Stabilization of a periodic trajectory for a chemostat with two species

We study a chemostat model with two organisms using Lyapunov function methods. Using a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration, we produce a locally exponentially stable oscillatory behavior for the species concentrations, meaning all trajectories of the chemostat that stay near the oscillatory reference trajectory are actually attracted to the reference trajectory exponentially fast. We also obtain a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. We demonstrate the efficacy of our method using a numerical simulation. © 2007 IEEE