OBSERVABILITY OF BAOUENDI-GRUSHIN-TYPE EQUATIONS THROUGH RESOLVENT ESTIMATES

In this article, we study the observability (or equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the 'degenerated directions' of the subelliptic structure. First, for any gamma >= 1, we establish a resolvent estimate for the Baouendi-Grushin-type operator Delta(gamma) = partial derivative(2)(x) + vertical bar x vertical bar(2 gamma) partial derivative(2)(y) which has step gamma+1. We then derive consequences for the observability of the Schrodinger- type equation i partial derivative(t)u-(-Delta(gamma))(s) u = 0, where s is an element of N. We identify three different cases: depending on the value of the ratio (gamma + 1)/s, observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations partial derivative(t)u + (-Delta(gamma))(s) u = 0 and establish a decay rate for the damped wave equation associated with Delta(gamma).