  We consider the optimal control problem of minimizing some quadratic
functional over all possible solutions of an internally controlled
multi-dimensional heat equation with a periodic terminal state constraint. This
problem has a unique optimal solution, which can be characterized by an
optimality system derived from the Pontryagin maximum principle. We define two
approximations of this optimal control problem. The first one is an impulse
approximation, and consists of considering a system of linear heat equations
with impulse control. The second one is obtained by the sample-and-hold
procedure applied to the control, resulting into a sampled-data approximation
of the controlled heat equation. We prove that both problems have a unique
optimal solution, and we establish precise error estimates for the optimal
controls and optimal states of the initial problem with respect to its impulse
and sampled-data approximations.
