Closed-loop control of a reaction-diffusion system

TL;DR

This paper studies a closed-loop control system for a thermodynamic process modeled by a reaction-diffusion PDE coupled with ODE inclusions, proposing a feedback law and proving solution existence.

Contribution

It introduces a novel feedback control law for a reaction-diffusion system with finite control devices and establishes the existence of solutions using a generalized Kakutani fixed point theorem.

Findings

Proposed a feedback law for controlling the reaction-diffusion system.

Proved the existence of solutions for the coupled PDE-ODE system.

Demonstrated the applicability of fixed point theorems in control system analysis.

Abstract

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.