On a constrained approximate controllability problem for the heat equation

In this work, we study an approximate control problem for the heat equation , with a nonstandard but rather natural restriction on the solution. It is well known that approximate controllability holds. On the other hand, the t otal mass of the solutions of the uncontrolled system is constant in time. Therefore, it is natural to analyze whether approximate controllability hol ds supposing the total mass of the solution to be a given constant along th e trajectory. Under this additional restriction, approximate controllabilit y is not always true. For instance, this property fails when Omega is a bal l. We prove that the system is generically controllable; that is, given an open regular bounded domain Omega, there exists an arbitrarily small smooth deformation u, such that the system is approximately controllable in the n ew domain Omega + u under this constraint. We reduce our control problem to a nonstandard uniqueness problem. This uniqueness property is shown to hol d generically with respect to the domain.