Nolwenn Ljll

Ma- Lika Garimé

Salima Larcher

Lounici

Maryse Desnous

Julien Guieu

Jules Lise

Giorgia Katia

Lucas

Juliette Léa

coach du Mathieu M Martin

Matthias Mathieu R Nelly

Juan Carlos

Lamberto

Marielle Lucie

Nicolas

Odile

Yan Walid

Keywords: optimization, mathematical modeling, hydrodynamics, dynamical system, resource allocation, periodic control, nonlinear problem, saint-venant equations, han model, microalgae optimisation, modélisation, hydrodynamique, système dynamique, allocation des ressources, contrôle périodique, problème nonlinéaire, équations de saint-venant, modèle de han, microalgues optimization, mathematical modeling, hydrodynamics, dynamical system, resource allocation, periodic control, nonlinear problem, saint-venant equations, han model, microalgae optimisation, modélisation, hydrodynamique, système dynamique, allocation des ressources, contrôle périodique, problème nonlinéaire, équations de saint-venant, modèle de han, microalgues

la modélisation et l'optimisation du couplage hydrodynamique-photosynthèse ", written by LU Liudi, completed on 7

Background and Motivation

The greenhouse effect is a process that warms up the Earth's surface, allowing life by maintaining Earth's temperature above freezing. However, this greenhouse effect is getting stronger as human activities release greenhouse gases in the atmosphere, trapping more and more heat and warming the climate of our planet. Since the pre-industrial period, it has been shown that Earth's global average temperature has increased by about 1 degree Celsius [START_REF] Hansen | Global surface temperature change[END_REF]. Among all the greenhouse gas emissions, carbon dioxide (CO 2 ) contributes up to 68 % of total emissions [START_REF] Brennan | Biofuels from microalgae-a review of technologies for production, processing, and extractions of biofuels and co-products[END_REF][START_REF] Ho | Perspectives on microalgal co2-emission mitigation systems -a review[END_REF]. According to a report by the Carbon Dioxide Information Analysis Center (CDIAC) 1 , CO 2 emissions worldwide have increased from 11 millions metric tons in 1751 to 30619 millions metric tons in 2006. The continuous increase in emissions will have significant effects on future climates, ocean temperature, as well as on economic systems. Therefore, it is imperative to develop an appropriate technology to reduce the emissions and accumulation of CO 2 . CO 2 fixation techniques can be divided into biological and physical methods [START_REF] Khoo | Life cycle energy and co2 analysis of microalgae-to-biodiesel: Preliminary results and comparisons[END_REF]. Between these two techniques, biological CO 2 fixation appears to be a promising economical and environmentally viable technology [START_REF] Ho | Perspectives on microalgal co2-emission mitigation systems -a review[END_REF][START_REF] Kumar | Enhanced co2 fixation and biofuel production via microalgae: recent developments and future directions[END_REF]. This techniques presents an attractive development option as plants or other photosynthetic organisms naturally capture and use CO 2 as part of their photosynthetic process. Although terrestrial plants are able to sequester vast amounts of CO 2 from the atmosphere, their growth is reduced due to the CO 2 availability. On the contrary, phytoplankton have faster growth rates and their CO 2 fixation efficiency is also between 10 and 30 times higher than the terrestrial plants [START_REF] Costa | Modelling of growth conditions for cyanobacterium spirulina platensis in microcosms[END_REF][START_REF] Langley | A critical evaluation of co2 supplementation to algal systems by direct injection[END_REF]. The biological mitigation of CO 2 by using phytoplankton could therefore offer huge advantages. The word phytoplankton originally comes from Greek where phytón means plant and planktós means drifter. These micro-organisms comprise eukaryotic cells (microalgae) and prokaryotic cells (cyanobacteria) which can all do photosynthesis. They can grow naturally in all aquatic environments, for instance oceans, lakes, ponds, and rivers. Despite the fact that phytoplankton biomass in the oceans corresponds to less than 2% of the total global plant carbon, these organisms collectively capture approximately 40% of the global carbon sequestration per year [START_REF] Brewer | Productivity of the ocean. present and past[END_REF][START_REF] Falkowski | Primary Productivity and Biogeochemical Cycles in the Sea[END_REF]. Therefore, the microalgae (synonym of phytoplankton in biotechnology and applied physiology) cultivation becomes an attractive alternative for reducing greenhouse gas emissions [START_REF] Yun | Carbon dioxide fixation by algal cultivation using wastewater nutrients[END_REF]. More precisely, no additional CO 2 is created during the photosynthetic process while biofuels could be produced from the lipids within the microalgae. Promising production potential of various secondary metabolites are also expected for numerous high added value commercial applications: pharmaceutical, cosmetics or food industries [START_REF] Eppink | From current algae products to future biorefinery practices: a review[END_REF][START_REF] Spolaore | Commercial applications of microalgae[END_REF].

Depending on the source of the light, these microorganisms are generally cultivated at industrial scale in open or closed photobioreactors. These devices vary from the most simple and cheapest open reactors to some high-tech closed photobioreactors. They can be run in batch, or continuously adding growth medium in the reactor. This so called chemostat was first introduced independently by Monod [START_REF] Monod | La technique de culture continue théorie et applications[END_REF] and Novick and Szilard [START_REF] Novick | Description of the chemostat[END_REF] in the 1950s for studying bacterial cultures. It is a perfectly mixed photobioreactor where the algae are permanently fed with a nutrient and simultaneously emptied such that the culture volume is kept constant. This classical device is often used to maintain a nutrient-limitation, which for microalgae can be modelled by the Droop model [START_REF] Droop | Vitamin b12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri[END_REF][START_REF] Droop | 25 years of algal growth kinetics a personal view[END_REF] as presented in Subsection 1.2.1. Since the late 1960s, the chemostat has been widely used in microalgae research [START_REF] Iehana | Kinetic analysis of the growth of spirulina sp. in batch culture[END_REF][START_REF] Lee | Accumulation of astaxanthin in haematococcus lacustris (chlorophyta)1[END_REF][START_REF] Molina Grima | Effect of growth rate on the eicosapentaenoic acid and docosahexaenoic acid content of isochrysis galbana in chemostat culture[END_REF][START_REF] Pirt | The photosynthetic efficiency of chlorella biomass growth with reference to solar energy utilisation[END_REF]. At larger scale, the open raceway pond firstly introduced by Oswald [START_REF] Oswald | Biological transformation of solar energy[END_REF] in the 1960s is currently the most frequently used and cheapest cultivation system for commercial production of microalgae. It is used worldwide including the United States, France, Thailand, China, Chile, and Israel. The pond is a shallow artificial hydraulic circuit divided into a rectangular grid, with each rectangle containing one oval-shaped channel. The water is kept in motion by a powered paddle wheel which also ensures the mixing. The industrial applications of microalgae for various industrial sectors, and especially using the cheaper raceway pond technologies, have motivated theoretical studies for modelling and optimization [START_REF] Bernard | A 2d model for hydrodynamics and biology coupling applied to algae growth simulations[END_REF][START_REF] Huggins | Analysis of sediment transport modeling using computational fluid dynamics (cfd) for aquaculture raceways[END_REF][START_REF] Huggins | Use of computational fluid dynamics (cfd) for aquaculture raceway design to increase settling effectiveness[END_REF][START_REF] Sukenik | Potential enhancement of photosynthetic energy conversion in algal mass culture[END_REF].

Optimization aspects and contributions

There is a strong motivation to improve microalgal productivity and make the involved processes more efficient. Theoretical developments are complementing the experimental works, in the idea of guiding the way towards more effective algal production systems.

Researches for enhancing this bio-process followed various directions ranging from determining growth conditions more efficient for the physiology of the microalgae to the design of more efficient algal cultivation devices. Different environmental conditions can affect microalgal growth such as temperature, nutrient availability, light condition and pH. For instance, authors in [START_REF] Ras | Temperature effect on microalgae: a crucial factor for outdoor production[END_REF] have investigated the impact of temperature for outdoor algal production. In [START_REF] Martínez | Theory of turbid microalgae cultures[END_REF], authors have studied the influence of the background turbidity in the photobioreactors on the algal production. Light intensity condition in reactors have also been investigated in [START_REF] Martínez | Maximizing microalgae productivity in a lightlimited chemostat[END_REF], where authors have provided an optimal strategy to maximize the productivity in a light-limited chemostat. The influence of both light condition and nutrient availability has been optimized to improve the algal production [START_REF] Masci | Microalgal biomass surface productivity optimization based on a photobioreactor model[END_REF]. The design of efficient cultivation devices has also been studied to optimize the industrial microalgal productivity. For instance, the authors in [START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF] have studied the algal productivity optimization on a algal rotating biofilm, where a flashing effect has been identified, meaning that the biofilm which rotates with an infinity velocity provides the maximal productivity. The coupling of biological system with hydrodynamics in raceways is studied [START_REF] Huggins | Use of computational fluid dynamics (cfd) for aquaculture raceway design to increase settling effectiveness[END_REF] in order to optimize the raceway design.

The contributions of this thesis mainly focus on the optimization problems in algal production. In Chapter 2, we study the influence of light distribution on algal productivity. An optimal medium turbidity has been identified for general photobioreactors, corresponding to that has been found in literature [START_REF] Grognard | Optimal strategies for biomass productivity maximization in a photobioreactor using natural light[END_REF][START_REF] Masci | Microalgal biomass surface productivity optimization based on a photobioreactor model[END_REF]. We then focus on a specific photobioreactor consisting of a raceway pond for the algal growth. We investigate the influence of the two main features of this device namely the paddle wheel and the topography of the raceway bottom. The idea is to bring more light resource to the algae at the lower part of the pond, since light is strongly Work in progress as of 7 th September, 2021 attenuated when passing through the algal culture. In Chapter 3, we focus on the optimal mixing of the paddle wheel. More precisely, we introduce a model to account for the mixing process and study the periodic regime of the algal growth (see also in [START_REF] Bernard | Optimizing microalgal productivity in raceway ponds through a controlled mixing device[END_REF]). The resulting problem consists of a specific allocation problem which gives rise to non trivial mixing strategies. We have then identified a periodic dynamical resource allocation problem, where we have developed a more complete theory which can be applied to problems other than the algal growth (see also in [START_REF] Bernard | Optimization of mixing strategy in microalgal raceway ponds[END_REF]). We then study the influence of the topography in Chapter 4. We introduce the methodology of optimization of the topography through a hydrodynamic-biologic system, where a non flat topography is observed for non periodic case (see also in [START_REF] Bernard | Controlling the bottom topography of a microalgal pond to optimize productivity[END_REF]). In order to better understand the impact of the topography, we then extend the previous work to a larger range of setting (see also in [START_REF] Bernard | Shape optimization of a microalgal raceway to enhance productivity[END_REF]). In some cases, theoretical results enable to identify the critical topography. However, the increase on algal growth remains very limited when using our optimal topographies, therefore we combine the topography optimization along with the optimization of the mixing strategy with the paddle wheel, and an enhancement on the algal growth is then obtained (see also in [START_REF] Bernard | Mixing strategies combined with shape design to enhance productivity of a raceway pond[END_REF]).

Biological and hydrodynamical models

In this section, we present some classical models related to the systems considered in this thesis. We start with the biological models to describe the algal growth, then we introduce the hydrodynamical models for the algal motion and end up with the light intensity modelling in the turbid medium which couple these two systems.

Biological model

Here, we give a brief review on some models often considered to describe algal systems. According to the application, these models account for describing different dynamical aspect of the growth of microalgae such as light, temperature, nutrient availability, salinity, pH, etc. We mainly focus on two models: the Droop model and the Han model.

Nutrient-Limited growth -The Droop model

One of the crucial elements which may limit microalgal growth is the nutrients. Co-limitation by two or several nutrients is more complex and will not be presented here, we refer to [START_REF] Bougaran | Modeling continuous cultures of microalgae colimited by nitrogen and phosphorus[END_REF][START_REF] Klausmeier | Phytoplankton growth and stoichiometry under multiple nutrient limitation[END_REF][START_REF] Lee | Integrated co-limitation kinetic model for microalgae growth in anaerobically digested municipal sludge centrate[END_REF][START_REF] Li | Effects of different nitrogen and phosphorus concentrations on the growth, nutrient uptake, and lipid accumulation of a freshwater microalga scenedesmus sp[END_REF] for further details. We rather focus on the case where only one nutrient is influencing microalgal growth, all the other being introduced in excess. The classical Monod kinetics [START_REF] Monod | The growth of bacterial cultures[END_REF], initially considered to model bacterial cultures, states that the growth rate denoted hereafter by µ [d -1 ] is directly related to the nutrient (inorganic nitrogen) concentration denoted by s [g N m -3 ] in the following way:

µ(s) = µ max s s + K s , with µ max [d -1
] the maximal growth rate, i.e. the growth rate reached at infinite substrate and K s [g N m -3 ] the half saturation constant for substrate uptake. The applicability of the Monod model has been shown in different works [START_REF] Haario | Reduced models of algae growth[END_REF][START_REF] Lee | Integrated co-limitation kinetic model for microalgae growth in anaerobically digested municipal sludge centrate[END_REF][START_REF] Li | Effects of different nitrogen and phosphorus concentrations on the growth, nutrient uptake, and lipid accumulation of a freshwater microalga scenedesmus sp[END_REF]. However, it has been firstly observed by Droop [START_REF] Droop | Vitamin b12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri[END_REF][START_REF] Droop | 25 years of algal growth kinetics a personal view[END_REF] that microalgae keep growing for some time even after nutrients have been totally depleted. This behaviour does not correspond to Monod kinetics which are therefore not suitable to predict microalgae growth under nutrient limitation. In order to take nutrientlimited growth into account, Droop suggests [START_REF] Droop | Vitamin b12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri[END_REF] to separate the nutrient uptake rate denoted by ρ [g N g C -1 d -1 ] from the growth rate µ which is related to the internal elemental nutrient

Work in progress as of 7 th September, 2021 quota (amount of intracellular nutrient). This model has ever since then been widely studied in [START_REF] Bernard | Transient behavior of biological loop models with application to the droop model[END_REF][START_REF] Lange | The attractiveness of the droop equations[END_REF][START_REF] Vatcheva | Experiment selection for the discrimination of semi-quantitative models of dynamical systems[END_REF] and thoroughly validated experimentally [START_REF] Bernard | Non-linear qualitative signal processing for biological systems: application to the algal growth in bioreactors[END_REF][START_REF] Droop | 25 years of algal growth kinetics a personal view[END_REF][START_REF] Sciandra | The steady states of continuous cultures with low rates of medium renewal per cell[END_REF][START_REF] Vatcheva | Experiment selection for the discrimination of semi-quantitative models of dynamical systems[END_REF].

In a continuous and homogeneous microalgae culture, the dynamics of nutrient concentration s, the biomass concentration X [g m -3 ], and the carbon-specific nitrogen quota q [g N g C

-1 ] of the cells are given by    ṡ = Ds in -ρ(s)X -Ds, Ẋ = (µ(q) -D) X, q = ρ(s) -µ(q)q,

where D [d -1 ] denotes the dilution rates and s in [g N m -3 ] is the nutrient concentration in the feed.

In this model, the growth rate functions µ is taken as a Droop function [START_REF] Droop | Vitamin b12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri[END_REF]:

µ(q) = µ max 1 - q 0 q ,
where q 0 [g N g C -1 ] stands for the minimal cell quota (µ(q 0 ) = 0), meaning that no algal growth can take place for cell quota below this level. By taking the initial condition q(0) ≥ q 0 , it can be proved that ∀t ≥ 0, q(t) ≥ q 0 (as shown in [START_REF] Bernard | Transient behavior of biological loop models with application to the droop model[END_REF]). As for the absorption functions ρ, it is generally taken as in the model proposed by Michaelis-Menten [START_REF] Caperon | Nitrogen-limited growth of marine phytoplankton-ii. uptake kinetics and their role in nutrient limited growth of phytoplankton[END_REF]:

ρ(s) = ρ max s s + K s ,
with ρ max the maximal uptake rate. Light intensity has a direct effect on algal growth through photosynthesis, on the same time nitrogen uptake can continue in the dark. However the classical Droop model cannot be used directly for varying light intensity. In order to account for this effect, an extension of the Droop model has recently been proposed by Bernard [18]. This approach consists in introducing light intensity, which will be denoted by I [µmol m -2 s -1 ], into the parameter µ max = µ max (I): µ(q, I) = µ max (I) 1 -q 0 q = µ max I I + K I 1 -q 0 q , with K I [µmol m -2 s -1 ] is the light half saturation coefficient. Note that in this case, the absorption function ρ also needs to depend on q as shown in [START_REF] Lehman | The assumptions and rationales of a computer model of phytoplankton population dynamics1[END_REF] ρ(s, q) = ρ max s s + K s q l -q q l -q 0 , with q l > q 0 .

Photosynthetic production and Photoinhibition -The Han model

The photosynthetic units (PSUs) are usually referred as a set of molecules including chlorophyll that capture photons to eventually oxidize a molecule of water (H 2 O), produce dioxygen (O 2 ) or generate a flux of energetic co-factors such as ATP or NADPH. The PSUs can be associated with two types of photosystems: photosystem-I (PSI) and photosystem-II (PSII) [START_REF] Han | Photosynthesis-irradiance response at physiological level: a mechanistic model[END_REF][START_REF] Golbeck | Structure, function and organization of the photosystem i reaction center complex[END_REF][START_REF] Sukenik | Light-saturated photosynthesis -limitation by electron transport or carbon fixation?[END_REF][START_REF] Wu | A model integrating fluid dynamics in photosynthesis and photoinhibition processes[END_REF] where PSII leads to a flux of electrons and PSI generates the energy-rich cell co-factors. Among these two systems, PSII is generally assumed to play the main role in the photosynthetic dynamics since it involves a specific molecule (D1) in the reaction centre of the photosystems that is easily damaged by an excess of energy. For large light intensities, there is a risk that some PSUs are Work in progress as of 7 th September, 2021 damaged, leading to reduced photosynthetic activity. This photoinhibition mechanism must be taken into account to accurately estimate photosynthetic production at high light intensity [START_REF] Brewer | Productivity of the ocean. present and past[END_REF][START_REF] Falkowski | The role of phytoplankton photosynthesis in global biogeochemical cycles[END_REF][START_REF] Khoo | Life cycle energy and co2 analysis of microalgae-to-biodiesel: Preliminary results and comparisons[END_REF][START_REF] Powles | Photoinhibition of photosynthesis induced by visible light[END_REF][START_REF] Long | Photoinhibition of photosynthesis in nature[END_REF]. Photoinhibition results in a decrease in photosynthesis rate in plants when they are exposed to high light intensity. In the last three decades, the understanding of photoinhibition mechanisms induced by photodamage to PSIIs has been largely improved motivated by the application of molecular biology [START_REF] Barber | Too much of a good thing: light can be bad for photosynthesis[END_REF][START_REF] Melis | Dynamics of photosynthetic membrane composition and function[END_REF][START_REF] Ohad | Mechanism of photoinhibition in vivo. a reversible light-induced conformational change of reaction center ii is related to an irreversible modification of the d1 protein[END_REF][START_REF] Sinclair | Target theory and the photoinactivation of photosystem ii[END_REF]. Therefore, quantitative representation of photoinhibition has received an increasing attention. As a consequence, number of models have been developed to represent and evaluate the effect of photoinhibition on photosynthesis [START_REF] Duarte | A mechanistic model of the effects of light and temperature on algal primary productivity[END_REF][START_REF] Duarte | Dynamic modelling of photosynthesis in marine and estuarine ecosystems[END_REF][START_REF] Eilers | Dynamic behaviour of a model for photosynthesis and photoinhibition[END_REF][START_REF] Han | Effect of photoinhibition on algal photosynthesis: a dynamic model[END_REF][START_REF] Han | A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii[END_REF][START_REF] Marshall | A mechanistic model of photoinhibition[END_REF][START_REF] Pahl-Wostl | Dyphora -a dynamic model for the rate of photosynthesis of algae[END_REF][START_REF] Zonneveld | Photoinhibition as affected by photoacclimation in phytoplankton: a model approach[END_REF]. Among all these models, the Han model [START_REF] Han | A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii[END_REF], which is initially inspired by the model of Eilers and Peeters [START_REF] Eilers | Dynamic behaviour of a model for photosynthesis and photoinhibition[END_REF], describes the effect of the light intensity and the dynamics of the reaction centres within the PSUs on microalgae growth. According to the Han model, the main contribution to photoinhibition is the damage of key proteins in PSUs. This model enables to describe complex photosynthetic processes only in terms of three possible physiological states of the PSUs, namely:

• A: open and ready to harvest a photon,

• B: closed while processing the absorbed photon energy,

• C: inhibited if several photons have been absorbed simultaneously.

The relations between these three states are schematically presented in Figure 1 Their evolution satisfies the following dynamical system

   Ȧ = -σ H IA + B τ , Ḃ = σ H IA -B τ + k r C -k d σ H IB, Ċ = -k r C + k d σ H IB. (1.1)
Here A, B and C are the relative frequencies of the three possible states with

A + B + C = 1, (1.2)
This model separates the dynamics into two parts:

• the photosynthetic production corresponds to the transition between open state A and closed state B. Excitation is assumed to occur at a rate of σ H I, with σ H [m 2 µmol -1 ] the effective cross-section of the PSUs, whereas deexcitation is assumed to occur at a rate of 1 τ , with τ [s] the turnover time of the electron transport chain. In particular, the photosynthesis rate is thus proportional to σ H IA.

Work in progress as of 7 th September, 2021

• the photo-inhibition mechanism occurs at high light intensity which corresponds to the transition from closed state B to inhibited state C. This process is assumed to occur at a rate of k d σ H I, with k d [-] a damage constant. Finally, the reverse transition from inhibited state C to closed state B accounts for the repair of damaged PSUs by enzymatic processes in the cell, a mechanism which is assumed to occur at a constant rate k r [s -1 ].

The system (1.1) can be then reduced to two equations by using (1.2)

Ȧ = -(σ H I + 1 τ )A + 1-C τ , Ċ = -(k r + k d σ H I)C + k d σ H I(1 -A), (1.3)
We then complete the system above with initial conditions

(A(0), C(0)) = (A 0 , C 0 ) ∈ {(x, y) ∈ R 2 + |x + y ∈ [0, 1]}.
The dynamics of the open state A reaches its steady state following a process whose speed is very high compared to the dynamics of the photoinhibition state C ). Note that we use an Euler Explicit scheme to solve the system (1.3) by using the parameters presented in Table 1.1. A log scale is also used for the time variable. is relatively small (see Table 1.1, where an example of possible values for the Han parameters is given). Since we usually focus on light variation at large time scale (larger than second) in real

k r 4.8 10 -4 [s -1 ] k d 2.99 10 -4 [-] τ 6.849 [s] σ H 0.0029 [m 2 µmol -1 ] k H 3.6467 10 -4 [-] I s 1500 [µmol m -2 s -1 ]
Table 1.1: Han model parameters taken from [START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF] life applications, we can then apply a slow-fast dynamics using singular perturbation theory [START_REF] Khalil | Nonlinear Systems[END_REF].

Work in progress as of 7 th September, 2021

More precisely, this consists in equating the first equation of (1.3) to zero and find the pseudo steady state of A as 1-C τ σ H I+1 . Replacing this into the second equation of (1.3), the previous two equations can finally be reduced to one equation, namely

Ċ = -α(I)C + β(I), (1.4)
where

α(I) = β(I) + k r , β(I) = k d τ (σ H I) 2 τ σ H I + 1 .
As mentioned above, the growth rate is proportional to σ H IA. Let us denote by k H [-] the growth rate coefficient such that the growth rate is defined by

µ(I) := k H σ H IA = k H σ H I τ σ H I + 1 (1 -C) = (1 -C)γ(I), (1.5)
where

γ(I) = k H σ H I τ σ H I + 1 .

Remark

The net growth rate is defined from the growth rate by removing the respiration rate R [d -1 ]:

µ(I) -R = -γ(I)C + ζ(I), ζ(I) = γ(I) -R. (1.6)
An interesting property of the Han model is that the open, closed and inhibited states can be computed analytically from (1.1) as a function of the light intensity I at steady state. For instance, the steady-state expression Ā corresponding to the open state A is given by

Ā = 1 k d kr τ (σ H I) 2 + τ σ H I + 1 .
Therefore, one can compute the growth rate at steady state of the system by

µ(I) = k H σ H I k d kr τ (σ H I) 2 + τ σ H I + 1
.

(1.7)

Remark

Note that (1.4) can be solved explicitly for a constant light flux I. Hence explicit computations can also be carried out in this case for the average growth rate or the productivity. However, a continuous time-varying signal I may rapidly increase the difficulty of solving (1.4).

Definition 1.2.1. A growth rate is said to be of Haldane type if it is of the form

µ(I) := µ max I I + µmax θ ( I I * -1) 2 , (1.8)
where θ is the initial slope of µ, µ max stands for the maximal value of µ and I * represents the optimal light intensity.

Work in progress as of 7 th September, 2021 Proof. Expanding (1.8), one has

µ(I) = θI I 2 I * 2 + ( θ µmax -2 I * )I + 1
.

Then it remains to identify the coefficients with (1.7). We find that:

θ = k H σ H , I * = k r k d τ σ 2 H , µ max = k H σ H τ σ H + 2 k d τ σ 2 H kr .
(1.9)

This concludes the proof.

Remark

The Haldane form highlights the significant parameters of the growth rate as illustrated by Figure 1.3. This so-called P I curve shows the maximum value of the growth rate µ max , the corresponding light intensity I * and the initial angle of the growth rate θ.

Hydrodynamical model

Research related to geophysical fluid dynamics often consists in understanding the coupled dynamics of atmosphere and ocean on Earth or on other planets. The analysis and modeling of these geophysical flows are usually complex and challenging especially for free surface flows. The main difficulties come not only from the discretization and the simulation of the Navier-Stokes equations, but also from the fact that the considered problems may involve dynamics associated with very different scales. Therefore models with reduced complexity but still capable of representing complex flows are required. Among all, the Saint-Venant system introduced in [START_REF] Barré De Saint-Venant | Théorie du mouvement non permanent des eaux avec applica-tions aux crues des rivières et à l'introduction des marées dans leur lit[END_REF] is a well known and efficient approximation of the Navier-Stokes system [START_REF] Stoker | Water Waves: The Mathematical Theory with Applications[END_REF][START_REF] Gerbeau | Derivation of viscous saint-venant system for laminar shallow water; numerical validation[END_REF][START_REF] Marche | Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects[END_REF][START_REF] Ferrari | A new two-dimensional shallow water model including pressure effects and slow varying bottom topography[END_REF][START_REF] Bouchut | Gravity driven shallow water models for arbitrary topography[END_REF] for a large class of problems such as dam break, flooding, debris flow, etc.

Work in progress as of 7 th September, 2021

The derivation of the Saint-Venant system from the Navier-Stokes equations is based on two main approximations namely:

• the vertical acceleration of the fluid can be neglected compared to the gravitational effects.

• the horizontal fluid velocity is approximated by its vertical average. One can see for instance in [START_REF] Gerbeau | Derivation of viscous saint-venant system for laminar shallow water; numerical validation[END_REF] for the derivation of this model.

There are two ways to describe the fluid namely the Eulerian description and the Lagrangian description. The Eulerian description focuses on the macroscopic aspect meaning that a control volume is defined as well as its related properties such as pressure, velocity or acceleration, whereas the Lagrangian description pay attention to the properties of individual fluid particles, such as their positions or velocities, described as a function of time. In this thesis, we are interested in the Lagrangian description of the flow to adapt to the microscopic behaviour of the photosynthetic process in a raceway pond.

In this way, we first introduce the smooth steady state solutions of the Saint-Venant equations. Such steady states are governed by the following partial differential equations:

∂ x (hu) = 0, (1.10)
∂ x (hu 2 + g h 2 2 ) = -gh∂ x z b , (1.11)
where h [m] is the water elevation, u [m s -1 ] is the horizontal averaged velocity of the water, the constant g [m s -2 ] stands for the gravitational acceleration, and z b [m] defines the topography.

The free surface η [m] is given by η := h + z b and the averaged discharge Q [m 2 s -1 ] is given by Q := hu. This system is presented in Figure 1.4. The z axis represents the vertical direction and Work in progress as of 7 th September, 2021

0 L z 0 x η(x) I s z b (x) h(x) u(x)

Remark

We assume that the solutions (h, u) are smooth such that we can carry out the following computation. Moreover, we focus only on the laminar regime of the flow.

Integrating (1.10), we get

hu = Q 0 , (1.12)
for a fixed positive constant Q 0 [m 2 s -1 ], which implies a constant discharge in space. Then (1.11) can be rewritten as

hu∂ x u + h∂ x gh + h∂ x gz b = 0. (1.13)

Remark

This equation is a nonlinear ordinary differential equation, some solutions in specific cases have been widely studied. For instance, by assuming h > 0 and Q 0 = 0, we obtain the steady state of a lake at rest (see [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF] and references therein), defined by

Q 0 = 0, h + z b = cst.
Let us assume that h is strictly positive. Dividing (1.13) by h and using (1.12) to eliminate u, we get

∂ x Q 2 0 2h 2 + g(h + z b ) = 0. This equation corresponds to Bernoulli's principle. Now let us consider two fixed constants h(0), z b (0) ∈ R. For all x ∈ [0, L], we obtain Q 2 0 2h 2 + g(h + z b ) = Q 2 0 2h 2 (0) + g(h(0) + z b (0)) =: M 0 ,
meaning that the topography z b satisfies

z b = M 0 g - Q 2 0 2gh 2 -h. (1.14)

Remark

Define the Froude number for the steady state by F r := u/ √ gh. The situation F r < 1 corresponds to the subcritical case (i.e. the flow regime is fluvial) while F r > 1 is to the supercritical case (i.e. the flow regime is torrential). In particular, the threshold value of h for F r = 1 is given by

h c := ( Q 2 0 g ) 1 3
.

Because of (1.14), h is the solution of a third order polynomial equation. The following result ensures that (1.14) has a unique solution.

Lemma 1.2.1 (Lemma 1 in [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF]). Given a smooth topography z b , if

h c + z b + Q 2 0 2gh 2 c - M 0 g < 0.
Work in progress as of 7 th September, 2021 there exists a unique positive smooth solution of (1.14) which satisfies the subcritical flow condition.

Let z(t) be the depth of a particle at time t in the raceway pond. We determine the Lagrangian trajectory of an algal cell which starts at a given position z(0) at time 0. From the incompressibility of the flow, one has ∇ • u = 0 with u = (u(x), w(x, z)). Here, u(x) is the horizontal velocity and w(x, z) is the vertical velocity. Note that u only depends on x since a laminar regime is assumed. This implies that

∂ x u + ∂ z w = 0.
(1.15)

Integrating (1.15) from z b to z gives:

0 = z z b ∂ x u(x) + ∂ z w(x, z) dz = ∂ x z z b u(x)dz + z z b ∂ z w(x, z)dz = ∂ x (z -z b )u(x) + w(x, z) -w(x, z b ) = (z -z b )∂ x u(x) -u(x)∂ x z b + w(x, z),
where we have used the kinematic condition at the bottom (i.e. w(x,

z b ) = u(x)∂ x z b ). It then follows from (1.14) that w(x, z) = ( M 0 g - 3u 2 (x) 2g -z)u (x).
The Lagrangian trajectory is characterized by the system

ẋ(t) ż(t) = u(x(t)) w(x(t), z(t))
.

Remark

Note that a Brownian motion can be included in these equations to take into account the fluid viscosity. In this case, the optimisation procedure requires a large set of simulations and an averaging strategy. Such questions exceed the scope of this thesis.

Remark

Since the raceway pond is a circuit basin, it is natural to impose h(0

) = h(L) (or z b (0) = z b (L)),
meaning that a space periodicity is assumed. One needs to account for this when dealing with the parameterization in the optimization part.

Light intensity modeling

The light intensity I plays an important role in algal growth, since it triggers photosynthesis. On the other hand, the position of the algae z(t) will influence the light perceived as well as the efficiency of the photosynthesis process. Therefore, the light intensity is the main connection which couples the hydrodynamic model and the physiological evolution of the algae. We will present in this section a well-known model to describe the vertical light attenuation.

Work in progress as of 7 th September, 2021

In most cases, the photobioreactors are illuminated from above by a light intensity denoted by I s hereafter. The subscript s emphasizes that I s is the light intensity at the illuminated surface, which is assumed to be a constant. It is often assumed for low-density cultures that all microalgae in the pond perceive the same light intensity as I s [START_REF] Jeon | Measurement of microalgal photosynthetic activity depending on light intensity and quality[END_REF]. However, in more general cases such as industrial applications and aquatic environments, a gradient of light intensity is observed when moving deeper into the culture medium. This is caused by light absorption and scattering by light-absorbing substances [START_REF] Bernard | Modelling of microalgae culture systems with applications to control and optimization[END_REF][START_REF] Borowitzka | Limits to growth[END_REF][START_REF] Kirk | Light and photosynthesis in aquatic ecosystems / John T.O. Kirk[END_REF]. In order to take into account this phenomenon, the Beer-Lambert law is often chosen as a first approximation to determine the light attenuation at any position in the medium. This law states that I satisfies

∂I ∂z = -εI,
where ε ≥ 0 [m -1 ] represents the light extinction coefficient and z ∈ [0, h] denotes the vertical distance from the illuminated surface to the position of the culture with h [m] the depth of the reactor. Since the microalgae culture is often assumed to be perfectly mixed, light extinction coefficient ε does not depend on the depth of the culture z. Therefore, we can integrate the previous equation to obtain:

I(z) := I s exp(-εz). (1.16)
The light extinction ε summarises the light absorption and diffusion in the system. In the case of a mono-culture, ε is mainly considered to be correlated to the biomass concentration X [START_REF] Kirk | Light and photosynthesis in aquatic ecosystems / John T.O. Kirk[END_REF]. In a first approximation, one can assume an affine relation between these parameters, i.e., ε(X)

:= α 0 X + α 1 , (1.17)
where α 0 > 0 [m 2 g -1 ] stands for the specific light extinction coefficient of the microalgae species and α 1 [m -1 ] defines the background turbidity due to all non-microalgae components i.e. suspended solids and dissolved colored material. Typical values of the coefficient α 0 for some microalgae species are given in [START_REF] Béchet | Modeling the effects of light and temperature on algae growth: State of the art and critical assessment for productivity prediction during outdoor cultivation[END_REF].

It is usually assumed that the biomass concentration X is homogeneous. However, in order to use (1.17), the density of the culture X must be small enough to guarantee that most of the photons are diffused at most once. For multi-diffusion regimes, the latter condition is generally not satisfied [START_REF] Fernández | A model for light distribution and average solar irradiance inside outdoor tubular photobioreactors for the microalgal mass culture[END_REF]. Hence, various empirical expressions have been developed to account for multiscattering, for instance, in [START_REF] Yun | Kinetic modeling of the light-dependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF] the authors have proposed the following expression:

ε(X) = β 0 X β 1 + X .
with β 0 and β 1 determined empirically.

Work in progress as of 7 th September, 2021

Remark

In this thesis, we focus only on light regime such that the light extinction can be described using (1.17).

As already mentioned, the light intensity I is the bridge to connect the hydrodynamical systems with the biological systems. In a raceway pond, it is often assumed that the photosynthetic units grow slowly such that the variations of biomass concentration X and background turbidity α 1 are negligible over one lap of the raceway. At such a time scale, these two quantities in (1.17) can be considered to be constant. Thus, the light extinction can be determined by measuring the light intensity at bottom of the reactor I b := I(h), so that we obtain for:

ε = 1 h ln( I s I b ).
From (1.16), one can compute the mean light intensity received by the algae culture

Ī = I s h 0 e -ε(X)z dz = I s ε(X) (1 -e -ε(X)h ).
The mean light intensity decreases with ε(X), which confirms the intuition that high biomass concentration X or background turbidity α 1 (see Equation (1.17)) leads to high attenuation and low value of the mean light.

Mixing device modeling

Except for keeping a constant motion in the raceway pond, the powered paddle wheel also modifies the elevation of algal cultures each time when they pass through this device. Studies have shown that this device mixes the algae position to prevent cell sedimentation, to ensure nutrient homogeneity and to influence light reception at the cell scale in the system [START_REF] Cuello | Comparison of continuous and day time only mixing on tetraselmis suecica (chlorophyta) in outdoor raceway ponds[END_REF][START_REF] Perner-Nochta | Simulations of light intensity variation in photobioreactors[END_REF]. However, modeling this complex device is challenging and involves stochastic behaviour when going through the device. In this thesis, we consider a generic mixing device and make a perfect mixing assumption, meaning that at each lap, the algae at the layer n 1 are supposed to be entirely transferred to the layer n 2 when passing through this mixing device. In this way, the mixing process can be modelled by a permutation matrix P ∈ P N , where P N denotes the set of permutation matrices of size N × N with N ∈ N. Denote also by σ ∈ S N the permutation corresponding to P where S N denotes the set of permutations of N elements. This mixing process is depicted schematically on an example in Figure 1.5.

Periodic Cauchy problem

In this section, we focus on the periodic Cauchy problem associated with the dynamics C (1.4). This will be useful to study periodic solutions of this biological model. Let us first consider the Cauchy problem associated with (1.4). Given I ∈ L 2 (0, T ; R + ) and an initial state

C 0 ∈ [0, 1]. Find C ∈ C(0, T ; [0, 1]) such that Ċ(t) = -α(I(t))C(t) + β(I(t)), t ∈ [0, T ] C(0) = C 0 (1.18)
The theorem below gives existence and uniqueness of a (weak) solution for this problem.

Work in progress as of 7 th September, 2021

0 L 0 L P z x 0 z 1 = z σ(4) z 2 = z σ(1)
z 3 = z σ(2) z 4 = z σ(3)
C(t) = C 0 + t 0 -α(I(s))C(s) + β(I(s))ds, (1.19
)

for all t ∈ [0, T ].
Proof. Since α(I) = k r + β(I), (1.4) can also be written as:

Ċ(t) = -k r C(t) -β(I(t))C(t) + β(I(t)).
Note that the light intensity I(t) ≥ 0 for all t ∈ [0, T ]. Since τ, σ H are all positive, then one has

β(I(t)) = k d τ (σ H I(t)) 2 τ σ H I(t) + 1 ≤ k d τ (σ H I(t)) 2 .
Consider now the mapping

Φ : C(0, T ; [0, 1]) → C(0, T ; [0, 1]) C → Φ(C)
where Φ(C) is defined for t ∈ [0, T ] by the Duhamel's formula:

Φ(C)(t) = e -krt C 0 + t 0 e -kr(t-s) -β(I(s))C(s) + β(I(s)) ds. (1.20)
Note first that Φ is well-defined: indeed since C ∈ C(0, T ; [0, 1]) and I ∈ L 2 (0, T ; R), s → -α(I(s))C(s) + β(I(s)) is Lebesgue integrable. Let us now prove that Φ is a contraction. Let C 1 and C 2 ∈ C(0, T ; [0, 1]). Since t ≥ s, then exp(-k r (t -s)) ≤ 1 which is a bounded function.

Work in progress as of 7 th September, 2021

Therefore, we obtain

Φ(C 2 ) -Φ(C 1 ) L ∞ (0,T ;[0,1]) = sup t∈[0,T ] |Φ(C 2 )(t) -Φ(C 1 )(t)| = sup t∈[0,T ] t 0 e -kr(t-s) β(I(s))(C 2 (s) -C 1 (s))ds ≤ k d τ σ 2 H sup t∈[0,T ] t 0 I 2 (s) |(C 2 (s) -C 1 (s))| ds ≤ k d τ σ 2 H C 2 -C 1 L ∞ (0,T ;[0,1]) T 0 I 2 (s)ds = k d τ σ 2 H C 2 -C 1 L ∞ (0,T ;[0,1]) I 2 L 2 (0,T ;R+) .
According to I L 2 (0,T ;R+) , two cases must be considered.

• If I L 2 (0,T ;R+) is small enough, the above inequality implies that Φ is a contraction. We deduce from the Banach fixed point theorem that Φ has a unique fixed-point, hence (1.18) admits a unique solution C ∈ C(0, T ; [0, 1]).

• If I L 2 (0,T ;R+) is not small, we consider the partition [0, T ] =

N -1 l=0 [T l , T l+1 ],
with T 0 = 0 and T N = T , such that I L 2 (T l ,T l+1 ;R) is small enough. Then by the Banach fixed-point theorem that Φ has a unique fixed point on each interval

[T l , T l+1 ] for all l ∈ {0, • • • , N -1}.
Finally, by applying Banach fixed-point of Φ, we remark that the right sides of (1.20) and (1.19) are almost everywhere differentiable and that their derivatives are almost everywhere equal. This concludes the proof.

We now consider a variant of the previous Cauchy problem, where the initial condition is unknown and must also satisfy the periodicity condition. Given I ∈ L 2 (0, T ; R + ), this problem reads:

Find (C 0 , C) ∈ [0, 1] × C(0, T ; [0, 1]) such that Ċ(t) = -α(I(t))C(t) + β(I(t)), t ∈ [0, T ] C(T ) = C 0 = C(0). (1.21)
To prove the existence and uniqueness of a (weak) solution of this problem, we need the following result.

Lemma 1.3.1. Given Z 0 ∈ [0, 1], let Z ∈ C(0, T ; [0, 1]) be the weak solution of

Ż(t) = -α(I(t))Z(t), t ∈ [0, T ] Z(T ) = Z 0 (1.22)
For t ∈ [0, T ]:

Z(t) 2 ≤ e -krt Z 0 2 . (1.23) Proof. Let t ∈ [0, T ],
as a weak solution of (1.22), the function Z is almost everywhere differentiable and a direct computation gives

d Z(t) 2 2 dt = 2 Z(t), -α(I(t))Z(t) ≤ -2k r Z(t) 2 2 , (1.24)
Work in progress as of 7 th September, 2021

where we have used the fact that α(I) = β(I) + k r , β(I) ≥ 0 and k r > 0. Defining f (t) := d Z(t)

2 2 dt + 2k r Z(t) 2
2 and multiplying both sides by e 2krt , we obtain:

de 2krt Z(t) 2 2 dt = e 2krt f (t),
which gives by integration over [0, t] that

Z(t) 2 2 = e -2krt Z 0 2 2 +
t 0 e 2kr(s-t) f (s)ds.

Then using (1.24), f (t) ≤ 0. This concludes the proof.

We can now state an existence and uniqueness result in the periodic case.

Theorem 1.3.2. There exists a unique couple

(C 0 , C) ∈ [0, 1] × C(0, T ; [0, 1]) satisfying C(t) = C 0 + t 0 -α(I(s))C(s) + β(I(s))ds, C(T ) = C 0 (1.25) for all t ∈ [0, T ].
Proof. Introduce the mapping Φ : [0, 1] → [0, 1] defined by

Φ(C 0 ) := C(T ),
where C is the weak solution of (1.18) obtained in Theorem 1.

3.1. Given C 1 0 ∈ [0, 1] and C 2 0 ∈ [0, 1], define Z 0 = C 2 0 -C 1 0 and Z(t) = C 2 (t) -C 1 (t)
, where C 1 and C 2 are the weak solutions obtained by Theorem 1.3.1 with C 0 = C 1 0 and C 0 = C 2 0 , respectively. Subtracting the corresponding weak representations (1. [START_REF] Bertsimas | Introduction to Linear Optimization[END_REF], we see that Z satisfies the assumptions of Lemma 1.3.1 such that (1.23) holds. As a consequence,

C 2 (T ) -C 1 (T ) 2 ≤ e -krT C 2 0 -C 1 0 2 ,
which implies that Φ is a contraction. Applying Banach fixed-point theorem, it follows that there exists a unique C 0 ∈ [0, 1] such that Φ(C 0 ) = C 0 . The corresponding weak solution C of (1.18) satisfies (1.25).

This result allows us to study the case when C is assumed to be periodic (see Subsection 4.3.2).

Adjoint methods for optimization problem

In this section, we briefly recall the basics of the adjoint method which will be used in Chapter 4 to solve the constrained optimization problem. The adjoint method originates from the theory of Lagrange multipliers in optimization which is presented in the following. Consider the problem optimize f (x), subject to g(x) = 0.

where f, g are functions

Ω ∈ R of class C 1 with Ω an open subset of R n .
Work in progress as of 7 th September, 2021

Adjoint methods for optimization problem

Theorem 1.4.1. If x * is a local minimum or local maximum of f subject to the constraint g = 0 and if ∇g(x * ) = 0, then there exists λ ∈ R such that the following system of equation is satisfied by x * and λ:

∇f (x * ) + λ∇g(x * ) = 0, g(x) = 0.
λ is so-called the Lagrange multiplier.

We refer to [START_REF] Ekeland | Convex Analysis and Variational Problems[END_REF]Chapter 3,Section 3] for more details about this theorem. The adjoint method is an extension of this theorem in the framework of optimal control and optimization. In this way, the variable x is the union of a state variable x and a control state u. The constraint g(x, u) = 0 gives the state x in terms of the control u. In such cases, the Lagrange multiplier λ is often called the adjoint state. This approach was first developed by Pontryagin [START_REF] Bittner | boltyanskii, r. v. gamkrelidze, e. f. mishechenko, the mathematical theory of optimal processes. viii + 360 s. new york/london 1962. john wiley & sons. preis 90[END_REF] to ordinary differential equations and by Lions [START_REF] Lions | Optimal Control of Systems Governed by Partial Differential Equations[END_REF] to partial differential equations.

We illustrate this method for the time-dependent problem which are often treated by semidiscretization on the spatial variable of a partial differential equation. Consider the following problem:

optimize T 0 f (x, a, t), subject to h(x, ẋ, a, t) = 0, g(x(0), a) = 0.
where a represents a vector of unknown parameters, x stands for a (often vector) function of time, h(x, ẋ, a, t) = 0 is an ordinary differential equation in implicit form and g(x(0), a) = 0 is the initial condition which may be a function of some of the unknown parameters. Note that the ordinary differential equation h may be the result of a spatial semi-discretization of a partial differential equation.

The gradient-based optimization algorithm requires to compute the total derivative

d a T 0 f (x, a, t)dt = T 0 (∂ x f d a x + ∂ a f ) dt.
where d a x denotes the total derivative of x with respect to the unknown parameters a which is challenging to compute in most cases. To overcome this difficulty, we first introduce the Lagrangian corresponding to the optimization problem:

L = T 0 f (x, a, t) + λ 1 h(x, ẋ, a, t)dt + λ 2 g(x(0), a),
where λ 1 , λ 2 are two Lagrange multipliers with λ 1 a function of time. Taking then the total derivative of the Lagrangian

d a L = T 0 ∂ x f d a x + ∂ a f + λ 1 (∂ x hd a x + ∂ ẋhd a ẋ + ∂h) dt + λ 2 ∂ x(0) gd a x + ∂ a g . (1.26)
Integrating by parts helps us to eliminate d a ẋ as

T 0 λ 1 ∂ ẋhd a ẋdt = [λ 1 ∂ ẋhd a x] T 0 - T 0 λ1 ∂ a h + λ 1 d t ∂ ẋh d a xdt.
Work in progress as of 7 th September, 2021

The next step consists in substituting the previous result into (1.26)

d a L = T 0 ∂ x f + λ 1 (∂ x h -d t ∂ ẋh) -λ1 ∂ ẋh d a x + ∂ a f + λ 1 ∂ a h + λ 1 ∂ ẋhd a x| T + (-λ 1 ∂ ẋh + λ 2 ∂ x(0) g)| 0 d a x(0) + λ 2 ∂ a g.
Since the two constraints h = 0 and g = 0 are always satisfied by construction, then we can set λ 1 (T ) = 0 and λ 2 = λ 1 ∂ ẋh| 0 ∂ -1 x(0) g. Finally we can avoid computing d a x for all t > 0 by setting

∂ x f + λ 1 (∂ x h -d t ∂ ẋh) -λ1 ∂ ẋh = 0,
one finds the total derivative as

d a T 0 f (x, a, t) = T 0 ∂ a f + λ 1 ∂ a hdt + λ 1 ∂ ẋh| 0 ∂ -1
x(0) g∂ a g.

Summary

In this section, we give a general summary of each contribution. This thesis can be separated into three parts:

• In Chapter 2, we focus on the optimal conditions to maximize the productivity in photobioreactors. More precisely, we investigate the influence of the light extinction and the background turbidity of the medium on growth and productivity. In this way, a general light attenuation function relying on biomass concentration is considered which also accounts for the background turbidity of the system. The concept of optical productivity is introduced to better understand the influence of the optical depth in the system dynamics. A global optimal optical depth is determined which depends only on the model settings.

From a biological point of view, this optimal condition consists in canceling the net growth rate of the algae at the bottom of the reactors. This optimal condition can then be used to characterize the optimization of the surface biomass productivity in some specific cases, whereas an asymptotic behaviour has been observed in more general case. In practise, this optimal condition helps to establish a relation between the algal biomass concentration and the reactor depth. We develop a nonlinear controller and prove the global asymptotic stability of the biomass concentration towards the desired optimal value. This part of the work corresponds to a submitted journal article [START_REF] Bernard | Optimal optical conditions for Microalgal production in photobioreactors[END_REF].

• Chapter 3 deals with a general class of problem where N resources are distributed to N activities, each activity then uses the assigned resource to evolve during a given time T > 0 after which the resources will be re-distributed. This problem is part of a large class of allocation problems, with the particularity that it includes a dynamical system. In such problems, the goal is to find the best allocation strategies to optimize the cost or the benefit of the system. We apply this theory to the industrial microalgal raceway where a mixing device, such as a paddle wheel, is considered to control the rearrangement of the depth of the algae cultures hence the light perceived at each lap. Non trivial permutation strategies are obtained with a significant increase in the algal growth. This part of the work corresponds to the theory we progressively developed in a conference proceeding [START_REF] Bernard | Optimizing microalgal productivity in raceway ponds through a controlled mixing device[END_REF] and a submitted journal article [START_REF] Bernard | Optimization of mixing strategy in microalgal raceway ponds[END_REF].

• Chapter 4 is dedicated to the analysis of the influence of the shape of the topography on al-Work in progress as of 7 th September, 2021 1.5. Summary gal growth in raceway ponds. A coupled biological-hydrodynamical model is considered to describe this complex dynamical system. For this problem, we apply an adjoint-based optimization scheme which includes the constraints associated with the shallow water regime.

On the contrary to a widespread belief, the flat topography is proved to be the optimal topography in a periodic regime, whereas non-trivial topographies can be obtained in other contexts, e.g., when the periodic assumption is removed or when an extra mixing strategy is included in the model. Note that in the examples considered in our numerical tests, such topographies slightly improved the biomass production, whereas a combination of the mixing strategies with the topographies can enhance this result. This chapter corresponds to the conference proceedings [START_REF] Bernard | Controlling the bottom topography of a microalgal pond to optimize productivity[END_REF][START_REF] Bernard | Mixing strategies combined with shape design to enhance productivity of a raceway pond[END_REF] and a submitted journal article [START_REF] Bernard | Shape optimization of a microalgal raceway to enhance productivity[END_REF].

Optimal condition for algal production

The biomass production in photobioreactors depends on the photosynthesis efficiency of the microalgae which mainly relies on the light radiation in the system. As shown in Subsection 1.2.3, the light intensity decreases rapidly along the photobioreactor depth. This phenomenon can be described by a Beer-Lambert law (1.16). Although a linear dependence of the biomass concentration has often been used in literature to describe the light extinction (as shown in (1.17)), experimental studies show that nonlinear behaviour can be obtained in some cases (hence a general nonlinear formulation must be considered). For instance, Morel calibrates in [START_REF] Morel | Optical modeling of the upper ocean in relation to its biogenous matter content (case i waters)[END_REF] the optical behaviour in the attenuation process of radiant energy based on oceanic water for which phytoplankton and their derivative play a predominant role in determining their optical properties. The background turbidity of the system may also have an influence on the average growth rate of the algae [START_REF] Martínez | Theory of turbid microalgae cultures[END_REF]. These two parameters are considered here, in order to describe the light extinction in photobioreactors. In the quest for optimising areal biomass productivity, authors in [START_REF] Masci | Microalgal biomass surface productivity optimization based on a photobioreactor model[END_REF] have found an optimal condition by using the Droop description (Subsection 1.2.1) of algal growth rate and a linear light extinction function. This condition expresses that respiration rate compensates exactly the algal growth rate at the bottom of the reactor. In Chapter 2, we extend [START_REF] Masci | Microalgal biomass surface productivity optimization based on a photobioreactor model[END_REF] by using a Haldane-type description (1.8) combined with a more general light extinction function: ε(X) := α 0 (s)X s + α 1 , where 0 < s ≤ 1, α 0 (s) > 0 stands for the specific light extinction coefficient and α 1 defines the background turbidity of the medium due to dissolved turbidity, bacteria, particulate matter and other microalgae.

In the first part, we introduce the concept of optical productivity P as the product between the average growth rate μ -R with the optical depth Y := ε(X)h (see (2.8)) and provide the following result to determine the optimal optical depth Y opt : Theorem 1.5.1. For a given light intensity I s , there exists an optimum Y opt to maximize the optical productivity P which satisfies µ (I(Y opt )) = R. This optimum can be computed explicitly as a function of the growth rate at the surface µ(I s ):

Y opt =            ln 2IsRµmax θI * 2 µmax-R+ 2Rµmax θI * -(µmax-R)(µmax-R+ 4Rµmax θI * ) , µ(I s ) > R, ln 2Is Rµmax θI * 2 µmax-R+ 2Rµmax θI * + (µmax-R)(µmax-R+ 4Rµmax θI * ) , µ(I s ) ≤ R,
where µ max , θ, I * are defined in Definition 1.2.1.

Work in progress as of 7 th September, 2021

The optimal optical depth Y opt represents the case where the growth rate at the bottom of the reactor equals the respiration rate R. Since the value of Y opt only depends on the choice of the model parameter (i.e. I s , R, µ max , θ, I * ), one can use it as a criterion to evaluate the optimum for different biomass concentrations or different reactor depths. Some results can be deduced directly from this theorem in particular one concerning the productivity Π (defined in (2.6)) with a fixed biomass concentration X.

Corollary 1.5.1. For a given biomass concentration X 1 , there exists a unique h 1 satisfied ε(X 1 )h 1 = Y opt which maximizes the productivity Π(X 1 , •).

On the contrary to the previous case, dealing with a fixed reactor depth h is more tricky since the light extinction function plays an important role. More precisely, we have found the following result.

Theorem 1.5.2. If α 1 > 0, there is no global optimum for the productivity

Π(•, •) in R + × R + .
We then define a sequence (X n , h n ) n>0 to study the asymptotic behaviour in this situation. For a given X 0 , this sequence is defined by:

h n = Y opt ε(X n-1 ) , X n := argmax X∈R+ Π(X, h n ).
Using this sequence, one has the following result on the productivity Π:

Theorem 1.5.3. The productivity Π for the sequence (X n , h n ) n>0 is given by

lim n→∞ Π(X n , h n ) = P (Yopt) α0 , s = 1, +∞, s < 1.
In practise, the reactor depth h is always a fixed constraint (e.g., h = 0.4 m for raceway ponds and h = 1 mm for biofilm reactors). In this way, choosing a biomass concentration to cancel the net growth rate at bottom may usually not be an optimal strategy to maximize the productivity Π. As shown in Figure 1.6, the blue point corresponds to the case when the net growth rate at the bottom is zero and the red point corresponds to the optimal biomass concentration for this given reactor depth h. In particular, the choice of such biomass concentration can have a huge impact on the productivity as shown on the right figure. In this case, one needs to have the Work in progress as of 7 th September, 2021 1.5. Summary specific formulation of the average growth rate μ (or the productivity Π) to locate the optimal biomass concentration.

Optimal mixing strategy for algal production

The classical resource allocation problem consists in finding an optimal strategy to assign a number of resources to the different activities. Due to its simple structure, this problem is encountered in a number of applications, for instance load scheduling [START_REF] Zhang | A model predictive control approach to the periodic implementation of the solutions of the optimal dynamic resource allocation problem[END_REF], manufacturing [START_REF] Zhang | A new manufacturing resource allocation method for supply chain optimization using extended genetic algorithm[END_REF], portfolio selection [START_REF] Hennessy | The use of archimedean copulas to model portfolio allocations[END_REF] and computational biology [START_REF] Akhmetzhanov | Optimal life-history strategies in seasonal consumer-resource dynamics[END_REF]. The optimization of these complex systems is challenging and the available works in the literature, especially when dealing with dynamics, are still rare.

In this way, we focus on the following problem: Given a period T , and initial time T 0 and a sequence (T k ) k∈N , with T k = kT + T 0 , we consider the following resource allocation problem: Consider N resources denoted by (I n ) N n=1 ∈ R N which can be allocated to N activities denoted by (x n ) N n=1 where x n consists of a real function of time. On a time interval [T k , T k+1 ), each activity uses the assigned resource and evolves according to a linear dynamics

ẋn = -α(I n )x n + β(I n ),
where α : R → R + and β : R → R + are given. At time T k+1 , the resources is re-assigned, meaning that x(T k+1 ) = P x(T - k+1 ) for some permutation matrix P ∈ P N . In order to evaluate the quality of the mixing associated with P , we define the benefit attached to each time period

[T k , T k+1 ) by f k := u, 1 T T k+1 T k x(t)dt ,
with u ∈ R N an arbitrary vector. Then the average benefit after K operations is given by

1 K K k=0 f k .
The first challenging issue that appears at this step is related to the periodicity of the permutation matrix P . Since we are interested in the average benefit for K operations, then it is natural to assume the periodicity on the system, meaning that x(T K ) = x(T 0 ). This assumption is very useful when it comes to e.g., crop harvesting or scheduling of appliances. A natural choice for K would be the order of the permutation associated with P . Indeed, in this case K is the minimal number of re-assignments required to recover the initial allocation. However, different permutation matrices may be associated with different order K, and some of these orders K can be very large such that computations will be time consuming and memory resource demanding. The following result helps us to overcome this difficulty. Theorem 1.5.4. (x(T k )) k∈N is a constant sequence and for all k ∈ N, we have

x(T k ) = (I N -P D) -1 P v.
This result shows that every KT -periodic evolution will actually be T -periodic. This property is decisive to formulate an optimization problem independent of K. In addition, the computations to solve the optimization problem will be reduced, since the CPU time required to assess the quality of a permutation will not depend on its order. In this way, let us consider the functional deduced from the average benefit as

J(P ) := u, (I N -P D) -1 P v .
Here lies the second challenge. Since the cardinal of the permutation set is N !, this problem Work in progress as of 7 th September, 2021 cannot be tackled in realistic cases where large values of N must be considered, e.g., to keep a good numerical accuracy. To overcome this difficulty, we propose an approximation of this problem whose optimum can be determined explicitly. For this purpose, we expand the functional J(P ) as follows u, (I -P D)

-1 P v = +∞ l=0 u, (P D) l P v = u, P v + +∞ l=1 u, (P D) l P v ,
and consider as an approximation of J(P ) the first term of this series, namely J approx (P ) := u, P v .

For this sub-problem, the optimal solution can be determined explicitly by using the following result. [START_REF] Borowitzka | Limits to growth[END_REF] and P + , P -∈ P N , the corresponding permutation matrices. Then P + = argmax P ∈P N J approx (P ), P -= argmin P ∈P N J approx (P ).

Lemma 1.5.1. Let σ + , σ -∈ S N such that v σ+(1) ≤ v σ+(2) • • • ≤ v σ+(N ) and v σ-(N ) ≤ v σ-(N -1) ≤ • • • ≤ v σ-
In other words, this result states that once u and v are given, P + will be the matrix which associates the largest coefficient of u with the largest coefficient of v, the second largest coefficient with the second largest, and so on. In the same way, P -is the matrix which associates the largest coefficient of u with the smallest coefficient of v, the second largest coefficient with the second smallest, and so on.

We then provide the next criterion to evaluate this optimal strategy corresponding to this sub-problem.

Theorem 1.5.5. Assume that u and v have positive entries and define

φ(m 1 ) := 1 s m 1 2 +∞ l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1 , (1.27)
where s and F + , F -are independent of the optimum of J(P ) and can be pre-computed explicitly (see (3.12) and (3.13) in Chapter 3 Section 3.2.2), d max := max n=1,...,N (d n ) and d min := min n=1,...,N (d n ). Assume that:

max m1≥2 φ(m 1 ) ≤ 1.
Then the problem max P ∈P N u, (I -P D) -1 P v (resp. min P ∈P N u, (I -P D) -1 P v ) and the problem max P ∈P N u, P v (resp. min P ∈P N u, P v ) have the same solution.

This periodic dynamical resource allocation theory is then applied to the optimization of a mixing strategy to enhance the growth rate in a microalgal raceway system. In [START_REF] Chiaramonti | Review of energy balance in raceway ponds for microalgae cultivation: Re-thinking a traditional system is possible[END_REF], authors have shown that the paddle wheel, which set the hydrodynamic system in motion in raceway ponds, homogenises the medium, ensures an equidistribution of the nutrients and guarantees that each cell will have regularly access to the light. In the present work, the paddle wheel is identified to a mixing device and modeled by a permutation. We then focus on the influence of the mixing strategy on the algal productivity in a flat raceway system with a constant average velocity for which the above resource allocation setting applies. Non-trivial optimal strategies have been observed, see, e.g., Figure 1.7 where the optimal matrix is represented by its stencil.

Work in progress as of 7 th September, 2021 As for the quality of the sub-problem and the criterion, one can see in Figure 1.8 that for large values of time duration T , the optimum approximation almost always coincides with the true optimum (purple stars). Nevertheless, we observe that the criterion (1.27) becomes less The red surface is obtained with P max and the blue surface is obtained with P + . The purple stars represent the cases where P max = P + or, in case of multiple solution, μN (P max ) = μN (P + ). The green circles represent the cases where the criterion (1.27) is satisfied. efficient for larger N in the sense that it is not satisfied even in the cases where two solutions coincide (green circles).

Optimal topography for algal production

Studies on the shape of the topography (or the bathymetry) have always been a challenging problem, since this problem usually involves the free-surface incompressible Navier-Stokes system.

Work in progress as of 7 th September, 2021

In particular, explicit computation some time may be difficult to provide due to the complexity of the chosen model. On the other hand, large computational resources are often needed to provide numerical simulations and experimental results. The goal of Chapter 4 is to investigate the influence of the topography of the raceway system. More precisely, we base on a common belief that some specific topographies can bring more light to the algae (at the lower part of the raceway), since they are closer to the surface when reaching the peak of these topographies. In this way, this chapter aims at answering the following question: Do non-flat topographies enable to increase the light perceived at the lower part of the raceway?

The first challenge to consider is to answer this question coming from the modeling aspect of this complex system. Indeed, this algal raceway system contains both a macroscopic aspect -the hydrodynamical behaviour of the fluid transporting the algae culture and a microscopic aspect -the photosynthesis within the chlorophyll cells. Therefore, finding a model which takes into account these two aspects is challenging. To overcome it, we limit ourselves to the 1D case and consider the Saint-Venant model (1.10)- (1.11) to describe the hydrodynamical behaviour and the Han model (1.1) to describe the photosynthetic process. The crucial element to couple these two models is the light intensity perceived by the algae culture which is described by the Beer-Lambert law (1.16). More precisely, we use the Saint-Venant model to compute the Lagrangian trajectories of the algae cells, which enables us to determine their depth. The trajectory is then used in the Beer-Lambert law to estimate the perceived light intensities . Finally, the Han model relates the light to the algal growth dynamics2 . Facing at this coupled model, the second challenge rises up as: How to optimize concretely the topography?

In a first study [START_REF] Bernard | Controlling the bottom topography of a microalgal pond to optimize productivity[END_REF], we consider one lap of the algal movement in the raceway pond and use the Lagrangian trajectory as an optimization constraint. For the sake of simplicity, we illustrate here the optimization problem with a single algal culture: where w stands for the vertical velocity of the algae cell and a is a parameter that encodes the topography and is supposed to be optimized. The general form of this problem can be found in Chapter 4. We then apply the adjoint-based method described in Section 1.4 to solve this optimization problem. Figure 1.9 represents the optimal topography obtained using a truncated Fourier series parameterization. The reason using this parameterization is presented in Subsection 4.4.2. The optimization procedure starts with a flat topography and ends up with a non-trivial topography as shown in this figure .   In a further step, a second analysis is given in [START_REF] Bernard | Shape optimization of a microalgal raceway to enhance productivity[END_REF] where we reduce the optimization constraints to a single constraint related to C by using the special form of the Lagrangian trajectory in the Saint-Venant description. The optimization problem for a single algal culture in this case

max a∈R N 1 V L 0 -γ(I(x, z; a))C + ζ(I(x, z; a))h(x; a)dx, C = -α(I(x, z; a))C + β(I(x, z; a)) Q 0 h(x; a).
This simplification consists in providing a time free formulation of the Lagrangian trajectories of the algal cells. It reduces the computational cost of the reconstruction of the hydrodynamical behaviour of the fluid since high order numerical schemes are often required to compute the Lagrangian trajectory as a function of time.

Another improvement with respect to the results presented in [START_REF] Bernard | Controlling the bottom topography of a microalgal pond to optimize productivity[END_REF] is related to the periodic behaviour of the biological state C which is a desirable property for industrial production. In this case, one can prove directly that the flat topography is a critical point for the previous optimization problem as stated in the following theorem. Theorem 1.5.6. Assume the volume of the system is fixed, then flat topography cancels the gradient of the net average growth rate (4.25) (defined in Chapter 4).

Note that this result is true only in the case where C is periodic. The numerical experiments show that flat topography is actually optimal as shown in Figure 1.10 where we use a random initial topography to proceed the optimization procedure.

In the last part of this work, we consider the case where depth can vary, so that the volume is also optimized. In this case, the biomass concentration X can no longer be considered as a constant. Therefore we apply the optimal condition found in Chapter 2 to established a relation between the biomass concentration X with the volume of the system V . In this framework, maximizing areal productivity is a relevant target. Productivity per unit of surface for a given biomass concentration X is given by:

Π := μX V S ,
Unlike the previous theorem, the flat topography does not cancel the gradient ∇Π.

Work in progress as of 7 th September, 2021 This result shows that no optimal solution can be found in this case.

In a third work [START_REF] Bernard | Mixing strategies combined with shape design to enhance productivity of a raceway pond[END_REF], we are interested in combining the mixing strategies presented in Chapter 3 with the topographies. The goal is to investigate numerically the optimal shape of the topographies when including permutation strategies. The optimization problem in this case reads max

P ∈P N max a∈R N 1 V Nz i=1 L 0 -γ(I(x, z i ; a))C P i + ζ(I(x, z i ; a))h(x; a)dx, C P i = -α(I(x, z i ; a))C P i + β(I(x, z i ; a)) Q 0 h(x; a).
Since N z ! permutation strategies needs to be tested in combination with the optimization of the topography, one can only test with a small number of N z .

In this case, the periodic result in [START_REF] Bernard | Controlling the bottom topography of a microalgal pond to optimize productivity[END_REF] is no longer valid since an extra permutation matrix is added into the optimization problem. Moreover, we have observed in the variable volume case that the optimization procedure stops because of the violation of the minimal water depth constraint (in practice 5 cm). Indeed, it appears that a small volume (or in other words a small water elevation) is better for increasing the productivity. This result is similar to the one found in Theorem 2.3.3 which shows that the productivity is increasing when considering high level of concentration and small water depth.

Work in progress as of 7 th September, 2021

Chapter 2

Optimal optical conditions for algal production in photobioreactors

Introduction

Most of the time, the light attenuation is considered to be described by a Beer-Lambert law where the light extinction rate varies with the process type and algal concentration. Some studies have more accurately represented the way light is attenuated in the process, especially to deal with the very dense multiscattering medium where a photon can be scattered several times before being eventually absorbed [START_REF] Morel | Optical modeling of the upper ocean in relation to its biogenous matter content (case i waters)[END_REF]. The influence of the background turbidity and the reactor depth on the average growth rate of the algae have also been considered in [START_REF] Martínez | Theory of turbid microalgae cultures[END_REF]. On the other hand, a range of biological models with different complexities describe various aspects of the growth of microalgae considering different mechanisms. For instance, authors in [START_REF] Masci | Microalgal biomass surface productivity optimization based on a photobioreactor model[END_REF] have used the Droop model [START_REF] Droop | Vitamin b12 and marine ecology. iv. the kinetics of uptake, growth and inhibition in monochrysis lutheri[END_REF][START_REF] Droop | 25 years of algal growth kinetics a personal view[END_REF] combining with the growth rate description in [START_REF] Bernard | A photobioreactor model in nitrogen limited conditions[END_REF] to give an optimal condition to maximize the surface biomass productivity. This condition consists in cancelling the net growth rate at bottom of the reactor and is much more convenient in practice to find the relation between the biomass concentration and the reactor depth, for instance authors in [START_REF] Bernard | Shape optimization of a microalgal raceway to enhance productivity[END_REF] have used this condition to study the algal growth in a raceway pond with a varying volume. Optimization of the microalgal productivity has also been studied intensively in the last decade [START_REF] Grognard | Optimal strategies for biomass productivity maximization in a photobioreactor using natural light[END_REF][START_REF] Bernard | Optimization of mixing strategy in microalgal raceway ponds[END_REF][START_REF] De La Hoz | Optimization of microalgal productivity using an adaptive, non-linear model based strategy[END_REF][START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF][START_REF] Cornet | Calculation of optimal design and ideal productivities of volumetrically lightened photobioreactors using the constructal approach[END_REF].

On top of light extinction and water depth which have been already deeply studied, in this chapter, we focus on understanding how the productivity in photobioreactors is also influenced by the background turbidity of the reactors. Our first contribution was to extend the work in [START_REF] Masci | Microalgal biomass surface productivity optimization based on a photobioreactor model[END_REF] by choosing a more realistic description of the algal growth dealing with photoinhibition. Our second contribution consisted in considering a general biomass dependent light extinction function accounting for the background turbidity of the system. The concept of optical depth productivity is introduced and a condition is derived on the optical depth for globally maximizing productivity. This optimum corresponds to the compensation condition, where the respiration compensates exactly the algal growth rate at the bottom of the reactors. We then use this optimal condition to characterize the optimization of the surface biomass productivity depending on the minimum achievable water depth. When the light extinction rate is affine with respect to the algal biomass, an upper limit to the productivity (obtained for an infinitely small depth) is given. A nonlinear controller is given and is proved to stabilize the evolution of the biomass towards the optimal desired value. The optimal behaviours are illustrated in different cases by numerical 29 experiments.

This chapter is organised as follows. In Section 2.2, we define the key concepts such as average growth rate and light distribution. We then study the optimization problem in Section 2.3. More precisely we investigate the global behaviour of the optical depth productivity and the optimal condition in Subsection 2.3.1. The optimal biomass concentration for a given reactor depth to maximize the surface biomass productivity is investigated in Subsection 2.3.2. A nonlinear controller is then introduced in Section 2.4 to stabilize the biomass concentration towards its optimal value. We illustrate and discuss the behaviour of the optima in different cases by some numerical experiments in Section 2.5.

Description of the model

For a given light intensity I, the growth rate of microalgae is defined by a Haldane-type description like one presented in Definition 1.2.1, which is recalled here:

µ(I) := µ max I I + µmax θ ( I I * -1) 2 , (2.1)
where θ is the initial slope of µ, µ max denotes the maximum value of µ and I * represents the optimal light intensity. This description results from a mechanical consideration of the light harvesting dynamics represented by the Han system [START_REF] Han | A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii[END_REF] (see Subsection 1.2.1 for more details). The light attenuation is described by a Beer-Lambert law

I(X, z) := I s exp ε(X)z , (2.2)
where X represents the biomass concentration, z ∈ [-h, 0] denotes the vertical position of the algae cells with h the depth and I s is the light intensity at the reactor surface. The light extinction ε, which summarises the light absorption and diffusion, is considered to be correlated to the biomass concentration X ε(X)

:= α 0 (s)X s + α 1 , (2.3)
where 0 < s ≤ 1, α 0 (s) > 0 stands for the specific light extinction coefficient of the microalgae species. It depends on the parameter s. The background turbidity α 1 is due to all non-microalgae components i.e. suspended solids and dissolved colored material. The dependence of s in α 0 will be omitted hereafter when no confusion may occur.

From (2.2) one can compute the mean light intensity received by the algae culture

Ī = I s 0 -h e ε(X)z dz = I s ε(X) 1 -e -ε(X)h .
This quantity is a decreasing function of ε(X), which confirms the intuition that a higher biomass concentration or a higher background turbidity leads to lower mean light received in the reactor, due to stronger light attenuation effect.

Replacing I by (2.2) in (2.1), one can see that the growth rate varies with depth of the reactor. Lower growth rate in the upper part of the reactor results from the photo-inhibition caused by high light density. Similarly, the growth rate is weak in the lower part of the reactor because of Work in progress as of 7 th September, 2021 low light intensity. The mean growth rate in the reactor is defined by

μ := 1 h 0 -h µ(I(X, z))dz. (2.4)
Applying then a change of the variable y = ε(X)z, it can be written as

μ := 1 ε(X)h ε(X)h 0 µ(I(-y))dy, (2.5)
so that the mean growth rate depends on the optical depth ε(X)h. This quantity is denoted by Y [-] hereafter. In this case, the average growth rate (2.5) can also be written as a function of Y (i.e. μ(X, h) = μ(Y )). Our aim is to optimize the surface biomass productivity (units:

g • m -2 • d -1
) which is defined by

Π := (μ -R)Xh. (2.6)

Remark

The evolution of the biomass concentration X is given by

Ẋ = (μ -R -D)X, (2.7)
where R is the respiration rate and D denotes the reactor dilution rate. At equilibrium, the biomass surface productivity Π is the product between dilution rate (D = μ -R) and surface biomass Xh.

Note that a nonlinear controller for D is introduced in section 2.4 to stabilize (2.7) to the value of X optimizing productivity.

Analysis of the optimal productivity

In this section, we investigate the optimization problem associated with the productivity Π. Note that the biomass concentration X and the depth h are both defined in R + .

Global optimality condition

First of all, let us define the optical depth productivity1 (units: d -1 ) by

P := (μ -R)Y. (2.8)

Remark

According to the definition of the optical depth productivity (2.8), a thin reactor with high biomass concentration is equivalent to a deep reactor with low biomass concentration as long as they both share the same optical depth Y . A low value of Y means a weaker photon harvesting since less light is absorbed. On the reverse, a too high Y means that light hardly reaches the bottom of the reactor, with an area where respiration (loss of CO 2 ) exceeds growth (fixation of CO 2 ). Hence, it is necessary to determine the optimal Y value maximizing the efficiency of the productivity P .

Theorem 2.3.1. Given a surface light intensity I s , there exists an optimum Y opt which maximizes the optical productivity P . This value satisfies µ (I(Y opt )) = R and can be computed explicitly according to the growth rate at the surface µ(I s ):

Y opt =            ln 2IsRµmax θI * 2 µmax-R+ 2Rµmax θI * -(µmax-R)(µmax-R+ 4Rµmax θI * ) , µ(I s ) > R, ln 2Is Rµmax θI * 2 µmax-R+ 2Rµmax θI * + (µmax-R)(µmax-R+ 4Rµmax θI * ) , µ(I s ) ≤ R.
(2.9)

Proof. For a given Y , the optical productivity P can be written from (2.5) and (2.8)

P (Y ) = Yopt 0 µ(I(-y)) -Rdy + Y Yopt µ(I(-y)) -Rdy =P (Y opt ) + Y Yopt µ(I(-y)) -Rdy, (2.10)
where Y opt is chosen according to (2.9). On the other hand, for the function

µ(I(-y)) = µ max I(-y) I(-y) + µmax θ ( I(-y) I * -1) 2
, there exists a y * > 0 with µ(I(-y * )) = µ max such that µ(I(-y)) is increasing from 0 to y * and decreasing from y * to 0. According to the value of µ(I(0)) (i.e. µ(I s )), two cases must be considered:

• if µ(I(0)) = µ(I s ) > R, the second term of (2.10) is always negative. Indeed, in the case where Y is smaller than Y opt , using the concavity of µ(I(-y)), one finds µ (I(-y)) > R, ∀y < Y opt . In other words, the second term of (2.10) removes the microalgae which grow more than they respire. Otherwise, Y is greater than Y opt , one finds µ (I(-y)) < R, ∀y > Y opt (for the same reason as above). This means that the second term of (2.10) adds the microalgae which respire more than their growth.

• if µ(I(0)) = µ(I s ) ≤ R, then there exists a ỹ ∈ [0, y * ) such that µ(I(-ỹ)) = R. Then if Y is greater than ỹ, the second term of (2.10) is negative for the same reason as above.

Otherwise, the productivity P (Y ) is negative.

In both cases, the second term of (2.10) is negative. Thus Y opt maximizes the quantity P .

Work in progress as of 7 th September, 2021

In order to compute Y opt , one needs to solve µ(I) = R, or equivalently:

Rµ max θI * 2 I 2 + (R -µ max - 2Rµ max θI * )I + Rµ max θ = 0.
The discriminant of this second order polynomial equation is given by ∆ = (µ max -R)(µ max -R + 4Rµmax θI * ) > 0, which implies that there exists two real roots. The sum and the product of two roots are both positive, hence both of these two roots are also positive. Finally Y opt can be determined by the growth rate at the surface µ(I s ):

• if µ(I s ) > R, then there exists one root in the interval (0, I s ) and one root in the interval (I s , +∞). In this case, one has

Y opt = ln   2IsRµmax θI * 2 µ max -R + 2Rµmax θI * -(µ max -R)(µ max -R + 4Rµmax θI * )   .
• if µ(I s ) ≤ R, then two roots both lie into the interval (0, I s ]. In this case, we choose the smaller one (since it represents the light at lower part of the reactors)

Y opt = ln   2IsRµmax θI * 2 µ max -R + 2Rµmax θI * + (µ max -R)(µ max -R + 4Rµmax θI * )   .
This concludes the proof.

Remark

As shown in (2.9), the value of Y opt only depends on the model parameters (θ, µ max , I * , R) and on the light intensity at the reactor surface I s . In other words, the cancellation of the net growth rate at the bottom of the reactor is the optimal strategy to maximize optical depth productivity (see in Figure 2.2 for illustrations).

Surface biomass productivity

In this section, we focus on the surface biomass productivity Π. From the definition of Π (2.6) and the definition of P (2.8), one has

Π = X ε(X) P. (2.11)
In general, it is not possible to apply the same strategy (as in the proof of Theorem 2.3.1) to optimize Π, since P and Π do generally not have the same behaviour. Only in the case where s = 1 and α 1 = 0, the factor X ε(X) simplifies, leading to the same optimum. Using then Theorem 2.3.1, we deduce directly the following results.

Corollary 2.3.1. If the light extinction function defined by (2.3) satisfies α 1 = 0 and s = 1, then Y opt defined by (2.9) maximizes the productivity Π and Y opt is the global optimum. Moreover, Ỹopt := Y opt /α 0 is the optimal surface biomass.

Work in progress as of 7 th September, 2021

Proof. Since α 1 = 0 and s = 1, Y = ε(X)h = α 0 Ỹ with Ỹ := Xh the surface biomass. Meanwhile, using (2.11), one has P (•) = α 0 Π(•), then following the same analysis, one finds that Y opt maximizes P (•), therefore the productivity Π(•). Finally, Ỹopt is given by Y opt /α 0 .

Corollary 2.3.2. If the objective is to reach a biomass concentration X 1 , there exists a unique reactor depth h 1 which satisfies ε(X 1 )h 1 = Y opt and maximizes the productivity Π(X 1 , •) for this target biomass.

Proof. Since X 1 is fixed, then using (2.11), one has directly that the optimum is given by Y opt . In this case, h 1 is defined by Y opt /ε(X 1 ).

In Corollary 2.3.2, we have studied the case with a fixed biomass concentration X. This result does not depend on the considered law ε(X). However, optimizing X is more tricky, Corollary 2.3.1 provides a result in the case with a specific value of α 1 and s. In more general case, the strategy used in the proof of Theorem 2.3.1 may fail when optimizing X. In this way, we focus on the case where the background turbidity α 1 is not zero in this section. According to the value of s, we separate the study into two parts.

The standard case s = 1

Let h 1 a given depth and

(X 1 , h 1 ) which satisfies Y opt = ε(X 1 )h 1 . For a biomass concentration X > X 1 , one has Y = ε(X)h 1 > Y opt . Applying then Theorem 2.3.1, one finds P (Y ) < P (Y opt ).
On the other hand, one has X ε(X) > X1 ε(X1) using the definition (2.3) for s = 1. According to (2.11), it is not clear to see if Π(X 1 , h 1 ) is larger than Π(X, h 1 ). Indeed, by using the explicit formulation given in Appendix 2.A, one can show that there always exists X > X 1 such that Π(X, h 1 ) > Π(X 1 , h 1 ) in the case s = 1.

According to Corollary 2.3.2, the couple (X 1 , h 1 ) satisfies ε(X 1 )h 1 = Y opt and corresponds to the optimum of Π(X 1 , •) for a given X 1 . However, this is not the optimal condition to optimize Π(•, h 1 ) for a given h 1 . We deduce then the next theorem.

Theorem 2.3.2. If α 1 > 0, there is no global optimum for the productivity Π(•,

•) in R + × R + .
Proof. Let us assume that there exists a global optimum for the productivity Π denoted by (X * , h * ). Since (X * , h * ) is a global optimum, in particular, this is an optimum in the direction of h. Using Corollary 2.3.2, we find ε(X * )h * = Y opt . However, there exists X * > X * such that Π( X * , h * ) > Π(X * , h * ). This contradicts the fact that (X * , h * ) is a global optimum. Therefore, the productivity Π(•, •) has no global optimum.

Since no global optimum for the productivity Π can be found when α 1 > 0, then we would like to study the asymptotic behaviour of Π. In the following, we focus on the optimum in the direction of X and in the direction of h separately. Given X 0 and consider the sequence

(X n , h n ) n∈N defined by h n = Y opt ε(X n-1 ) , X n := argmax X∈R+ Π(X, h n ).
(2.12)

From the definition above, the sequence (X n-1 , h n ) n>0 corresponds to the optimum in the direction of h for X n-1 , whereas the sequence (X n , h n ) n>0 corresponds to the optimum in the direction of X for h n . In plain words, these two sequences defined by (2.12) aims at searching the local optima by optimizing in the direction of h and in the direction of X alternately. Let us provide some more information about the sequence (X n , h n ) n>0 .

Work in progress as of 7 th September, 2021

Lemma 2.3.1. The sequence (X n , h n ) verifies lim n→∞ ε(X n )h n = Y opt ,
and the growth rate at the reactor bottom satisfies lim n→∞ µ(I(X n , h n )) = R.

Proof. From the definition of the sequence

(X n-1 , h n ) n>0 , one has ε(X n-1 )h n = Y opt , which means lim n→∞ ε(X n )h n = Y opt lim n→∞ ε(X n-1 ) ε(X n ) = Y opt .
Denoting by

Y n = ε(X n )h n , since µ • I(-y) is a continuous function with respect to y in R + , one has lim n→∞ µ(I(X n , h n )) = lim n→∞ µ(I(Y n )) = µ(I(Y opt )) = R.
This concludes the proof.

This lemma enables us to prove the next theorem, without constraint on the minimal reactor depth.

Theorem 2.3.3. lim n→∞ X n = ∞, lim n→∞ h n = 0 and lim n→∞ Π(X n , h n ) = P (Yopt) α0
.

Proof. Since (X n ) n∈N is a strictly increasing sequence, the sequence (h n ) n∈N * is strictly decreasing by its construction (2.12). Since for each n ∈ N * , h n > 0, then this sequence converges to a limit that we denote by h lim . Assume that h lim > 0, then from (2.12), one has

h lim = lim n→∞ h n = lim n→∞ Y opt ε(X n-1 ) = Y opt lim n→∞ 1 α 0 X n-1 + α 1 ,
which means that lim n→∞ X n =: X lim < ∞. Then (X lim , h lim ) is a global optimum, hence a contradiction with Theorem 2.3.2. Therefore h lim = 0, which means that X lim = ∞.

On the other hand, by the construction of these two sequences

(X n-1 , h n ) n>0 , (X n , h n ) n>0 , one has Π(X n-1 , h n ) < Π(X n , h n ) < Π(X n , h n+1 ).
Using (2.11), one has Π(X n-1 , h n ) = Xn-1 ε(Xn-1) P (Y opt ) and Π(X n , h n+1 ) = Xn ε(Xn) P (Y opt ). Taking the limit similarly as in the above inequalities gives lim n→∞ Π(X n , h n ) = P (Yopt) α0

. This concludes the proof.

Case s < 1

In the case s < 1, one can still use the explicit formula of Π given in Appendix 2.A to show that the couple (X 1 , h 1 ) satisfying ε(X 1 )h 1 = Y opt is not the optimum in the direction of X for a given h 1 . There exists X > X 1 such that Π(X, h 1 ) > Π(X 1 , h 1 ). Therefore, one can keep the same definition of the sequences (2.12). Lemma 2.3.1 remains true in this case by the definition of the sequence (X n , h n ). However the last limit in Theorem 2.3.3 will be different as shown in the next theorem.

Theorem 2.3.4. lim n→∞ X n = ∞, lim n→∞ h n = 0 and lim n→∞ Π(X n , h n ) = +∞.
Proof. Following the similar strategy as in Theorem 2.3.3, one can show that lim n→∞ X n = ∞, lim n→∞ h n = 0.

On the other hand, by the construction of the two sequences

(X n-1 , h n ) n>0 , (X n , h n ) n>0 , one has Π(X n-1 , h n ) < Π(X n , h n ) < Π(X n , h n+1 ).
Work in progress as of 7 th September, 2021

Using (2.11) and passing to the limit, one finds

lim n→∞ Π(X n-1 , h n ) = lim n→∞ X n-1 ε(X n-1 ) P (Y opt ) = P (Y opt ) α 0 lim n→∞ X 1-s n-1 = +∞, lim n→∞ Π(X n , h n+1 ) = lim n→∞ X n ε(X n ) P (Y opt ) = P (Y opt ) α 0 lim n→∞ X 1-s n = +∞. Therefore, lim n→∞ Π(X n , h n ) = +∞.
Note that for real reactors, there is a constraint on the minimal reactor depth h lim (below which mixing is no more possible). An optimal solution can then be found in this case. Indeed, as shown in Theorem 2.3.4 (or Theorem 2.3.3), a higher productivity can be obtained for higher biomass concentration and smaller reactor depth. Considering the minimal reactor depth, one can find the optimal biomass concentration maximizing the productivity.

Optimal control implementation in closed loop

As shown in previous section, there exists optimal biomass concentration for a given reactor depth h. In this section, let us show that the evolution of the biomass concentration X (defined in (2.7)) can be stabilized to a desired biomass concentration by applying an appropriate controller. More precisely, we consider the dilution rate D in (2.7) as a controller. Let us denote by X ∈ (0, X(0)) the desired biomass concentration. Assumption 2.4.1 (H1). We assume that: a. the quantity Φ := (μ(X, h) -R)X is measured on-line from the plant, b. the growth rate for the influent light intensity is larger than the respiration (i.e. µ(I s ) > R), c. the maximal dilution rate D max is larger than the maximal growth rate µ max . The quantity Φ denotes the average oxygen production which is available from the reactor. Indeed, oxygen sensors or numerical estimators can be applied to obtain the quantity Φ. In the sequel, we assume that (H1) holds. Then we have the following result.

Proposition 2.4.1. The control law

D = D max X ≥ X (μ(X, h) -R) X X X < X (2.13)
globally stabilizes equation (2.7) towards the positive point X .

Remark

X > X is chosen to determine the area where the control will be at its maximum rate. It is defined so that

(µ max -R) X X < D max .
Proof. From the definition of (2.13), the control variable D is positive. On the other hand, μ(0, h) > R, lim X→∞ μ(X, h) = 0 and μ(•, h) is continuously decreasing with respect to X. If the initial state X(0) ≥ X, then replacing D = D max into (2.7) gives

Ẋ = (μ(X, h) -R -D max )X.
Work in progress as of 7 th September, 2021

In particular, μ(X(0), h) -R -D max < 0, hence there exists a time t 1 > 0 such that the state X decreases from 0 to t 1 and X(t 1 ) = X.

When t > t 1 , D = Φ X . Replacing D = Φ X into (2.7) gives Ẋ = (μ(X, h) -R) X X (X -X) = Φ X (X -X). (2.14)
Note that the system is now in the positive invariant region X < X and cannot come back again to X ≥ X. Moreover, 0 < μ(X, h) -R < µ max -R. Then, integrating (2.14) gives ∀t ≥ t 1 , 0 < X ≤ X(t) ≤ X(0).

In the case the initial state X(0) ∈ (0, X), the control variable D = Φ X and the evolution equation (2.7) once again becomes (2.14), hence we follow the small strategy as above.

Finally we find in both two cases that

∀t ≥ 0, 0 < X ≤ X(t) ≤ X(0).
Therefore, the state X is globally exponentially stable for the evolution equation (2.7) by using the control law (2.13).

Numerical results

In this section, we will illustrate some optimal conditions to maximize the algal productivity. In this way, we first introduce an algorithm to compute the sequences defined in (2.12). We then give the parameters that we use for the numerical experiments and show some numerical results.

Numerical algorithm

In practise, one can use the next algorithm to compute two sequences (X n-1 , h n ) n>0 and (X n , h n ) n>0 defined by (2.12). Set n = n + 1.

Algorithm 1 Search Optimum

6:

Compute h n = Y opt /ε(X n-1 ).

7:

Compute X n such that dΠ X (X n , h n ) = 0. 8: end while

Parameter settings

The Han model parameters are taken from [START_REF] Grenier | Worldwide perspectives of rotating algal biofilm up-scaling[END_REF] and recalled in Table 2.1. Then one can use (1.9) to find µ max , α, I * . The considered surface light intensity is set to be I s = 2000 µmolm -2 s -1 . For s = 1, we take from Table 1.2 the specific light extinction coefficient for the species Chlorella pyrenoidosa α 0 = 0.2 m 2 • g -1 and the background turbidity α 1 = 10 m -1 . Note that for the case where s < 1, we compute the coefficient α 0 (s) to find the one providing an extinction coefficient as close as possible to the reference linear case which is generally the one measured:

α 0 (s) = argmax X∈[Xmin,Xmax] |α 0 (1)X -α 0 (s)X s | . (2.15)
Work in progress as of 7 th September, 2021 s -1

Numerical study

In this section, we provide some numerical tests to illustrate the influence of the water depth h, the biomass concentration X and the light extinction function ε(X) on algal productivity.

Evaluation of different light extinction coefficient

As mentioned in the previous section, the light extinction coefficient α 0 needs to be better estimated when s < 1, in comparison with the reference case s = 1. For this reason, let us set biomass concentration X in a range of [0, 1000] (g • m -3 ). We then use (2.15) to find α 0 that provide the same average extinction rate.

Global optimum of optical depth

The optimal optical depth (Y opt ) can be computed explicitly using (2.9) once the light intensity at the reactor surface I s and the model parameters (θ, µ max , I * , R) are fixed. Work in progress as of 7 th September, 2021 In the same way, for a given biomass concentration X, Corollary 2.3.2 provides a condition to determine the optimal depth to maximize the surface biomass productivity Π. one can see that this optimal satisfies µ (I(X, h * )) = R. It is worth remarking that the range of the productivity Π changes for different value of s and α 1 , this motivates to better understand how these parameters affect growth.

Influence of the background turbidity and s

Here we study the influence of the background turbidity α 1 and the value of s on the productivity Π. We keep the biomass concentration value X = 50 g • m -3 and compute h by using the relation ε(X)h = Y opt for different values of α 1 and s. Note that the depth h computed in this way is the optimum to maximize the productivity for the given biomass concentration. Figure 2.4 represents the optimal surface biomass productivity Π with respect to the background turbidity.

Work in progress as of 7 th September, 2021

As we can expect intuitively, the larger the background turbidity is, the smaller the productivity is. Furthermore, the productivity increases with the value of s for a fixed value of turbidity α 1 .

Local optima in the case s = 1

In real life applications, the depth h depends on the type reactors, for instance, h = 0.1 m -0.5 m for raceway ponds, h = 1 cm-10 cm for tubular photobioreactors and h = 0.1 m-1 mm for biofilm reactors (where the microalgal biomass is fixed on a support). By knowing the lowest bound admissible for the reactor depth (depending on the process type), we only need to optimize the productivity in the direction of X. Note that the turbidity α 1 may change the optimal condition to maximize the surface biomass productivity Π. Indeed, Figure 2.5 illustrates this for a reactor depth h = 0.15 m. Note that X 0 satisfies the relation ε(X 0 )h = Y opt which also means that the net growth rate at the bottom of the reactor is zero (see the blue point in these two figures). On the other hand, the red point (X 1 , h) is the optimum which maximizes the surface biomass productivity Π for this given depth h. One can see that X 0 = X 1 = 158.427 g • m -3 in the case the background turbidity is zero in the system (Left), meaning that the optimum Work in progress as of 7 th September, 2021 is the point which cancels the net average growth rate at the reactor bottom as mentioned in Corollary 2.3.1. However, as mentioned in Subsection 2.3.2, that by taking account into the background turbidity (Right), these two points are no longer the same and the optimum then satisfies X 1 = 204.190 g • m -3 > X 0 = 108.427 g • m -3 .

The global behaviour of the surface biomass productivity Π is represented on Figure 2.6, for h ∈ (0, 1] and X ∈ (0, 1000]. To discuss the influence of the background turbidity, we consider two possible values, α 1 = 0 m -1 and α 1 = 10 m -1 . Note that the blue points in the left figure Figure 2.6: Global behaviour of productivity (Π) with respect to depth (h) and biomass concentration (X). Left: α 1 = 0 m -1 . The blue stars represent the optimal couple (X, h) such that Π finds its global maximum. Right: α 1 = 10 m -1 . The red circles represent the suboptimal couple (X, h) where Π finds its maximum in the direction of h for a given X. The black squares represent the suboptimal couple (X, h) where Π finds its maximum in the direction of X for a given h.

(X, h) satisfy the relation ε(X)h = Y opt which is also the global optimum. However, by taking account into the background turbidity (see figure on the right), no global optimum exists in this case as mentioned in Theorem 2.3.2. Instead, for a given biomass concentration, the optimal depths can still be found using the relation ε(X)h = Y opt (represented by the red circles in the right figure). For a given water depth, the optimal concentrations are obtained by cancelling the derivative of Π(•, h) (represented by the black squares in the right figure). Furthermore, one can observe that this two suboptima become closer when X increases and h decreases, meanwhile the productivity also increases in this direction.

Let us set X 0 = 50 g • m -3 , α 1 = 10 m -1 and n max = 10 4 . Figure 2.7 illustrates the proprieties of these two sequences constructed by Algorithm 1. Starting from the figure on the top, the blue point and the yellow point are the first-two element of the sequence (X n-1 , h n ) n>0 , the red point and the purple point are the first-two element of the sequence (X n , h n ) n>0 . Recall that the sequence (X n-1 , h n ) n>0 always satisfies ε(X n-1 )h n = Y opt and the net growth rate at the reactor bottom is always 0. We then only study the asymptotic behaviour of the sequence (X n , h n ) n>0 . As shown in bottom left figure, the surface biomass X n h n converges to Yopt α0 and the optical depth ε(X n )h n converges to Y opt , as proved in Lemma 2.3.1. The productivity Π(X n , h n ) converges to P (Y opt )/α 0 , see bottom right figure as proved in Theorem 2.3.3. Finally, the net growth rate at the reactor bottom converges to zero, which is the global optimum condition in the case where the background turbidity is 0 (see Corollary 2.3.1). In particular, since (X n , h n ) are the optima in the direction of X for each h n , one can see that the net growth rate at the reactor bottom for these optima are always negative, meaning that the compensation condition is only satisfied Work in progress as of 7 th September, 2021 and optical depth (ε(X n )h n ) for the sequence (X n , h n ) n>0 . Down Right: Productivity Π(X n , h n ) and net growth rate at the reactor bottom µ(X n , h n ) -R for the sequence (X n , h n ) n>0 .

Work in progress as of 7 th September, 2021 asymptotically.

Local optima in the case s < 1

We start with a similar study as in Figure 2.5 in the case s < 1. Recall that the depth of the reactor is given by h = 0.15 m and two background turbidity values are given by α 1 = 0 m -1 and α 1 = 10 m -1 . Figure 2.8 illustrates the results for s = 0.365. Recall that the blue point (X 0 , h) satisfies the relation ε(X 0 )h = Y opt which also means that the net growth rate at the reactor bottom of the reactor is zero, and the red point (X 1 , h) represents the optimum which maximizes the productivity for this depth h. In the case α 1 = 0 m -1 , we find X 0 = 13.327 g • m -3 and X 1 = 1149.298 g • m -3 , whereas we obtain X 0 = 4.715 g • m -3 and X 1 = 1064.574 g • m -3 in the case α 1 = 10 m -1 . These two points do not coincide even when the background turbidity is zero, which is different from the case s = 1.

Figure 2.9 shows the global behaviour of surface biomass productivity in the case s = 0.365. Unlike for the case s = 1 (Figure 2.6), the influence of the background turbidity becomes smaller when s < 1. However, similarly to this s = 1 case (Right), the productivity becomes larger when the biomass concentration X increases and the water depth h decreases. Furthermore, Figure 2.10 shows the divergence of the productivity Π, as proved in Theorem 2.3.4.

Controller test

We present the efficiency of the controller D designed in Proposition 2.4.1. Let us set h = 0.1 m, s = 1, X(0) = [2500, 50]g m -3 , D max = 10µ max and keep other parameter settings. Figure 2.11 illustrates the behaviour of the biomass concentration X under our controller D. Note that the desired biomass concentration X = argmax X∈R+ Π(X, h). One can see that the evolution of the biomass concentration X in closed loop converges to the desired optimal biomass concentration (after five days).

Conclusion

The concept of optical productivity P has been defined and a global optimum Y opt has been found to maximize P . This condition corresponds to a situation where the net growth rate at Work in progress as of 7 th September, 2021 Figure 2.9: Global behaviour of productivity (Π) with respect to depth (h) and biomass concentration (X). Left: α 1 = 0 m -1 . Right: α 1 = 10 m -1 . The red circles represent the suboptimal couple (X, h) where Π finds its maximum in the direction of h for a given X. The black squares represent the suboptimal couple (X, h) where Π finds its maximum in the direction of X for a given h. Work in progress as of 7 th September, 2021 the reactor bottom is zero. This optimum can be used to characterize the optimal water depth which maximizes surface biomass productivity Π for a target biomass concentration value. When the light extinction rate is affine with respect to the biomass concentration, an upper limit to the productivity is given which is obtained for an infinitely small depth and an infinitely large biomass concentration.

The proposed nonlinear controller stabilizes the biomass concentration to its optimal value X . It could be improved by integrating an extremum seeking strategy [START_REF] Liu | Introduction to extremum seeking[END_REF][START_REF] Guay | Adaptive extremum seeking control of continuous stirred tank bioreactors with unknown growth kinetics[END_REF] to automatically target the desired biomass without identifying it in advance.

2.A Explicit computations for average growth rate

In this section, we provide details about the computation on μ. As mentioned in Subsection 1.2.1, the growth rate described by Han model can be easily written in the Haldane description. We choose to present the corresponding computation hereafter with Han model parameters. From the definition of μ (2.4), one has

μ = 1 h 0 -h µ(I(X, z))dz = k r kσ hε(X) Is I b 1 k d τ (σI) 2 + k r (τ σI + 1) dI,
where

I b = I s exp(-ε(X)h). Define a = k d τ σ 2 , b = k r τ σ, c = k r .
According to the sign of the discriminant ∆ of equation aI 2 + bI + c, three cases must be considered.

• ∆ > 0 : Then there exists two reel roots denoted by

d 1 = -b+ √ b 2 -4ac 2a and d 2 = -b- √ b 2 -4ac 2a . Hence one has μ = k r kσ ahε(X) e 1 ln I s -d 1 I b -d 1 + e 2 ln I s -d 2 I b -d 2 .
with • ∆ < 0 : Then one has

aI 2 + bI + c = 4ac -b 2 4a I + b 2a 4ac-b 2 4a 2 2 + 1 .
Applying a change of variable by setting y =

I+ b 2a 4ac-b 2 4a 2
, one gets

μ = k r kσ hε(X) 2 √ 4ac -b 2 arctan( 2aI s + b √ 4ac -b 2 ) -arctan( 2aI b + b √ 4ac -b 2 ) .
Work in progress as of 7 th September, 2021

Since one has the explicit formulation of the average growth rate μ, the surface biomass concentration Π can be then computed explicitly including its derivatives. This is at the basis of the determination of the optimal biomass concentration X for a given depth h.

Work in progress as of 7 th September, 2021

Chapter 3

Periodic resource allocation in dynamical systems, application to raceway microalgal production

Introduction

Considering a fixed amount of the resources and a set of activities, we look for a distribution strategy which optimizes a given objective function. This is the so-called resource allocation problem [START_REF] Ibaraki | Resource allocation problems[END_REF]. Due to its simple structure, this problem is encountered in a number of applications including load scheduling [START_REF] Zhang | A model predictive control approach to the periodic implementation of the solutions of the optimal dynamic resource allocation problem[END_REF], manufacturing [START_REF] Zhang | A new manufacturing resource allocation method for supply chain optimization using extended genetic algorithm[END_REF], portfolio selection [START_REF] Hennessy | The use of archimedean copulas to model portfolio allocations[END_REF] and computational biology [START_REF] Akhmetzhanov | Optimal life-history strategies in seasonal consumer-resource dynamics[END_REF]. Periodic versions have also been considered. The periodic scheduling problem was first addressed in [START_REF] Liu | Scheduling algorithms for multiprogramming in a hard-realtime environment[END_REF] in the framework of operation research. Later on, the concept of proportionate fairness constraint has been introduced [START_REF] Baruah | Proportionate progress: A notion of fairness in resource allocation[END_REF] to design allocation algorithms which schedule the resources in proportion to task weight. Periodic resource allocation problems are also used in ecology, e.g., in [START_REF] Cominetti | Asymptotic convergence of optimal policies for resource management with application to harvesting of multiple species forest[END_REF][START_REF] Piazza | About optimal harvesting policies for a multiple species forest without discounting[END_REF] where the authors investigate long-term behaviour of harvesting policies for a forest composed of multiple species with different maturity ages. In such problems, the state can also be described in terms of dynamical systems. As an example, hospital resources (hospital beds) continuous allocation is studied in [START_REF] Abdelrazec | Modeling the spread and control of dengue with limited public health resources[END_REF] as a strategy to control the dengue fever, associated with a patient recovery rate. In the same way, a population of a single species with logistic growth in a patchy environment is considered in [START_REF] Nagahara | Maximizing the total population with logistic growth in a patchy environment[END_REF]. The problem here consists of the maximization of the total population by re-distributing the limited resources among the patches.

In general, resource allocation problems are related to the assignment of a resource to a sequence of two or more tasks at the same time. However, we focus in this chapter on problems where N resources are assigned to N tasks over some time intervals. Additionally, we consider permanent regimes which are often relevant in the case of long term processes, as, e.g., crop harvesting, scheduling of appliances, etc. Moreover, here we also account for the dynamical evolution of the system between two re-allocations, further increasing the difficulty of the analysis. In this way, our work is related to the fields of switched systems [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF], impulse control [START_REF] Bensoussan | Contrôle impulsionnel et inéquations quasi variationnelles[END_REF][START_REF] Hespanha | Switching between stabilizing controllers[END_REF] and to periodic control [START_REF] Colonius | Optimal periodic control[END_REF]. These techniques are usually used to tackle stabilization issues. In this chapter, we consider them in view of optimization issues.

In order to model the allocation process in the periodic system, we study the following resource allocation problem : Consider a system with N resources and N activities, each activity uses the allocated resource to evolve during a given time T > 0. At time T , the resources will be re-allocated according to a given permutation. It is proven that if the dynamics of the system is periodic, then it is one period corresponding to one allocation process whatever the order of the considered permutation matrix is. A nonlinear objective functional to be optimized is then introduced in order to find best allocation strategies. Since N ! permutations need to be tested in the general case, it can be numerically solved only for a limited number of N . To overcome this difficulty, we propose a second optimization problem -a typical assignment problem -associated with a suboptimal solution of the initial problem, which can be determined explicitly. In addition, a sufficient condition to characterize cases where the two problems have the same solution is given.

For the sake of concreteness, we illustrate our theory by an industrial application, namely the mixing of microalgae cells in a cultivation set-up. Outdoor algal cultivation is mainly carried out in open raceway ponds exposed to solar radiation. This hydrodynamic system is set in motion by a paddle wheel which homogenises the medium for ensuring an equidistribution of the nutrients and guarantees that each cell will have regularly access to the light [START_REF] Chiaramonti | Review of energy balance in raceway ponds for microalgae cultivation: Re-thinking a traditional system is possible[END_REF]. Microalgae then grow between two re-distributions depending on the light intensity received in their layer. Different strategies have been proposed to optimize the production of the biomass in this algal raceway system [START_REF] De La Hoz | Optimization of microalgal productivity using an adaptive, non-linear model based strategy[END_REF][START_REF] Cuaresma | Luminostat operation: A tool to maximize microalgae photosynthetic efficiency in photobioreactors during the daily light cycle?[END_REF][START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF][START_REF] Cornet | Calculation of optimal design and ideal productivities of volumetrically lightened photobioreactors using the constructal approach[END_REF][START_REF] Amos | Principles for attaining maximal microalgal productivity in photobioreactors: an overview[END_REF][START_REF] Bernard | Controlling the bottom topography of a microalgal pond to optimize productivity[END_REF][START_REF] Bernard | Shape optimization of a microalgal raceway to enhance productivity[END_REF][START_REF] Bernard | Mixing strategies combined with shape design to enhance productivity of a raceway pond[END_REF]. However, the mixing policy has not been investigated up to now. In the current study, the paddle wheel is identified to a mixing device and modeled by permutation. We then focus on the influence of the mixing strategy on the algal productivity in a flat raceway system with a constant average velocity.

This chapter is organized as follows. We introduce our periodic resource allocation problem and the considered dynamical system in Section 3.2. More precisely, the optimization problem together with a simpler version based on an approximated functional are introduced in Subsection 3.2.2. Some technical lemmas are given in Subsection 3.2.3 and a criterion to guarantee that the original problem and its approximation share the same solution is given in Subsection 3.2.4. Some implementation remarks conclude this section in Subsection 3.2.5. Section 3.3 is devoted to the application to algal production. We present the models associated with the biological and the mixing device in a raceway pond in Subsection 3.3.1. The considered parameters are given in Subsection 3.3.2. We illustrate the performance of our strategies by numerical experiments in Subsection 3.3.4. Finally, we conclude with some perspectives of our work in Section 3.4.

Notation. In what follows, N denotes the set of non-negative integers, P N denotes the set of permutation matrices of size N × N with N ∈ N and S N denotes the associated set of permutations of N elements. The cardinal of a set E is denoted by #E. Given a matrix M , we denote by ker(M ) its kernel and by M i,j its coefficient (i, j). In the same way, W n denotes the n-th coefficient of a vector W . •, • denotes the scalar product in R 2 .

Description of the problem and optimization

Given a period T , and initial time T 0 and a sequence (T k ) k∈N , with T k = kT + T 0 , we consider the following resource allocation problem: Consider N resources denoted by (I n ) N n=1 ∈ R N which can be allocated to N activities denoted by (x n ) N n=1 where x n consists of a real function of time. On a time interval [T k , T k+1 ), each activity uses the assigned resource and evolves according to a linear dynamics ẋn = -α(

I n )x n + β(I n ), (3.1)
where α : R → R + and β : R → R + are given. At time T k+1 , the resources is re-assigned, meaning that x n1 (T k+1 ) = x n2 (T - k+1 ) for some n 1 , n 2 ∈ {1, . . . , N }. In this way, k ∈ N represents the number of re-assignments and T - k represents the moment just before re-assignment. This problem can also be defined using the permutation matrix P ∈ P N . In particular, the assignment process can be formulated by

x(T k ) = P x(T - k ).
Work in progress as of 7 th September, 2021

Assume that the resource (I n ) N n=1 are constant with respect to time, then the evolution equation (3.1) can be computed explicitly. Indeed, for a given initial vector of states (x n (T 0 )) N n=1 , we have

x(t) = D(t)x(T k ) + v(t), t ∈ [T k , T k+1 ), (3.2)
where D(t) is a time dependent diagonal matrix with D nn (t) := e -α(In)(t-T k ) and v(t) is a time dependent vector with

v n (t) := β(I n ) α(I n ) (1 -e -α(In)(t-T k ) ). (3.3)
The detail about the computations giving rise to (3.2) is presented in Appendix 3.A. For the sake of simplicity, we write hereafter D, v instead of D(T ), v(T ).

Let u ∈ R N an arbitrary vector. Define

f k := u, 1 T T k+1 T k x(t)dt , (3.4)
the benefit attached to the time period [T k , T k+1 ) after k times of re-assignment. Then the average benefit after K operations is given by

1 K K k=0 f k .

Remark

Such a formalization is used in the context of forest maintenance and exploitation in [START_REF] Cominetti | Asymptotic convergence of optimal policies for resource management with application to harvesting of multiple species forest[END_REF]. In this work, an infinite sum is considered to study the total benefit of all the operations.

Periodic regime

Consider P ∈ P N the permutation matrix describing the resource assignment process. According to (3.2) and by the definition of P , we have

x(T k+1 ) = P (Dx(T k ) + v). (3.5)
Before studying the sequence (x(T k )) k∈N , let us give a first simple result. We denote by I the identity matrix of size N .

Lemma 3.2.1. Given k ∈ N and P ∈ P N , the matrix I N -(P D) k is invertible.

Proof. Assume I N -P D is not invertible, then there exists a non-null vector X ∈ ker(I N -P D), which means X = P DX. Let us denote d n = D nn , n = 1, . . . , N . Denoting by σ the permutation associated with P , we find that (DX) n = d n X n and X n = (P DX) n = d σ(n) X σ(n) . In the same way, we have

X n = (P D) k X n = d σ k (n) . . . d σ(n) X σ k (n) ,
where σ k (n) denotes the k-times repeated composition of σ with itself. Denoting by K the order of σ, we have

X n = (P D) K X n = d σ K (n) . . . d σ(n) X σ K (n) = d σ K (n) . . . d σ(n) X n .
Work in progress as of 7 th September, 2021 50CHAPTER 3. Periodic resource allocation in dynamical systems, application to raceway microalgal production Since, 0 < d n < 1 for n = 1, . . . , N , then 0 < d σ K (n) . . . d σ(n) < 1. This implies that X n = 0, which contradicts our assumption. Therefore, I N -P D is invertible. That I N -(P D) k is invertible can be proved in much the same way.

Assume now that the state x is KT -periodic in the sense that after K times of re-assignment, the state of each activity returns to its initial state x n (T K ) = x n (T 0 ). A crucial property of (x(T k )) k∈N is given in the next proposition. Theorem 3.2.1. We keep the notation of the previous lemma and assume that the state x is KT -periodic. Then (x(T k )) k∈N is a constant sequence and we have for all k ∈ N x(T k ) = (I N -P D) -1 P v.

Proof. Thanks to Lemma 3.2.1, there exists a unique x satisfying x = P (Dx + v). Let us then define the sequence (e k ) k∈N by e k := x(T k ) -x.

We have e k+1 = (P D)e k .

Since x is assumed to be KT -periodic, we have

e 0 = e K = (P D) K e 0 .
According to Lemma 3.2.1, I N -(P D) K is invertible, meaning that e 0 = 0. It follows that e k = 0, for k ∈ N. The result follows.

A natural choice for K would be the order of the permutation associated with P . Indeed, in this case K is the minimal number of re-assignments required to recover the initial state of x. The previous result shows that every KT -periodic evolution will actually be T -periodic. In the next section, we show that this property is decisive to formulate an optimization problem. In addition, the computations to solve the optimization problem will be reduced, since the CPU time required to assess the quality of a permutation will not depend on its order.

Objective function

Let us take the benefit (3.4) and replace x(t) by its solution (3.2), one has

f k = 1 T u, Dx(T k ) + ṽ = 1 T Du, x(T k ) + u, ṽ ,
where Dnn =

T k+1 T k D nn (t)dt and ṽn = T k+1 T k v n (t)dt.
The only term depends on the reassignment process is x(T k ). On the other hand, using Theorem 3.2.1, one finds

Du, x(T k ) = Du, x(T 0 ) = Du, (I N -P D) -1 P v ,
meaning that the benefit is the same for each re-assignment process.

Without loss of generality, let us still denote Du by u. We now consider both minimization and maximization problems associated with the functional

J(P ) := u, (I N -P D) -1 P v , (3.6)
Work in progress as of 7 th September, 2021

where P ∈ P N . Since #S N = N !, this problem cannot be tackled in realistic cases where large values of N must be considered, e.g., to keep a good numerical accuracy. To overcome this difficulty, we propose in this section an approximation of this problem whose optimum can be determined explicitly. For this purpose, we expand the functional (3.6) as follows u, (I N -P D)

-1 P v = +∞ l=0 u, (P D) l P v = u, P v + +∞ l=1 u, (P D) l P v ,
and consider as an approximation the first term of this series, namely

J approx (P ) := u, P v . (3.7)
Without loss of generality (see Appendix 3.B for the details), we assume that the entries of u are sorted in an ascending order, meaning that u 1 ≤ . . . ≤ u N . Note that optimizing J approx amount to solving an assignment problem [START_REF] Burkard | Assignment Problems[END_REF]. Indeed, we have for example

min P ∈P N J approx (P ) = min σ∈S N N n=1 u n v σ(n) .
This reads as an assignment problem associated with the matrix [u i v j ] (i,j=1,...,N ) . To make our exposition self-contained, we give the solution of this problem in Section 3.2.4.

Remark

A fairly common approach to deal with permutation optimization is to relax the problem by extending the optimization to the set of bistochastic matrices. As an example, this technique corresponds to the Kantorovitch relaxation considered in optimal transport [START_REF] Kantorovich | On the transfer of masses (in russian)[END_REF], see also [START_REF] Bertsimas | Introduction to Linear Optimization[END_REF] for a more general presentation of the linear case, and [START_REF] Lieb | Variational principle for many-fermion systems[END_REF] for a similar strategy in the context of quantum chemistry. This approach allows the optimization to be performed by gradienttype methods. At the theoretical level, the goal is then to prove that the convergence takes place towards extremal points, i.e. permutation matrices. We have tested this approach to the nonlinear problem (3.6). Our results indicate that the obtained limits are indeed permutation matrices. However, we have observed that the obtained matrices are not always optimal, which leads us to conjecture the existence of local non-global minima for this extended form of J.

Some technical lemmas

Let us state some preliminary properties about the permutation set S N that we will use in the next section. Given k ∈ N, and two arbitrary permutations σ, σ ∈ S N , let us define

E k (σ, σ) := n = 1, . . . , N | σ k (n) = σ k (n) , G k (σ, σ) :={n = 1, . . . , N | ∀k ≤ k, σ k (n) = σ k (n)}, and
m k := #E k (σ, σ).
We have the following result.

Lemma 3.2.2. For k ∈ N, we have m k ≤ km 1 and #G k (σ, σ) ≥ max(N -km 1 , 0).
Work in progress as of 7 th September, 2021 52CHAPTER 3. Periodic resource allocation in dynamical systems, application to raceway microalgal production Proof. To shorten notation, we write in this proof

E k instead of E k (σ, σ), E k+1 instead of E k+1 (σ, σ), G k instead of G k (σ, σ), etc.
From the definition of E k , we have:

E k+1 = (({1, . . . , N } \ E 1 ) ∩ E k+1 ) ∪ (E 1 ∩ E k+1 ).
The first subset in the right hand side satisfies

σ (({1, . . . , N } \ E 1 ) ∩ E k+1 ) = σ (({1, . . . , N } \ E 1 ) ∩ E k+1 ) ⊂ E k , so that # (({1, . . . , N } \ E 1 ) ∩ E k+1 ) ≤ #E k =: m k . On the other hand, (E 1 ∩ E k+1 ) ⊂ E 1 , hence #(E 1 ∩ E k+1 ) ≤ m 1 . As a consequence, m k+1 ≤ m k + m 1 . This implies m k ≤ km 1 .
As for G k , we have:

G k = (G k+1 ∩ G k ) ∪ (σ -k (E 1 ) ∩ G k ). (3.8) Indeed, let n ∈ G k , i.e, σ k (n) = σ k (n). If σ k+1 (n) = σ k+1 (n), then n ∈ G k+1 . Otherwise, σ k+1 (n) = σ k+1 (n), meaning that σ k+1 (n) = σ(σ k (n)) which implies σ k (n) = σ k (n) ∈ E 1 , so that n ∈ σ -k (E 1
). This proves (3.8), and we get as a by-product

(G k+1 ∩ G k ) ∩ (σ -k (E 1 ) ∩ G k ) = ∅. Moreover, since G k+1 ⊂ G k , we get G k+1 ∩ G k = G k+1 . It follows that #G k = #G k+1 + #{σ -k (E 1 ) ∩ G k }. Since #{σ -k (E 1 ) ∩ G k } ≤ #E 1 = m 1 , we obtain #G k+1 ≥ #G k -m 1 . The result follows.
In what follows, a transposition in S N between two elements i = j is denoted by (i j). By abuse of notation, (n n) denotes the identity for all n = 1, . . . , N . Given a permutation σ ∈ S N , we consider the sequence of permutations (σ n ) n=0,...,N defined by

σ 0 = σ σ n = (n σ n-1 (n)) • σ n-1 .
(3.9)

For all n ≤ N , it immediately follows from this definition that

σ n | {1,...,n} = Id| {1,...,n} and σ N -1 = σ N = Id,
where Id denote the identity permutation. Let us give two additional properties of this sequence.

Lemma 3.2.3. Let σ ∈ S N and (σ n ) n=1,...,N -1 defined by (3.9). One has:

{i = 1, . . . , N | σ(i) = i} = {i = 1, . . . , N | ∀n = 1, . . . , N -1, σ n (i) = i} .
Proof. Given i with 1 ≤ i ≤ N , such that σ(i) = i, let us prove that σ n (i) = i by induction on n. Since σ 0 = σ, the result holds for n = 0. Suppose it holds at a rank n -1, meaning that σ n-1 (i) = i. By definition of (σ n ) n=1,...,N , one has:

σ n (i) = (n σ n-1 (n)) • σ n-1 (i) = (n σ n-1 (n))(i). If i = n, then (n σ n-1 (n))(i) = σ n-1 (n) = σ n-1 (i) = i. If i = σ n-1 (n), then i = σ n-1 (i) = σ n-1 (
n) and i = n, so that we conclude as in the previous case. In the other cases, σ n (i) = σ n-1 (i) = i. The result follows.

Work in progress as of 7 th September, 2021 Lemma 3.2.4. Let i, j ∈ {1, . . . , N }, with i < j. Let σ ∈ S N σ = (i j) • σ , where (i j) and σ have disjoint supports, i.e., σ (i) = i and σ (j) = j. One has: σ j = σ j-1 .

Proof. From the definition of (σ n ) n=1,...,N , one has

σ j = (j σ j-1 (j)) • σ j-1 .
We need to prove that σ j-1 (j) = j. Since σ and (i j) are disjoint, then for n < i, σ n = (i j) • σ n , where σ n is defined by (3.9), with the initial term σ 0 = σ . In particular, σ n (i) = j for n < i. In the case n = i, one has

σ i = (i σ i-1 (i)) • σ i-1 = (i j) • σ i-1 = (i j) • (i j) • σ i-1 = σ i-1 .
In particular, σ i (j) = j.

Finally, since σ i-1 (i) = i, we find that σ i = σ i-1 , and it follows by induction that for n > i, σ n = σ n , which means σ n (j) = j. In particular σ j-1 (j) = j. This concludes the proof.

The sequence (σ n ) n=0,...,N can be used to decompose J(I) -J(P ) for an arbitrary P ∈ P N , as stated in the next Lemma. Lemma 3.2.5. Let σ ∈ S N and P ∈ P N the associated permutation matrix, we have:

u, (I N -P )v = N -1 n=1 (u n -u σ -1 n-1 (n) )(v n -v σn-1(n) ).
Proof. Given j ∈ {0, . . . , N }, define S j = N n=1 u n v σj (n) . Since σ j (n) and σ j-1 (n) might only differ for n = j and n = σ -1 j-1 (j), we have

S j -S j-1 = N n=j u n (v σj (n) -v σj-1(n) ) = u j (v σj (j) -v σj-1(j) ) + u σ -1 j-1 (j) (v σj (σ -1 j-1 (j)) -v σj-1(σ -1 j-1 (j)) ) = u j (v j -v σj-1(j) ) + u σ -1 j-1 (j) (v σj-1(j) -v j ) = (u j -u σ -1 j-1 (j) )(v j -v σj-1(j) ).
The result then follows from u, (I N -P )v = S N -1 -S 0 .

Solutions of the optimization problems

The previous lemma enables us to solve the problems max P ∈P N J approx (P ) and min P ∈P N J approx (P ).

Recall that the entries of u are sorted in an ascending order.

Lemma 3.2.6. Let σ + , σ -∈ S N such that v σ+(1) ≤ v σ+(2) • • • ≤ v σ+(N ) and v σ-(N ) ≤ v σ-(N -1) ≤ • • • ≤ v σ-(1)
and P + , P -∈ P N , the corresponding permutation matrices. Then P + = argmax P ∈P N J approx (P ), P -= argmin P ∈P N J approx (P ).

Work in progress as of 7 th September, 2021 54CHAPTER 3. Periodic resource allocation in dynamical systems, application to raceway microalgal production Proof. Let P ∈ P N and σ ∈ S N the associated permutation, we have u, (P + -P )v = u, (I N -P P -1

+ )w = N -1 n=1 (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ), (3.10)
where w = (w n ) N n=1 := (v σ+(n) ) N n=1 and σ n is the sequence defined by (3.9) with σ := σ -1 + • σ the permutation associated with P P -1 + . Since (w n ) N n=1 by its definition is an increasing sequence, σ n-1 (n) ≥ n and (σ n-1 ) -1 (n) ≥ n, we find that u, (P + -P )v ≥ 0. The proof of the problem min P ∈P N u, P v is similar.

We immediately deduce from this lemma that once u and v are given, the matrix P + , P -of Lemma 3.2.6 can be determined explicitly. More precisely, P + is the matrix corresponding to the permutation which associates the largest coefficient of u with the largest coefficient of v, the second largest coefficient with the second largest, and so on. In the same way, P -is the matrix corresponding to the permutation which associates the largest coefficient of u with the smallest coefficient of v, the second largest coefficient with the second smallest, and so on.

Remark

The optimal matrices P + and P -are not unique as soon as either u or v contains at least two identical entries.

We focus now on the case where u as well as v have entries with a constant sign. Since the results in this section hold both for minimization and maximization problems, we can assume without loss of generality that u, v are both positive. Using the properties given in the previous section, we will show that in some cases, the problem max P ∈P N J(P ) (resp. min P ∈P N J(P )) and max P ∈P N J approx (P ) (resp. min P ∈P N J approx (P )) have the same solution.

We keep the notation of Lemma 3.2.6. Define for n = 1, . . . , N , p n := min i,j=1,...,N,i =n,j =n

|(u n -u i )(v σ+(n) -v σ+(j) )|. (3.11)
Denote by i n and j n the solutions of the previous problem. Since u n , v σ+(n) are sorted in an ascending order, we find immediately that if n = 1 (resp. N ), then i n = j n = 2 (resp. i n = j n = N -1). Otherwise, i n = n -1 or i n = n + 1, and the same result holds for j n . Sort ( p n ) N n=1 and denote by (p n ) N n=1 the resulting sequence, i.e., p 1 ≤, . . . , ≤ p N . Define then for m = 1, . . . , N

s m := m n=1 p n , (3.12)
and

F - m := min(m,N ) n=1 u n v σ-(N -m+n) , F + m := N n=max(1,N -m+1) u n v σ+(n) . (3.13)
From the definition of these sequences, we have F + m ≥ F - m . See Appendix 3.C for the case where u or v negative. We are now in a position to give the main result of this section.

Work in progress as of 7 th September, 2021 Theorem 3.2.2. Assume that u and v have positive entries and define

φ(m 1 ) := 1 s m 1 2 +∞ l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1 , (3.14)
where m 1 refers to the notation in Lemma 3.

From the definition E k (σ + , σ) and G k (σ + , σ) , we have in particular

E 1 (σ + , σ) G 1 (σ + , σ) = {1, . . . , N }.
Let us denote by (w n ) N n=1 = (v σ+(n) ) N n=1 and by σ n the sequence defined by (3.9) with σ 0 := σ -1 + • σ. From the definition of E 1 (σ + , σ) and G 1 (σ + , σ), we have σ(G 1 (σ + , σ)) = σ + (G 1 (σ + , σ)) and σ(E 1 (σ + , σ)) = σ + (E 1 (σ + , σ)), which implies σ 0 (E 1 (σ + , σ)) = E 1 (σ + , σ), and for any i ∈ G 1 (σ + , σ), σ 0 (i) = i. Using these properties and (3.10), we have u, (P

+ -P )v = N -1 n=1 (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ) = n∈E1(σ+,σ) (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ) + n∈G1(σ+,σ) (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ) = n∈E1(σ+,σ) (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ). (3.18)
In the case where there exists a transposition (i i ) with i < i in σ , Lemma 3.2.4 implies that u (σ i -1 ) -1 (i ) = u i and w σ i -1 (i ) = w i . The maximum number of transpositions in σ 0 is m1 2 if m 1 is even, m1-3 2 otherwise. Hence, the smallest number of non-zero terms present in the last sum of (3.18) is given by m

1 -m1 2 = m1 2 if m 1 is even, m1-1 2
otherwise. In other words, there

Work in progress as of 7 th September, 2021 56CHAPTER 3. Periodic resource allocation in dynamical systems, application to raceway microalgal production exists at least m1 2 non zero terms in the last sum of (3.18), which implies u, (P

+ -P )v = n∈E1(σ+,σ) (u n -u (σ n-1 ) -1 (n) )(w n -w σ n-1 (n) ) ≥ s m 1 2 . (3.19)
For n ∈ {1, . . . , N } and l ∈ N * , let us denote by

d σ,l,n := d σ l (n) d σ l-1 (n) • • • d σ(n) .
Considering now the second term of the right hand side of (3.17), we get

< u, (P D) l P v >= N n=1 u n d σ l (n) d σ l-1 (n) • • • d σ(n) v σ l+1 (n) = N n=1 u n d σ,l,n v σ l+1 (n) .
Using this notation and Lemma 3.2.2, we find

u, (P + D) l P + v -(P D) l P v = N n=1 u n (d σ+,l,n v σ l+1 + (n) -d σ,l,n v σ l+1 (n) ) = n ∈G l+1 (σ+,σ) u n (d σ+,l,n v σ l+1 + (n) -d σ,l,n v σ l+1 (n) ) = n ∈G l+1 (σ+,σ) u n d σ+,l,n v σ l+1 + (n) - n ∈G l+1 (σ+,σ) u n d σ,l,n v σ l+1 (n) ≤d l max n ∈G l+1 (σ+,σ) u n v σ+(n) -d l min n ∈G l+1 (σ+,σ) u n v σ-(n) ≤d l max F + (l+1)m1 -d l min F - (l+1)m1 . (3.20)
This result combined with (3.15), gives

+∞ l=1 u, (P + D) l P + v -(P D) l P v ≤ +∞ l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1
≤s m 1 2

.

Considering now (3.19), we obtain

| u, (P + -P )v | ≥ +∞ l=1 u, (P + D) l P + v -(P D) l P v .
It follows that the first term of (3.17) dominates the second one. As a consequence, the former has the same sign as (3.16). The result follows.

Work in progress as of 7 th September, 2021

Implementation Remarks

Let us conclude with some Remarks on the computation of the function φ(m 1 ), more precisely of the sum in (3.14). Given m 1 ∈ {2, . . . , N }, define by l * such that

l * := N m 1 -1.
We have

+∞ l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1 = l * l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1 + +∞ l=l * +1 d l max F + (l+1)m1 -d l min F - (l+1)m1 = l * l=1 d l max F + (l+1)m1 -d l min F - (l+1)m1 + d l * +1 max 1 -d max F + N - d l * +1 min 1 -d min F - N .
As for the evaluation of s

Application to algal production

In this section, we apply the theory developed in the previous section to an algal production case and provide some numerical results to evaluate the efficiency of the mixing strategies and their approximation.

Raceway modeling

We consider the Han model [START_REF] Han | A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii[END_REF] to describe the dynamics of the photosynthetic units characterizing the growth of the algae after each allocation. Each light harvesting unit is assumed to have three different states: open and ready to harvest a photon (A), closed while processing the absorbed photon energy (B), or inhibited if several photons have been absorbed simultaneously leading to an excess of energy (C). Their dynamics is described by the following system where

       Ȧ = -σ H IA + 1 τ B, Ḃ = σ H IA - 1 τ B + k r C -k d σ H IB, Ċ = -k r C + k d σ H IB.
α(I) := k d τ (σ H I) 2 τ σ H I + 1 + k r , β(I) := k d τ (σ H I) 2 τ σ H I + 1 .
The net specific growth rate is obtained by balancing photosynthesis and respiration, which gives

µ(C, I) := -γ(I)C + ζ(I),
where

γ(I) := k H σ H I τ σ H I + 1 , ζ(I) := k H σ H I τ σ H I + 1 -R.
Here, R denotes the respiration rate and k H is a factor which relates the photosynthetic activity to the growth rate.

We assume that the system is perfectly mixed such that the biomass concentration (X) in (4.4) is homogeneous. Furthermore, it is often assumed that the photosynthetic units grow slowly so that the variations of biomass concentration (X) and background turbidity (α 1 ) are negligible over one lap of the raceway. As a consequence, the turbidity (α 1 ) and the biomass concentration (X) is supposed to be constant over the considered time scale. In this framework, we use the Beer-Lambert law to describe the light attenuation as a function of the depth z, i.e.

I(z) = I s exp(εz), (3.22)
where I s is the light intensity at the free surface and ε is the light extinction coefficient. The average net specific growth rate over the domain is then defined by

μ := 1 T T 0 1 h 0 -h µ C(t, z), I(z) dzdt, (3.23)
where h is the depth of the raceway pond and T is the average duration of one lap of the raceway pond.

Let us now see how this model can be included in the framework of Section 3.2. In order to compute numerically (3.23), we introduce a vertical discretization of the fluid. More precisely, we consider N layers uniformly distributed on a vertical grid. The depth of the layer n is given by

z n := - n -1 2 N h, n = 1, . . . , N. (3.24)
For a given initial photo-inhibition state C n (0), let C n (t) be the solution of (3.21) at time t.

In this semi-discrete setting, the average net specific growth rate in the raceway pond can be defined by

μN := 1 T T 0 1 N N n=1 µ(C n (t), I n )dt, (3.25)
where I n is the light intensity received in the layer n. The solution of (3.21) can be computed explicitly to get a formula that takes the form of (3.2). It follows that (3.25) satisfies

μN = 1 N T Γ, C(0) + 1, Z , (3.26)
Work in progress as of 7 th September, 2021 where 1 is a vector of size N whose coefficients are equal to 1, and Γ, Z are two vectors such that

Γ n := γ(I n ) α(I n ) (e -α(In)T -1), Z n := γ(I n )β(I n ) α(I n ) 2 (1 -e -α(In)T ) - γ(I n )β(I n ) α(I n ) T + ζ(I n )T.
The detail about the computations giving rise to (3.26) is presented in Appendix 3.A. In this framework, the light intensity at different layers represents the resource I n in the system and the photosynthesis at each layer represents the activity. Suppose now that at each lap, the algae at the layer n 1 are supposed to be entirely transferred to the layer n 2 when passing through the mixing device. During a lap, the photosynthesis of the algae follows a (3.1) type of evolution and the goal is to find the optimal resource allocation strategy to increase the productivity of the system. This mixing process is depicted schematically on an example in Figure 3.1. The interest of such a device is to mix the algae to better balance their exposure to 0 T 0 T

Layer four

Layer three

Layer two

Layer one light and increase the production. Note that in actual raceway ponds, this device is generally a paddle wheel (see for example [START_REF] Demory | How do microalgae perceive light in a high-rate pond? towards more realistic lagrangian experiments[END_REF]). Focusing on permanent regimes, we assume the system to be KT -periodic. From Theorem 3.2.1, we get immediately that C(0) = (I N -P D) -1 P V . Our goal is to maximize the average growth rate μN . Since only C(0) in (3.26) depends on the permutation matrix P , we find that the objective function has the form of (3.6) with u = Γ and v = V .

P z 0 -h z 1 = z σ(4) z 2 = z σ(1) z 3 = z σ(2) z 4 = z σ(3)

Parameter settings

Consider a raceway whose water elevation h = 0.4 m, which corresponds to typical raceway pond setting. All the numerical parameters values considered in this section for Han's model are taken from [START_REF] Grenier | Worldwide perspectives of rotating algal biofilm up-scaling[END_REF] and recalled in Table 3.1. Recall that I s is the light intensity at the free surface. In order to fix the value of the light extinction coefficient ε in (3.22), we assume that only a fraction q of I s reaches the bottom of the raceway pond, meaning that I b = qI s , where q ∈ [0, 1] and I b Work in progress as of 7 th September, 2021 60CHAPTER 3. Periodic resource allocation in dynamical systems, application to raceway microalgal production is the light intensity at the bottom. It follows that ε can be computed by

ε = (1/h) ln(1/q).
In practice, this quantity can be implemented in the experiments by adapting the biomass harvesting frequency, or the dilution rate for continuous cultivation. In what follows, the varying parameters are I s , the ratio q and T . We consider I s ∈ [0, 2500]µmol m -2 s -1 , q ∈ [0.1%, 10%] and T ∈ [1, 1000]s. The number of layers N remains small as we need to test numerically N ! permutation matrices for each triplet (I s , q, T ).

Examples of optimal devices

In this section, we present some examples of optimal solution of (3.6). Set N = 11 the number of layers, meaning that we test numerically N ! (i.e. 39916800) permutation matrices. The light intensity at the free surface is set to be I s = 2000 µmol m -2 s -1 which corresponds to a maximum value during summer in the south of France.

Let us start with a series of tests with the average time duration for one lap of the raceway pond T = 1000 s. When the light attenuation ratio q = 10%, we find that P max = I. When q = 1%, the optimal permutation matrix P max is given by (3.27).

P max =                  
0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

                  . ( 3

.27)

When q = 0.1%, the optimal permutation matrix P max is given by (3.28). For all three cases, Work in progress as of 7 th September, 2021

P + = P max . P max =                  
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

                  . (3.28)
We next study a much extreme case where the time duration of one lap T = 1 s. When the ratio q = 10%, we find the optimal matrix P max = I. When q = 1%, the optimal permutation matrix P max is a two-block matrix consisting of a block of identity and a block of anti-diagonal matrix with one as entries. This matrix is shown in (3.29).

P max =                  
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

                  . ( 3.29)
When q = 0.1%, we find the optimal matrix P max as an anti-diagonal matrix with one as entries. For all three cases, P + is an anti-diagonal matrix with one as entries. Note that among all the previous matrices, some of them are easier to achieve in real life application than others, for instance the anti-diagonal matrix or the identity/anti-diagonal two-block matrices (3.29).

Further numerical tests

As shown in previous section, Problem (3.6) admits non trivial optimal permutation strategies which may significantly change according to the parameter settings. In this section, we study and compare the true and the approximated solutions as well as their efficiency with respect to the average net specific (3.23). We start by investigating some properties of the items defined in the previous sections. Recall that the two sequences u, v used in Section 3.2.2 correspond in our application to Γ, V respectively. We consider N = 20 layers and two parameters triplets, namely (I s , q, T ) = (2000, 5%, 1000) and (800,0.5%,1). Figure 3.2 shows the evolution of these two quantities as a function of I. Note that in both cases, V is positive with sorted entries, as it can be seen in (3.3). On the contrary, the discretized Γ is negative and not necessarily sorted. We refer to Appendix 3.A for more details about V and Γ.

We then study the behaviour of the sequences F + m , F - m , s m and φ(m 1 ) defined in Section 3.2.4 for the same two parameters triplets. Note that since Γ is negative, F - m and F + m are defined as Work in progress as of 7 th September, 2021 ) with respect to m 1 . One can see in Figure 3.4 that the maximal value of φ(m 1 ) is always obtained for m 1 = 2, and that the maximal value φ(m 1 ) appears to be an increasing function of N . This makes the criterion given in Section 3.2.4 less efficient for a large number of layers N . Further analysis is required to obtain a criterion that does not depend on N .

The next test is devoted to the convergence of the average growth rate μN with respect to the number of layers N . We keep the two triplets of parameters of the previous test. Due to the limit of the computer memory, the computation of μN (P max ) is tractable for small values of N , in our case lower than or equal to N = 11. Such an issue does not occur in the case of μN (P + ). Figure 3.5 presents the behaviour of μN . For the parameter triplet (2000,5%,1000), the criterion is satisfied until N = 7 (green circle), which is confirmed in Figure 3.4 (Left) where the maximal value of φ(m 1 ) is already close to 1. Though the criterion is not satisfied for N > 7, we observe that P + = P max from N = 2 to N = 11 . As for the triplet (800,0.5%,1), one can see that P + = P max until N = 3. Figure 3.6 shows the optimal matrices for these two different parameter triplets in the case N = 11 and N = 100. It can be observed that for the parameter triplet (2000,5%,1000), the two matrices P + , P max have the same form for N = 11 and N = 100 (Figure 3.6 Top). Hence, one can expect P max = P + for larger N . However, this may not be the case for (800,0.5%,1) since P + , P max have already different forms for N = 11 (Figure 3.6 Bottom).

In the following tests, we focus only on two special cases: large lap duration time (T = 1000 s) and small lap duration time (T = 1 s). In practise, the former corresponds to typical time required to complete one lap in a raceway pond system, whereas the latter rather corresponds to the optimal case in a photobioreactor, for instance see [START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF]. In the small lap duration time case, we observe the so-called flashing effect. This phenomenon corresponds to the fact that the higher the light exposition frequency is, the larger the growth rate is. In other words, algae exposed to high frequency flashing, in our case, a small time duration for one lap, have a better growth. It can be observed in Figure 3.7, where μN (P max ) decreases with respect to T for all considered light intensities. This phenomenon has already been reported in literature, see, e.g. [START_REF] Hartmann | The effect of photosynthesis time scales on microalgae productivity[END_REF][START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF].

Work in progress as of 7 th September, 2021 To assess the influence of the light intensity at the free surface I s and the light attenuation ratio q, we compute μN for the optimal strategy associated with P max . We consider again N = 7 layers, I s ∈ [0, 2500], q ∈ [0.1%, 10%] and T ∈ [START_REF] Borowitzka | Limits to growth[END_REF]1000]. The results are shown in Figure 3.8. We observe that for a fixed light intensity at surface (I s ), the influence of the time duration (T ) Figure 3.8: Average net specific growth rate μN (P max ) for q ∈ [0.1%, 10%] when I s = 2000 µmol m -2 s -1 (Left) and for I s ∈ [0, 2500] when T = 1 s (Right). is very weak. Moreover, there exists an optimal value for % of the transmitted light (q) which is around 3%. We also find that for small values of q, there exists a non-trivial optimal light intensity at surface, e.g., I s ≈ 500 µmol m -2 s -1 for q = 0.1%.

We aim at now studying the influence of permutation strategies on the average of net specific growth rate μN . More precisely, we compute μN for the next four strategies: the optimal matrix P max that solves Problem (3.6), the worst matrix P min which minimizes (3.6), the no permutation case where P = I and the matrix P + which solves the approximate Problem (3.7). In our test, we consider N = 7 layers, I s ∈ [0, 2500], and q ∈ [0.1%, 10%]. Figure 3.9 presents the results for T = 1 s, T = 500 s and T = 1000 s.

We see that the original problem (3.6) and the approximated problem (3.7) coincide much more often for large values of the lap duration time T . In fact, the four surfaces become closer one to the others for large values of T .

Work in progress as of 7 th September, 2021 optimum. Nevertheless, we observe that the criterion (3.15) becomes less efficient for larger N . Note that the case corresponding to I s = 0 µmol m -2 s -1 is particular since no light is available in the system, implying that Γ, V equal to zero. In this case the value of the functionals do not depend on P . Hence μN (P max ) = μN (P + ) when

I s = 0 µmol m -2 s -1 .
Work in progress as of 7 th September, 2021 where P min ∈ P N is the matrix that minimizes J, (see (3.6)), i.e., that corresponds to the worse strategy. We consider N = 9 layers. Figure 3.11 presents the results for T = 1 s and T = 1000 s.

Better performance is in most cases obtained for a small time duration T = 1 s. In this way, we observe that the relative improvement between the best and the no mixing strategy may reach 15%, whereas the relative improvement between the worst and the best strategy may reach 30%.

In both two cases, a better improvement can be obtained with high values of I s and low values of q.

To compare the efficiency of the approximation P + with respect the true optimal mixing strategy P max , we define two extra ratios:

r1 := μN (P + ) -μN (I) μN (I) , (3.33)
r2 := μN (P + ) -μN (P min ) μN (P min ) .

Figure 3.12 presents the results for T = 1 s and T = 1000 s. As already mentioned, for a large lap duration time, the optimization problem (3.7) provides a good approximation. This can be observed with the blue and red surface in Figure 3.11 (Right) and in Figure 3.12 (Right), both surfaces have the same behaviours. As expected, the approximation becomes less efficient in the case of short lap duration time. This can be observed in Figure 3.11 (Left) and in Figure 3.12 (Left). However, the maximal values of r 1 , r 2 are still preserved by their approximations r1 , r2 . This is due to the fact that the maximal value is obtained at the extreme Work in progress as of 7 th September, 2021 case (i.e. I s = 2500 µmol m -2 s -1 and q = 0.1%) where μN (P max ) = μN (P + ) as shown in Figure 3.10 (Top).

Conclusion

We have presented a periodic resource allocation problem combined with a linear dynamical system. The periodicity of the problem enables us to reduce the computation to one assignment process. A significant computational effort is still required when dealing with larger number of N . We overcome this difficulty by defining a second optimization problem which has an explicit solution that coincide with the true solution when a given criterion is satisfied.

This developed theory is then applied to a microalgal production system with a mixing device. Non-trivial optimal mixing strategies can be obtained and the proposed second optimization problem provides a reliable approximation for large time duration T . Meanwhile, our experimental results show the significance of the choice of the mixing strategy: the relative ratio between the best and the worst case reaches 30% in some cases. We also observe a flashing effect meaning that better results are obtained when T goes to zero.

Further works will be devoted to the improvement of the function φ used in Theorem 3.2.2 in order to improve our approach for large number of N . More complicated dynamical system can also be considered including nonlinearities.

3.A Explicit Computations

In this appendix, we provide the computational details to solve (3.1) and (3.25) for an arbitrary number n ∈ {1, . . . , N }. Given two points t 1 , t 2 ∈ [0, T ]. Since I n is constant, Equation (3.1) can be integrated and becomes

x n (t 2 ) = e α(In)(t1-t2) x n (t 1 ) + β(I n ) α(I n ) (1 -e α(In)(t1-t2) ). (3.35)
Work in progress as of 7 th September, 2021 70CHAPTER 3. Periodic resource allocation in dynamical systems, application to raceway microalgal production

The time integral in (3.25) can be computed by

T 0 µ(C n (t), I n )dt = T 0 -γ(I n )C n (t) + ζ(I n )dt = -γ(I n ) T 0 C n (t)dt + ζ(I n )T.
Replacing x n by C n , t 2 by t and t 1 by 0 in (3.35) and integrating t from 0 to T gives

T 0 C n (t)dt = T 0 e -α(In)t C n (0) + β(I n ) α(I n ) (1 -e -α(In)t ) dt = C n (0) α(I n ) (1 -e -α(In)T ) + β(I n ) α(I n ) T - β(I n ) α 2 (I n ) (1 -e -α(In)T ).
Using notations given in Section 3.3.1, we have

Γ = γ(I) α(I) (e -α(I)T -1), V = β(I) α(I) (1 -e -α(I)T ).
From the definition of α(I), β(I), γ(I), we find

β(I) α(I) = β(I) β(I) + k r = k d τ (σ H I) 2 k d τ (σ H I) 2 + k r τ σ H I + k r , γ(I) α(I) = k H σ H I k d τ (σ H I) 2 + k r τ σ H I + k r .
Remark that Γ and V always have the opposite sign. Note also that I → β(I) α(I) is increasing on [0, +∞), which is not the case for I → γ(I) α(I) . It follows that V increases on R + and Γ is not monotonic on R + (see Figure 3.2).

3.B Optimization problem with arbitrary vectors

Let ũ, v ∈ R N two arbitrary vectors. Let Q ∈ P N such that u := Qũ has entries sorted in an ascending order. Since Q is a permutation matrix, we have Q T = Q -1 . For any P ∈ P N , let us denote by P := Q -1 P Q, we have P ∈ P N a permutation matrix. Let us denote by ṽ := Q -1 v and by D = Q -1 DQ. Note that D is still a diagonal matrix with a different order of the diagonal coefficients. Using this notation, we find for the objective function (3.6) satisfies

J(P ) := u, (I N -P D) -1 P v = ũ, Q -1 (I N -P D) -1 QQ -1 P QQ -1 v = ũ, Q -1 (I N -P D)Q -1 P ṽ = ũ, (Q -1 Q -Q -1 P QQ -1 DQ) -1 P ṽ = ũ, (I N -P D) -1 P ṽ .
For the objective function (3.7), we get

J approx (P ) := u, P v = ũ, Q -1 P QQ -1 v = ũ, P ṽ .
Therefore, these problems can still be treated similarly in the general case.

Work in progress as of 7 th September, 2021

3.C. Remark on F + m , F - m 71 3.C Remark on F + m , F - m Let u, v ∈ R N such
that the entries of u are sorted in an ascending order. One should be careful when defining the two sequences F + m and F - m in Section 3.2.4, since the sign of u and v plays an important role in the definition of these two sequences. For instance, assume that u is now negative and v is positive. Let ũ := -u, since u is assumed to be sorted in an ascending order, ũ is positive and sorted in a descending order. Using the definition in (3.13), one has

F + m := min(m,N ) n=1 ũn v σ+(n) , F - m := N n=max(1,N -m+1) ũn v σ-(2N -m-n+1) ,
where

v σ+(1) ≥ v σ+(2) ≥, . . . , ≥ v σ+(N ) and v σ-(1) ≤ v σ-(2) ≤, . . . , ≤ v σ-(N )
. Let us define by σ + := σ-and σ -:= σ+ . One has

F + m = - min(m,N ) n=1 u n v σ-(n) , F - m = - N n=max(1,N -m+1) u n v σ+(2N -m-n+1) .
Therefore, in this case we can define F + m and F - m by

F - m := min(m,N ) n=1 u n v σ-(n) , F + m := N n=max(1,N -m+1) u n v σ+(2N -m-n+1) .
The case where u is positive and v is negative, or both u, v are negative can be treated in a similar way.

Work in progress as of 7 th September, 2021

Chapter 4

Shape optimization of a microalgal raceway to enhance productivity

Introduction

Studies on the shape of the topography (or the bathymetry) have always been a challenging problem, since this problem usually involves the free-surface incompressible Navier-Stokes system, for instance see [START_REF] Bouharguane | Minimization principles for the evolution of a soft sea bed interacting with a shallow[END_REF][START_REF] Cocquet | Optimization of bathymetry for long waves with small amplitude[END_REF][START_REF] Van Dongeren | Beach wizard: Nearshore bathymetry estimation through assimilation of model computations and remote observations[END_REF][START_REF] Mohammadi | Optimal dynamics of soft shapes in shallow waters[END_REF]. In particular, explicit computation some time may be difficult to provide due to the complexity of the chosen model, meanwhile large computational resources are needed to provide numerical simulations and experimental results. The goal of this chapter is to investigate the influence of the topography in the raceway ponds. More precisely, we base on a common belief that some specific topographies can bring more light to the algae (at the lower part of the raceway), since they are closer to the surface when reaching the peak of these topographies.

In order to study this complex system, we develop a coupled model to describe the growth of algae in a raceway pond, accounting for the light that they receive. More precisely, this coupled model consists in combining the Han photosynthesis equations with an hydrodynamic law based on the Saint-Venant equations. This approach enables us to formulate an optimization problem where the raceway topography is designed to maximize the productivity. For this problem, we present an adjoint-based optimization scheme which includes the constraints associated with the Saint-Venant regime. On the contrary to a widespread belief, we prove that the flat topography is optimal in a periodic case for productivity in laminar regime. However, non-trivial topographies can be obtained in other contexts, e.g., when the periodic assumption is removed or an extra mixing strategy is included in the model. Note that in the examples considered in our numerical tests, such topographies only slightly improved the biomass production. Furthermore, some well chosen mixing strategies can boost this production.

The outline of this chapter is as follows. In Section 4.2, we present the biological and hydrodynamic models underlying our coupled model. Section 4.3 is devoted to the optimization problem and a corresponding numerical optimization procedure in different cases. We then provide some numerical results obtained with our approach in Section 4.4 and conclude in Section 4.5 with some perspectives opened by this work.

Coupling hydrodynamic and biological models

Our approach is based on a coupling between the hydrodynamic behavior of the particles and the evolution of the photosystems driven by the light intensity they received when traveling across the raceway pond.

Modeling the photosystems dynamics

We consider the Han model [START_REF] Han | A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii[END_REF] which describes the dynamics of the reaction centres. These subunits of the photosynthetic process harvest photons and transfer their energy to the cell to fix CO 2 . In this compartmental model, the photosystems can be described by three different states: open and ready to harvest a photon A, closed while processing the absorbed photon energy B, or inhibited if several photons have been absorbed simultaneously C. Their evolution satisfies the following dynamical system

   Ȧ = -σIA + B τ , Ḃ = σIA -B τ + k r C -k d σIB, Ċ = -k r C + k d σIB.
Here A, B and C are the relative frequencies of the three possible states with A + B + C = 1, and I is the photon flux density, a continuous time-varying signal. The other parameters are σ, that stands for the specific photon absorption, τ which is the turnover rate, k r which represents the photosystem repair rate and k d which is the damage rate. As shown in Subsection 1.2.1, one can reduce this system to a single evolution equation:

Ċ = -α(I)C + β(I), (4.1)
where α(I) = β(I) + k r with β(I) = k d τ (σI) 2 τ σI+1 . The net specific growth rate is obtained by balancing photosynthesis and respiration, which gives

µ(C, I) = -γ(I)C + ζ(I), (4.2)
where ζ(I) = γ(I) -R with γ(I) = kσI τ σI+1 . Here, k is a factor that relates received energy with growth rate and the term R represents the respiration rate. In particular for a constant light intensity I, the growth rate µ associated with the steady state of (4.1) is given by

µ(I) = -γ(I)C * (I) + ζ(I),
where

C * (I) = β(I) α(I) . (4.3)
In this framework, the dynamics of the biomass concentration X in an open system where a fraction of the biomass is continuously sampled and is derived from the growth rate µ:

Ẋ = (μ -D)X, (4.4)
where D is the dilution rate and μ is the average net growth rate which will be defined later.

Work in progress as of 7 th September, 2021

Steady 1D Saint-Venant equations

The Saint-Venant system is one of the most popular model for describing geophysical flows, which is derived from the free surface incompressible Navier-Stokes equations (see for instance [START_REF] Gerbeau | Derivation of viscous saint-venant system for laminar shallow water; numerical validation[END_REF]). In the current study, we focus on the smooth steady state solutions of the Saint-Venant equations in a laminar regime. Such steady states are governed by the following partial differential equations:

∂ x (hu) = 0, (4.5)
∂ x (hu 2 + g h 2 2 ) = -gh∂ x z b , (4.6)
where h is the water elevation, u is the horizontal averaged velocity of the water, the constant g stands for the gravitational acceleration, and z b defines the topography. The free surface η is given by η = h + z b and the averaged discharge Q = hu. This system is presented in Figure 4.1.

The z axis represents the vertical direction and the x axis represents the horizontal direction.

0 L z 0 x η(x) I s z b (x) h(x) u(x)
= Q 0 , (4.7)
for a fixed positive constant Q 0 , which implies a constant discharge in space. Then (4.6) can be rewritten as

hu∂ x u + h∂ x gh + h∂ x gz b = 0. (4.8)
Let us assume that h is strictly positive. Dividing (4.8) by h and using (4.7) to eliminate u, we get

∂ x Q 2 0 2h 2 + g(h + z b ) = 0.

This equation corresponds to the Bernoulli's principle. Now let us consider two fixed constants

Work in progress as of 7 th September, 2021

h(0), z b (0) ∈ R. For all x ∈ [0, L], we obtain Q 2 0 2h 2 + g(h + z b ) = Q 2 0 2h 2 (0) + g(h(0) + z b (0)) =: M 0 ,
meaning that the topography z b satisfies

z b = M 0 g - Q 2 0 2gh 2 -h. (4.9)

Remark

Define the Froude number for the steady state by F r = u/ √ gh. The situation F r < 1 corresponds to the subcritical case (i.e. the flow regime is fluvial) while F r > 1 is to the supercritical case (i.e. the flow regime is torrential). In particular, the threshold value of h for F r = 1 is given by

h c := ( Q 2 0 g ) 1 3
.

Because of (4.9), the water depth h is the solution of a third order polynomial equation. Given a smooth topography z b , if

h c + z b + Q 2 0 2gh 2 c - M 0 g < 0.
there exists a unique positive smooth solution of (4.9) which satisfies the subcritical flow condition (see [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF]Lemma 1]). The velocity u can then be obtained thanks to (4.7).

Lagrangian trajectories of the algae and captured light intensity

Let z(t) be the depth of a particle at time t in the raceway pond. We first determine the Lagrangian trajectory of an algal cell that starts at a given position z(0) at time 0. From the incompressibility of the flow, we have ∇ • u = 0 with u = (u(x), w(x, z)). Here, w(x, z) is the vertical velocity. This implies that

∂ x u + ∂ z w = 0. (4.10)
Integrating (4.10) from z b to z gives:

0 = z z b ∂ x u(x) + ∂ z w(x, z) dz = ∂ x z z b u(x)dz + z z b ∂ z w(x, z)dz = ∂ x (z -z b )u(x) + w(x, z) -w(x, z b ) = (z -z b )∂ x u(x) -u(x)∂ x z b + w(x, z),
where we have used the kinematic condition at the bottom (i.e. w(x, z b ) = u(x)∂ x z b ). It then follows from (4.9) that

w(x, z) = ( M 0 g - 3u 2 (x) 2g -z)u (x).
Work in progress as of 7 th September, 2021

The Lagrangian trajectory is characterized by the system

ẋ(t) ż(t) = u(x(t)) w(x(t), z(t)) . (4.11)

Remark

The geometry of the raceway pond with small dissipation and shear effects (reduced wall friction and viscosity) justifies a laminar flow modelled by a shallow water model, such as the Saint-Venant system. This regime also minimizes the mixing energy, hence is favoured at industrial scale. A higher mixing energy would lead to a turbulent regime. A possible way to enrich the representation of the Lagrangian trajectories in this case would consists in adding a Brownian motion to the definition (4.11). However, getting time-free expressions of the trajectories (as in Equations (4.12) and (4.16)) in this case is much more challenging so that such a strategy optimisation procedure would require a large set of simulations together with an averaging strategy.

The previous expression of the Lagrangian trajectory (4.11) is the general formulation which is still available even if we change the hydrodynamical model. However based on the special form of the Saint-Venant system that we have chosen, we can find a simpler formulation of the Lagrangian trajectory. More precisely, we denote by z(x) the depth of a particle at the position x. From (4.11), we get

z := ż ẋ = ( M 0 g - 3u 2 2g -z) u u . (4.12)
Note that from (4.7) and (4.9), we have

η = h + z b = M0 g -u 2 2g
, which implies η = -uu /g. Multiplying (4.12) both sides by u and using the formulation of η, η gives

z u + zu = (η - u 2 g )u = ηu + η u,
which implies that u(z -η) = 0. We then obtain

z(x) = η(x) + u(0) u(x) (z(0) -η(0)).
Using (4.7), then the equation above can be re-written as

z(x) = η(x) + h(x) h(0) (z(0) -η(0)). (4.13)
The computation of trajectories in the Saint-Venant system can be carried out with this formula, where h(0), z(0) and η(0) = h(0) + z b (0) are given as data of the system. On the contrary, h(x), η(x) will depend on the parameterization of the system. Indeed, if we choose to parameterize the topography z b (x), then the water depth h(x) can be found naturally by (4.9) and it will then be used to compute the free surface η

(x) = h(x) + z b (x).
Work in progress as of 7 th September, 2021

Remark

Since Q 0 is chosen to be positive, h is necessarily positive and so does u from (4.7). Moreover, from (4.13), if z(0) belongs to [z b (0), η(0)], then z(x) belongs to [z b (x), η(x)]. In particular, choosing z(0) = z b (0) in (4.13) and using (4.7) gives z(x) = z b (x). In the same way, we find that z(x) = η(x) when z(0) = η(0).

We assume that the system is perfectly mixed such that the biomass concentration X in (4.4) is homogeneous. Furthermore, it is often assumed that the photosynthetic units grow slowly so that the variations of biomass concentration X and background turbidity α 1 are negligible over one lap of the raceway. As a consequence, the turbidity α 1 and the biomass concentration X are supposed to be constant over the considered time scale. In this framework, we use the Beer-Lambert law to describe the light intensity observed on the trajectory z:

I(x, z) = I s exp -ε(η(x) -z) . (4.14)
Here ε is the light extinction coefficient. Combining (4.14) with (4.13), we get the following expression for the captured light intensity along the trajectory z

I(x, z(x)) = I s exp -ε h(x) h(0) (η(0) -z(0)) . (4.15)
It follows that in this approach, the light intensity I couples the hydrodynamic model and Han model: the hydrodynamical movement provides the trajectories of the algae and then define the received light intensity, which is used in the photosystem dynamics.

We can then derive the equation satisfied by C. Indeed, repeating the reasoning done to get (4.12) with (4.1), we find a time-free reformulation, namely

C := Ċ ẋ = - α(I) Q 0 hC + β(I) Q 0 h, (4.16)
where all functions on the right-hand side only depend on x.

Optimization problem

In this section, we define the optimization problem associated with the biological-hydrodynamical model. We first introduce our procedure in the case of one single layer, and then extend this to a multiple layers system. The volume of our 1D system V [m 2 ] is defined by

V = L 0 h(x)dx. (4.17)
And as we have mentioned in Subsection 4.2.2, a given topography z b corresponds to a unique water depth h in the subcritical case. Therefore, we choose to parameterize h by a vector a ∈ R N , which will be the variable to be optimized, in order to handle the volume of our system. In this way, we assume the volume is constant with respect to the parameter a. Given a vector a and the associated h, the optimal topography can be obtained by means of (4.9).

Work in progress as of 7 th September, 2021

Remark

In usual Saint-Venant solver, equations of type (4.9) are usually consider to compute water depth h for a given topography z b in the simulations. Here, we use this equation in the opposite way, i.e., to recover the topography z b from water depth h.

Optimization problem for non periodic case with time dependent description

In this section, we study the optimization problem for the time dependent description. Note that in this case the Lagrangian trajectory is defined by (4.11) and the photoinhibition state is defined by (4.1). For sake of clarity, we first present the formulation of the optimization problem for a single trajectory, then we generalize it for multiple trajectories. For simplicity, we omit t in the notations.

One single trajectory

Our goal here is to optimize the topography to maximize the average net growth rate for one trajectory μ1 which is defined by The optimal control problem then reads: Find a solving the maximization problem:

max a∈R N μ1 (a). (4.19)
In order to solve this optimization problem, we apply the technique presented in Subsection 1. where p 1 , p 2 and p 3 are the Lagrange multipliers associated with the constraints (4.18). The optimality system is obtained by cancelling all the partial derivatives of L. Differentiating L with respect to p 1 , p 2 , p 3 and equating the resulting terms to zero gives the corrected model equations (4.18). Integrating the terms p 1 Ċdt, p 2 żdt and p 3 ẋdt on the interval [0, T ] by Work in progress as of 7 th September, 2021 parts and differentiating L with respect to C, z, x, C(T ), z(T ), x(T ) gives rise to

                   ∂ C L = -1 T γ(I) + ṗ1 -α(I)p 1 ∂ z L = 1 T -γ (I)C + ζ (I) + p 1 -α (I)C + β (I) ∂ z I + ṗ2 + p 2 ∂ z w ∂ x L = 1 T -γ (I)C + ζ (I) + p 1 -α (I)C + β (I) ∂ x I + p 2 ∂ x w + ṗ3 + p 3 ∂ x u ∂ C(T ) L = p 1 (T ) ∂ z(T ) L = p 2 (T ) ∂ x(T ) L = p 3 (T ).

Remark

Note that in one trajectory case, all the formulations above are presented in continuous form which may not be the case for multiple trajectories.

Multiple trajectories Problem

We now extend the previous procedure to deal with multiple layers. Let us denote by N z the number of layers and C i the photoinhibition state associated with the trajectory z i . We consider the semi-discrete average net growth rate over the domain as the objective function, namely:

μNz (a) = 1 N z Nz i=1 1 T T 0 µ(C i , I i (a))dt, (4.22)
where C i , x, z i verify the constraints (4.18) for i = 1, . . . , N z .

Remark

Note that the average net growth rate over the domain is defined by:

μ∞ := 1 T T 0 1 h (x(t)) η(x(t)) z b (x(t))
µ C (x(t), z(t)) , I (x(t), z(t)) dzdt.

(4.23)

Our approach consequently consists in considering a vertical discretization of μ∞ , which gives (4.22). This discretization should not give rise to any problem and is left to Subsection 4.3.2 for further details.

Work in progress as of 7 th September, 2021

The Lagrangian of the problem is defined by where p 1,i , p 2,i , p 3 the associated Lagrange multipliers with C i , z i , x respectively. Similar computations as in the previous section give rise to the following systems

L(C i , z i , x, p 1,i , p 2,i ,
                                   ∂ Ci L = -1 T Nz γ(I i ) + ṗ1,i -α(I i )p 1,i ∂ zi L = 1 T Nz -γ (I i )C i + ζ (I i ) + p 1,i -α (I i )C i + β (I i ) ∂ zi I i + ṗ2,i + p 2,i ∂ z w ∂ x L = Nz i=1 1 T N z -γ (I i )C i + ζ (I i ) + p 1,i -α (I i )C i + β (I i ) ∂ x I i + Nz i=1 p 2,i ∂ x w i + ṗ3 + p 3 ∂ x u ∂ Ci(T ) L = p 1,i (T ) ∂ zi(T ) L = p 2,i (T ) ∂ x(T ) L = p 3 (T )
where I i := I(x, z i ; a) and w i := w(x, z i ; a). Finally, the gradient ∇μ Nz (a) is given by

∂ a L = Nz i=1 T 0 1 T N z -γ (I i )C i + ζ (I i ) + p 1,i -α (I i )C i + β (I i ) ∂ a I i dt + Nz i=1 T 0 p 2,i ∂ a w i dt + T 0 p 3 ∂ a udt.

Optimization problem for periodic case with time independent description

In this section, we study the optimization problem for the time independent description. Note that in this case the Lagrangian trajectory is defined by (4.13) and the photoinhibition state is defined by (4.16). We first present the formulation of the optimization problem then we add periodic assumption to investigate the optimal topography in this case.

Optimality system

The parameter-dependent average net growth rate over the domain is defined by Work in progress as of 7 th September, 2021

Note that this formulation is actually the same as (4.23) (see Appendix 4.B). In order to tackle numerically this optimization problem, let us consider a vertical discretization. Let us keep N z as the number of particles, and consider a uniform vertical discretization of their initial position:

z i (0) = η(0) - i -1 2 N z h(0), i = 1, . . . , N z .
From (4.13), we obtain

z i (x) -z i+1 (x) = 1 N z h(x), i = 1, . . . , N z ,
meaning that the distribution of particles remains uniform along the trajectories. To simplify notations, we write I i (a) instead of I(x, z i (x); a) hereafter. Then the semi-discrete average net specific growth rate in the raceway pond can be defined from (4.24) by

μNz (a) := 1 V L 0 Nz i=1 1 N z h(a)µ(C i , I i (a))dx = 1 V N z Nz i=1 L 0 µ(C i , I i (a))h(a)dx. (4.25)
where C i satisfies the following parameterized version of (4.16)

C i = (-α (I i (a)) C i + β (I i (a))) h(x; a) Q 0 . (4.26)

Remark

One of the advantages of the time free formulation is that this expression helps us to reduce the computational cost. Indeed, in this case the optimization constraint (4.26) reduces to one single constraint comparing to the constraints (4.18) in time dependent formulation.

In this case, the optimization problem reads:

Find a * solving the maximization problem:

max a∈R N μNz (a). (4.27)
The Lagrangian of the problem is then defined by

L(C i , a, p i ) = Nz i=1 L 0 1 V N z -γ(I i (a))C i + ζ(I i (a)) h(a) -p i C i + α(I i (a)) -β(I i (a)) Q 0 h(a) dx
where p i the associated Lagrange multipliers. Similar computations as in the previous section give rise to the following systems

∂ Ci L = -1 V Nz γ(I i )h + p i -α(Ii) Q0 hp i , ∂ Ci(T ) L = p i (T ).
(4.28)

As we have mentioned above, one can also see that the optimality system has been much simplified comparing to (4.20).

Work in progress as of 7 th September, 2021

Finally, the partial derivative ∂ a L (hence the gradient ∇μ Nz (a)) is then given by

∂ a L = Nz i=1 L 0 1 V N z -γ (I i ) C i + ζ (I i ) + p i -α (I i )C i + β (I i ) Q 0 h∂ a I i dx + Nz i=1 L 0 1 V N z -γ (I i ) C i + ζ (I i ) + p i -α(I i )C i + β(I i ) Q 0 ∂ a hdx. (4.29)

Periodic problem

We now consider a variant of our problem, where the photoinhibition state C is periodic, with a period corresponding to a raceway lap. This situation occurs, e.g., when an appropriate harvest is performed after each lap. To describe the corresponding model, let us first consider a variant of the usual Cauchy problem for (4.16):

Given I ∈ L 2 (0, L; R), find (C 0 , C) ∈ [0, 1] × C(0, L; [0, 1]) such that C (x) = -α(I(x))C(x)+β(I(x)) Q0 h(x), x ∈ [0, L] C(L) = C(0) = C 0 .
(4.30)

A similar theorem as Theorem 1.3.2 gives the existence and uniqueness of a (weak) solution of the problem (4.30). The proof follows the same technique given in Subsection 1.3. The optimality system in this case is obtained as shown in the previous sections.

Remark

Note that the periodicity of C i implies that p i is also periodic. Indeed, since C i (L) = C i (0), then by differentiating L with respect to C i (L), we have ∂ Ci(L) L = p i (L) -p i (0), so that equating the above equation to zero gives the periodicity for p i .

Next, we state a result in the case (C i ) i=1,...,Nz are periodic. Theorem 4.3.2. Assume the volume of the system V is constant. Then ∇μ Nz (0) = 0.

Proof. Replacing the formulation for z i (4.13) in (4.14) gives

I i (a) = I s exp(-ε i -1 2 N z h(a)). (4.32)
The partial derivative of I with respect to a gives

∂ a I i = -ε i -1 2 N z I i ∂ a h.
Work in progress as of 7 th September, 2021

Consider now the vector a f for representing the parameter in the case where the topography is flat. In this way, the water depth h(a f ) hence the light intensity I i (a f ) are all constant with respect to x that denote by h f and I f i respectively. For a given C(0), (4.26) gives

C i (x) = e -α(I i (a f )) Q 0 h(a f )x C i (0) + β(I i (a f )) α(I i (a f )) (1 -e -α(I i (a f )) Q 0 h(a f )x ). (4.33)
Since C i are periodic (i.e. C i (L) = C i (0)), we get from the previous equation that

C i (0) = β(Ii(a f ))
α(Ii(a f )) . Inserting this value in (4.33), we find

C i (x) = β(I i (a f )) α(I i (a f )) , ∀x ∈ [0, L],
which corresponds to the steady state (4.3), that we denote by C f i hereafter. Similar operations applying on the dynamics of p i

p i = 1 V N z γ(I i )h + α(I i ) Q 0 hp i , gives p i (x) = - Q 0 γ(I i (a f )) V N z α(I i (a f )) , ∀x ∈ [0, L],
that denote by p f i hereafter. On the other hand, since the volume of the system is assumed to be constant with respect to a, one has

0 = ∂ a V = ∂ a L 0 h(x; a)dx = L 0 ∂ a hdx.
Replacing a by a f in (4. [START_REF] Chiaramonti | Review of energy balance in raceway ponds for microalgae cultivation: Re-thinking a traditional system is possible[END_REF] gives

∇μ Nz (a f ) = Nz i=1 L 0 1 V N z -γ I f i C f i + ζ I f i + p f i -α (I f i )C f i + β (I f i ) Q 0 h f -ε i -1 2 N z I f i ∂ a hdx + Nz i=1 L 0 1 V N z -γ I f i C f i + ζ I f i + p f i -α(I f i )C f i + β(I f i ) Q 0 ∂ a hdx = Nz i=1 1 V N z -γ I f i C f i + ζ I f i + p f i -α (I f i )C f i + β (I f i ) Q 0 h f -ε i -1 2 N z I f i + 1 V N z -γ I f i C f i + ζ I f i + p f i -α(I f i )C f i + β(I f i ) Q 0 L 0 ∂ a hdx =0.
This concludes the proof.

Work in progress as of 7 th September, 2021

Remark

This theorem consequently shows that in the case C periodic, the flat topography is a critical point for Problem (4.27). On the contrary, in the case C non periodic, (4.33) implies that C depends also on the space variable x, hence the computations above will no longer hold. In other words, the flat topography is not an optimum for the case C non periodic, These has been confirmed by our numerical tests. In particular, we have observed that the flat topography is actually optimal in the periodic case (see Subsection 4.4.3).

Optimization for non-constant volume problem

In this section, we focus on the case where the volume of the system also changes hence can be optimized. Let us define a volume related parameter a 0 as the average depth of the raceway system:

a 0 := 1 L L 0 h(x)dx = V L . (4.34)
In this way, the biomass concentration changes with the system volume, therefore the light extinction ε can no longer be assumed as a constant. Hence, we consider the general form of ε as shown in (1.17). As mentioned in Chapter 2, one of the solution to find the relation between the biomass concentration X with the system volume V is to use the optimal optical depth Y opt defined by (2.9). This consists in regulating the biomass concentration X to a value such that the steady state value of the net growth rate µ at the average depth a 0 is 0, i.e.

µ(I a0 ) = 0. (4.35)
This gives us the following relation:

Y opt = ε(X)a 0 = (α 0 X + α 1 )a 0 ,
or in other words

X = Yopt a0 -α 1 α 0 .

Optimize areal productivity

In this framework, maximizing areal productivity is a relevant target. Productivity per unit of surface for a given biomass concentration X is given by:

Π := μX V S ,
where S is the ground surface of the raceway pond and μ is defined in (4.24). Note that this is a general form of the surface biomass productivity defined in (2.6). Note that the water depth h now depends both on the parameter a and a 0 , which we will denote by the extended parameter vector ã = [a 0 , a] ∈ R N +1 hereafter. Moreover, since we consider a 1D framework, we have S = L. Let us consider the parameter-dependent objective function:

Π(ã) = Y opt -α 1 a 0 α 0 1 V N z Nz i=1 L 0 µ(C i , I i (ã))h(ã)dx,
Work in progress as of 7 th September, 2021 where C i satisfies (4.26) with a replacing by ã.

The corresponding optimization problem reads: Find ã * solving the maximization problem:

max ã∈R N +1 Π(ã).
The Lagrangian associated with this problem reads as

L(C i , p i , ã) = Y opt -α 1 a 0 V N z α 0 Nz i=1 L 0 -γ(I i (ã))C i + ζ(I i (ã)) h(ã)dx - Nz i=1 L 0 p i C i + α(I i (ã))C i -β(I i (ã)) Q 0 h(ã) dx,
where p i is the Lagrangian multiplier associated with the constraint (4.26) for C i . By computations similar to that of the previous section, we find the optimality system as

∂ Ci L = p i -p i α(I) Q0 h -γ(I)h Yopt-α1a0 V Nzα0 ∂ Ci(L) L = p i (L).
However, there is an extra element in the gradient Π(ã), namely the derivative with respect to a 0 , meaning that ∇Π(ã

) := [∂ a0 L, ∂ a L],
where

∂ a0 L = Nz i=1 L 0 Y opt /a 0 -α 1 LN z α 0 -γ (I i ) C i + ζ (I i ) + p i -α (I i )C i + β (I i ) Q 0 h∂ a0 I i dx + Nz i=1 L 0 Y opt /a 0 -α 1 LN z α 0 -γ (I i ) C i + ζ (I i ) + p i -α(I i )C i + β(I i ) Q 0 ∂ a0 hdx - Nz i=1 L 0 Y opt /a 2 0 LN z α 0 -γ (I i ) C i + ζ (I i ) hdx.
(4.36)

∂ a L = Nz i=1 L 0 Y opt -α 1 a 0 V N z α 0 -γ (I i ) C i + ζ (I i ) + p i -α (I i )C i + β (I i ) Q 0 h∂ a I i dx + Nz i=1 L 0 Y opt -α 1 a 0 V N z α 0 -γ (I i ) C i + ζ (I i ) + p i -α(I i )C i + β(I i ) Q 0 ∂ a hdx.. (4.37)

Periodic problem

Note that the water depth h for a flat topography is a constant (meaning that vector a = 0).

From the definition of a 0 (4.34), one has in particular

h(x; [a 0 , 0]) = a 0 , ∀x ∈ [0, L],
which means that there exists a function h 1 such that h(x; ã) := a 0 + h 1 (x; a),

∂ a0 h = 1, h(ã f ) = a 0 . Consider i ∈ {1, • • • , N z }. Since I i (ã) = I s exp - Y opt a 0 i -1 2 N z h(ã) ,
we find that

∂ a0 I i (ã f ) = -(- Y opt a 0 + Y opt a 0 ) i -1 2 N z I i (ã f ) = 0.
Hence the first sum in (4.36) is zero. Keeping the notation of Theorem 4.3.2, and using that

I i (ã f ) = I f i , we obtain that C i := C f i . In particular -α(I f i )C f i + β(I f i ) = 0, so that (4.36) gives ∂ a0 L(ã f ) = - Nz i=1 L 0 α 1 LN z α 0 -γ I f i C f i + ζ I f i dx.
Since µ(I s ) > 0, µ(I a0 ) = 0 and µ is concave with respect to I, one has µ(

I f i ) = -γ I f i C f i + ζ I f i > 0.
In other words, ∂ a0 L(ã f ) = 0. This concludes the proof.

Remark

Note that the coefficient a 0 considered in the previous theorem needs to satisfy h c ≤ a 0 to guarantee that the system remains in a subcritical regime (see Remark 4.2.2).

Optimization problem in periodic case for topographies combining with mixing strategies

In this section, we present the optimization problem by combining the topography with the mixing strategy P ∈ P N . As shown in Chapter 3 in a periodic case, the period is actually one for whatever the permutation matrix is. Therefore, we only focus on the periodic case with period one in the following. In particular, we cannot keep the proof of the critical topography in Theorem 4.3.1 since an extra matrix P is added into the system so that the solution of the state C i and p i in periodic case can no longer be constant.

Constant reactor volume

In this way, we first define the objective function which is similar as (4.25) for the constant volume case:

μP Nz (a) := 1 V N z Nz i=1 L 0 µ(C P i , I i (a))h(a)dx, (4.39)
Work in progress as of 7 th September, 2021

where C P i satisfies the following parameterized version of (4.16) with a periodic condition

(C P i ) = -α(I i (a))C i + β(I i (a)) h(a) Q0 , P C P i (L) = C P i (0). (4.40)
Our optimization problem then reads: Find a permutation matrix P max and a parameter vector a * solving the maximization problem:

max P ∈P N max a∈R N μP ∆ (a).
For a given permutation matrix P ∈ P N , the Lagrangian of (4.39) can then be written by

L(C i , p i , a) = 1 V N z Nz i=1 L 0 -γ(I i (a))C P i + ζ(I i (a)) h(a)dx - Nz i=1 L 0 p P i (C P i ) + α(I i (a))C P i -β(I i (a)) h(a) Q 0 dx,
where p P i is the Lagrange multiplier associated with the constraint (4.40). The optimality system is obtained by cancelling all the partial derivatives of L. Differentiating L with respect to p P i and equating the resulting expression to zero gives (4.40). Integrating the terms p P i (C P i ) dx on the interval [0, L] by parts enables to differentiate L with respect to C P i and C P i (L). Equating the result to zeros gives rise to

(p P i ) -p P i α(I i (a)) h(a) Q0 -h(a) V Nz γ(I i (a)) = 0, p P i (L) -p P i (0)P = 0. (4.41)
Given a vector a, let us still denote by C P i , p P i the corresponding solutions of (4.40) and (4.41). The gradient ∇μ P Nz (a) is obtained by ∇μ P Nz (a) = ∂ a L, where

∂ a L = Nz i=1 L 0 1 V N z -γ (I i )C P i + ζ (I i ) + p P i -α (I i )C P i + β (I i ) Q 0 h∂ a I i dx + Nz i=1 L 0 1 V N z -γ(I i )C P i + ζ(I i ) + p P i -α(I i )C P i + β(I i ) Q 0 ∂ a hdx.

Variable reactor volume

In this section, we focus on the case where the reactor volume can also vary. Consider the extend parameter vector ã := [a 0 , a] ∈ R N +1 . From (4.39), the objective function is given by

Π P (ã) := Y opt -α 1 a 0 V N z α 0 Nz i=1 L 0 µ(C P i , I i (a))h(a)dx. (4.42)
The corresponding optimization problem reads: Find a permutation matrix P max and a parameter vector ã * solving the maximization problem:

max P ∈P N max ã∈R N +1 Π P (ã).
Work in progress as of 7 th September, 2021

Let us denote by L the Lagrangian associated to (4.42). We follow the same optimization procedure presented in previous sections. Note that an extra element appears in this gradient, which is the partial derivative of L with respect to the variable a 0 . More precisely, we have

∇Π P (ã) = [∂ a0 L, ∂ a L] where ∂ a0 L = Nz i=1 L 0 Y opt /a 0 -α 1 LN z α 0 -γ (I i )C P i + ζ (I i ) + p P i -α (I i )C P i + β (I i ) Q 0 h∂ a0 I i dx + Nz i=1 L 0 Y opt /a 0 -α 1 LN z α 0 -γ(I i )C P i + ζ(I i ) + p P i -α(I i )C P i + β(I i ) Q 0 ∂ a0 hdx - Nz i=1 L 0 Y opt /a 2 0 LN z α 0 -γ(I i )C P i + ζ(I i ) hdx
and ∂ a L is similar to the constant volume case.

Numerical Experiments

In this section, we will illustrate some optimal topographies obtained in various frameworks as presented in Section 4.3.

Numerical Methods

We start with an algorithm to solve the optimization problem and the numerical solver that we use for our numerical tests.

Gradient Algorithm

We detail now a gradient-based optimization procedure and apply it to Problem (4.27). The procedure is given in Algorithm 2.

Algorithm 2 Gradient-based optimization algorithm 1: Input: Tol> 0, ρ > 0. Compute I i by (4.15).

7:

Set C i as the solution of (4.26).

8:

Set p i as the solution of (4.28).

9:

Compute the gradient ∇J by (4.29). Set err := ∇J . 12: end while In addition to a numerical tolerance criterion on the magnitude of the gradient, we need to take into account a constraint on the water depth h to guarantee that the simulated flow remains in a subcritical regime (min x∈[0,L] h(x) > h c ), see Remark 4.2.2 (and in the range of industrial Work in progress as of 7 th September, 2021 constraints, see [START_REF] Chiaramonti | Review of energy balance in raceway ponds for microalgae cultivation: Re-thinking a traditional system is possible[END_REF]). Remark that since no interaction between trajectories is considered, the gradient computation can be partially parallelized when computing I i (hence C i ) and p i .

Numerical Solvers

To solve our optimization problem numerically, we introduce a supplementary space discretization with respect to x. In this way, let us take a space increment ∆x, set N x = [L/∆x] and x nx = n x ∆x for n x = 0, . . . , N x . We choose to use the Heun's method for computing C i via (4.26). Following a first-discretize-then-optimize strategy, we get that the Lagrange multiplier p i is also computed by a Heun's type scheme. Note that this scheme is still explicit since it solves a backward dynamics starting from p i (L) = 0.

Parameter settings

In this section, we provide the parameter used in the numerical test and the parameterization of the water depth h.

Parameter for the models

The spatial increment is set to ∆x = 0.01 m so that the convergence of the numerical scheme has been ensured and the numerical tolerance is set to Tol= 10 -10 . We set the averaged discharge Q 0 = 0.04 m 2 s -1 and z b (0) = -0.4 m to stay in standard ranges for a raceway. The freefall acceleration g = 9.81 m s -2 . We choose I s = 2000 µmol m -2 s -1 which approximates the maximum light intensity, e.g., at summer in the south of France. All the numerical parameters values for Han's model are taken from [START_REF] Grenier | Worldwide perspectives of rotating algal biofilm up-scaling[END_REF] and recalled in Table 4.1. In order to determinate the light extinction coefficient ε, two cases must be considered:

• constant volume: let us assume that only 1% of light can be captured by the cells at the bottom of the raceway, i.e. for z = z b , meaning that I b = 0.01I s . Then ε can be computed by ε = (1/h(0; a)) ln(I s /I b ).

• non-constant volume: in this case, the volume related parameter a 0 is also to be optimized and one needs to use (1.17) to take into account of the variation of the volume. We take from [START_REF] Martínez | Theory of turbid microalgae cultures[END_REF] the specific light extinction coefficient of the microalgae specie α 0 = 0.2 m 2 • g and the background turbidity α 1 = 10 m -1 .

Work in progress as of 7 th September, 2021

Parameterization

In order to describe the bottom of an optimal raceway pond, we choose to parameterize h by a truncated Fourier series for our numerical tests. More precisely, h reads:

h(x) = a 0 + N i=1 a i sin(2nπ x L ). (4.43)
The parameter to be optimized is then the Fourier coefficients a := [a 1 , . . . , a N ]. Although other parameterizations can also be considered, in this chapter, we focus only on this parameterization based on the following reasons :

• We consider a hydrodynamic regime where the solutions of the Saint-Venant equations are smooth and hence the water depth can be approximated by (4.43). Hence, the truncated Fourier number N is set to be small and limit situations where N → +∞ are not considered in what follows.

• One has naturally h(0) = h(L) under this parameterization, which means that we have accomplished one lap of the raceway pond.

• We assume a constant volume of the system V , which can be achieved by fixing a 0 . Indeed, under this parameterization and using (4.17), one finds V = a 0 L.

From (4.7) and (4.9), the velocity u and the topography z b read also as functions of a. Once we find the vector a maximizing the functional μNz , we then find the optimal topography of our system. Note that in constant volume case, a 0 is fixed to be 0.4, and in non constant volume case, we keep a 0 = 0.4 as the initial guess.

Numerical results

In this section, we provide the numerical support for the optimization problems presented in Section 4.3.

Convergence of the vertical discretization

The first test consists in studying the influence of the vertical discretization number N z . For this purpose, let us set the length L = 10 m and the truncated Fourier number N = 5. We take 100 random initial guesses of a. Note that the choice of a should respect the subcritical condition.

For N z varies from 1 to 80, we compute the average value of μNz for each N z . The results are shown in Figure 4.2. We observe numerical convergence when N z grows, showing the convergence towards continuous model. In view of these results, we take N z = 40 for the successive studies.

Influence of the truncated Fourier value

The second test is given to study the influence of the truncated Fourier value N on the optimal average growth rate μNz . Let us set N = [0, 5, 10, 15, 20] and set flat topography as the initial guess, meaning that a is set to 0. Meanwhile, we keep the length L = 10 m as in previous test. Table 4.2 shows the optimal value of our objective function μNz for different values of N .

There is a slight increase of the optimal value of the average growth rate μNz when N becomes larger. However, the corresponding values of μNz remain close to the one associated with a flat topography.

Work in progress as of 7 th September, 2021 We focus now on the shape of the optimal topography in the case C non periodic. We choose N = 5 the truncated Fourier series and keep the length L = 10 m as an example to show the shape of the optimal topography. We consider the flat topography as the initial guess. The optimal shape of the topography is shown in Figure 4.3. In particular, the resulting optimal topography is not flat and the a * for the final iteration reads a * = [0.1043, 0.0503, 0.0333, 0.0250, 0.0201].

Optimal topography shape in periodic case

In this test, we focus on the optimization problem in the case C periodic. We keep the same parameter settings as in previous case and consider a random topography as initial guess. The optimal shape of the topography is shown in Figure 4.4. As shown in this figure, the resulting optimal topography is flat in this case, meaning that the flat topography, proved to be a critical topography in Theorem 4.3.1, is actually the optimal topography.

Example with a mixing device

The next test aims to simulate a more realistic raceway pond situation where a mixing device is considered in the system. More precisely, we simulate several laps, with a mixing device that mix up the algae after each laps. The mixing device is modelled by a permutation matrix P that rearrange the trajectories at each lap. In our test, P is an anti-diagonal matrix with the entries one. This choice actually corresponds to an optimum in some parameter settings and has been Work in progress as of 7 th September, 2021 Work in progress as of 7 th September, 2021

shown in [START_REF] Bernard | Optimizing microalgal productivity in raceway ponds through a controlled mixing device[END_REF], where other choices will be investigated. Meanwhile, this permutation matrix P corresponds to the permutation π which is given by

π = (1 N z )(2 N z -1)(3 N z -2) • • • ,
where we use the standard notation of cycles in the symmetric group. Note that π is of order two. The photo-inhibition state C is then set to be 2-periodic (i.e C 1 (0) = P C 2 (L)). The details about the optimization procedure are given in Appendix 4.C. We keep the truncated Fourier number N = 5 and set the length L = 100 m. The initial guess a is still zero. Figure 4.5 presents the shape of the optimal topography and the evolution of the photoinhibition state C for two laps. Note that here we are still in a periodic regime, unlike the previous test the resulting optimal topography in this case is not flat. However, the increase in the optimal value of the objective function μNz compared to a flat topography is around 0.216%, and compare to a flat topography with non permutation (i.e. P = I Nz ) case is around 0.263%, in both cases, the increase remains small. On the other hand, we observe that the state C is actually 1-periodic for each laps. This result is actually proved in [START_REF] Bernard | Optimizing microalgal productivity in raceway ponds through a controlled mixing device[END_REF] in the case of a flat topography. In particular, this result enables us to perform the computation for any permutation matrix by doing one lap of the raceway. In this way, we provide the next test.

Constant volume with mixing device

The current test is dedicated to study the optimal permutation matrix and the associated shape of the topography for constant volume. To evaluate the efficiency of the corresponding mixing strategy, define:

r 1 := μPmax Nz (a * ) -μPmax Nz (0) μPmax Nz (0) , r 2 := μPmax Nz (a * ) - μI Nz Nz (0) μI Nz Nz (0) . ( 4

.44)

Here r 1 defines the gain of the optimal permutation strategy with the optimal topography compare to the optimal permutation strategy with a flat topography, and r 2 defines the gain of the optimal permutation strategy with the optimal topography compare to no permutation strategy with a flat topography. Let us consider two raceway pond length L = 100 m and L = 1 m re-Work in progress as of 7 th September, 2021 spectively. The optimal matrices P max for different L are denoted by P L max and given in (4.45) with the associated optimal topographies presented in Figure 4.6.

P 100 max =          
0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0

          , P 1 max =          
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0

          (4.45)
A non flat topography associated with a non trivial permutation matrix has been observed. In particular, these optimal matrices corresponds to the optimal matrices obtained with a flat topography under the same parameter settings in Chapter 3. More precisely, the left hand side matrix of (4.45) is the same type as (3.27) and the right hand side matrix of (4.45) is the same type as (3.29). The two ratios defined in (4.44) are r 1 = 0.148%, r 2 = 1.070% and r 1 = 0.001% r 2 = 3.453% respectively.

As we observed in the test above, the length of raceway has a potential influence on the objective function and the gain, we then provide a test for different values of the length L. Figure (4.7) shows the objective function μNz and the two ratios r 1 , r 2 as a function of the length L. Note that the objective function decreases when L increases except in the neighbourhood of L = 12.5 m, on the same time, we observe that the influence of topography is very limited comparing to the influence of the permutation strategies.

Varying volume with mixing device

We consider now that the volume of the system (V ) can also vary, meaning that the volume related coefficient a 0 is also a parameter to be optimized. Let us define two ratios similar Work in progress as of 7 th September, 2021

r1 := Π Pmax (ã * ) -Π Pmax (ã f ) Π Pmax (ã f ) , r2 := Π Pmax (ã * ) -Π I Nz (ã f ) Π I Nz (ã f ) , (4.46)
where ãf := [ã * 0 , 0, • • • , 0] and ã * 0 is the optimal volume related value. We keep the same length setting as in the previous test, the optimal matrices P L max are given in (4.47) and the associated optimal topographies are presented in Figure 4.8.

P 100 max =          
0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0

          , P 1 max =          
0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0

          (4.47)
The optimization algorithm stops before finding an optimum since it is limited by the constraint subcritical flow, one can see that the water depth becomes very small. This corresponds to the result that we have mentioned in Chapter 2 where smaller water depths provide better biomass surface productivity. The two ratios defined in (4.46) are r1 = 0.918%, r2 = 9.284% and r1 = 0.00003%, r2 = 12.714% respectively. As shown experimentally in the previous test, the influence of the topographies remain limited, at the same time, non trivial permutation strategies P L max are obtained for different raceway length L, in particular these strategies are also different from the case with a fixed volume. Moreover, these strategies have a better improvement when the volume is also optimized. Figure 4.9 shows the objective function Π P and the two ratios r1 , r2 as a function of the length L. Note that the average growth rate Π P increase when L goes to 0. This flashing effect corresponds to the fact that the algae exposed to high frequency flashing have a better growth. This phenomenon has already been reported in literature e.g. [START_REF] Bernard | Optimizing microalgal productivity in raceway ponds through a controlled mixing device[END_REF][START_REF] Lamare | Gradient-based optimization of a rotating algal biofilm process[END_REF].

Work in progress as of 7 th September, 2021 On the other hand, if we just focus on the increase provided by the optimization of the volume as shown in Figure 4.10. One can see that comparing with the initial state which is a flat topography with 0.4 m water depth, the value of the objective function Π at its final state is doubled. Therefore a shallow raceway may provide a more considerable gain comparing a non flat topography or a mixing strategy.

Conclusion and future works

A non flat topography slightly enhances the average growth rate. However the gain remains very limited and it is not clear if the difficulty to design such a pattern could be compensated by the increase in the process productivity.

A flat topography cancels the gradient of the objective functional in many situations where C is assumed to be periodic. However, including a mixing device gives rise to an optimal non flat topography with a slight gain of the average growth rate. Adapting the shape of the raceway to an original mixing system is an innovative strategy to boost the algal process productivity. To realize in practice the ideal mixing a system more elaborated than a paddle wheel is required.

If the condition is allowed, a shallow raceway pond may then provide considerable gain comparing with the topography or mixing strategy design.

4.A Proof of Theorem 4.3.1

Let us give the details about the proof of Theorem 4.3.1. To prove the existence and uniqueness of a (weak) solution of this problem, we need the following result.

4.A.1. Given Z 0 ∈ R, let Z ∈ C(0, L; R) be the weak solution of Z (x) = -α(I(x))h(x) Q0 Z(x), x ∈ [0, L] Z(L) = Z 0 (4.48) For x ∈ [0, L]: Z(x) 2 ≤ e -kr hc Q 0 x Z 0 2 . (4.49)
Work in progress as of 7 th September, 2021

Proof. Let x ∈ [0, L], as a weak solution of (4.48), the function Z is almost everywhere differentiable and a direct calculation gives

d Z(x) 2 2 dx = 2 Z(x), - α(I(x))h(x) Q 0 Z(x) ≤ -2 k r h c Q 0 Z(x) 2 2 , (4.50)
where we have used the fact that α(I) = β(I) + k r , β(I) ≥ 0 and h > h c (in subcritical case).

Defining f (x) := d Z(x) x , we obtain: (s-x) f (s)ds.

de 2 kr hc Q 0 x Z(x)
Because of (4.50), f (x) ≤ 0, this concludes the proof.

Let us now give the proof of Theorem 4.3.1.

Proof. Introduce the mapping Φ : R → R defined by

Φ(C 0 ) := C(L),
where C is the weak solution of (4.16). Given C1 0 ∈ R and C2 0 ∈ R, define Z 0 = C2 0 -C1 0 and Z(x) = C2 (x) -C1 (x), where C1 and C2 are the weak solutions of (4.16) with initial conditions C1 0 and C2 0 , respectively. Subtracting the corresponding weak representations, we obtain that Z satisfies the assumptions of Lemma 4.A.1 so that (4.49) holds. As a consequence,

C2 (L) -C1 (L) 2 ≤ e -kr hc Q 0 L C2 0 -C1 0 2 ,
which implies that Φ is a contraction. Applying Banach fixed-point theorem, it follows that there exists a unique C 0 ∈ R such that Φ(C 0 ) = C 0 . The corresponding weak solution C of (4.16) satisfies (4.31). Therefore we find the expression (4.24).

4.B Relation between two growth rate definitions

4.C System with a paddle-wheel

Let us denote by P the permutation matrix associated with π i.e., 1 as entries on the antidiagonal. Let us denote by C 1 (resp. C 2 ) the photo-inhibition state for the first (resp. second) lap of the raceway. We then assume that the state C is 2-periodic, meaning that C 1 (0) = P C 2 (L). From (4.25), we define the objective function by In general, the average growth rate for multiple laps is defined by replacing 2 in (4.51) with the number of the laps and follow the same computation as above. As one may have noticed, this computation becomes fastidious especially when the number of the laps increase. Therefore the periodic result like one presented in Theorem 3.2.1 is very necessary to reduce the computational cost.

Work in progress as of 7 th September, 2021

Conclusion

Main contributions

In this thesis, we provide a deeper insight into different aspects on dynamics of microalgae growth with three main contributions.

Optimal optical conditions

In Chapter 2, we have extended the results of [START_REF] Masci | Microalgal biomass surface productivity optimization based on a photobioreactor model[END_REF] and [START_REF] Grognard | Optimal strategies for biomass productivity maximization in a photobioreactor using natural light[END_REF] where the authors have identified an optimal strategy for biomass productivity maximization in photobioreactors. This so-called compensation condition consists in canceling the net growth rate at the bottom of the photobioreactors to maximize the surface biomass productivity. Based on their works, we investigate the influence of the optical condition for microalgal production. More precisely, we consider a general light attenuation function relying on biomass concentration and the background turbidity of the system. We introduce the notion of optical depth productivity which is much more convenient to study the optimal condition of the optical depth of the light intensity. A global optimal condition for the optical depth is determined which depends only on the parameter of the model. This optimal condition also coincides with the compensation condition found in literature in other contexts. Then, we study the classical areal biomass productivity maximization problem. For a given biomass concentration, it is shown that an optimal water depth exists and can be determined explicitly using the optimal optical depth. However, optimizing biomass concentration for a given reactor depth is more tricky, since the biomass concentration determined by the compensation condition is not always an optimum. In this case, we find an asymptotic behaviour both for areal biomass productivity and net growth rate at the bottom of the reactors. A nonlinear controller is designed to stabilize the evolution of the biomass concentration to its desired value. In practise, this optimal optical depth helps in establishing a relation between the biomass concentration and the reactor depth as shown in variable volume case in Chapter 4.

Optimal mixing strategies

In Chapter 3, we identify a class of problem where N resources are distributed to N activities, each activity then uses the assigned resource to evolve during a given time T > 0 after which the resources are redistributed. In light of a bibliographic review, this type of allocation problem is not much documented, especially when dealing with a dynamical system. However, this type of problem often arises in real life applications. In this way, we develop a complete theory that can be apply to various applications. As classical resource allocation problems, we study the allocation strategies which optimize the cost or the benefit generated by the system. For this purpose, we introduce an objective function related to the average benefit or cost which, in addition, takes into account a periodic condition associated with the dynamics. We show that the periodicity of the resulting problem does not depend on the order of the permutation associated with the assignment. We then propose a suboptimal problem whose solution can be determined explicitly a priori and provide a criterion to compare it with the original non linear problem. We apply finally this theory to the raceway microalgal production where a mixing device, such as a paddle wheel, is considered to control the rearrangement of the depth of the algae cultures hence the light perceived at each lap. Non trivial permutation strategies are observed and the approximation quality of the suboptimal problem has also been examined.

Optimal topographies

In Chapter 4, we study the influence of the shape of the topography on algal growth in raceway ponds. An intuitive idea is that non-flat topographies could ensure a better distribution of the light intensity, in particular at the lower part of the pond, by bringing up the algae closer to the surface. The optimization of this complex system is challenging since it contains both a hydrodynamical movement and biological evolution. For this purpose, we consider a coupled biological-hydrodynamical model which enables us to describe this complex dynamical system, and use it to optimize the topography. We then apply an adjoint-based optimization scheme which includes the constraints associated with the shallow water regime. On the contrary to a widespread belief, the flat topography is proved to be the optimal topography in a periodic regime, whereas non-trivial topographies can be obtained in other contexts, e.g., when we remove the periodic assumption or when an extra mixing strategy is included in the model. Note that in the examples considered in our numerical tests, such topographies only slightly improved the biomass production.

Discussion and Perspectives

Microalgal industrial production is still challenging, especially in full-scale systems subjected to permanent fluctuations of light intensity. In order to obtain an optimal productivity in the reactors, an appropriate light intensity must be supplied to guarantee a continuous growth. Note that low light intensity will limit the growth, especially in dense algal cultures case due to selfshading. On the other hand, too high levels may lead to photoinhibition of the algal cell which will also limit their growth. In our work, we mainly focus on the light condition aspect and provide an optimal condition. The co-limitation problem could be considered in future work to account for both light and nutrients limitation and test this optimal condition. On the other hand, other controllers can be investigated for instance by using the extremum seeking strategy in order to find automatic controller without identifying it in advance.

As for the raceway ponds design, we work on a simplified coupling model to provide some optimization results in Chapter 3 and Chapter 4. One of the advantages of our coupling system is that this allows us to provide some explicit computations on this complex system as well as some theoretical proofs about optimal topographies and mixing strategies. A second order study on the optimization problem could be investigated to prove that the flat topography is the local optimal solution in the periodic case. Other investigations could consists of numerical tests dealing with coupled models closer to real life situations, e.g., in [START_REF] Bernard | A 2d model for hydrodynamics and biology coupling applied to algae growth simulations[END_REF], where a coupled multilayer model (2D multilayer Saint-Venant and Droop) is considered. One can also use the optimal control strategies (for instance Bocop code) to investigate this coupling optimization problem.

From a hydrodynamical point of view, our study is based on the laminar flow assumption, therefore future work may account for the turbulent flow where particles can change their vertical positions. Such a dynamic play the role of a mixing device acting continuously along the Work in progress as of 7 th September, 2021 trajectories. Since our work in Chapter 3 shows that a mixing device could enhance the algal growth, this working direction may provide another insight on the industrial raceway ponds design. As a preliminary test, one could include a Brownian motion component into the Lagrangian trajectories to simulate this turbulence as mentioned in Remark 1.2.2.

As for the periodic dynamical resource allocation problem that we present in Chapter 3, we provide an application to the algal production. Other applications would also be interesting by applying our theory in other contexts, as mentioned in Remark 3.2. As an example, we could consider the case where the dynamical system is associated with the growth of different trees species, as done in [START_REF] Cominetti | Asymptotic convergence of optimal policies for resource management with application to harvesting of multiple species forest[END_REF]. Meanwhile, one can also consider other strategies of solving the resource allocation problems presented in this chapter, for instance by using the mixed-integer optimization techniques.

From a theoretical point of view, although our suboptimal problem provides a faithful approximation for real life simulation cases, the proposed criterion can still be improved. For instance, other strategies to regroup the permutations can be used to establish a new criteria, or an improvement on the two bounds provided in the proof of Theorem 3.2.2. Meanwhile, providing an approximation for small time duration problem is also an interesting working direction. Although this may be difficult to accomplish in real life algal production, the productivity in this limit case is still attractive, and this corresponds to the flashing effect observed for other types of photobioreactors. The main challenge in this case is that the diagonal matrix D in (3.6) tends to the identity matrix for which (I N -P D) -1 is no longer invertible, therefore it is much more difficult to identify it and propose an approximation for it. Numerically, we have tested it with some algorithms which can not always find the optimal permutation matrices. Therefore, this could be a challenging working direction.

Work in progress as of 7 th September, 2021

Approches Lagrangiennes pour la modélisation et l'optimisation du couplage hydrodynamique-photosynthèse Abstract

Microalgae are photosynthetic micro-organisms whose potential has been highlighted in the last decade.

Applications can be found from to renewable energy production and wastewater treatment to some high added value commercial products e.g., food, pharmaceutical, cosmetics. Nevertheless, finding optimal growth conditions for full-scale cultivation of microalgae remains challenging in practice. Mathematical models are therefore of great help to better manage this complex, nonlinear dynamical system. The aim of this thesis is to better understand how different factors affect microalgal growth. In a first part, we study the influence of the light attenuation and the optimal condition to maximize the productivity. In this way, we introduce an optical productivity which enables us to determine the optimal condition for general light extinction function. A global optimal optical depth is found which consists in canceling the algal net growth rate at the bottom of the reactors to maximize the optical productivity. It can be used to characterize the optimization of the areal productivity in some specific cases, whereas an asymptotic behaviour has been observed in more general case.

We then limit ourselves to a specific reactor -the raceway pond, which is an outdoor circuit basin combining with a paddle wheel. We start by investigating a resource allocation problem issuing from the re-distribution of the light resource to the algae by the paddle wheel. A generic mixing device is considered to assign at each lap the light resource to the algae layers in the raceway. We determine the optimal allocation strategies to maximize the algal growth.

In a third part, we show how the shape of the topography affects (or not) the algal growth in raceway ponds. In this way, we consider a hydrodynamical-biological coupled model and introduce an optimization problem associated with the topography to maximize the algal growth. We also combine the optimization of the topographies with the previous allocation strategies to investigate their influence on algal production. Non-trivial topographies are obtained numerically to enhance the algal growth.

The mathematical study of these optimization problems leads to new interesting working directions, improves and clarifies the understanding of influence by different factors on algal growth. We conclude with some discussions and perspectives of this work. Mots clés : optimisation, modélisation, hydrodynamique, système dynamique, allocation des ressources, contrôle périodique, problème nonlinéaire, équations de saint-venant, modèle de han, microalgues

1 Introduction 1 . 1 Contents 3 71 4 1 Introduction 1

 11137111 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Optimization aspects and contributions . . . . . . . . . . . . . . . . . . . 1.2 Biological and hydrodynamical models . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Biological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Hydrodynamical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Light intensity modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Mixing device modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Periodic Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Adjoint methods for optimization problem . . . . . . . . . . . . . . . . . . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Optimal condition for algal production . . . . . . . . . . . . . . . . . . . . 1.5.2 Optimal mixing strategy for algal production . . . . . . . . . . . . . . . . 1.5.3 Optimal topography for algal production . . . . . . . . . . . . . . . . . . . 2 Optimal optical conditions for algal production in photobioreactors 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Analysis of the optimal productivity . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Global optimality condition . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Surface biomass productivity . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Optimal control implementation in closed loop . . . . . . . . . . . . . . . . . . . 2.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Numerical algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Parameter settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.A Explicit computations for average growth rate . . . . . . . . . . . . . . . . . . . ix x Periodic resource allocation in dynamical systems, application to raceway microalgal production 47 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Description of the problem and optimization . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Periodic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.2 Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.3 Some technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.4 Solutions of the optimization problems . . . . . . . . . . . . . . . . . . . . 53 3.2.5 Implementation Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 Application to algal production . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.1 Raceway modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.2 Parameter settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.3 Examples of optimal devices . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.4 Further numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.A Explicit Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.B Optimization problem with arbitrary vectors . . . . . . . . . . . . . . . . . . . . 70 3.C Remark on F + m , F - m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shape optimization of a microalgal raceway to enhance productivity 73 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Coupling hydrodynamic and biological models . . . . . . . . . . . . . . . . . . . 74 4.2.1 Modeling the photosystems dynamics . . . . . . . . . . . . . . . . . . . . . 74 4.2.2 Steady 1D Saint-Venant equations . . . . . . . . . . . . . . . . . . . . . .75 4.2.3 Lagrangian trajectories of the algae and captured light intensity . . . . . . 76 4.3 Optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.1 Optimization problem for non periodic case with time dependent description 79 4.3.2 Optimization problem for periodic case with time independent description 81 4.3.3 Optimization for non-constant volume problem . . . . . . . . . . . . . . . 85 4.3.4 Optimization problem in periodic case for topographies combining with mixing strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.5 Variable reactor volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.2 Parameter settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.5 Conclusion and future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.A Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.B Relation between two growth rate definitions . . . . . . . . . . . . . . . . . . . . 99 4.C System with a paddle-wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Conclusion 103 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Optimal optical conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Optimal mixing strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Optimal topographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Discussion and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Bibliography 107 Work in progress as of 7 th September, 2021 Chapter

Figure 1 . 1 :

 11 Photon IPhoton I

Figure 1 . 2 :

 12 Figure 1.2: Evolution for open state A and photoinhibition state C with the initial condition (A(0), C(0)) = (1, 0). Note that we use an Euler Explicit scheme to solve the system (1.3) by using the parameters presented in Table1.1. A log scale is also used for the time variable.

Figure 1 . 3 :

 13 Figure 1.3: Growth rate for light intensity.

Figure 1 . 4 :

 14 Figure 1.4: Representation of the hydrodynamic model.

Figure 1 . 5 :

 15 Figure 1.5: Schematic representation of the mixing process over two laps. Here, N = 4 and P corresponds to the cyclic permutation σ = (1 2 3 4).

Figure 1 . 6 :

 16 Figure 1.6: Productivity (Π) with respect to biomass concentration (X) for h = 0.15 m. Blue point: ε(X 0 )h = Y opt Red point: X 1 = argmax X∈R+ Π(X, h).

Figure 1 . 7 :

 17 Figure 1.7: Optimal matrix P max for N = 11 obtained with functional J(P ) (Left) and P + for N = 100 obtained with functional J approx (P ) (Right). The blue points represent non-zero entries, i.e., entries equal to 1.

Figure 1 . 8 :

 18 Figure 1.8: Average net specific growth rate μN for T = 1000 s. Left: N = 5. Right: N = 9.The red surface is obtained with P max and the blue surface is obtained with P + . The purple stars represent the cases where P max = P + or, in case of multiple solution, μN (P max ) = μN (P + ). The green circles represent the cases where the criterion (1.27) is satisfied.

  I(x, z; a))C + ζ(I(x, z; a))dt, Ċ = -α(I(x, z; a))C + β(I(x, z; a)), ẋ = u(x; a), ż = w(x, z; a).

Figure 1 . 9 :

 19 Figure 1.9: The initialization (Left) and the final iteration (Right) of the optimization procedure for truncated Fourier number N = 5. The red thick curve represents the topography (z b ), the blue thick curve represents the free surface (η), and all the other curves between represent the trajectories for different layers.

Figure 1 . 10 :

 110 Figure 1.10: The initialization (Left) and the final iteration (Right) of the optimization procedure for truncated Fourier number N = 5 in the case C periodic.

Figure 2 .

 2 1 shows ε(X) for different values of s when the background turbidity α 1 > 0.

Figure 2 . 1 :

 21 Figure 2.1: ε(X) with respect to X for s ∈ {0.2, 0.4, 0.6, 0.8, 1}.

  Figure 2.2 presents the evolution of the growth rate µ and optical depth productivity (P ) with respect to y for different value of s and α 1 . One can see that the optimum is obtained with Y opt = 6.337, which also satisfies numerically µ(I(Y opt )) = R in our case. Moreover, as mentioned in Remark 2.3.1, Y opt does not change for other values of α 1 and s.

Figure 2 . 2 :

 22 Figure 2.2: Growth rate µ and optical depth productivity P with respect to y. Left: s = 1 and α 1 = 10 m -1 . Right: s = 0.365 and α 1 = 0 m -1 .

  Figure 2.3: Productivity (Π) and net growth rate (µ net (X, h)) with respect to depth (h) for X = 50 g • m -3 . Left: s = 1 and α 1 = 10 m -1 . Right: s = 0.365 and α 1 = 0 m -1 .

Figure 2 . 4 :

 24 Figure 2.4: Optimal surface biomass productivity with respect to the background turbidity α 1 for X = 50 g • m -3 and different value of s.

Figure 2 . 5 :

 25 Figure 2.5: Productivity (Π) with respect to biomass concentration (X) for h = 0.15 m. Left: α 1 = 0 m -1 . Right: α 1 = 10 m -1 .

Figure 2 . 7 :

 27 Figure 2.7: Up: First-two elements of these two sequences. Down Left: Surface biomass (X n h n )and optical depth (ε(X n )h n ) for the sequence (X n , h n ) n>0 . Down Right: Productivity Π(X n , h n ) and net growth rate at the reactor bottom µ(X n , h n ) -R for the sequence (X n , h n ) n>0 .

Figure 2 . 8 :

 28 Figure 2.8: Productivity (Π) with respect to biomass concentration (X) for h = 0.15 m. Left: α 1 = 0 m -1 . Right: α 1 = 10 m -1 .

Figure 2 .

 2 Figure 2.10: Productivity Π(X n , h n ) for the sequence (X n , h n ) n>0 .

Figure 2 . 11 :

 211 Figure 2.11: Evolution of the biomass concentration X in closed loop for two initial conditions.

e 1 +

 1 e 2 = 0 and e 1 d 2 + e 2 d 1 = -1, i.e. e 1 = 1 d1-d2 = a √ b 2 -4ac and e 2 = 1 d2-d1 = -a √ b 2 -4ac . Using e 2 = -e 1 , we find μ = k r kσe 1 ahε(X) ln (I s -d 1 )(I b -d 2 ) (I b -d 1 )(I s -d 2 ) . • ∆ = 0 : Then there exists a unique root denoted by d = -b 2a . And one has μ = k r kσ hε(X) I s -I b (I s -d)(I b -d)

2 . 2 ,φ(m 1 ) ≤ 1 . ( 3 . 15 )

 2211315 d max := max n=1,...,N (d n ) and d min := min n=1,...,N (d n ). Assume that: max m1≥2 Then the problem max P ∈P N u, (I -P D) -1 P v (resp. min P ∈P N u, (I -P D) -1 P v ) and the problem max P ∈P N u, P v (resp. min P ∈P N u, P v ) have the same solution. Proof. We keep the notation in Section 3.2.3 and give the proof in the case of the maximization problem. The case of the minimization problem can be handled in the very same way. Let P ∈ P N and σ ∈ S N the associated permutation, we have u, (I -P + D) -1 P + v -u, (I -P D) -1 P v (3.16) = +∞ l=0 u, (P + D) l P + -(P D) l P v = u, (P + -P )v + +∞ l=1 u, (P + D) l P + -(P D) l P v .

m 1 2 , only N 2 terms

 22 need to be computed. Examples of behaviour of s m and F + m , F - m are presented in Figure 3.3, whereas examples of behaviour of the function (3.14) with respect to m 1 are shown in Figure 3.4.

  Here A, B and C are the relative frequencies of the three possible states with A + B + C = 1, and I is a continuous time-varying signal representing the photon flux density. The coefficient σ H stands for the specific photon absorption, τ is the turnover rate, k r represents the photosystem repair rate and k d is the damage rate. As shown in Subsection 1.2.1, one can use a fast-slow approximation and singular perturbation theory to reduce this system to a single evolution Work in progress as of 7 th September, 2021 58CHAPTER 3. Periodic resource allocation in dynamical systems, application to raceway microalgal production equation on the photo-inhibition state C: Ċ = -α(I)C + β(I), (3.21)

Figure 3 . 1 :

 31 Figure 3.1: Schematic representation of the mixing process over two laps. Here, N = 4 and P corresponds to the cyclic permutation σ = (1 2 3 4).

Figure 3 . 2 :

 32 Figure 3.2: Γ and V with respect to the light intensity I (Blue curve). Discretisation points (Red point) chosen for (I s , q, T ) = (2000, 5%, 1000) (Left) and (800,0.5%,1) (Right).

Figure 3 .

 3 [START_REF] Amos | Principles for attaining maximal microalgal productivity in photobioreactors: an overview[END_REF] shows the values of the function φ(m 1 ) (see(3.14)

Figure 3 . 3 :

 33 Figure 3.3: Example of sequences F + m , F - m (Top) and s m (Bottom) with respect to m for the two parameters triplets. Left: N = 7. Right: N = 20.

Figure 3 . 4 :Figure 3 . 5 :

 3435 Figure 3.4: Example of behaviour of φ(m 1 ) with respect to m 1 for two parameters triplets and two different N . Left: N = 7. Right: N = 20.

Figure 3 . 6 :

 36 Figure 3.6: Optimal matrix P max for Problem (3.6) and N = 11 (Left) and P + for Problem (3.7) and N = 100 (Right) for the two parameters triplets. The blue points represent non-zero entries, i.e., entries equal to 1.

Figure 3 . 7 :

 37 Figure 3.7: Average specific growth rate in the case q = 0.1% and N = 7 for four different light intensities I s .

Figure 3 . 9 :

 39 Figure 3.9: Average net specific growth rate μN for I s ∈ [0, 2500] and q ∈ [0.1%, 10%]. In each figure, the red surface is obtained with P max , the dark blue surface is obtained with P min , the green surface is obtained with I and the light blue surface is obtained with P + . The black stars represent the cases where P max = I and the purple circles represent the cases where P max = P + . Top: for T = 1 s. Middle: for T = 500 s. Bottom: for T = 1000 s.

68CHAPTER 3 .

 3 Periodic resource allocation in dynamical systems, application to raceway microalgal production We finally evaluate the efficiency of various mixing strategies. Define r 1 := μN (P max ) -μN (I) μN (I) , (3.30) r 2 := μN (P max ) -μN (P min ) μN (P min ) , (3.31) r 3 := μN (I) -μN (P min ) μN (I) , (3.32)

Figure 3 . 11 :

 311 Figure 3.11: Three ratios (3.30)-(3.32) for T = 1 s (Left) and for T = 1000 s (Right). In each figure, the red surface represents r 1 , the blue surface represents r 2 and the green surface represents r 3 .

Figure 3 . 12 :

 312 Figure 3.12: Two ratios (3.33)-(3.34) for T = 1 s (Left) and for T = 1000 s (Right). In each figure, the red surface represents r1 , the blue surface represents r2 .

Figure 4 . 1 :

 41 Figure 4.1: Representation of the hydrodynamic model.

  I(x, z; a))C + ζ(I(x, z; a))dt, where C, x, z satisfy    Ċ = -α(I(x, z; a))C + β(I(x, z; a)) ẋ = u(x; a) ż = w(x, z; a).

4 .T 0 - 0 p 1 Ċ 0 p 2 ż 0 p 3 ẋ

 40010203 More precisely, define the Lagrangian of Problem (4.19) by L(C, z, x, p 1 , p 2 , p 3 , a) := 1 T γ(I(x, z; a))C + ζ(I(x, z; a)) dt -T + α(I(x, z; a))C -β(I(x, z; a)) dt -T -w(x, z; a) dt -T -u(x; a) dt.

( 4 . 20 ) 1 T

 4201 Given a vector a, let us still denote by C, x, z, p 1 , p 2 , p 3 the corresponding solutions of (4.18) and (4.20). The gradient ∇μ 1 (a) is obtained by∇μ 1 (a) = ∂ a L,where∂ a L = T 0 -γ (I)C + ζ (I) + p 1 -α (I)C + β (I) ∂ a I + p 2 ∂ a w + p 3 ∂ a udt. (4.21)

p 3 , 0 p 3 ẋ

 303 I(x, z i ; a))C i + ζ(I(x, z i ; a))dt -Ċi + α(I(x, z i ; a))C i -β(I(x, z i ; a)) dt i żi -w(x, z i ; a) dt -T -u(x; a) dt.

  x, z), I(x, z; a) dzdx,(4.24)

Theorem 4 . 3 . 1 .

 431 Given I ∈ L 2 (0, L; R), there exists a unique couple (C 0 , C) ∈ [0, 1]×C(0, L; [0, 1]) satisfying C(x) = C 0 + x 0 -α(I(s))C(s)+β(I(s)) Q0 h(s)ds , C(L) = C 0 (4.31)for all x ∈ [0, L].

L 0 h 1 Theorem 4 . 3 . 3 .

 01433 (x; a)dx = 0, ∀a ∈ R N .(4.38) Work in progress as of 7 th September, 2021 Unlike Theorem 4.3.2, the flat topography does not cancel the gradient ∇Π. Assume that the net growth rate at the surface satisfies µ(I s ) > 0 and that I a0 solves (4.35). Let ãf = [a 0 , a f ] the flat topography with an average depth a 0 , then ∇Π(ã f ) = 0.Proof. Computations similar to that in the proof of Theorem 4.3.2 gives ∂ a L(ã f ) = 0. It follows from(4.34), that

Figure 4 . 2 :

 42 Figure 4.2: The value of μNz for N z = [1, 80].

Figure 4 . 3 :

 43 Figure 4.3: The optimal topography at the first iteration (Left) and the final iteration (Right) for the truncated Fourier number N = 5. The red thick curve represents the topography (z b ), the blue thick curve represents the free surface (η), and all the other curves between represent the trajectories for different layers.

Figure 4 . 4 :

 44 Figure 4.4: The optimization process at the first iteration (Left) and the final iteration (Right) for the truncated Fourier number N = 5. The red thick curve represents the topography (z b ), the blue thick curve represents the free surface (η), and all the other curves between represent the trajectories for different layers.

Figure 4 . 5 :

 45 Figure 4.5: The optimal topography (Left) and the evolution of the photoinhibition state C (Right) for two laps.

Figure 4 . 6 :

 46 Figure 4.6: The optimal topographies and the associated trajectories for the permutation matrices (4.45). Left: L = 100 m. Right: L = 1 m.

Figure 4 . 7 :

 47 Figure 4.7: The optimal value of the objective function μNz (Left) and the two ratios r 1 , r 2 (Right) for L = 100/2 {0,...,10} .

Figure 4 . 8 :

 48 Figure 4.8: The optimal topographies and the associated trajectories for the permutation matrices (4.47). Left: L = 100 m. Right: L = 1 m.

Figure 4 . 9 :

 49 Figure 4.9: The optimal value of the objective function Π P (Left) and the two ratios r1 , r2 (Right) for L = 100/2 {0,...,10} .

Figure 4 . 10 :

 410 Figure 4.10: The optimization process for length L = 100 m and P = P 100 max in (4.45).

Lemma

2 2

 2 and multiplying both sides by e 2 kr hc Q 0

1 u

 1 Let us start with the definition of the average growth rate in time(4.23). Note that the time duration of one lap of the raceway pond T = L 0 dx and hu = Q 0 because of (4.7). By doing a Work in progress as of 7 th September, 2021 change of variable x = x(t) and using the fact that dx = udt, one has b (x(t)) µ C (x(t), z(t)) , I (x(t), z(t)) x, z) , I (x, z) dzdx.

  I i (a))C j i + ζ(I i (a)) Q 0 h(a)dx.(4.51)Let us still denote by L the Lagrangian associated with this optimization problem. It can be written asL(C, p, a) I i (a))C j i + ζ(I i (a)) 2V N z h(a) -p j i C j i + α(I i (a))C j i -β(I i (a)) Q 0 h(a) dx,where p j i is the Lagrangian multiplier associated with the constraint (4.26) for C j i . Integrating the terms p j i C j i dx on the interval [0, L] by parts, we get By definition of our paddle wheel model, C 2 (0) = P C 1 (L). Differentiating L with respect to C 1 (L) and C 2 (L), we find∂ C 1 (L) L = p 1 (L) -p 2 (0)P, ∂ C 2 (L) L = p 2 (L) -p 1 (0)P.Then differentiating L with respect to C j i gives the evolution of p j i by∂ C j i (L) L = p j ii (a))h(a).Work in progress as of 7 th September, 2021Finally, the partial derivative ∂ a L (hence the gradient) is given by∂ a L = I i (a))C j i + ζ (I i (a)) 2V N z + p j i -α (I i (a))C j i + β (I i (a)) Q 0 h(a)∂ a I i (a)dx + I i (a))C j i + ζ(I i (a)) 2V N z + p j i -α(I i (a))C j i + β(I i (a)) Q 0 ∂ a h(a)dx.

  Keywords: optimization, mathematical modeling, hydrodynamics, dynamical system, resource allocation, periodic control, nonlinear problem, saint-venant equations, han model, microalgae Laboratoire Jacques-Louis Lions Sorbonne Université -Campus Pierre et Marie Curie -4 place Jussieu -75005 Paris -France Résumé Les microalgues sont des micro-organismes photosynthétiques dont le potentiel a été mis en évidence au cours de la dernière décennie. Des applications peuvent être trouvées dans la production d'énergie renouvelable ou dans le traitement des eaux usées par exemple. Elles peuvent être utilisées dans beaucoup de produits commerciaux à haute valeur ajoutée comme par exemple dans l'alimentation, la pharmacie ou les cosmétiques. Néanmoins, trouver des conditions optimales pour la production des microalgues à grande échelle reste un défi en pratique. Les modèles mathématiques sont donc d'une grande aide pour mieux gérer ce système dynamique complexe et non linéaire. L'objectif de cette thèse est de mieux comprendre comment différents facteurs affectent la croissance des microalgues. Dans un premier temps, nous étudions l'influence de l'atténuation lumineuse et obtenons une condition d'optimalité pour maximiser la productivité. De cette façon, nous introduisons une productivité optique qui nous permet de caractériser la fonction d'extinction de la lumière optimale dans un cadre général. On trouve une profondeur optique optimale globale qui consiste à annuler le taux de croissance net des algues au fond des réacteurs pour maximiser la productivité optique. Cette étude nous permet de caractériser la productivité surfacique optimale dans certains cas particuliers, et de décrire le comportement asymptotique des autres cas dans certains régimes. On se limite ensuite à un réacteur spécifique, le raceway pond, qui est un bassin de circuit extérieur associé à une roue à aubes. Nous commençons par étudier un problème d'allocation de ressources issu de la redistribution de la ressource lumineuse aux algues par la roue à aubes. Un dispositif de mélange générique est envisagé pour affecter à chaque tour la ressource lumineuse aux algues qui se situent sur differentes couches dans le raceway. Nous déterminons les stratégies d'allocation optimales pour maximiser la croissance des algues. Dans une troisième partie, nous montrons comment la forme de la topographie affecte (ou non) la croissance des algues dans le raceway. De cette façon, nous considérons un modèle hydrodynamiquebiologique couplé et introduisons un problème d'optimisation associé à la topographie pour maximiser la croissance des algues. Nous combinons également l'optimisation des topographies avec les stratégies d'allocation précédentes pour étudier leur influence sur la production d'algues. Des topographies non triviales sont obtenues numériquement pour améliorer la croissance des algues. L'étude mathématique de ces problèmes d'optimisation conduit à de nouvelles directions de travail, améliore et clarifie la compréhension de l'influence de différents facteurs sur la croissance des algues. Nous concluons par quelques discussions et perspectives de ce travail.

Table 1

 1

	.2.
	Extinction coefficient [m 2 g -1 ]	Algae species
	0.214	Monodus subterraneu
	0.175	Spirulina platensis
	0.150	Porphyridium cruentum
	0.200	Chlorella pyrenoidosa

Table 1 .

 1

2: Values of the light extinction coefficient α 0 for different microalgae species see in

Table 2 .

 2 1: Parameter values for Han Model.

	k r	6.8 10 -3	s -1
	k d	2.99 10 -4	-
	τ	0.25	s
	σ	0.047	m 2 • (µmol) -1
	k	8.7 10 -6	-
	R 1.389 10 -7

Table 3 .

 3 1: Parameter values for Han Model

	k r	6.8 10 -3	s -1
	k d	2.99 10 -4	-
	τ	0.25	s
	σ H	0.047	m 2 µmol -1
	k H	8.7 10 -6	-
	R	1.389 10 -7	s -1

Table 4 . 1 :

 41 Parameter values for Han Model.

	k r	6.8 10 -3	s -1
	k d	2.99 10 -4	-
	τ	0.25	s
	σ	0.047	m 2 • (µmol) -1
	k	8.7 10 -6	-
	R 1.389 10 -7	s -1

Table 4 . 2

 42

	0	1.061	-
	5	1.0754	-10.0161
	10	1.0763	-10.0248
	15	1.0766	-10.0282
	20	1.0768	-10.0297
	Optimal topography shape in non periodic case

: The value of μNz for different N N μNz (unit d -1 ) log 10 ( ∇μ Nz )

https://cdiac.ess-dive.lbl.gov/ftp/ndp030/global.1751_2008.ems

Note that one can also use this approach for more complicated hydrodynamical model as soon as the Lagrangian trajectory can be computed.Work in progress as of 7 th September, 2021 1.5. Summary

also called "optical productivity" Work in progress as of 7 th September,

Work in progress as of 7 th September, 2021

Remerciements

Remerciements à Alexandre P. pour toutes les discussions et aussi les conseils sur le manuscrit. Je souhaite Work in progress as of 7 th September, 2021 Bibliography