Mobile Point Controls Versus Locally Distributed Ones for the Controllability of the Semilinear Parabolic Equation

It is well known now that a rather general semilinear parabolic equation with globally Lipschitz nonlinear term is both approximately and exactly null-controllable in L2 (Ω)$, when governed in a bounded domain by the locally distributed controls. In this paper we intend to show that, in fact, in one space dimension ($Ω= (0, 1)$) the very same results can be achieved by employing { at most} two mobile point controls with support on the curves properly selected within an arbitrary subdomain of QT = (0,1) × (0,T). We will show that such curves can be described by a certain differential inequality and the explicit examples are provided. We also discuss some extensions of our main results to the superlinear terms and to the case of several dimensions.