,x)$ is described by an integrodifferential parabolic equation containing a heat source of the form $f(t)z_0(x)$. We formulate an initial and boundary value control problem based on a feedback device located on $\Gamma$ and prescribed by means of a quite general memory operator. Assuming both $u$ and the source factor $f$ unknown, we study the corresponding inverse and control problem on account of an additional information. We prove a result of existence and uniqueness of the solution $(u,f)$.