Null controllability with vanishing energy

Linear, null controllable systems, for which an arbitrary initial state can be transferred to the origin with arbitrarily small energy, are characterized. Theorems are stated in terms of an associated algebraic Riccati equation and in terms of the spectrum of the linear part of the system. The results so obtained allow us to determine Ornstein–Uhlenbeck operators for which the Liouville theorem about bounded harmonic functions holds.