Section: New Results
Analysis and supervision of bioprocesses
Models development and identification
Participants : Yessmine Daoud, Jérôme Harmand, Nesrine Kalboussi, Guilherme Pimentel, Alain Rapaport.
Membrane bioreactors combine a filtration process (with a membrane) and a suspended growth rate bioreactor. This recent technology present many advantages compared to conventional ones, but is more sophisticated and requires refined control because of possible problems related to the risk of membrane fouling. After the PhD by Amine Charfi defensed in 2014 we continue to work on the modeling and control of membrane bioreactors.

Within this framework, new results have been obtained and a new model including several fouling mechanisms has been proposed [22] .

In the scope of the PhD of Guilherme Pimentel (defensed in February 2015, [11] ), we have proposed a simple three time scales model in view of the control of the cake formation [14] , [33] . This model has been validated on real data from a pilot plant at Univ. Mons (Belgium).

The PhD thesis by Nesrine Kalboussi (ENITLAMSIN, Tunis, Tunisia) has just begun. It is dedicated to the early detection and control of membrane fouling. At present time, Nesrine is working on the bibliography about modeling and control of membrane bioreactors.
In many bioprocesses models, the loss of nutrient used for the maintenance of bacteria is neglected compared to the important nutrient supply. In poor environment, such as natural one in the oceans, this is no longer verified. In our collaboration with the LOMIC lab (BanyulssurMer), we have shown that the consideration of a maintenance term in the chemostat model allows to fit the data observed in experimental chemostats, and moreover that the level of maintenance is correlated to the activities of bacteria under the presence of light [23] . This gives a possible explanation of the variable yield observed in the bacterial compartment of marine ecosystems.
Synthesis of control laws
Participants : Térence Bayen, Walid Bouhafs, Amel Ghouali, Jérôme Harmand, Claude Lobry, Guilherme Pimentel, Alain Rapaport, Victor Riqueleme.
We investigate two kinds of bioprocesses to be controlled, arising in industrial biotechnology (digesters, wastewater purification...) or in the bioremediation of natural environments (lakes, landfill...).
Industrial biotechnology
In the framework of the PhD of Guilherme Pimentel [11] and the pilot plant at Mons, a nonlinear predictive law based on the model exposed in 7.2.1 has been tested and validated for piloting the process[34] .
Control of biological reactors are still of great interest, notably but not only with respect to anaerobic digestors that can be destabilized due to the accumulation of intermediate metabolites that can inhibate the growth of some bacteria.

Amel Ghouali (Cotutelle Univ. Montpellier and Univ. of Tlemcen, Algeria) who has defensed her PhD in December has developed an optimal control strategy to optimize the production of biogas over a given period of time [12] . In particular, she has solved an original optimal control problem using the maximum principle of Pontryagin [25] .

Within the scope of the PhD thesis by Walid Bouhafs (ENITLAMSIN, Tunis, Tunisia), we have proposed a new optimal control strategy for systems in which two specific substrates are degraded by two different bacterial consortia, one being limited by the oxygen while the other is inhibited. Walid will defense his PhD in next February.
The minimal time criterion is of particular interest in biotechnology, as it leads to timeindependent feedback controllers.

The paper [19] is devoted to the study of the minimal time problem of a fedbatch reactor, under the presence of a saturation point on the singular locus (this typically occurs whenever the growth rate function is of Haldane type and when typically the maximum input flow rate is not high enough to maintain the substrate concentration constant). This brings nonintuitive issues for the optimal synthesis (existence of switching curve and point of prior saturation).

In the work [47] , we study the minimal time control to drive a chemostat model to a target point. Such a problem finds application typically in the case where the input substrate concentration changes yielding in a new steady state. Converging fast towards this new reference point is much desired in practice. One essential feature of the present work is that the system takes into account a recirculation of biomass (as it is more and more often the case in modern biotechnology). We depict an optimal synthesis and provide an optimal feedback control by using the Pontryagin Maximum Principle and geometric control theory for both Monod and Haldane kinetics.
Bioremediation of natural environments
In the scope of the associated team with Chile (see 9.3.1.1 ) and the cosupervision of the PhD of Victor Riquelme, we have carried on the study of optimal syntheses for the minimal time treatment of natural water reservoirs (such as lakes) [41] . We have proved that the minimal time strategy consists in a mostrapid approach to homogeneous concentrations, even though the optimal control problem is non convex. Moreover, we have shown that a large diffusion increases the treatment time when the resource is everywhere highly polluted, while it can at the opposite be beneficial when only part of the resource is polluted (paper under revision for SIAM J. Cont. & Optim.). This feature should serve the practitioners in the choice of pumps positioning in a originally clean water resource that is suddenly affected by a local pollution. Moreover, we have shown, in collaboration with A. Rousseau (EPI LEMON), how these analytic feedback laws obtained on a oversimplified representation of the spatial heterogeneity behave quite satisfactorily when simulated [17] . This year we have started to study to problem of treating two different pollutants, with a anaerobic/aerobic process in series.
Also in the scope of the associated team with Chile, we have characterized the optimal strategy to treat as fast as possible a landfill with the recirculation as a manipulated variable [40] , [29] , based on a model that we have proposed last year. In presence of singular arcs that are nonadmissible (in the sense that the upper bound of the recirculation pump does not allow to stay on the singular arc), we have shown that a kind of anticipation law is necessary before operating optimally the switching. This analyses reveals several subdomains for which the optimal policy requires different kind of measurements. Knowing in which subdomain the initial stage of landfill could be inform then the practitioners about which concentration (leachate or solubilized or both) should be ideally measured. This primarily work has led to the codevelopment of a software mockup with Chilean partner (see 6.1 ), in order the study the consideration of spatial heterogeneity in landfill, with the approach exposed in 7.1.1 .
This year, again in the scope of the associated team with Chile and Inria Chile, we have begun a new investigation on modeling and control strategy for the regulation of a lagoon that communicates temporarily with the sea and whose water is exploited by pumping.
Theoretical development
The time crisis is an interesting criterion that measures the time spent by a system in a “bad” zone or in “danger”. Typically, when a desired species is under a given (low) threshold, one can consider that this defines a crisis domain. For controlled system, the minimal time crisis has already been proposed in the literature [56] . Nevertheless, only sufficient conditions (i.e. characterization of the solutions of the associated HamiltonBellmanJacobi equation) have been given, and no necessary conditions have been yet proposed, due to the lack of continuity of the integrand cost. We have proposed a regularization of this problem by a family of optimal control problems for which the usual necessary conditions can be derived, and studied the convergence [20] . Practically, this allows to use classical software, such as Bocop, to approximate the optimal solutions. In the internship of C. Romero (U. Chile), this technique has been successfully applied on the LotkaVolterra model with a control on the predator, and a threshold on the prey.