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We prove observability and null-controllability for quadratic parabolicdifferential equations. The sensor set is allowed to be sparse and have finitevolume if the generator has trivial singular space $S$. In the case ofgenerators with singular space $S \neq \{0\}$ the sensor set is permitted todecay in directions determined by $S$. The proof is based on dissipationestimates for the quadratic differential operator with respect to spectralprojections of partial harmonic oscillators and corresponding uncertaintyrelations.

Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density