241 research outputs found
Contrôle non linéaire et Applications [Nonlinear Control and Applications] , Les Cours du CIMPA
No full textExtrait de la préface : Il s'agit d'un livre passionnant et très complet, qui devrait intéresser aussi bien les ingénieurs que les chercheurs débutants ou confirmés travaillant en théorie du contrôle ou cherchant à s'initier à ce domaineThis book on control theory, edited by Tewfik Sari, contains the courses which was given in the CIMPA scholl (Tlemcen, Algeria, 2003). There are six chapters. Chapter 1 : Introduction à la théorie du contrôle; Chapter 2 : An Introduction to Optimal Control; Chapter 3 : Controllability of Partial Differential Equations; Chapter 4: Singular Perturbations in Control Theory; Chapter 5 : Théorie du Contrôle et Equations Algébriques de Ricatti; Chapter 6 : Equations différentielles à second membre discontinu. The authors are Ugo Boscain (Trieste), Claude Lobry (Nice), Sorin Micu (Craiova), Benedetto Picolli (Rome), Gauthier Sallet (Metz), Tewfik Sari (Mulhouse) and Enrique Zuazua (Madrid). The preface is by Jean-Michel Coron (Paris)
A model of a syntrophic relationship between two microbial species in a chemostat including maintenance
No full textMany microbial ecosystems can be seen as microbial ‘food chains’ where the different reaction steps can be seen as such: the waste products of the organisms at a given reaction step are consumed by organisms at the next reaction step. In the present paper we study a model of a two-step biological reaction with feedback inhibition, which was recently presented as a reduced and simplified version of the anaerobic digestion model ADM1 of the International Water Association (IWA). It is known that in the absence of maintenance (or decay) the microbial ‘food chain’ is stable. In a previous study, using a purely numerical approach and ADM1 consensus parameter values, it was shown that the model remains stable when decay terms are added. However, the authors could not prove in full generality that it remains true for other parameter values. In this paper we prove that introducing decay in the model preserves stability whatever its parameters values are and for a wide range of kinetics
Global dynamics of the chemostat with variable yields
No full text7 pagesIn this paper, we consider a competition model between species in a chemostat including both monotone and non-monotone response functions, distinct removal rates and variable yields. We show that only the species with the lowest break-even concentration survives, provided that additional technical conditions on the growth functions and yields are satisfied. LaSalle's extension theorem of the Lyapunov stability theory is the main tool
Best Operating Conditions for Biogas Production in Some Simple Anaerobic Digestion Models
No full textInternational audienceWe consider one-step and two-step simple models of anaerobic digestion which are able to adequately capture the main dynamical behavior of the full anaerobic digestion model ADM1 and has the advantage that a complete analysis for the existence and local stability of their steady states is available. We describe the best operating conditions for biogas production in these simple anaerobic digestion models. We study also the best operating conditions for biomass production in the simple one-step model. We provide the subsets of best operating conditions in the operating diagram of the model. This set gives a clear graphical description of the best operating conditions. Our models incorporate biomass decay terms, corresponding to maintenance. The growth functions are general and are characterized by their qualitative properties. Numerical plots with specied growth functions and biological parameters illustrate the obtained results
A Lyapunov function for the chemostat with variable yields
No full textInternational audienceIn this Note, we give a global asymptotic stability result for the competition mathematical model between several species in a chemostat, by using a new Lyapunov function. The model includes both monotone and non-monotone response functions, distinct removal rates for the species and variable yields, depending on the concentration of substrate. We obtain, as corollaries of our result, three global stability theorems which were considered in the literature.Dans cette Note on propose une nouvelle fonction de Lyapunov pour l'étude de la stabilité asymptotique globale dans un modèle mathématique de compétition entre esspèces dans le chemostat. Le modèle inclut des fonctions de croissance monotones ou non monotones, des taux de mortalité différents pour chaque espèce et des taux de rendement variables, fonctions de la concentration en substrat. On obtient, comme corollaires de notre résultat, trois théorèmes de stabilité globale qui ont été considérés dans la litérature
Competitive Exclusion for Chemostat Equations with Variable Yields
No full text[Departement_IRSTEA]Ecotechnologies [TR1_IRSTEA]INSPIREIn this paper, we study the global dynamics of a chemostat model with a single nutrient and several competing species. Growth rates are not required to be proportional to food uptakes. Our approach is based on the construction of Lyapunov functions. The Lyapunov functions extend those used by Hsu (SIAM J. Appl. Math. 34:760-763, 1978) and by Wolkowicz and Lu (SIAM J. Appl. Math. 52:222-233, 1992) in the case when growth rates are proportional to food uptakes. Our result generalizes a large variety of previous results obtained by Lyapunov techniques
Comments on "Limit cycles in the chemostat with constant yields" Mathematical and Computer Modelling 45 (2007) 927-932
No full textInternational audienceSeveral elements of the 2007 paper "Limit cycles in the chemostat with constant yields" are incorrect and in contradiction to well established results of the literature. In particular the claim that limit cycles can exist in the chemostat with two competitors for a single nutrient and constant yields is utterly false. It is well known that in this model the competitive exclusion principle holds
Croissance et compétition des espèces microbiennes dans un chemostat
No full textInternational audienc
Averaging in Hamiltonian systems with slowly varying parameters
No full textThe aim of this paper is to describe the general averaging principle and to discuss the particular case of single-frequency systems, the case of systems with constant frequencies and the case of Hamiltonian systems. We show how the stroboscopic method, which is a method of the nonstandard perturbation theory of differential equations, can be used in this kind of problems. We give various examples which illustrate the simplicity and the effectiveness of the method
Analysis of anaerobic digestion models : Applications to the modeling and the control of bioreactors
No full textCette thèse porte sur l’analyse mathématique de différents modèles de la digestion anaérobie. Dans la première partie, nous étudions un modèle à quatre étapes avec dégradation enzymatique du substrat (matière organique) qui peut être sous forme solide. Nous étudions l’effet de l’hydrolyse sur le comportement du processus de la digestion anaérobie et de la production du biogaz (méthane et hydrogène). Nous considèrons, dans un premier modèle, que l’hydrolyse se fait d’une manière enzymatique, alors que dans un second, nous supposons qu’elle est réalisée par un compartiment microbien. Les modèles considérés incluent l’inhibition de croissance des bactéries acétogènes, méthanogènes hydrogénétrophes et acétoclastes par plu- sieurs substrats. Pour étudier l’effet de ces inhibitions en présence de l’étape de l’hydrolyse, nous étudions dans un premier temps un modèle sans inhibition. Nous déterminons les équilibres et nous donnons des conditions nécessaires et suffisantes pour leur stabilité. L’existence et la stabilité des équilibres sont illustrées avec des diagrammes opératoires. Nous montrons que le modèle avec hydrolyse enzymatique change la production du méthane et d’hydrogène. En outre, l’introduction du com- partiment hydrolytique microbien donne de nouveaux équilibres et affecte les régions de stabilité. Nous prouvons que la production de biogaz est maximale en un seul point d’équilibre selon les paramètres opératoires et nous déterminons le taux maxi- mal de biogaz produit, dans chaque cas. Dans la deuxième partie, nous nous sommes intéressés à un modèle à deux étapes décrivant les phases de l’acétogénèse et de la méthanogénèse hydrogénotrophe. Le modèle représente une relation de syntrophie entre deux espèces microbiennes (les bactéries acétogènes et méthanogènes hydro- génotrophes), avec deux substrats à l’entrée (l’acide gras volatile et l’hydrogène), incluant les termes de mortalité et l’inhibition de croissance des bactéries acéto- gènes par un excès d’hydrogène dans le système. L’analyse de l’existence et de la stabilité des équilibres du modèle donne naissance à un nouvel équilibre qui peut être stable selon les paramètres opératoires du système. En utilisant les diagrammes opératoires, on remarque que, quelle que soit la région de l’espace considérée, il existe un seul équilibre localement exponentiellement stable. Cette étude est géné- ralisée dans le cas où la croissance des bactéries méthanogènes hydrogénotrophes est inhibée. Ce modèle donne naissance à deux équilibres strictement positifs et une bistabilité. Nous illustrons, en utilisant les diagrammes opératoires l’effet de cette inhibition sur la réduction des régions de coexistence et l’émergence de régions de bistabilité.This PhD thesis focuses on the mathematical analysis of different anaerobic digestion (AD) models. In a first part, we study a 4-step model with enzymatic degradation of the substrate (organic matter) that can partly be under a solid form. We investigate the effects of hydrolysis on the behavior of the AD process and the production of biogas (namely, the methane and the hydrogen). We consider, in a first model, that the microbial enzymatic activity is constant, then we take into consideration an explicit hydrolytic microbial compartment for the substrate biodegradation. The considered models include the inhibition of acetogens, hydroge- notrophic methanogens and acetoclastic methanogens growth bacteria. To examine the effects of these inhibitions in presence of a hydrolysis step, we first study an inhibition-free model. We determine the steady states and give sufficient and neces- sary conditions for their stability. The existence and stability of the steady states are illustrated by operating diagrams. We prove that modeling the hydrolysis phase by a constant enzymatic activity affects the production of methane and hydrogen. Furthermore, introducing the hydrolytic microbial compartment yields new steady states and affects the stability regions. We prove that the biogas production occurs at only one of the steady states according to the operating parameters and state variables and we determine the maximal rate of biogas produced, in each case. In the second part, we are interested in a reduced and simplified model of the AD pro- cess. We focus on the acetogenesis and hydrogenetrophic methanogenesis phases. The model describes a syntrophic relationship between two microbial species (the acetogenic bacteria and the hydrogenetrophic methanogenic bacteria) with two in- put substrates (the fatty acids and the hydrogen) including both decay terms and inhibition of the acetogenic bacteria growth by an excess of hydrogen in the sys- tem. The existence and stability analysis of the steady states of the model points out the existence of a new equilibrium point which can be stable according to the operating parameters of the system. By means of operating diagrams, we show that, whatever the region of space considered, there exists only one locally exponentially stable steady state. This study is generalized to the case where the growth of the hydrogenetrophic methanogens bacteria is inhibited. This model exhibits a rich be- havior with the existence of two positive steady states and bistability. We illustrate by means of operating diagrams the effect of this inhibition on the reduction of the coexistence region and the emergence of a bistability region
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