Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method

Abstract

In this article we compute the exact value of the minimal null control time for the Grushin equation controlled on a strip. This result is already known from a recent work by K. Beauchard, J. Dardé and S. Ervedoza but we propose a different approach based on the moments method. Our approach requires a careful spectral analysis of a truncated harmonic oscillator. We have also extended known results on biorthogonal families to real exponentials in the absence of a gap condition in order to obtain uniform estimates with respect to the asymptotic beahviour of the associated counting function. As a byproduct of our approach we also characterize the minimal null control time for systems of coupled Grushin equations with a distributed control on a strip or with a boundary control.

Null-controllability properties of the generalized two-dimensional Baouendi–Grushin equation with non-rectangular control sets

“…When ω is a vertical strip of the form (a, b) × T, with a > 0, as in [BCG14], the precise value of the critical time T * was obtained independently in the works [ABM21, BDE20, LL23]. More precisely, in [ABM21], using new estimates for biorthogonal sequences to real exponentials and the moments method, the authors prove that in the case q(x) = x, the critical time is a 2 2 . In [BDE20], with a function q satisfying the assumptions of Theorem 1.6, the authors use a Carleman strategy to obtain that eq.…”

“…To end this overview on controllability issues for the parabolic Baouendi-Grushin equation, we point out that partial controllability results are known in some multidimensional configurations [BDE20] while precise results are known for cascade systems of two-dimensional Baouendi-Grushin equations with one control, in the case q(x) = x [ABM21].…”