Approximate Control of Parabolic Equations with On-off Shape Controls by Fenchel Duality

We consider the internal control of linear parabolic equations through on-offshape controls, i.e., controls of the form M(t)χ_ω(t) with M(t)≥ 0 and ω(t) with a prescribed maximal measure. We establishsmall-time approximate controllability towards all possible final statesallowed by the comparison principle with nonnegative controls. We manage tobuild controls with constant amplitude M(t) ≡ M. In contrast, if themoving control set ω(t) is confined to evolve in some region of thewhole domain, we prove that approximate controllability fails to hold for smalltimes. The method of proof is constructive. Using Fenchel-Rockafellar dualityand the bathtub principle, the on-off shape control is obtained as thebang-bang solution of an optimal control problem, which we design by relaxingthe constraints. Our optimal control approach is outlined in a rather generalform for linear constrained control problems, paving the way forgeneralisations and applications to other PDEs and constraints.