Null controllability in large time of a parabolic equation involving the Grushin operator with an inverse-square potential

Abstract

We prove the null controllability in large time of the following linear parabolic equation involving the Grushin operator with an inversesquare potential ut − Δxu − |x| 2 Δyu − μ |x| 2 u = v1ω in a bounded domain Ω = Ω1 × Ω2 ⊂ R N 1 × R N 2 (N1 ≥ 3, N2 ≥ 1) intersecting the surface {x = 0} under an additive control supported in an open subset ω = ω1 × Ω2 of Ω.

Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity

“…for all large s ≥ s 1 , λ ≥ λ 1 and z = θv ε . In order to eliminate the boundary term, we introduce a cut-function χ ∈ C 2 (I) such that      χ(x, y) = 0, (x, y) ∈ ω (1) , 0 < χ(x, y) < 1, (x, y) ∈ ω (2) \ω (1) , χ(x, y) = 1, (x, y) ∈ I\ω (2) .…”

“…Obviously, the system (1.1) is not only degenerate, but also singular on boundary {x = 0} × I y . Further, the degeneracy is weak if 0 < γ < 1 2 and strong if γ ≥ 1 2 . This paper focus on the Carleman estimates for stochastic Grushin equation with singular potential and then apply them to study the following null controllability and inverse source problem.…”

“…Theorem 1.1. Let γ > 0, 0 ≤ σ < 1 4 and α, β ∈ L ∞ F (0, T ; L ∞ (I)). Then for any u 0 ∈ L 2 (Ω, F 0 , P; L 2 (I)), there exists a pair (g, G) ∈ L 2 F (0, T ; L 2 (ω)) × L 2 F (0, T ; L 2 (I)) such that the corresponding solution u of (1.2) satisfies u(T ) = 0 in I, P-a.s. for any T > 0.…”

“…Remark 1.1. Condition 0 ≤ σ < 1 4 is used to guarantee well-posedness issues linked to the use of the following Hardy inequality [9]…”

“…However, a coming flaw with such a control domain is that the null controllability could not hold for σ = 1 4 . This is because that we need 1 4 − σ > 0 to prove the Cacciopoli inequality (3.47), when our control domain ω touches the line {x = 0}.…”