2020/03/04 by Camille Laurent, Laurent, Camille, Matthieu Léautaud +1
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Nonlinear Partial Differential Equations #Mathematical Biology Tumor Growth
paper · doi:10.48550/arxiv.2003.02147
We consider a transport equation by a gradient vector field with a small viscous perturbation --εΔg. We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit ε→ 0+, that is, the possibility of having a uniformly bounded observation constant (resp. control cost). We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation ε= 0. We also prove that the two minimal times coincides for positive solutions. The proofs rely on a semiclassical reformulation of the problem together with (a) Agmon estimates concerning decay of eigenfunctions in the classically forbidden region [HS84] (b) fine estimates of the kernel of the semiclassical heat equation [LY86].