Section: New Results

Theoretical results

Control of continuous bioreactors

Participants : Jérôme Harmand, Frédéric Mazenc, Alain Rapaport.

The team maintains a regular research activity on the automatic control of continuous stirred bioreactors, with several objectives:

• the stabilization about a reference target, despite model uncertainties, disturbances and partial measurements;

• the determination of optimal strategies for reaching a target, with minimal time as a usual criterion;

• the tracking of prescribed trajectories, for instance periodic regimes allowing the coexistence of several species in competition on a single resource.

This year, the team has begun to study a new kind of optimal control problems for the preservation of water quality in natural environment. In natural sites, such as lake or water table, to be depolluted from toxic substrates, the introduction of micro-organisms in order to collect the pollution is not suited because of the eutrophisation attempting to the preservation of other living organisms (fish, plankton,...). A typical problem is the control of the input flow of a bioreactor pumping water from the site, treating it and rejecting it cleanup, after biomass filtration. Under the consideration that the dynamics of the bioreactor is much faster than the one of the average concentrations in the lake or the water table, because of the large differences of volumes, we have shown that a non-constant optimal feedback could provide significant gains in time for the depollution of the site. With the help of the Maximum Principle, we have derived optimal feedback laws, considering the concentrations diffusion between the pumping and rejecting locations [43] [50] . This work is conducted in cooperation with the CMM (Santiago de Chile) and UTFSM (Valparaiso) within the INRIA/CONICYT program.

The team has pursued its work on the problem of ensuring the persistence of several species through control. In [27] , we have designed feedback controllers for chemostats with two species and one limiting substrate so that a positive equilibrium (with arbitrary prescribed species concentrations) becomes globally asymptotically stable. The design uses a new, global, explicit, strict Lyapunov function construction. By taking advantage of the Lyapunov function we have proposed, we have quantified the effects of disturbances using the input-to-state stability paradigm. The control we use requires only the measure of a linear combination of the species concentrations: this feature of our result is a crucial advantage from a practical point of view.

In the literature, all control strategies aiming at ensuring persistence of species for chemostats described by classical equations ensure only the persistence of two species. For the first time, in the paper [40] , we presented a technique that ensures the stable persistence of an arbitrary number of species competing for a single limiting substrate. This technique is based on an appropriate choice for the substrate input concentration and for the dilution rate. The control laws we obtained, are given by explicit, nonlinear formulas, are time-varying, positive everywhere and of class C¹ . They require the measurement of all the species concentrations. They stabilize a periodic trajectory. We gave a local version of this result in the situation where only the substrate concentration is available for feedback design. The stabilization result of [40] applies only when the growth functions satisfy some conditions. We show that when they are violated, persistence by trajectory-stabilization cannot be achieved.

Designs under Matrosov's conditions and robust adaptive control

Participant : Frédéric Mazenc.

In [26] the design of explicit, global, strict Lyapunov functions is performed under conditions of Matrosov-type. The advantages of [26] are (a) the results are simpler than the known constructions relying on the Matrosov's approach; (b) the Lyapunov functions are locally lower bounded by positive definite, quadratic functions for a large class of systems; (c) they only require a non-strict positive definite function whose derivative along the trajectories is non-positive instead of a (radially unbounded) non-strict Lyapunov function. The motivation for (c) is that for biological models, one can frequently find non-strict Lyapunov-like functions which are not proper. Another useful property of the construction of [26] is that it yields some robustness results. We used them to prove robustness for a waste-water treatment process stabilized through adaptive feedback. This illustrates the value added by strict Lyapunov functions for biological models.

One difficulty in applying the known Matrosov theorems is that one needs to know some special functions, called auxiliary functions, to build the global Lyapunov function. This was overcome in [25] for the special case of the tracking-error dynamics for adaptively controlled, nonlinear systems that are affine in the unknown parameter. The main assumption was a classical persistence-of-excitation condition.

The contribution of [26] involved (a) constructing explicit auxiliary functions for adaptive controlled error dynamics; (b) extending the resulting global Lyapunov function construction to cases where the unknown parameter also has additive, time-varying uncertainty. This made it possible to explicitly quantify the effects of the uncertainty using the celebrated input-to-state stability paradigm, provided the regressor satisfies an additional affine growth condition. The results in [26] apply under general adaptation laws that could include, for example, projection operators, least-squares estimators, and prediction-error-based estimators. The practical interest of this work is that it leads to uniform, global, asymptotic stability of the error dynamics by constructing explicit, global, strict Lyapunov functions.

In [28] , we have presented two new strict Lyapunov function constructions, based on transforming non-strict Lyapunov functions into strict ones, under Lie-derivative conditions. The main novelty of the first construction is that it allows us to cope with the difficult case of periodic time-varying systems. It applies even when the higher-order Lie derivatives of the weak Lyapunov function vanish at some points outside the equilibrium, on some time intervals, provided a suitable persistence-of-excitation property is satisfied. The simplicity of the construction is also one of its advantages. The second result uses the Matrosov approach. We already mentioned that, in general, Matrosov's method can be difficult to apply, because one needs to find the necessary auxiliary functions. In addition, one needs to select the auxiliary functions so that the resulting strict Lyapunov function has the most desirable properties. Here we gave simple sufficient conditions leading to a systematic design of auxiliary functions. Another important feature of our work is that it applies to cases where the state space of the system is a general subset of Euclidean space, instead of the whole Euclidean space. This is desirable for biological systems, whose state spaces are often restricted by the requirement that physical quantities need to be non-negative. We have chosen to illustrate our approach using an error dynamics associated with the celebrated Lotka-Volterra system.

Systems with delays

Participant : Frédéric Mazenc.

We worked on three distinct problems related to the presence of delays in a model. In this section, we describe two of them, and in the next section, we will explain the last one because it is related to the design of interval observers.

A central result of the theory of the chemostat is the Competitive Exclusion Principle. It states that for a chemostat model with several species with increasing growth rates at most one competitor can survive when there is a single limiting resource. However, this result is valid only in the absence of delay in the equations while they naturally occur in biological models; in particular, chemostats models with delays in the dynamics of the species concentrations are more realistic than models without delays. In [17] it is proved that, for models of this type, the competitive exclusion principle still holds true, provided that the delays are smaller than an upper bound for which an explicit expression is given. The proof is established through the construction of Lyapunov-Krasovskii functional.

Quantized control systems are systems in which the control law is a piece-wise constant function of time taking values in a finite set. For a family of these systems, which contains nonlinear systems, we have used Lyapunov-Krasowskii functionals to design quantized continuous-time control laws in the presence of time-invariant point-wise delays in the input. Our quantized control laws are implemented via hysteresis which allows us to avoid chattering. Our analysis applies to a fairly large class of systems, namely the class of the stabilizable nonlinear systems and for any value of the quantization density. The quantized feedbacks we obtained are parametrized with respect to the quantization density. Moreover, the maximal allowable delay tolerated by the system is characterized as a function of the quantization density.

Interval observers

Participant : Frédéric Mazenc.

The interval-observer method is a recent state-estimation technique. It is used in particular in biological contexts, where taking into account the presence of uncertainties is essential. We have completed the theory of the linear interval observers in several works.

The contribution of the work [24] (see also [39] ) is twofold. A first part of our work is devoted to the problem of exhibiting necessary and sufficient conditions which guarantee that, for a time-invariant linear system of dimension two, a time-invariant, linear and exponentially stable interval observer can be constructed. In the second part of the work, we have shown that when these conditions are violated, one can still construct exponentially stable, linear interval observers, but these interval observers have the remarkable feature of being time-varying . Thus, we managed to give a complete picture of the difficulties and of the solutions which can be given for systems of dimension two. To illustrate the power of our approach, we have applied it to a chaotic system which is known to be highly sensitive to uncertainties in the initial conditions.

In [42] , we have investigated the problem of constructing interval observers for exponentially stable, linear systems with point-wise delays. First, we have proved that classical interval observers for systems without delays are not robust with respect to the presence of delays that appear in a specific structure location, no matter how small the delay is. Next, we have shown that, in general, for linear systems classical interval observers endowed with a point-wise delay are not satisfactory because they are exponentially unstable. Finally, we have designed interval observers of a new type. Our construction relies on framers that incorporate distributed delay terms. These framers are interval observers when the delay is smaller than an upper bound that we have estimated.

Optimal control of fed-batch reactors

Participants : Jérôme Harmand, Alain Rapaport.

In industrial biotechnology, it is not always possible to operate bioreactors in continuous mode like the chemostat. The batch mode, that consists in cycles of sequential feeding and emptying the tank, can be used instead. But each sequence has to be optimized with respect to the occupancy time. The initialization phase of continuous bioreactor of very large volume may also present an issue as far as the time necessary for filling the reactor up to reference values of the concentrations is considered.

The problem of feeding in minimal time a batch reactor with one reaction involving one substrate and one biomass has been originally solved by J. Moreno in 1999 using a technique based on Green's theorem. Recently, the team has contributed to extensions with several species and impulse controls, using techniques based on Hamilton-Jacobi-Bellman equation and Maximum Principle, instead of the one proposed by Moreno that is suited for planar dynamics only.

When the growth function is non monotonic, the optimal control synthesis may present singular arcs. The case of growth functions with only one maximum, such as the Haldane law, has been completely solved by the Moreno's approach and presents a single singular arc. But for complex non monotonic kinetics, characterized for instance by the combination of two Haldane models, the minimal time problem exhibits several candidate singular arcs, and the technique based on the Green's theorem provides only local optimal conditions. and the problem exhibits several candidate singular arcs. The analytical determination of which singular arc is optimal, and for how long, appears to be a tricky problem. We have proposed an approximation procedure, that considers a family of approximate optimal control problems with smooth controls (i.e. without singular arc), based on former idea of C. Lobry. This has led us, very recently, to a new numerical method for determining the optimal selection of singular arcs.

Modelling and identification of batch processes

Participants : Miled El Hajji, Jérôme Harmand, Alain Rapaport.

With INRA Dijon, we conduct experiments of batch cultures of micro-organisms collected in soil ecosystems, in a modelling perspective. Before mixing several species in experiments for the investigation of their interactions, we have first revisited and fitted on the data the usual models in microbiology. Most often, theses models are well suited for the growth phase but not longer for situations with restricted nutrient availability, as it happens in soil ecosystems.

Last year, we have proposed a model of batch reaction with an explicit compartment of inactive (or dead) cells and an additional nutrient recycling term in the substrate dynamics. The dynamical behavior of this model fits qualitatively well the experimental data collected at INRA Dijon, in the framework of El Hajji's PhD thesis. The on-line observation made on this system are the concentration S and an optical density that provides a measurement of the total biomass, and not the proportion of viable cells. A particularity of this model is to be non-identifiable and non-observable at steady state. So we cannot use the usual techniques of on-line reconstruction, that requires the global observability of the system. We have developed a decomposition technique into cascade of observers in different time scales. Each sub-system is observable on its own time scale. But one of the time scale is bounded (i.e. the time does not go toward infinite), which freezes the observer before the system approaches the steady state. Nevertheless, this technique has allowed us to derive and prove the practical convergence of the cascade of observers.

Theory of competition for a substrate

Participants : Claude Lobry, Frédéric Mazenc, Denis Dochain, Miled El Hajji, Alain Rapaport, Bart Haegeman, Jérôme Harmand, Tewfik Sari.

A microbial ecosystem is a complex ecosystem where multiple interactions are initiated involving, in addition to the substrate consumption , the production of substrate through the degradation by enzimes produced by bacteria, “consumption” of bacteria by viruses , the chemotaxis and “quorum sensing” (Some species of microorganism produce at a constant rate a specific molecule which is spread in the environment. The concentration of the chemical is proportional to the population size ; the individuals have a receptor for the specific molecule and, thus, are informed about the size of their own species.), the mutations and a variable environment in time . For each of these specific traits we construct a “toy model” which reproduces it ; the model is analyzed mathematically from the point of view of bio-diversity. This is a “reductionist approach” that we assume but that does not prevent us from taking advantage of more “global” approaches. Below are the topics that we considered in 2009.

• Coexistence through oscillations We considered “substrate-dependent” models but in the case of a three trophic level system: substrate-bacteria-bacteriophage . We produced models which show through simulation the following picture:

  • there are three competing species: A, B, and C. If the three species are present then there is coexistence through oscillations.

  • Species B and C alone do not coexist and species B eliminates C. Thus species A appears to organize the coexistence.

  By itself the possibility of coexistence through periodic solution is not original, it goes back to the seventies ! Actually, it was the main emphasis of the seminal paper of Armstrong and Mc Gehee [53] but our rationale is different and our model has more than two competitors. This model was the subject of a public presentation within the network TREASURE in Tlemcen. Karim Yadi, a doctoral student of T. Sari in Tlemcen mathematically proved that the observed simulations are really a property of the model. This is a rather technical work - it uses the techniques of singular perturbations with two parameters - which is a part of the thesis of Yadi. The work was also presented at the school “Modeling of complex biological systems in the context of genomics”, La Colle sur Loup 30/03-3/04/09 (http://epigenomique.free.fr/fr/orateurs.php ) and at the workshop “System theory in Chemical and Life Science” (03-05/06/09),Centre Interfacultaire Bernoulli, EPFL (Lausanne). A paper, Coexistence of three predators competing for a single biotic resource by Lobry, Sari and Yadi is submitted for publication in a Book to be edited by the Bernoulli center.

• Slowly varying environment Imagine a microbial ecosystem subject to a “slow” variation of its environment. For example, a seasonal cycle is slowly varying compared to time constants of cellular division (of the order of the day). One might wonder about coexistence when all species are favored at a given time. We have recently shown a mathematical technique (based on earlier work by some members of the project) that can predict and understand the coexistence on the basis of knowledge of growth rates of various species. Notice also that there are some previous results when the environment varies on a periodic basis, but our technique also works when the forcing term is not periodic. The work is largely in progress. Some results are published [38] or exposed [49] .

• Coexistence during transients In these papers we consider that in many circumstances we do not observe the asymptotic behavior but a transient. We started to investigate the conditions for “practical coexistence”, that is to say during a long transient. We have some results published in 2008 and 2009 [16] , [32] .

• Mutualism Bacteria might produce chemicals that can be substrates for other species. A case of such an “obligate mutualism” is when the species 1 produces the substance B necessary for growth of species 2 that produces the substance A necessary for the growth of species 1. We have published a model of this situation [15] .

• Systems with viruses We compare classical mathematical work [65] to recent experimental work where the technique of “fingerprints” povides insight into the changes in concentrations of various species along time. One can observe, in a constant environment, a dynamic equilibrium where species dominates one after another. Biologists believe that this phenomenon is explained through the relationship “bacterium-virus” in the form of “kill the winner” mechanism: each species of bacteria B_i is accompanied by a species of virus V_i , when the species B_i becomes sufficiently large and begins to dominate this favors the virus V_i who growths to the detriment of B_i . We can reproduce this phenomenon in simulation and the mathematical justification is in progress.

• The effect of a delay is studied in [17] .

• The paper [51] is a contribution to an old open mathematical problem related to Competitive Exclusion Principle . It is submitted.

Neutral community models for microbial ecology

Participant : Bart Haegeman.

Hubbell's neutral model [59] describes the dynamics of an ecological community in terms of random birth, death and immigration events, attributing equivalent characteristics to all species. Although the absurd simplicity of these assumptions, remarkable agreement between neutral model predictions (species-abundance distributions and species-area relationships) and empirical observations has been reported for some, mostly rather diverse, ecological communities.

There is some evidence that also certain aspects of microbial communities can be well described by the neutral model. Highly diverse microbial communities have been difficult to deal with using more traditional modelling approaches from community ecology. The neutrality assumption could lead to an effective global description, without requiring quantitative species data (growth characteristics, interaction strengths, etc). We are actively participating in the development of neutral community models, with a focus on microbial systems (e.g., [19] ). This is joint work with R. Etienne of Univeristy of Groningen, The Netherlands.

We are especially interested in the description of microbial microplate experiments. Microplates consists of a large number of tiny batch reactors, in each of which a microbial community grows on the substrate available. Due to the small dimensions of these systems, many (identical or different) experimental conditions can be studied in parallel. For example, the same microbial community can be grown on a large number of substrates; or the performance of microbial communities with a different past can be measured on a given substrate. Communities from different reactors can be mixed together, thus creating a spatial network of reactors. Together with D. Vanpeteghem, KULeuven, Belgium, we are developping a theoretical framework for this type of microbial systems [33] . Experiments are conducted by J. Hamelin at INRA-LBE.

The modelling framework of neutral community theory is close in spirit to statistical mechanics. Many individual contributions (organisms in ecology, particles in physical systems) yield some global, averaged system behavior (a community in ecology, a gas or a solid in physics). The model outcome on this global level is often rather insensitive to the modelling assumptions on the detailed level, justifying an oversimplified microscopic description. This mapping between global and detailed level can be formulated as a so-called entropy maximization problem, also known as the MaxEnt algorithm.

Entropy-maximization ideas could be particularly useful to infer community structure characteristics based on a limited number of global, community-averaged properties. This could lead to ecological models of drastically reduced complexity. We have studied the entropy-maximization ideas for two ecological problems:

• First, to predict abundances of the species present in an ecological community [20] . This is joint work with M. Loreau of McGill University, Montreal;

• Second, to predict the spatial distribution of an ecological community [18] . This is joint work with R. Etienne of University of Groningen, The Netherlands.

Both studies consider ecological communities in general, but might be particularly relevant for microbial communities. Indeed, due to the large number of species and individuals in such communities, model complexity reduction is both necessary (the full complexity is impossible to deal with) and effective (due to averaging effects over many components).

Individual-based modeling

Participants : Fabien Campillo, Marc Joannides.

Individual-based modeling (IBM) has been very active over the past fifteen years. It allows to account for the dynamics of complex ecological systems. Although there are many computer realizations of such models, there is a strong need for their mathematical representations. Such a framework is given by birth-death and branching Markov processes, it allows to analyze IBMs at different scales. Indeed, the asymptotic analysis in large population size bridges individual-based models (at microscropic scale) to aggregate models (at macroscopic scale). These last models are usually of integro-differential type. This approach allows one to propose rigorous Monte Carlo simulation algorithms.

We have extended the work of Fournier and Méléard [58] for terrestrial populations to the case of individuals with explicit zone of influence. We have proposed a complete study of the underlying Markov process, its Monte Carlo simulation and its limiting behavior in large populations size [47] . Also as part of the ANR MODECOL in collaboration with Nicolas Champagnat (EPI TOSCA, Sophia Antipolis) and Pierre Del Moral (EPI ALEA, Bordeaux), we have proposed IBMs for terrestrial population dynamics. We propose a description of the underlying Markovian dynamics and of its Monte Carlo simulation. Here the dynamics of individuals is coupled with the dynamics of resources. We also describe the large population size approximation of this process as an integro-differential/partial differential system [34] , [46] .