Mobile point controls versus the locally distributed ones for the controllability of the semilinear parabolic equation

It is well known that a rather general semilinear parabolic equation with globally Lipschitz nonlinear term is both approximately and exactly -controllable in L² (Ω), when governed in a bounded domain by locally distributed controls. We show that, in fact, in one space dimension 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 Q_T = (0,1) × (0,T). We show that such curves can be described by certain differential inequalities and provide explicit examples.