2023/09/14 by Paul Alphonse, Alphonse, Paul, Armand Koenig +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2309.07533
openalex publication_date 2023/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the null-controllability properties of heat-like equations posed on the whole Euclidean space \mathbb Rn. These evolution equations are associated with Fourier multipliers of the form ρ(\vert Dx\vert), where ρ\colon[0,+∞)→\mathbb C is a measurable function such that \Reρ 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 ρ(ξ) = ξs in the regime s∈(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 ρ, in particular assuming that ρ(ξ) = o(ξ), 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 ρ.Finally, using Smith-Volterra-Cantor sets, we provide examples of non-trivial control supports that satisfy these necessary or sufficient conditions.