Null controllability of the parabolic spherical Grushin equation

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 $\mathbb S^2$. This is the natural generalization of the Grushin operator $\mathcal G = \partial_x^2 + x^2\partial_y^2$ on $\mathbb R^2$ to this curved setting, and presents a degeneracy at the equator of $\mathbb S^2$. We prove that the null controllability is verified in large time when the control acts as a source term distributed on a subset $\barω = \{ (x_1,x_2,x_3)\in \mathbb S^2\mid α<|x_3|<β\}$ for some $0\leα<β\le 1$. More precisely, we show the existence of a positive time $T^{*}>0$ such that the system is null controllable from $\barω$ in any time $T\ge T^{*}$, and that the minimal time of control from $\barω$ satisfies $T_{min}\ge\log(1/\sqrt{1-α^2})$. Here, the lower bound corresponds to the Agmon distance of $\barω$ 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.

No reviews yet.

No comments yet.