A GENERALIZATION OF THE KALMAN RANK CONDITION FOR TIME-DEPENDENT COUPLED LINEAR PARABOLIC SYSTEMS

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Local null controllability of the three-dimensional Navier–Stokes system with a distributed control having two vanishing components (2012, background): “The natural question is then: Can we remove another component of the control, which would be an optimal result with respect to the number of controls? Reducing the number of components of the control is important for applications, and have already been studied many times for linear or parabolic syst”
Indirect controllability of some linear parabolic systems of m equations with m − 1 controls involving coupling terms of zero or first order (2016, background): “Moreover, in [7], some results are obtained in the case of time-dependent coefficients under a sharp sufficient condition which is similar to the sharp Silverman-Meadows Theorem in the finite-dimensional case.”
RECENT RESULTS ON THE CONTROLLABILITY OF LINEAR COUPLED PARABOLIC PROBLEMS: A SURVEY (2011, unknown): “In [5] the authors prove a better result: a Carleman type inequality for the system:  −∂tφ+ L∗(t)φ = A∗(t)φ+ F0 +∇ · F in QT , φ = 0 on ΣT , φ(·, T ) = φ in Ω, (33)”