Approximate control of parabolic equations with on-off shape controls by Fenchel duality

Abstract

We consider the internal control of linear parabolic equations through on-off shape controls, i.e., controls of the form $M(t){\chi }_{\omega (t)}$ with $M(t)\ge 0$ and $\omega (t)$ with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative controls. We manage to build controls with constant amplitude $M(t)\equiv \bar{M}$. In contrast, if the moving control set $\omega (t)$ is confined to evolve in some region of the whole domain, we prove that approximate controllability fails to hold for small times. The method of proof is constructive. Using Fenchel–Rockafellarduality and the bathtub principle, the on-off shape control is obtained as the bang-bang solution of an optimal control problem, which we design by relaxing the constraints. Our optimal control approach is outlined in a rather general form for linear constrained control problems, paving the way for generalisations and applications to other PDEs and constraints.