  We consider a semi-linear heat equation with Dirichlet boundary conditions
and globally Lipschitz nonlinearity, posed on a bounded domain of R^N (N $\in$
N *), assumed to be an unknown perturbation of a reference domain. We are
interested in an insensitizing control problem, which consists in finding a
distributed control such that some functional of the state is insensitive at
the first order to the perturbations of the domain. Our first result consists
of an approximate insensitization property on the semi-linear heat equation. It
rests upon a linearization procedure together with the use of an appropriate
fixed point theorem. For the linear case, an appropriate duality theory is
developed, so that the problem can be seen as a consequence of well-known
unique continuation theorems. Our second result is specific to the linear case.
We show a property of exact insensitization for some families of deformation
given by one or two parameters. Due to the nonlinearity of the intrinsic
control problem, no duality theory is available, so that our proof relies on a
geometrical approach and direct computations.
