Chair of Computational Mathematics Seminar: Minimal controllability time for the heat equation under state constraints

Minimal controllability time for the heat equation under state constraints

Jérôme Lohéac
Institut de Recherche en Communications et Cybernétique de Nantes

Abstract
The free heat equation is well known to preserve the non-negativity of
solutions. On the other hand, due to the infinite velocity of propagation, the
heat equation is null-controllable in an arbitrary small time. The following
question then arises naturally: Can the heat dynamics be controlled under a
positivity constraint on the state, requiring that the state remains non-negative
all along the time dependent trajectory?
I will show that, if the control time is large enough, constrained controllability
holds. I will also show that it fails to be true if the control time is too short. In
other words, despite of the infinite velocity of propagation, under the natural
positivity constraint on the state, controllability fails when the time horizon is
too short.