2021/01/27 by Tamekue, Cyprien
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2101.11447
We investigate the null controllability property of the parabolic equation associated with the Grushin operator defined by the canonical almost-Riemannian structure on the 2-dimensional sphere \mathbb S2. This is the natural generalization of the Grushin operator \mathcal G = ∂x2 + x2∂y2 on \mathbb R2 to this curved setting, and presents a degeneracy at the equator of \mathbb S2. We prove that the null controllability is verified in large time when the control acts as a source term distributed on a subset ω = \ (x1,x2,x3)∈ \mathbb S2| α0 such that the system is null controllable from ω in any time T≥ T*, and that the minimal time of control from ω satisfies Tmin≥log(1/√(1-α2)). Here, the lower bound corresponds to the Agmon distance of ω from the equator. These results are obtained by proving a suitable Carleman estimate by using unitary transformations and Hardy-Poincaré type inequalities to show the positive null-controllability result. The negative statement is proved by exploiting an appropriate family of spherical harmonics, which concentrates at the equator, to falsify the uniform observability inequality.