Description:
<jats:p>We prove observability and null-controllability for quadratic parabolic differential equations. The sensor set is allowed to be sparse and have finite volume if the generator has trivial singular space <jats:italic>S</jats:italic>. In the case of generators with singular space <jats:italic>S</jats:italic> ≠ {0} the sensor set is permitted to decay in directions determined by <jats:italic>S</jats:italic>. The proof is based on dissipation estimates for the quadratic differential operator with respect to spectral projections of partial harmonic oscillators and corresponding uncertainty relations.</jats:p>