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

Abstract

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 S. In the case of generators with singular space S ≠ {0} the sensor set is permitted to decay in directions determined by S. 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.