Output-Feedback Stabilization in Prescribed-Time of a Class of Reaction-Diffusion PDEs With Boundary Input Delay

Time-varying prescribed-time (PT) controllers use growing gains not only to achieve convergence in desired time but to reduce state peaking during stabilization and to also reduce the control effort by distributing it more evenly over the time interval of convergence. In this article, we consider a 1-D reaction-diffusion system with boundary input delay and propose a general method for studying the problem of PT boundary stabilization. To achieve this objective, we first reformulate the system as a PDE-PDE cascade system (i.e., a cascade of a linear transport partial differential equation (PDE) with a linear reaction-diffusion PDE), where the transport equation represents the effect of the input delay. We then apply a time-varying infinite-dimensional backstepping transformation to convert the cascade system into a prescribed-time stable (in short PTS) target system. The stability analysis is conducted on the target system, and the desired stability property is transferred back to the closed-loop system using the inverse transformation. The effectiveness of the proposed approach is demonstrated through numerical simulations.