Controllability of Underactuated Coupled Parabolic Systems
This thesis studies the null controllability of a system of coupled parabolic PDEs. Moreover, our work specializes to an important subclass of these control problems, where systems are underactuated and are coupled by first and zero-order couplings. 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, which allows us to recover null controllability by means of internal controls which appear on only a few of the equations. By solving these control problems concurrently, we resolve the original problem (after some technical verifications on the regularity of the controls in the analytic system).