Null controllability of a parabolic equation involving the Grushin operator in some multi-dimensional domains

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

“…The controllability for degenerate parabolic equations in dimension one has been studied widely in recent years by many authors (see e.g., [2,[7][8][9][10][11]18,19]). The null controllability of parabolic equations involving the Grushin operator has been studied first in dimension two [4], and then in some multi-dimensional domains [3,5]. On the other hand, the controllability results of parabolic equations with an inverse-square potential were obtained in [14,19] for the case of internal singularity, and in [13] for the case of boundary singularity.…”

“…×(0,T ) ξC 14 Me −2Mσ (t(T − t)) 3 |∇w| 2 dxdt ≤ − Ω 1 ×(0,T )Gn,μw C 14 Mwξe −2Mσ (t(T − t))3 dxdt+ Ω 1 ×(0,T ) C 14 M |w| 2 e −2Mσ 2(t(T − t)) 3 Δξ − 4M ∇ξ • ∇σ + ρ 4M 2 |∇σ| 2 − 2M Δσ dxdt + Ω 1 ×(0,T ) C 14 Mξ|w| 2 e −2Mσ (t(T − t)) 3 μ |x| 2 − 2Mσt − 3 T − 2t (t(T − t)) 4 dxdt ≤ 1 (0,T ) |e −Mσ Gn,μw| 2 dxdt + ω 1 ×(0,T ) C 15 M 3 e −2Mσ (t(T − t)) 9 |w| 2 dxdtfor some positive constant C 15 = C 15 (β, ξ). Here, we have used the fact that 0 R N 1 / ∈ω 1 and supp(ξ), supp(Δξ), supp(∇ξ) ⊂ ω 1 .…”

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) .…”

“…x, y, t)dB(t), (x, y, t) ∈ Q T , u(x, y, t) = 0, (x, y, t) ∈ Σ T , u(x, y, 0) = 0, (x, y) ∈ I, When no singular term was involved, the null controllability of deterministic Grushin equation with I = (−1, 1) × (0, 1) was studied in [2,3]. The null controllability for Grushintype equations was obtained for any time T > 0 and for any degeneracy γ > 0, with a control that acts on one strip, touching the degeneracy line {x = 0} in [5].…”

“…As its applications to deterministic differential equations, we refer to [10,18,19,20,33,36] for inverse problems, [7,29,30,34] for unique continuation problems, [17,14,26,11] for control theory. For Carleman estimates related to deterministic Grushin equation, we refer to [2,3,28,21]. In recent years, many efforts have been devoted to studying the Carleman estimate for stochastic partial differential equations, for example [6,22,31,35] for stochastic heat equation, [38] for stochastic wave equation, [13] for stochastic KdV equation, [15] for stochastic Kuramoto-Sivashinsky equation, [25] for stochastic Schrödinger equation, and so on.…”