| Contents |
| The model to be studied is the heat equation with memory in the smooth domain $\Omega \subset \mathbb{R}^m$
$$
\frac{\partial w}{\partial t} (t,x) = \Delta_x w(t,x) + \int^t_0 M(t-s) \Delta_x w(s,x) ds
$$
subject to initial conditions
$$
w(0,x) = \xi (x), \quad x \in \Omega
$$
and with a boundary control on $\Gamma \subset \partial \Omega$
$$
w(t,x) = \left\{\begin{array}{cl} v(t,x), & x \in \Gamma, \; t \in [0,T] \\ \noalign{\medskip} 0, & x \in \partial\Omega - \Gamma, \; t \in [0,T]. \end{array} \right.
$$
or with an interior control
$$
\frac{\partial w}{\partial t} (t,x) = \Delta_x w(t,x) + \int^t_0 M(t-s) \Delta_x w(s,x) ds+u(t,x)\chi_{\omega}, \omega\subset\Omega
$$
Under smoothness hypothesis on the kernel $M$, approximate and null controllability will be investigated using the reduction to a moment problem. |
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