Closed-loop control of a reaction-diffusion system
This work addresses the theoretical foundation for controlling reaction-diffusion systems with limited sensors and actuators, but it is incremental as it extends existing fixed-point methods to a specific PDE-ODE system.
The paper studies a closed-loop control problem for a thermodynamic process governed by the Allen-Cahn reaction-diffusion model, proposing a feedback law using a finite number of control devices and measurement points. It proves the existence of solutions via a generalization of the Kakutani fixed point theorem.
A system of a parabolic partial differential equation coupled with ordinary differential inclusions that arises from a closed-loop control problem for a thermodynamic process governed by the Allen-Cahn diffusion reaction model is studied. A feedback law for the closed-loop control is proposed and implemented in the case of a finite number of control devices located inside the process domain basing on the process dynamics observed at a finite number of measurement points. The existence of solutions to the discussed system of differential equations is proved with the use of a generalization of the Kakutani fixed point theorem.