arXiv subjects

Alain Rapaport

Publications and source records attributed to Alain Rapaport.

29 records · Page 2Linked to original sources

arXiv2017

Equivalence of finite dimensional input-output models of solute transport and diffusion in geosciences

We show that for a large class of finite dimensional input-output positive systems that represent networks of transport and diffusion of solute in geological media, there exist equivalent multi-rate mass transfer and multiple interacting continua representations, which are quite popular in geo-sciences. Moreover, we provide explicit methods to construct these equivalent representations. The proofs show that controllability property is playing a crucial role for obtaining equivalence. These results contribute to our fundamental understanding on the effect of fine-scale geological structures on the transfer and dispersion of solute, and, eventually, on their interaction with soil microbes and minerals.

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arXiv2015

Optimal feedback synthesis and minimal time function for the bioremediation of water resources with two patches

This paper studies the bioremediation, in minimal time, of a water resource or reservoir using a single continuous bioreactor. The bioreactor is connected to two pumps, at different locations in the reservoir, that pump polluted water and inject back sufficiently clean water with the same flow rate. This leads to a minimal-time optimal control problem where the control variables are related to the inflow rates of both pumps. We obtain a non-convex problem for which it is not possible to directly prove the existence of its solutions. We overcome this difficulty and fully solve the studied problem by applying Pontryagin's principle to the associated generalized control problem. We also obtain explicit bounds on its value function via Hamilton-Jacobi-Bellman techniques.

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arXiv2014

The buffered chemostat with non-monotonic response functions

We study how a particular spatial structure with a buffer impacts the number of equilibria and their stability in the chemostat model. We show that the occurrence of a buffer can allow a species to persist or on the opposite to go extinct, depending on the characteristics of the buffer. For non-monotonic response functions, we characterize the buffered configurations that make the chemostat dynamics globally asymptotically stable, while this is not possible with single, serial or parallel vessels of the same total volume and input flow. These results are illustrated with the Haldane kinetic function.

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arXiv2013

Extremum seeking via continuation techniques for optimizing biogas production in the chemostat

We consider the chemostat model with the substrate concentration as the single measurement. We propose a control strategy that drives the system at a steady state maximizing the gas production without the knowledge of the specific growth rate. Our approach separates the extremum seeking problem from the feedback control problem such that each of the two subproblems can be solved with relatively simple algorithms. We are then free to choose any numerical optimization algorithm. We give a demonstration for two choices: one is based on slow-fast dynamics and numerical continuation, the other is a combination of golden-section and Newton iteration. The method copes with non-monotonic growth functions.

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arXiv2013

A method for the reconstruction of unknown non-monotonic growth functions in the chemostat

We propose an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We then show how one can use this control law to trace out (reconstruct) the whole graph of the growth function. The process of tracing out the graph can be performed either continuously or step-wise. We present and compare both approaches. Even in the case of two species in competition, which is not directly accessible with our approach due to lack of controllability, feedback control improves identifiability of the non-dominant growth rate.

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arXiv2012

About Transgressive Over-Yielding in the Chemostat

We show that for certain configurations of two chemostats fed in parallel, the presence of two different species in each tank can improve the yield of the whole process, compared to the same configuration having the same species in each volume. This leads to a (so-called) "transgressive over-yielding" due to spatialization.

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arXiv2012

Extensions of the chemostat model with flocculation

In this work, we study a model of the chemostat where the species are present in two forms, isolated bacteria and under an aggregated form like attached bacteria or bacteria in flocks. We show that our general model contains a lot of models which were previously considered in the literature. Assuming that flocculation and deflocculation dynamics are fast with respect to the growth of the species, we construct a reduced chemostat-like model in which both the growth functions and the apparent dilution rate depend on the density of the species. We also show that such a model involving monotonic growth rates may exhibit bistability, while it may only occur in the classical chemostat model when the growth rate in non monotonic.

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arXiv2011

Comparison of numerical simulations of reactive transport and chemostat-like models

The objective of the paper is to evaluate the ability of reactive transport models and their numerical implementations (such as MIN3P) to simulate simple microbial transformations in conditions of chemostat or gradostat models, that are popular in microbial ecology and waste treatment ecosystems. To make this comparison, we first consider an abstract ecosystem composed of a single limiting resource and a single microbial species that are carried by advection. In a second stage, we consider another microbial species in competition for the same limiting resource. Comparing the numerical solutions of the two models, we found that the numerical accuracy of simulations of advective transport models performed with MIN3P depends on the evolution of the concentrations of the microbial species: when the state of the system is close to a non-hyperbolic equilibrium, we observe a numerical inaccuracy that may be due to the discretization method used in numerical approximations of reactive transport equations. Therefore, one has to be cautious about the predictions given by the models.

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arXiv2011

Minimal-time bioremediation of natural water resources

We study minimal time strategies for the treatment of pollution of large volumes, such as lakes or natural reservoirs, with the help of an autonomous bioreactor. The control consists in feeding the bioreactor from the resource, the clean output returning to the resource with the same flow rate. We first characterize the optimal policies among constant and feedback controls, under the assumption of a uniform concentration in the resource. In a second part, we study the influence of an inhomogeneity in the resource, considering two measurements points. With the help of the Maximum Principle, we show that the optimal control law is non-monotonic and terminates with a constant phase, contrary to the homogeneous case for which the optimal flow rate is decreasing with time. This study allows the decision makers to identify situations for which the benefit of using non-constant flow rates is significant.

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arXiv2010

Effects of spatial structure and diffusion on the performances of the chemostat

Given hydric capacity and nutrient flow of a chemostat-like system, we analyse the influence of a spatial structure on the output concentrations at steady-state. Three configurations are compared: perfectly-mixed, serial and parallel with diffusion rate. We show the existence of a threshold on the input concentration of nutrient for which the benefits of the serial and parallel configurations over the perfectly-mixed one are reversed. In addition, we show that the dependency of the output concentrations on the diffusion rate can be non-monotonic, and give precise conditions for the diffusion effect to be advantageous. The study encompasses dead-zone models

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arXiv2010

Minimal time control of fed-batch processes with growth functions having several maxima

We address the issue of minimal time optimal control of fedbatch reactor in presence of complex non monotonic kinetics, that can be typically characterized by the combination of two Haldane models. The optimal synthesis may present several singular arcs. Global optimal trajectory results are provided on the basis of a numerical approach that considers an approximation method with smooth control inputs.

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