NULL-CONTROLLABILITY FOR WEAKLY DISSIPATIVE HEAT-LIKE EQUATIONS

We study the null-controllability properties of heat -like equations posed on the whole Euclidean space R-n. These evolution equations are associated with Fourier multipliers of the form rho(|D-x|), where rho: [0, +infinity) -> C is a measurable function such that Re rho is bounded from below. We consider the "weakly dissipative" case, a typical example of which is given by the fractional heat equations associated with the multipliers rho(xi) = xi(s) in the regime s is an element of (0, 1), for which very few results exist. We identify sufficient conditions and necessary conditions on the control supports for the null-controllability to hold. More precisely, we prove that these equations are null-controllable in any positive time from control supports which are sufficiently thick at all scales. Under assumptions on the multiplier rho, in particular assuming that rho(xi) = o(xi), we also prove that the null-controllability implies that the control support is thick at all scales, with an explicit lower bound of the thickness ratio in terms of the multiplier rho. Finally, using Smith-Volterra-Cantor sets, we provide examples of non -trivial control supports that satisfy these necessary or sufficient conditions.