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

TL;DR

This paper investigates the null controllability of coupled parabolic PDE systems with multiple underactuations, introducing a novel framework that combines analytic and algebraic methods to achieve controllability with fewer controls.

Contribution

It develops a new approach dividing the control problem into analytic and algebraic parts, enabling null controllability with minimal control inputs in coupled parabolic systems.

Findings

Established null controllability for coupled parabolic systems with underactuations.

Introduced a combined analytic and algebraic framework for control design.

Achieved controllability with controls on only a subset of equations.

Abstract

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.