Prescribed-time tracking for triangular systems of reaction-diffusion PDEs

Abstract

Approximate controllability of systems of coupled parabolic partial differential equations has been of interest for a few decades, where the existence of open-loop control laws performing approximate state transitions within a finite time is studied. In this work, we specialize to systems of reaction-diffusion equations where the connectivity structure is triangular in the reaction parameters and the controls appear at the boundary. We first generate controllers by combining a decoupling backstepping approach with differential flatness that allow us to generate admissible trajectories for system outputs from a given initial condition. As a byproduct of our approach, we achieve approximate state transitioning for the system within a finite terminal time. We enhance our control law by introducing time-varying error feedback controllers which reject variations in initial conditions within the terminal time. The resulting control law not only performs the approximate control task but also output trajectory tracking, all within the terminal time which can be prescribed independently of initial conditions.