Closed-loop control of a reaction-diffusion system

A system of a parabolic partial di erential equation coupled with ordinary di erential inclusions that arises from a closed-loop control problem for a thermodynamic process governed by the Allen-Cahn di usion reaction model is studied. A feedback law for the closed-loop control is proposed and implemented in the case of a nite number of control devices located inside the process domain basing on the process dynamics observed at a nite number of measurement points. The existence of solutions to the discussed system of di erential equations is proved with the use of a generalization of the Kakutani xed point theorem. Acknowledgement: Grzegorz Dudziuk was supported by the International Ph.D. Projects Programme of Foundation for Polish Science operated within the Innovative Economy Operational Programme 200720013 funded by European Regional Development Fund (Ph.D. Programme: Mathematical Methods in Natural Sciences). AMS Subject Classi cation: 35Q79, 35Q93.