Controllability of coupled parabolic systems with multiple underactuations: parts I and II

This work studies the null controllability of a system of coupled parabolic PDEs. In particular, our work specializes to an important subclass of these control problems which are coupled by first and zero-order couplings and are, additionally, underactuated. We pose our control problem in a fairly new framework which divides the problem into interconnected parts: we refer to the first part as the analytic control problem, where we use slightly non-classical techniques to prove null controllability by means of internal controls appearing on every equation; we refer to the second part as the algebraic control problem, where we use an algebraic method to invert a linear partial differential operator that describes our system; this allows us to recover null controllability by means of internal controls which appear on only a few of the equations. We establish a null controllability result for the original problem by solving these control problems concurrently.