Stabilization of chemostats using feedback linearization and reduction of dimension

Stabilization via feedback linearization of models of competition between two species of micro-organisms for two essential resources based in the chemostat is considered, extending previous work recently done by the authors; see and references therein. We show that though the full four-dimensional system is not stabilizable due to the dynamical properties of the system, it is possible to achieve the goal in modified form by pursuing a process of dimensional reduction prior to feedback linearization that results in replacing the four dimensional analysis with one in three dimensions. This technique has appeared in the literature applied to a similar system in the seminal papers of Hoo and Kantor. However, in that case it appears that the authors thought of the method as a matter of convenience, and apparently did not realize that their original (higher dimensional) system was not stabilizable without utilizing the reduction procedure. (That is, that the reduction process was necessary in order to achieve the stabilization goal.) In this paper we show how the problem and its solution are very similar for both our model and that of Hoo and Kantor. This suggests that the dimensional reduction method could be rather generally applicable.