  Dynamic phenomena in social and biological sciences can often be modeled by
employing reaction-diffusion equations. When addressing the control of these
modes, from a mathematical viewpoint one of the main challenges is that,
because of the intrinsic nature of the models under consideration, the
solution, typically a proportion or a density function, needs to preserve given
lower and upper bounds. Controlling the system to the desired final
configuration then becomes complex, and sometimes even impossible. In the
present work, we analyze the controllability to constant steady-states of
spatially homogeneous semilinear heat equations, with constraints in the state,
and using boundary controls, which is indeed a natural way of acting on the
system in the present context. The nonlinearities considered are among the most
frequent: monostable and bistable ones. We prove that controlling the system to
a constant steady-state may become impossible when the diffusivity is too small
(or when the domain is large), due to the existence of barrier functions. When
such an obstruction does not arise, we build sophisticated control strategies
combining the dissipativity of the system, the existence of traveling waves and
some connectivity of the set of steady-states. This connectivity allows
building paths that the controlled trajectories can follow, in a long time,
with small oscillations, preserving the natural constraints of the system. This
kind of strategy was successfully implemented in one-space dimension, where
phase plane analysis techniques allowed to decode the nature of the set of
steady-states. These techniques fail in the present multi-dimensional setting.
We employ a fictitious domain technique, extending the system to a larger ball,
and building paths of radially symmetric solution that can then be restricted
to the original domain.
