We study periodic solutions of the chemostat model under an integral constraint, either on the flow rate (Pb. 1) or on the substrate concentration (Pb. 2). We give conditions on the growth kinetics for which it is possible to improve the averaged water quality (Pb. 1) or the total quantity of treated water (Pb. 2) over a given time period, compared to steady-state. When this is possible, we characterize optimal periodic solutions and show a duality between the two optimization problems. The results are illustrated on four types of growth kinetics, given by Monod, Haldane, Hill and Contois functions
