Nonlinear Analysis: Real World Applications 12 (2011) 487–500 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Dynamical study and robustness for a nonlinear wastewater treatment model M. Serhani a,∗ , J.L. Gouze b , N. Raissi c a Equipe TSI, Mathematics and Computer Department, Faculty of Sciences, University Moulay Ismail, BP 11201, Zitoune, Meknes, Morocco b Projet COMORE, INRIA-Sophia Antipolis, B.P. 93, France c EIMA, Mathematics and Computer Department, Faculty of Sciences, University Ibn Tofail, B.P. 133, Kenitra, Morocco article info Article history: Received 4 June 2010 Accepted 27 June 2010 This paper is dedicated to Serhani Khadija. Keywords: Dynamical system Asymptotical stability Lyapunov function Robustness Wastewater treatment abstract In this work we deal with a wastewater treatment by using the activated sludge process. The problem is formulated as a nonlinear dynamical system. Firstly, we develop the dynamical study of the model when all parameters are well known. Hence, basic properties of invariance and dissipation are established and, under a suitable condition on parameters, a globally asymptotically stable equilibrium point occurs. Secondly, when the bacterium growth function and the substrate concentration in the feed stream are not well known, the robustness analysis provides the existence of an attractor domain to which all trajectories of the system converge. Finally, we prove that we can reduce the size (volume) of this attractor domain by increasing the recycle rate to a maximum fixed level. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction In this paper, we study a problem of wastewater treatment, namely that of the activated sludge process in a depura- tion station. The mathematical model is formulated as a nonlinear ordinary differential system. The working principle is described thoroughly in the literature (see for example [1–6]). Basically, the process can be summarized as follows: the wastewater is discharged into an aerator with a flow Q in and a concentration s in in the feed stream. The phase of biological oxidation of the polluted water (substrate), by a blend of a bacterial population in an aerobic reaction consuming the oxy- gen, begins in the aerator and completes in the settler tank. At this stage, due to gravity, the solid components will settle and concentrate at the bottom, while the sedimentation of soluble organic matter is assumed to be not significant. Part of the bacteria biomass is recycled into the aerator in order to stimulate the oxidation. A schematic of the process is shown in Fig. 1, where s, x and x r are the state variables representing respectively the substrate biomass (pollutant), the bacteria biomass and the recycled bacteria biomass concentrations. Q in , Q out , Q r , Q w are the influent, effluent, recycle and waste flow rates, respectively. V a and V s represent the aerator and settler volumes and s in corresponds to the substrate concentrations in the feed stream. Although many works showed interest in a similar problem, where biological aspects and aspects of observability [2–8], estimation, adaptive control, computer simulation [6,9–13], and numerical optimal control [5,14] were studied, there appears to be no work focused upon the dynamical behavior of a model involving a nonlinear bacterium growth function, Our goal in this work is to carry out a rigorous dynamical study for two cases: firstly, when all parameters of the model are known, and secondly, when some parameters are not known. Hence, in the first part of this paper, the basic properties of invariance and dissipation are established and we prove that under some conditions on parameters, there ex- ists a globally asymptotically stable interior equilibrium point. In the second part, our attention is focused on the case where ∗ Corresponding author. E-mail addresses: mserhani@hotmail.com (M. Serhani), Jean-Luc.Gouze@sophia.inria.fr (J.L. Gouze), n.raissi@mailcity.com (N. Raissi). 1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.06.034