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 S-2. This is the natural generalization of the Grushin operator G - partial derivative(2)(x) + x(2)partial derivative(2)(y) on R-2 to this curved setting and presents a degeneracy at the equator of 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 (omega) over bar = {(x(1), x(2), x(3)) is an element of S-2 vertical bar alpha < vertical bar x(3)vertical bar < beta} for some 0 <= alpha < beta <= 1. More precisely, we show the existence of a positive time T* > 0 such that the system is null controllable from (omega) over bar in any time T >= T*, and that the minimal time of control from (omega) over bar satisfies T-min >= log(1/root(1 - alpha(2))) . Here, the lower bound corresponds to the Agmon distance of (omega) over bar from the equator. These results are obtained by proving a suitable Carleman estimate using unitary transformations and Hardy-Poincare type inequalities to show the positive null-controllability result. The negative statement is proved by exploiting an appropriate family of spherical harmonics, concentrating at the equator, to falsify the uniform observability inequality.