Jawhar Ghommam

Ahmed Chemori

Faiçal Mnif

Boubaker Ch008 Tex

Finite-time stabilization of underactuated mechanical systems in the presence of uncertainties: application to the cart-pole system

Introduction

Underactuated mechanical systems [START_REF] Choukchou-Braham | Analysis and Control of Underwatcuated Mechanical Systems[END_REF][START_REF] Fantoni | Non-linear Control for Underactuated Mechanical Systems[END_REF][START_REF] Jadlovska | Modeling and optimal control of nonlinear underactuated mechanical systems a survey[END_REF] are those systems with less control inputs than generalized coordinates (called also degrees of freedom), i.e. they have unactuated generalized coordinates. For such systems, the unactuated generalized coordinates may indirectly be controlled by the actuated coordinates through the dynamic coupling, inherent to these systems [START_REF] Blajer | Modeling of underactuated mechanical systems in partly specified motion[END_REF]. This coupling is often nonlinear, resulting in generally nonintegrable dynamic constraints and therefore second-order nonholonomic.

Many examples of such systems exist, and mainly in robotics. They include, among others, Inverted pendulums [START_REF] Freidovich | Shaping stable periodic motions of inertia wheel pendulum: theory and experiments[END_REF][START_REF] Jadlovska | Application of the state-dependent riccati equation method in nonlinear control design for inverted pendulum systems[END_REF][START_REF] Jadlovska | A complex overview of modeling and control of the rotary single inverted pendulum system[END_REF], underactuated robot manipulators [START_REF] Bergerman | Dynamics and Control of Underactuated Manipulators[END_REF], gymnast robots and particularly the acrobot [START_REF] Spong | The swing up control problem for the acrobot[END_REF] and the pendubot [START_REF] De Luca | Stabilization of underactuated planar 2r manipulator[END_REF], undersea vehicles [START_REF] Fossen | Nonlinear Modelling and Control of Underwater Vehicles[END_REF], PVTOL (Planar Vertical Takeoff and Landing) aircrafts [START_REF] Poulin | Minimum energy oriented global stabilizing control of the PVTOL aircraft[END_REF] and mobile robots [START_REF] Ge | Autonomous Mobile Robots[END_REF].

Underactuation in these mechanical systems is generally introduced intentionally at the design level to reduce the manufacturing cost, the weight and/or failure rate; consequently, the obtained systems may be able to perform complex tasks with a reduced number of actuators. The source of underactuation may also be unintentionally due to failure of one or more actuators in a fully actuated system.

Underactuated systems may require new approaches/techniques to design effective control strategies; therefore, they constitute a good framework for nonlinear control problems regarding both theoretical and practical aspects. For these reasons, they are attracting more and more attention of researchers from nonlinear control community as well as from many research fields in robotics.

In the literature, many research efforts have been made in control aspects [START_REF] Lee | Control algorithms for stabilizing underactuated robots[END_REF][START_REF] De Luca | Stabilization of underactuated planar 2r manipulator[END_REF][START_REF] Nakamura | Nonlinear behavior and control of nonholonomic free-joint manipulator[END_REF][START_REF] Reyhanoglu | Discontinuous feedback control of a special class of underactuated mechanical systems[END_REF][START_REF] Spong | The swing up control problem for the acrobot[END_REF][START_REF] Spong | Energy based control of a class of underactuated mechanical system[END_REF][START_REF] Spong | Underactuated mechanical systems[END_REF], giving rise to multiple control solutions; however, control problem of such systems is still open.

For instance in [START_REF] Bortoff | Pseudolinearization of the acrobot using spline functions[END_REF], a technique of pseudolinearization using spline functions is proposed to control the acrobot underactuated system. In [START_REF] Spong | Energy based control of a class of underactuated mechanical system[END_REF], a design technique combining partial feedback linearization and Lyapunov methods based on saturation functions, switching and energy shaping has been proposed. In [START_REF] Spong | Underactuated mechanical systems[END_REF], the author discuss a survey on some existing results such as geometric nonlinear control and passivity-based control for stabilization and tracking control of such systems. Three control algorithms have been proposed in [START_REF] Lee | Control algorithms for stabilizing underactuated robots[END_REF] for an underactuated two-link robot, namely an optimal LQ controller, a partial feedback-linearizing regulator and a sliding mode controller. In [START_REF] De Luca | Stabilization of underactuated planar 2r manipulator[END_REF], the testbed is also a two-link underactuated robot but the actuator is located at the base (in contrast to the one proposed in [START_REF] Lee | Control algorithms for stabilizing underactuated robots[END_REF] where the actuator is located at the joint between the two links), and the authors propose a method based on partial feedback linearization [START_REF] Spong | Partial feedback linearization of underactuated mechanical systems[END_REF], nilpotent approximation and an iterative stabilization procedure. Roughly speaking, they propose an open-loop controller able to steer the system closer to the desired equilibrium point in finite time, and apply it iteratively, which provides a robust exponential convergence to the equilibrium. In [START_REF] Nakamura | Nonlinear behavior and control of nonholonomic free-joint manipulator[END_REF], the authors propose to study the nonlinear behavior of the same robot proposed in [START_REF] De Luca | Stabilization of underactuated planar 2r manipulator[END_REF] through a graphical tool based on Poincaré map. The study in [START_REF] Reyhanoglu | Discontinuous feedback control of a special class of underactuated mechanical systems[END_REF] concerns underactuated mechanical systems with several actuated degrees of freedom and a single unactuated degree of freedom, where a discontinuous nonlinear feedback controller is proposed, allowing the closed-loop equilibrium at the origin to be globally attractive. In [START_REF] Touati | Predictive control for the stabilization of a fast mechatronic system: from simulation to real-time experiments[END_REF], a generalized predictive controller has been proposed, based on the linearized dynamics of the system. In [START_REF] Chemori | Nonlinear predictive control of under-actuated mechanical systems, application: the ecp 505 inverted pendulum[END_REF], a nonlinear predictive controller is proposed for both stabilization and stable-limit cycle generation with an application to the ECP 505 inverted pendulum. In [START_REF] Spong | Control Using Logic-Based Switching[END_REF], a technique based on switching and saturation control is proposed. In [START_REF] Khraief | External disturbance rejection in IDA-PBC controller for underactuated mechanical systems: from theory to real-time experiments[END_REF], a passivity-based approach has been proposed to control the inertia wheel inverted pendulum. In [START_REF] Freidovicha | Periodic motions of the pendubot via virtual holonomic constraints: theory and experiments[END_REF], the technique of virtual holonomic constraints has been proposed with an application to the pendubot for generation of periodic motions, and to the Inertia Wheel Pendulum in [START_REF] Freidovich | Shaping stable periodic motions of inertia wheel pendulum: theory and experiments[END_REF]. In [START_REF] Andary | Control of the underactuated inertia wheel inverted pendulum for stable limit cycle generation[END_REF], the proposed control techniques are based on partial feedback linearization, combined with the optimization of the reference trajectories to stabilize the internal dynamics of the system. In [START_REF] Andary | A dual model-free control of non-minimum phase systems for generation of stable limit cycles[END_REF] and [START_REF] Andary | A dual model-free control of underactuated mechanical systems, application to the inertia wheel inverted pendulum[END_REF], a dual-model-free control technique is proposed. The control architecture includes two model-free controllers, the first one is used to track the reference trajectories on the actuated degrees of freedom of the system, whereas the second one is used to stabilize the internal dynamics. In [START_REF] Jadlovska | Application of the state-dependent riccati equation method in nonlinear control design for inverted pendulum systems[END_REF], a technique based on Riccati and nonlinear control is proposed for inverted pendulums. Output feedback control with an extended high gain observer was recently proposed in [START_REF] Khalid | Output feedback control of a class of underactuated nonlinear systems using extended high gain observer[END_REF].

To deal with model uncertainties, a time-scale approach along with the Lyapunov design has been proposed in [START_REF] Ravichandran | Robust stabilization of a class of underactuated mechanical systems using time scaling and Lyapunov redesign[END_REF]. The authors in [START_REF] Park | Swing-up and stabilization control of inverted pendulum systems via coupled sliding-mode control method[END_REF] used a sliding mode technique for the cart-pole system to stabilize the system in presence of disturbances. Model uncertainties however have not been tackled in this work. In [START_REF] Adhikary | Integral backstepping sliding mode control for underactuated systems: swing-up and stabilization of the cart-pendulum system[END_REF], disturbances and model uncertainties have been fully considered in a technique that involves a backstepping procedure combined with sliding mode, applied to the inverted pendulum system after the system was converted into a normal form. Integral sliding mode control was also applied in [START_REF] Xu | Design and implementation of integral sliding mode control on an underactuated two-wheeled mobile robot[END_REF] to deal with uncertainties in the two-wheeled mobile Finite-time stabilization of underactuated mechanical systems 167 inverted pendulum. The control law being designed is based on the linearized system dynamics which resulted in a small region of attraction for the equilibrium. The problem becomes more challenging when ever nonparametric uncertainties and unknown time-varying disturbances are considered in the design of effective and robust control law for the underactuated system.

AQ1

An important research issue which has attracted the attention of the control community is the finite-time convergence and finite-time stabilization [START_REF] Amato | Finite-time control of linear systems subject to parametric uncertainties and disturbances[END_REF][START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF][START_REF] Haimo | Finite-time controllers[END_REF][START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF][START_REF] Hong | Stabilization of uncertain chained form systems within finite settling time[END_REF][START_REF] Huang | Global finite-time stabilization of a class of uncertain nonlinear systems[END_REF], that ensures faster convergence rate, better disturbance rejection and robustness against uncertainty. Although myriad approaches have been proposed in the literature that tackle the problem of finite-time stabilization, they are not effective in dealing with time-varying and state-depending uncertainties. Few attempts have been concentrated in the design of adaptive schemes [START_REF] Hong | Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties[END_REF][START_REF] Hong | Adaptive finite-time control of nonlinear systems with parametric uncertainty[END_REF] to resolve the problem of state dependent and time-varying uncertainties. However, the adaptive scheme has only been applied to a class of p-normal forms [START_REF] Cheng | On normal form of nonlinear systems[END_REF]. To the best of the authors'knowledge, few have dealt with finite-time stabilization of underactuated systems subject to time-varying and state-dependent uncertainties. This chapter focuses on the design of a novel nonlinear controller for a class of underactuated system; we would like to introduce a new strategy that addresses an adaptive robust finite-time stabilization law derived from adding a power integrator (API) [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF] technique to a class of single input dynamics described as a normal form system [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF]. The idea is to generate coordinate transformations that decouple the underactuated dynamics into an upper-triangular form allowing for only one control input governing the whole system to be designed, whereby a continuous finite-time stabilizing control law is derived from the API technique, and an adaptive compensator is proposed to compensate for the matched and unmatched uncertainties collected throughout the steps of the design. As a consequence, the resulting control law is able to stabilize the underactuated mechanical system in a finite time.

The reminder of this chapter is organized as follows. Section 8.2 discusses the Lagrangian model of mechanical underactuated systems and the coordinate transformation that transforms this model into a normal-form representation. Section 8.3 formulates the finite-time stabilization problem for the nth-order underactuated system. Section 8.4 gives a background on finite-time stability and provides some useful lemmas needed throughout the paper. The control design is developed in Section 8.5. Simulation results with an application to the cart-pole system are presented and discussed in Section 8.6. Finally, Section 8.7 draws a conclusion for this chapter.

Dynamic modeling of underactuated mechanical systems: Lagrangian formalism

The Lagrangian formalism is one among the powerful mathematical tools based on the vibrational method to model a large class of mechanical systems. Underactuated mechanical systems are branches of this class. They can be modeled using the Euler-Lagrange equations. Let Q be an n-dimensional configuration manifold, and q ∈ Q is a vector of generalized coordinates. The Euler-Lagrange equations for any mechanical system of n degrees of freedom (DOF) are given by [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] d dt

∂L(q, q) ∂ q - ∂L(q, q) ∂q = B(q)u (8.1)
where u ∈ R m and B(q) is the external forces matrix. L(q, q) is the Lagrangian associated to the mechanical system and expressed by the difference between kinetic and potential energies:

L(q, q) = K -V = 1 2 q M (q)q -V (q) (8.2)
where K is the kinetic energy, V (q) denotes the potential energy and M (q) = M (q) ∈ R n×n is a positive definite inertia matrix. Let m ik ∈ R + denote the entries of the inertia matrix M with i = 1, . . . , n and k = 1, . . . , n. Based on (8.1) and (8.2), the equations of motion can be deduced as follows:

k m ik (q)q k + i,k ik (q)q i qk + g i (q) = e B(q)u, = 1, . . . , n (8.3)
where e is the th standard basis in R n , g i = ∂ qi V (q), and ik (q) is the Christoffel symbol defined by

ik (q) = 1 2 ∂m k (q) ∂q i + ∂m i (q) ∂q k - ∂m ik (q) ∂q (8.4)
In a matrix form, (8.3) can be rewritten as M (q)q + C(q, q)q + G(q) = B(q)u (8.5)

The term C(q, q) includes two distinct terms, in particular, the first term q i qk involves the centrifugal forces (i = k), and the second is the Coriolis forces for (i = k), and G(q) represents the gravity vector. It is interesting to point out that the mechanical system described by (8.5) satisfies the skew symmetric property of the matrix Ṁ (q) -2C(q, q). It is important to distinguish two categories of mechanical systems governed by the equation of motion (8.5). A mechanical system described by (8.5) is said to be fully actuated if m = n, which is equivalent to say that the matrix B(q) ∈ R n×m is invertible. In other words, it can be roughly inferred that the mechanical system is fully actuated, if the number of actuators is equal to the dimension of its configuration manifold. On the other hand, if m < n, the mechanical system is said to be underactuated, which is equivalent to have fewer independent inputs than the number of degree of freedom to be controlled. This implies that the matrix B(q) becomes noninvertible and consequently the underactuated mechanical system as opposed to fully actuated systems is not exact feedback linearizable. Because the number of inputs is less than the number of degree of freedom, the controllability of the mechanical underactuated system is not always satisfied. In this chapter, however, we will assume that the Finite-time stabilization of underactuated mechanical systems 169 considered mechanical systems are controllable. Furthermore, we suppose that the equation of motion (8.5) can be rearranged so that the matrix B(q) can be written as:

B(q) = 0 (n-m)×m I m (8.6)
where the first nm equations of (8.5) represent the unactuated dynamics while the last m equations are the actuated dynamics. The generalized coordinates q can further be partitioned as follows:

q = q u q a (8.7)
where q u ∈ R (n-m) is the vector of unactuated configuration variables, and q a ∈ R m is the vector of actuated configuration variables. Then, it is straightforward to rewrite the equation of motion (8.5) as follows:

m 11 (q) m 12 (q) m 21 (q) m 22 (q) qu qa + c 11 (q) c 12 (q) c 21 (q) c 22 (q) qu qa + g 1 (q) g 2 (q) = 0 u 2 (8.8)
The partitioned equation of motion (8.8) can also be represented in component form as follows:

m 11 (q)q u + m 12 (q)q a + h 1 (q, q) = 0 (8.9)

m 21 (q)q u + m 22 (q)q a + h 2 (q, q) = u 2 (8.10)
where h 1 (q, q) = c 11 (q)q u + c 12 (q)q a + g 1 (q) and h 2 (q, q) = c 21 (q)q u + c 22 (q)q a + g 2 (q). It is worth to notice that there is no control input in (8.9); therefore, it is not possible to feedback linearize the system (8.9) and (8.10). It is however possible to partially feedback linearize the system using a change of control. The procedure of linearization of the actuated dynamics is called collocated partial linearization which is credited to Spong [START_REF] Spong | Underactuated mechanical systems[END_REF] and consists in finding a global invertible change of control of the form u 2 = α(q)v + β(q, q) that partially linearizes the dynamics (8.8) such as, after partial linearization, the new control v appears in the nonlinear-subsystem dynamics as well as in the linearized subsystem dynamics as follows:

qu = f 1 (q, q) + g 1 (q, q)v (8.11) qa = v (8.12)
It can be observed that the unactuated part (8.11) is still complex, while the actuated part (8.12) is linearized into a double-integrator dynamics. The new control input v is present in both subsystems, which largely increases the difficulty in the control design. The author in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] introduced a global change of coordinates that decouples these two subsystems and transforms the dynamics of the system into a cascaded form called the normal form, where the control input of the original actuated subsystem does not appear in the unactuated subsystem. This significantly simplifies the control design for the underactuated mechanical system. However, with this approach, there is an inherent requirement that the model parameters of the mechanical system must be perfectly known to be able to design controller for the transformed normal form.

Due to lack of knowledge on the system parameters and the inability to measure the whole system state, the controller has to cope with these uncertainties. This situation justifies the need for the development of a global transformation that decouples the underactuated mechanical system without passing by the partial feedback linearization of the system. Consider the dynamic (8.9), resolve for qu , it gives qu = -m 11 (q) -1 (m 12 (q)q a + h 1 (q, q)) (8.13) Substituting (8.13) into (8.10), after some manipulation, we obtain

qa = f 3 (q, q) + b 3 (q)u 2 (8.14)
Again substituting the resulting dynamics (8.14) into the dynamics of (8.10), then solving for qu leads to the following dynamics: qu = f 4 (q, q) + b 4 (q)u 2 (8.15) where

f 3 (q, q) = (m 22 -m 21 m -1 11 m 12 ) -1 [m 21 m -1 11 h 1 (q, q) -h 2 (q, q)] =, b 3 (q) = (m 22 - m 21 m -1 11 m 12 ) -1 , f 4 (q, q) = -m -1 11 m 12 f 4 (q, q -m -1
11 h 1 (q, q)) and b 4 (q) = -m -1 11 m 12 b 3 (q). Regrouping the dynamic (8.14) and (8.15), one can obtain a state space representation of the system as follows:

ẋ1 = x 2 ẋ2 = f 3 (q, q) + b 3 (q)u 2 ẋ3 = x 4 (8.16) ẋ4 = f 4 (q, q) + b 4 (q)u 2
where the state space vector x has been defined as [x 1 , x 2 , x 3 , x 4 ] = [q u , qu , q a , qa ] . Note the structural difference between (8.12), (8.11) and (8.16) is that the actuated dynamics in (8.12) is a double integrator, while in (8.16), the actuated dynamics has some nonlinearities.

To decouple the nonlinear system represented by (8.16), we may use the global transformation proposed in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] as follows:

z 1 = x 1 - x 3 0 b 3 (s) b 4 (s) ds, z 2 = x 2 - b 3 (x) b 4 (x) , z 3 = x 3 (8.

Finite-time stabilization of underactuated mechanical systems 171

The equation dynamics in the Z-space is found by taking the time derivative of (8.17), leading to

ż1 = z 2 + b 3 (z) b 4 (z) z 4 - d dt z 3 0 b 3 (s) b 4 (s) ds ż2 = f 3 (z) - b 3 (z) b 4 (z) f 4 (z) - d dt b 3 (z) b 4 (z) z 4 ż3 = z 4 (8.18) ż4 = f 4 (z) + b 4 (z)u 2
Clearly, the dynamics of the underactuated system in Z-space, is in a special cascade form with only one control input appearing in the last subsystem. To ease the representation, let us define

d 1 (z) = b 3 (z) b 4 (z) z 4 -d dt z 3 0 b 3 (s) b 4 (s) ds, d 2 (z) = -z 3 + f 3 (z) -b 3 (z) b 4 (z) f 4 (z) - d dt b 3 (z)
b 4 (z) z 4 , d 3 (z) = 0, then the system (8.18) can be rewritten as

ż1 = z 2 + d 1 (z) ż2 = z 3 + d 2 (z) ż3 = z 4 + d 3 (z) (8.19) ż4 = f 4 (z) + b 4 (z)u 2
where d 1 (z), d 2 (z) and d 3 (z) are regarded as unmatched uncertainties to be compensated by the control u 2 . In a similar manner, an nth-order underactuated system can be represented in the state space form as

ẋ1 = x 2 ẋ2 = f 3 (q, q) + b 3 (q)u ẋ3 = x 4 ẋ4 = f 4 (q, q) + b 4 (q)u . . . (8.20) ẋn-1 = x n ẋn = f n (q, q) + b n (q)u
To decouple the nth order system (8.20), the global transformation as proposed in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF] can be extended to nth order recursively as follows:

z 1 = x 1 - x 3 0 b 3 (s) b 4 (s) ds, z 2 = x 2 - b 3 (x) b 4 (x) , z 3 = x 3 - x 5 0 b 4 (s) b 5 (s) ds z 4 = x 4 - b 4 (x) b 5 (x) . . . (8.21) z n-3 = x n-3 - x n-1 0 b n-2 (s) b n (s) ds z n-2 = x n-2 - b n-1 (x) b n (x) z n-1 = x n-1 z n = x n
Hence, the system dynamics in Z-space can similarly be obtained, taking the time derivative of (8.21), we have

ż1 = z 2 + b 3 (z) b 4 (z) z 4 - d dt z 3 0 b 3 (s) b 4 (s) ds ż2 = f 3 (z) - b 3 (z) b 4 (z) f 4 - d dt b 3 (z) b 4 (z) z 4 ż3 = z 4 + b 4 (z) b 6 (z) z 6 - d dt z 5 0 b 4 (s) b 6 (s) ds ż4 = f 4 (z) - b 4 (z) b 6 (z) f 6 (z) - d dt b 4 (z) b 6 (z) z 6 . . . (8.22) żn-3 = z n-2 + b n-2 (z) b n (z) z n - d dt z n-1 0 b n-2 (s) b n (s) ds żn-2 = f n-2 (z) - b n-2 (z) b n (z) f n - d dt b n-2 (z) b n (z) z n żn-1 = z n żn = f n (z) + b n (z)u
Let us now define the following transformation: Finite-time stabilization of underactuated mechanical systems 173

d 1 (z) = b 3 (z) b 4 (z) z 4 - d dt z 3 0 b 3 (s) b 4 (s) ds d 2 (z) = -z 3 + f 3 (z) - b 3 (z) b 4 (z) f 4 - d dt b 3 (z) b 4 (z) z 4 . . . ( 8
d n-3 = b n-2 (z) b n (z) z n - d dt z n-1 0 b n-2 (s) b n (s) ds d n-2 = -z n-1 + f n-2 (z) - b n-2 (z) b n (z) f n - d dt b n-2 (z) b n (z) z n d n-1 = 0 d n = f n (z)
The dynamics in (8.23) can be similarly written in a standard form like in (8.18) as follows:

żi = z i+1 + d i (z), 1 ≤ i ≤ n -1 żn = b n (z)u + d n (z) (8.24) y = h(z)
where d i (z) ∈ R, i = 1, 2, . . . , n are regarded as matched and unmatched uncertainties. It can be noticed that by construction, the variation bounds of the uncertainties d i (z), i = 1, . . . , n are not assumed to be known and also cannot be assumed linearly parameterizable into a multiplication of a known regressor and a vector of unknown constant parameters. Also for the simplicity of the control design, let us consider the following two assumptions:

Control problem formulation

In this chapter, we consider a class of underactuated systems of the form (8.20), which can be transformed without partial feedback linearization and through the global change of coordinates (8.21) into the cascaded form (8.24). Where

z = [z 1 , z 2 , . . . , z n ] ∈ , where is a compact set of R n , u ∈ R is the control input and d i (z) ∈ R, i = 1, . . . ,

Preliminaries

In this section, a review of some basic concepts and lemma related to the concept of finite-time stability and the corresponding Lyapunov theorem are presented; they will be useful for most of the stability analysis of this chapter.

Notions of finite stability

Theorem 8.1. [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF]: Consider the non-Lipschitz continuous autonomous system ẋ = f (x), f (0) ∈ R. Assume, there are C 1 function V (x) defined on a neighborhood D ⊂ R of the origin, and real numbers c > 0 and 0 < α < 1 such that

1. V (x) is positive definite on D 2. V (x) + cV α ≤ 0, ∀x ∈ D
Then, the origin x is locally finite-time stable. If D = R, and V (x) is radially unbounded, then the origin x = 0 is globally finite-time stable. Moreover, it can be verified that the settling time being dependent on the initial state x(0) = x 0 satisfies T x (x 0 ) ≤ V (x 0 ) 1-α c(1-α) for all x 0 in some open neighborhood of the origin.

Lemma 8.1. [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF] For any x i ∈ R, i = 1, . . . , n and 0 < p ≤ 1, the following inequality holds

(|x 1 | + • • • + |x n |) p ≤ |x 1 | p + • • • + |x n | p ≤ n 1-p (|x 1 | + • • • + |x n |) p . When 0 < p = p 1 p 2 ≤ 1
, where p 1 and p 2 are positive odd integers, then the following holds |x py p | ≤ 2 1-p |x -y| p , for any x, y ∈ R. Lemma 8.2. [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF] For any x ∈ R, y ∈ R, c > 0, d > 0 and γ (x, y) > 0, a real-valued function, the following holds:

|x| c |y| d ≤ (cγ (x, y)/(c + d))|x| c+d + (dγ (x, y) -c/d /(c + d))|y| c+d .

Proposed control design

The purpose of this section is to devise a control law that stabilizes the system (8.24) in finite time. Given the fact that the Z-system (8.24) contains unmatched uncertainty, traditional adaptive techniques like the backstepping technique [START_REF] Khalil | Nonlinear Systems[END_REF] and the multiple-surface sliding [START_REF] Qian | Hierarchical sliding mode control for a class of SIMO under-actuated systems[END_REF] cannot be used, because the variation bounds of d i , i = 1, . . . , n are not assumed to be known. To solve the problem of finite-time stabilization with unmatched uncertainties, we propose a novel robust finite-time control scheme whereby a continuous recursive finite-time stabilizing control law is derived from the adding power integrator (API) technique [START_REF] Qian | A continuous feedback approach to global strong stabilization of nonlinear systems[END_REF] and an adaptive compensator.

Finite-time stabilization of underactuated mechanical systems 175

To develop the finite-time controller for (8.24), we first introduce the following tracking error e 1 and the virtual errors e m , m = 2, . . . , n as follows: where q m = (4 n + 3 -2m)/(4 n + 1) > 0, ( ∈ Z + ); z * m , m = 2, . . . , n is the virtual control given by z * md = -β m-1 e qm m-1 with β m-1 being positive constant to be determined later. One main feature for the use of the parameter in the fraction power q m is that it allows adjusting the control precision and finite-time convergence of the state variables.

e 1 = z 1 -z *
The design procedure consists in n steps, throughout the n -1 steps, the virtual controllers z * (m-1)d will be designed. Upon the completion of step n, the fast control term as well the adaptive compensator are designed.

Step 1: Consider the following Lyapunov candidate function:

V 1 (z) = 1 1 + S e 1+S 1 (8.28)
where S = (4 n -1)/(4 n + 1). The time derivative of (8.28) gives

V1 (z) = e S 1 ė1 = e S 1 z * 2 + e S 1 (z 2 -z * 2d ) + e S 1 d 1 (z) (8.29)
By introducing the virtual control z * 2d = -β 1 e S 1 , where β 1 is a design parameter, V1 (z) rewrites:

V1 (z) = -β 1 e 2S 1 + e S 1 (z 2 -z * 2d ) + e S 1 d 1 (z) (8.30)
Using Lemmas 8.1 and 8.2, the second term of (8.30) can be upper-bounded as follows:

e S 1 (z 2 -z * 2d ) ≤ 2 1-S |e 2 | S |e 1 | S ≤ 1 2 |e 1 | 2S + C 2 |e 2 | 2S (8.31)
where C 2 is a positive constant. In light of (8.31), the time derivative of V 1 (z) can be written as follows:

V1 (z) ≤ -β 1 e 2S 1 + 1 2 |e 1 | 2S + C 2 |e 2 | 2S + e S 1 d 1 (z) (8.32)
Step 2: According to [START_REF] Huang | Global finite-time stabilization of a class of uncertain nonlinear systems[END_REF], a C 1 and positive definite Lyapunov candidate function is constructed as

V 2 ((z)) = V 1 (z) + W 2 (z 1 , z 2 ) (8.33) with W 2 (z 1 , z 2 ) = z 2 z * 2d χ 1 q 2 -z * 1 q 2 2d 1+S-q 2 dχ (8.34)
then it is easy to show that the Lyapunov function candidate V 2 (z) is positive definite and satisfies V 2 (z) ≤ max{ 1 1+S , 2}(e 1+S

1 + e 1+S 2 ). Note that the function W 2 (z 1 , z 2 ) has the following properties:

∂W 2 (.) ∂z 2 = e 1+S-q 2 2 (8.35) ∂W 2 (.) ∂z 1 = -(1 + S -q 2 ) ∂z * 1 q 2 2d ∂z 1 z 2 z * 2d χ 1 q 2 -z * 1 q 2 2d S-q 2 dχ = - ∂z * 1 q 2 2d ∂z 1 (z 2 -z * 2d ) (8.36)
Hence, the derivative of W 2 (.) can be computed as follows:

Ẇ2 (.) = e 2 ż2 -((z 2 -z * 2d ) ∂z * 1 q 2 2d ∂z 1 ż1 ) = e 2 (z 3 -z * 3d ) + e 2 z * 3 + e 2 d 2 (z) -(z 2 -z * 2d ) ∂z * 1 q 2 2d ∂z 1 ż1 (8.37)
using the fact that

(z 2 -z * 2d ) ∂z * 1 q 2 2d ∂z 1 ż1 ≤ 2 1-q 2 e q 2 2 ∂z * 1 q 2 2d ∂z 1 z 2 + 2 1-q 2 e q 2 2 ∂z * 1 q 2 2d ∂z 1 d 1 (z) (8.38)
Let the first lumped unmatched uncertainty be

τ 1 = e 2 d 2 (z) + 2 1-q 2 β 1 q 2 1 e q 2 2 d 1 (z), it is then straightforward to obtain Ẇ2 ≤ e 2 (z 3 -z * 3d ) + e 2 z * 3d + 2 1-q 2 β 1 q 2 e q 2 2 |z 2 | + τ 1 (8.39)
According to Lemma 8.1, the third term of the right hand side of (8.39) can be bounded such that

|z 2 | ≤ |e 2 + z * 1 q 2 2d | q 2 ≤ |e 2 | q 2 + β 1 |e 1 | q 2 (8.40)
which implies that

|e| q 2 2 2 1-q 2 β 1 q 2 |z 2 | ≤ 2 1-q 2 β 1 q 2 |e 2 | 2q 2 + 2 1-q 2 β 1+ 1 q 2 |e 2 | q 2 |e 1 | q 2 ≤ 2 2(1-q 2 )-1 β 2(1+ 1 q 2 ) |e 2 | 2q 2 + 1 2 |e 1 | 2q 2 + 2 1-q 2 β 1 q 2 |e 2 | 2q 2 = 1 2 |e 1 | 2q 2 + C2 |e 2 | 2q 2 (8.41)
where C2 = 2 2(1-q 2 ) β 2(1+ 1 q 2

) + 2 1-q 2 β 1 q 2 . Combining (8.32), (8.39) and (8.41) yields the following:

V2 (.) ≤ -(β 1 -1)|e 1 | 2S + (C 1 + C2 )|e 2 | 2S + e 3 z * 3d + e 2 (z 3 -z * 3d ) + τ 1 (8.42)

Finite-time stabilization of underactuated mechanical systems 177

At this stage, it is worth to note that q 3 = q 2 -2 4 n+1 , which implies that q 3 = 1 + 2q 2 = 1 + 2S. Therefore, by selecting β 1 > n -1 -κ, where κ > 0 is a design parameter, and the virtual control

z * 3d = -β 2 z q 3 2d with β 2 ≥ n -2 + κ + C 2 + C2 , we obtain V2 ≤ -(n -2 + κ)|e 1 | 2S -(n -2 + κ)|e 2 | 2S + e 2 (z 3 -z * 3d ) + τ 1 (8.43)
Step k (k: = 3 . . . n -1): We proceed to the derivation of the virtual control by using an inductive argument. Suppose at step k -1, that there exists a C 1 Lyapunov candidate function V k-1 (z 1 , . . . , z k-1 ), positive definite and verifies

V k-1 (.) ≤ max{ 1 1 + S , 2} k-1 m=1 e 1+S m (8.44) such that Vk-1 (.) ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + e 1+S-q k-1 k-1 (z k -z * kd ) + k-2 m=1 τ m (8.45)
Now let us claim that (8.44) and (8.45) hold as step k. To prove this claim, consider the following Lypunov candidate function:

V k (z 1 , . . . , z k ) = V k-1 (.) + W k (z 1 , . . . , z k ) (8.46)
with

W k (.) = z k z * dk (χ 1 q k -z * 1 q k dk ) 1+S-q k dχ (8.47)
From the previous step, it can be observed that W k (.) has the following properties:

∂W k (.) ∂z k = e 1+S-q k (8.48) ∂W k (.) ∂z m = -(1 + S -q k ) ∂z * 1 q k dk ∂z m z k z * dk (χ 1 q k -z 1 q k dk ) S-q k dχ (8.49)
also, it is easy to show that V k (.) is C 1 and positive definite, which verifies

V k (.) =≤ max 1 1+S , 2 k m=1 e 1+S m . The time derivative of V k (.) satisfies Vk (.) ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + e 1+S-q k-1 k-1 (z k -z * dk ) + e 1+S-q k k z k+1 + e 1+S-q k k d k (z) + k-1 m=2 ∂W k (.) ∂z m żm + k-1 m=1 τ m ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + e 1+S-q k-1 k-1 (z k -z * dk ) + e 1+S-q k k z k+1 + k-1 m=2 ∂W k (.) ∂z m z m+1 + e 1+S-q k k d k (z) + k-1 m=2 ∂W k (.) ∂z m d m (z) + k-1 m=1 τ m (8.50)
If the kth lumped unmatched uncertainty is defined as

τ k = e 1+S-q k k d k (z) + k-1 m=2 ∂W k (.)
∂zm d m (z), then (8.50) rewrites:

Vk (.) ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + e 1+S-q k-1 k-1 (z k -z * dk ) + e 1+S-q k k z k+1 + k-1 m=2 ∂W k (.) ∂z m z m+1 + k m=1 τ m (8.51)
Next, we bound the second and the fourth terms of the right hand side of (8.51). First according to Lemma 8.2, it holds that

|e 1+S-q k-1 k-1 (z k -z * dk )| ≤ 2 1+S-q k-1 |e k-1 | 1+S-q k-1 |e k | q k ≤ |e k-1 | 2S 2 + C k |e 2S k | (8.52)
with C k a positive constant. As for the fourth term, it is easy to obtain the following:

k-1 m=2 ∂W k (.) ∂z m z m+1 ≤ (1 + S -q k )2 1-q k |e k | S k-1 m=2 ∂z * dk ∂z m z m+1 (8.53)
To further bound the fourth term in (8.51), we need to conduct the analysis by inductive argument and assume that at step k -1, the following holds

k-2 m=2 ∂z * dk ∂z m z m+1 ≤ k-1 m=1 γ (k-1)m e S m (8.54)
where γ (k-1)m ≥ 0, then show the inequality also holds for k. Therefore, we have m-1 e m-1 , and therefore, the following inequality holds

k-1 m=2 ∂z * dk ∂z m z m+1 ≤ -β 1 q k k-1 k-1 m=2 ∂e k-1 ∂z m z m+1 ≤ β 1 q k k-1 z 1 q k -1 k-1 q k-1 z k + k-2 m=2 ∂z * 1 q k -1 k-1 ∂z m z m+1 (8.55)
|z m | ≤ |e m + z * 1 qm m | qm ≤ |e m | qm + β m-1 |e m-1 | qm (8.56)
Applying (8.56) to the inequality (8.55) and using the assumption of the inequality (8.54) yields

k-1 m=2 ∂z * dk ∂z m z m+1 ≤ β 1 q k k-1 1 q k-1 |e k-1 | 1-q k-1 + β 1 q k-1 -1 k-2 e 1-q k-1 k-2 |e k | q k + β k-1 |e k-1 | q k + k-2 m=2 ∂z * dk ∂z m z m+1 ≤ k m=1 γ km |e m | S (8.57)
with γ km being a positive constant. Therefore,

k-1 m=2 ∂W k (.) ∂z m z m+1 ≤ (1 + S -q k )2 1-q k |e k | S k m=1 γ km |e m | S ≤ 1 2 k-1 m=1 e 2S m + Ck |e k | 2S (8.58)
with Ck ≥ 0 being a positive constant. Substituting (8.52) and (8.58) into (8.51) leads to AQ2

Vk (.) ≤ -(n -k + 1 + κ) k-1 m=1 e 2S m + (C k + Ck )|e k | 2S + |e k-1 | 2 + 1 2 k-1 m=1 e 2S m + k m=1 τ k + e 1+S-q k k (z k+1 -z d(k+1) ) * + e 1+S-q k k z * k+1 ≤ -(n -k + κ) k-1 m=1 e 2S m + (C k + Ck )|e k | 2S + e 1+S-q k k (z k+1 -z * d(k+1) ) + e 1+S-q k k z * k+1 + k m=1 τ k (8.59) By introducing the virtual control z * (k+1)d = -β k e q k+1 k with β k selected such that β k ≥ n -k + κ + C k + Ck > 0, we get Vk (.) ≤ -(n -k + κ) k m=1 e 2S m + e 1+S-q k k (z k+1 -z * (k+1)d ) + k m=1 τ k (8.60)
Step n: This is the final stage of the design where the real control input appears in the dynamics. For this step, consider the nth part of the Lyapunov candidate function

V n (z 1 , . . . , z n ) = V n-1 (.) + W n (z 1 , . . . , z n ) (8.61)
where

W n (.) = zn z * nd (χ 1 qn -z 1 qn n ) 1+S-qn dχ (8.62)
then, it is obvious to conclude that V n (.) is C 1 , positive definite and satisfies V n (.) ≤ max{ 1 1+S , 2} n m=1 e 1+S m . Furthermore, let G = G -G be the estimate error of the matched and unmatched uncertainty, and G is the total lumped uncertainty to be defined later. From the above inductive argument, one can conclude that The main result of this chapter can be summarized in the following theorem. Consider the nth order underactuated system in the X -space represented by (8.20), through the coordinate transformation (8.21) and under Assumptions 8.1 and 8.2, the finite-time convergence objective specified in (8.25) is achieved by the control input u consisting of (8.64) and the compensator (8.65) with the virtual control z * md , m = 1, . . . , n being applied.

Vn ≤ -κ n-1 m=1 e 2S m + e 1+S-qn n żn + (C n + Cn )e 2S n + n-1 m=1 τ m ≤ -κ n-1 m=1 e 2S m + e 1+S-qn n (b n u + d n (z)) + (C n + Cn )e 2S
Proof. Consider the Lyapunov function with μ = κ max{ 1 1+S ,2} . From (8.70), it is straightforward to conclude from Theorem 8.1 that the solutions of the closed-loop system consisting of (8.24) and the control input (8.64) along with the compensator (8.65) are finite-time stable, and therefore, the limit in (8.25) is satisfied. This completes the proof.

V = V n (.). Since bn 2 G 2 is positive definite, then V ≤ V n (.) + bn 2 G 2 .
+ b n σ G -b n G ˙ G ≤ -κ n-1 m=1 e 2S m -(b n β n -C n -Cn )e 2S n + b n (Gσ -G|σ |) -b n G|σ | (8.67) If β n is chosen such that β n b max -C n -Cn > κ,

Numerical simulations: the case of the cart-pole system

The cart-pole inverted pendulum depicted schematically in Figure 8.1 consists of a pole mounted on a cart by means of a pivot in such a way that the pole can freely swing in the x-y plane. The equations of motion can be obtained by neglecting friction in the pivot and by applying the Euler-Lagrange formulation. The system model can be represented as follows [START_REF] Park | Swing-up and stabilization control of inverted pendulum systems via coupled sliding-mode control method[END_REF]:

(M + m sin 2 θ )ẍ -m sin θ (l θ 2 -g cos θ) = u (8.71) (M + m sin 2 θ )l θ + ml θ 2 sin θ cos θ -(M + m)g sin θ = -cos θ(u)
where M is the mass of the cart, m is the mass of the pendulum, l is half the length of the pendulum, i.e., the distance from the pivot to the center of mass of the pendulum, g is the acceleration due to gravity. The cart-pole system has two equilibrium points, one of which is known as the stable vertically downward position where θ = π , and the other one being the unstable vertically upward position where θ = 0. F(t) is the horizontal force being applied to the cart to swing the pendulum until it reaches the desired unstable position where θ = 0, θ = 0 and ẋ = 0. As can be seen from the equations of motion (8.71), only one control u appears in both dynamic equations of the cart-pole system; therefore, the whole system is underactuated. Meaning that it has fewer control inputs than degrees of freedom. Let . The state space representation (8.72) is in the form (8.20) with n = 4. The system can be readily transformed from the X -space into Z-space using the coordinate transformation (8.21) as follows:

x = [x, ẋ, θ , θ] = [x 1 , x 2 , x 3 , x 4 ] ∈ R 4 ,
z 1 = x 1 - x 3 0 b 2 (s) b 4 (s) ds = x 1 + l ln( sec x 3 + tan x 3 ) z 2 = x 2 - b 2 (x) b 4 (x) x 4 = x 2 + lx 4 sec x 3 z 3 = x 3 (8.73) z 4 = x 4
with the change of coordinate (8.73), the system dynamics in the Z-space becomes

ż1 = z 2 + d 1 (z) ż2 = z 3 + d 2 (z) ż3 = z 4 + d 3 (z) (8.74) ż4 = b 4 (z)u + d 4 (z)
where d 1 (z) = d 3 (z) = 0 and

d 2 (z) = -z 3 -lz 2 4 sec z 3 tan z 3 + (M + m)g tan z 3 -mg sin z 3 cos z 3 M + m sin 2 z 3 d 4 (z) = (M + m)g sin z 3 -mlz 2 4 sin z 3 cos z 3 Ml + ml sin 2 z 3 (8.75) b 4 (z) = -cos z 3 Ml + ml sin 2 z 3
The objective is then to control the transformed system (8.74) such that all states z 1 , z 2 , z 3 and z 4 converge in finite time to zero regardless of the internal uncertainties and external disturbances. For such system, the virtual controllers and the actual

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z * 1d = 0, e 1 = z 1 q 1 1 -z * 1 q 1 1d z * 2d = -β 1 e q 2 1 , e 2 = z 1 q 2 2 -z * 1 q 2 2d z * 3d = -β 2 e q 3
2 , e 3 = z

1 q 2 3 -z * 1 q 2 3d z * 4d = -β 3 e q 4
3 , e 4 = z the controller is able to steer the system to its desired values. The control effort after the application of the disturbances is shown in Figure 8.6, clearly the amplitude of the control input even with the external perturbation remains with reasonable size. The convergence of the state variables in Z-space is shown in Figure 8.7. From these figures, it can be seen that our control design is robust to tolerate significant variation of the system parameters as well as the external disturbances.

Conclusion

In this paper, the problem of global finite-time stabilization was addressed for the control of a class of uncertain underactuated mechanical systems. By integrating a fractional power feedback control method with a compensator, the uncertainties in the system can be effectively handled and finite-time stabilization is achieved. Future work is to extend the current design technique to a class of chained form systems perturbed by external disturbances as well as experiments on the inertia wheel inverted pendulum.

Assumption 8 . 1 .Assumption 8 . 2 .

 8182 The terms d i (z), i = 1, . . . , n are all unknown functions, whose bounds are also unknown. The uncertain control coefficient b n (z) is bounded as b min ≤ b n (z) ≤ b max . and b n (z) is sign-definite such that sign(b n (z)) = +1. Further, it is convenient to assume that b n (z) = 0.

Finite-time stabilization 1 ,

 1 of underactuated mechanical systems 179In light of the definition of the tracking error, it is worth recalling that e m = z it can then be inferred that z * 1

-

  m + b n e 1+S-qn n u + (C n + Cn )e 2S n + b n e 1+S-qnwhere C n > 0 and Cn > 0 are positive constants. The actual control law can therefore be designed as follows:u = -β n e q n+1n Gsign(σ ) (8.64)with the compensator being designed as follows:˙ G = |σ |, σ = e 1+S-qn (8.65)where G is the estimation of the total uncertainty function G defined as G = τ m , and β n is a positive design parameter chosen arbitrarily by the designer.

Finite-time stabilization of underactuated mechanical systems 181 Theorem 8 . 2

 18182

  Taking the derivative of V along (8.63), yieldsV ≤ Vnb n G ˙ G m + b n e 1+S-qn n u + (C n + Cn )e 2S n + b n e 1+S-qn n d n (z) m + b n e 1+S-qn n u + (C n + Cn )e 2S n + b n e 1+S-qn n Gb n G ˙ G (8.66) Substituting the control law (8.64) and the update law (8.65) in (8.66) leads to mb n β n e 1+S-qn+q n+1 n + (C n + Cn )e 2S nb n σ Gsign(σ )

Figure 8 . 1 ẋ1 = x 2 ẋ2 = f 2 (x) + b 2 x 3 -mg sin x 3 cos x 3 M +m sin 2 x 3 , b 2 (x) = 1 M +m sin 2 x 3 , f 4 ( 3 Ml+ml sin 2 x 3

 81222333213433 Figure 8.1 Schematic view of the cart-pole inverted pendulum

Figure 8 . 2

 82 Figure 8.2 Trajectories of the cart position and pendulum angle versus time

4 - 1 . 1 :Figure 8 . 3 Figure 8 . 4 Figure 8 . 5 Figure 8 . 6 2 . 2 :Figure 8 . 7

 411838485862287 Figure 8.3 Evolution of the control effort versus time

  .23)

	Boubaker CH008.tex	August 16, 2017	18: 53	Page 173

  n -1 are unmatched uncertainties, while d n is the matched uncertainty. Under Assumptions 8.1 and 8.2, the idea is to design the controller u such that the system state z tracks in finite time the desired trajectoryz * d = [z * 1d , z * 2d , . . . , z * nd ] ∈ dwhere d is a compact set of . This means that for any finite initial condition z i (0), i = 1, . . . , n, there exists a finite time T such that The reference input for the Z-subsystem z * 1d = 0 implies stabilization of the state variable x 1 to zero.

	lim t→T	z i -z * id = 0, i = 1, . . . , n	(8.25)
	Remark 8.1. The control design satisfying the limit (8.25) guarantees stabilization

in finite time of the underactuated mechanical system of the form

(8.20)

to a desired reference input x d . In most of the cases, stabilization around the origin of the first state x 1 is desired.

National Institute of Applied Sciences and Technology (INSAT), Tunisia

Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM), France

College of Engineering, Sultan Qaboos University, Sultanate of Oman

Chapter 8

Finite-time stabilization of underactuated mechanical systems

Author Queries AQ1: As per style, the running heads should be of 65 characters (including space). Hence, we have shortened the title of the chapters in the book, wherever needed, so as to match the requirement. Please confirm if these shortened titles are fine. AQ2: An extra closing parenthesis has been deleted in Equation (8.59). Please check, and correct if necessary. AQ3: As Equation (8.68) contains label without equation content, we have removed the equation number and renumbered remaining equations. Please check and amend if necessary. AQ4: Please supply the name of the city of publication for Reference [START_REF] Ge | Autonomous Mobile Robots[END_REF]. AQ5: References [START_REF] Huang | Global finite-time stabilization of a class of uncertain nonlinear systems[END_REF] and [START_REF] Huang | Global finite-time stabilization of a class of uncertain nonlinear systems[END_REF] are almost similar. Please check and correct if necessary.


Carlos Martínez
email: carlos.martinez@inria.fr

Francis Mairet
email: francis.mairet@ifremer.fr

Olivier Bernard
email: olivier.bernard@inria.fr

Theory of turbid microalgae cultures

Keywords: Photobioreactor, Optical depth, Photoinhibition, Light limitation, Turbidity, Modelling

come

Introduction

Microalgae are photosynthetic microorganisms whose biotechnological potential has been highlighted in the last decade. They grow naturally in all the aquatic environments, and can also be cultivated industrially, for production of compounds such as food complements, colorants, antioxydants, pharmaceuticals, or molecules for green chemistry, including biofuels [START_REF] Norsker | Microalgal productiona close look at the economics[END_REF]. They can be mass cultivated in closed or open reactors named photobioreactors 10 (PBR). Depending on the applications, they are grown with artificial or natural light. absorption and scattering by algal cells. Thus, phytoplankton cells perceive a different light intensity depending on their position in the PBR. In the case of wastewater treatment or aquatic systems, the light extinction is exacerbated by the high 20 content of highly diffusive particulate matter and the presence of colored substances [START_REF] Borowitzka | Limits to Growth[END_REF][START_REF] Kirk | Light and Photosynthesis in Aquatic Ecosystems[END_REF]. Experimental evidence in perfectly mixed PBRs shows that photosynthetic efficiency increases with depth [START_REF] Huisman | Principles of the light-limited chemostat: theory and ecological applications[END_REF], consequently phytoplankton cells respond [START_REF] Yun | Kinetic modeling of the lightdependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF] almost intantaneously to all light intensities within the culture medium and not to an average light intensity. Thus, an appropriate way to model the growth rate in phytoplankton population models, consists in accounting for the differences in local 30 growth rates to obtain the average growth rate (AGR). This way of computing the average growth rate is a trade-off between simple models and complicate models accounting for photosynthesis dynamics [START_REF] Demory | How do microalgae perceive light in a high-rate pond? towards more realistic lagrangian experiments[END_REF] which are more difficult to handle 35 and to analyse. [START_REF] Huisman | Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model[END_REF] [START_REF] Huisman | Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model[END_REF] used the AGR for constructing the theory of light-limited chemostats. [START_REF] Gerla | Photoinhibition 555 and the assembly of light-limited phytoplankton communities[END_REF] [START_REF] Gerla | Photoinhibition 555 and the assembly of light-limited phytoplankton communities[END_REF] and Hsu et al. (2011) [START_REF] Hsu | Dynamics of phytoplankton communities under photoinhibition[END_REF] extended the work of Huisman by including photoinhibition i.e. a decrease in the photosynthetic rate due to high light intensities. Some other works are devoted to optimization of the microalgae productivity [START_REF] Mairet | Adaptive control for optimizing microalgae production[END_REF][START_REF] Mairet | The photoinhibistat: Operating 565 microalgae culture under photoinhibition for strain selection[END_REF][START_REF] Martínez | Maximizing microalgae productivity by shading outdoor cultures[END_REF]. The main objective of this paper is to understand how the AGR is affected by different factors; incident light intensity, form of the PBR, microalgae concentration, background turbidity, and light path-length of the reactor (or distance between the illuminated surface and the darkest point).

Most of the mathematical models describing the microalgae growth in PBRs assume that the system is perfectly mixed. Even if real systems are not always perfectly mixed, these models have shown good performances [START_REF] Huesemann | A validated model to predict microalgae growth in outdoor pond cultures subjected to fluctuating light intensities and 575 water temperatures[END_REF][START_REF] Quinn | Microalgae bulk growth model with application to industrial scale systems[END_REF][START_REF] Blanken | Predicting microalgae growth[END_REF][START_REF] Béchet | Full-scale validation of an algal productivity model including nitrogen limitation[END_REF]]. An important property in perfectly mixed phototrophic cultures in vertical cylindrical PBRs illuminated from above (e.g. race-way photobioreactors or panel-type reactors), is that the AGR is completely determined by the incident light intensity and the light intensity at the bottom of the culture. This property has been used by the works cited above. It allows to study the dynamics of microalgae populations, to scale-up theoretical results, and to set-up experiments for measuring the growth rate.

In the first part of this work, we extend this result to PBRs with a flat light-exposed surface and to cylindrical PBRs radially enlightened.

An important variable in our study is the optical depth.

This dimensionless variable describes how much absorption occurs when light travels through the PBR. It includes the effects of the light path-length, the background turbidity, and the microalgae population density. In the second part of this work, we study how the AGR varies with the incident light and the optical depth in flat-plate reactors (or in general vertical cylindrical PBRs illuminated from above). We assume that phytoplankton can suffer from photoinhibition.

Theoretical works [START_REF] Gerla | Photoinhibition 555 and the assembly of light-limited phytoplankton communities[END_REF][START_REF] Hsu | Dynamics of phytoplankton communities under photoinhibition[END_REF] show that photoinhibition can lead algae cultures to the washout, while experimental results show that photoinhibition may cause a loss in biomass productivity even in high dense outdoor cultures [START_REF] Qiang | Physiological characteristics of spirulina platensis (cyanobacteria) cultured at ultrahigh cell densities1[END_REF][START_REF] Vonshak | Photoadaptation, photoinhibition and productivity in the blue-green alga, spirulina platensis grown outdoors[END_REF]. How much microalgae are affected by photoinhibition strongly depends on temperature [START_REF] Talbot | De la Noüe, A 595 comparative study and mathematical modeling of temperature, light and growth of three microalgae potentially useful for wastewater treatment[END_REF] and photoacclimation [START_REF] Neidhardt | 600 Photosystem-ii repair and chloroplast recovery from irradiance stress: relationship between chronic photoinhibition, light-harvesting chlorophyll antenna size and photosynthetic productivity in dunaliella salina (green algae)[END_REF][START_REF] Anning | Photoacclimation in the marine diatom skeletonema costatum[END_REF].

The study of the AGR can lead to some inter-90 esting discussions as shown in the last part of the paper. We study the AGR as a function of the microalgae population density, which gives conditions for the occurrence of an Allee effect. Then, we discuss conditions such that the incident light 95 intensity (sunlight) ostensibly limits or inhibits the AGR in outdoor cultures. Then, based on the results of section 2, we compare the AGR of flat plate PBRs with that of other PBRs. Finally, in order to compare the performance of the different 100 PBRs, we evaluate numerically the productivity that can be reached by each PBR. This paper is organized as follows. In section 2, we define the average growth rate and we deter-105 mine which factors determine its value of the AGR. In section 3, we describe in details the properties of the average growth rate in the case of a vertical cylindrical PBR illuminated from above. Then, in section 4, we discuss some implications of the 110 analysis of the growth rate in PBRs and compare productivities depending on PBR design.

Average growth rate (AGR)

This section begins with the study of PBRs with a flat light-exposed surface. These PBRs serve to 115 introduce the law of Lambert-Beer, the concept of optical depth, and the definition of AGR. Then, we briefly describe the cylindrical PBR radially enlightened. These PBRs are described by simple models which allow to obtain theoretical results.

120

The main results of this section are given by Theorems 2 and 3. They state that the AGR is completely described by two factors; the incident light intensity and the optical depth.

Geometry of PBRs with a flat light-exposed surface.

Let us consider a perfectly mixed PBR having a flat surface illuminated perpendicularly with an unidirectional light field. We can described the region Ω occupied by the PBR (more precisely the volume of the culture) by

Ω = {(x, y, z) ; (x, y) ∈ Ω 0 , z ∈ [0, h(x, y)]},
with Ω 0 ⊂ R 2 (compact set1 ) the illuminated flat-125 surface of the PBR that receives the same light intensity at any point denoted I in (incident light intensity) moving in the positive direction of axis z (see Fig. 1), and h(x, y) a continuous function that represents the depth of the PBR at different 130 points (x, y) ∈ Ω 0 . In this way we can describe different PBRs [START_REF] Ugwu | Photobioreactors 610 for mass cultivation of algae[END_REF]; a flat plate, a vertical-column (illuminated in one base), or an open pond PBR. A flat-plate PBR is described by a constant function h. By choosing adequately Ω 0 and h we can describe other kind of PBRs like a (horizontal cylindrical) triangular PBR (see Fig. 2A) or a (horizontal cylindrical) semicircular PBR (see Fig. 2B). In the following we will denote by L the maximum depth of the PBR that corresponds to the distance from the light-exposed surface to the lowest point of the reactor (i.e. L := max (x,y)∈Ω0 h(x, y)).

Light gradient. Light decreases progressively in moving deeper into the culture medium due to 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[END_REF]. As a first approximation, the law of Lambert-Beer can be used to determine the light intensity I at any position in the PBR. This means that multi-diffusion effects are not explicitly represented. Thus

∂I ∂z = -ξI, (1)
with ξ ≥ 0 an extinction coefficient that depends on the medium. Since the culture is perfectly mixed, ξ does not depend on z. Thus, we can easily integrate Eq.( 1) to obtain I(ξ, I in , z) = I in e -ξz , for all (x, y, z) ∈ Ω. [START_REF] Borowitzka | Limits to Growth[END_REF] Here, z corresponds to the distance from the illuminated surface of the PBR to the position (x, y, z).

A key variable in our study is the light intensity in the lowest (or the darkest) part of the culture, that is

I out (I in , ξL) := I in e -ξL . (3)
We emphasize the fact that I out depends only on the product ξL, and not on L and ξ separately.

145

In microalgae cultures illuminated with natural light, the incident light is in general not perpendicular to the surface. The incident flux must be computed accounting for loss due to reflection. The transmitted light fraction is then refracted into the 150 medium. Due to the very diffusive character of the microalgae culture, we assume that it is rapidly anisotropic and therefore, we keep the Lambert-Beer approximation, assuming now an incident flux ηI in with η ∈ [0, 1].

155

Extinction coefficient. In the case of a monoculture, ξ is mainly correlated to the microalgae population density X [START_REF] Kirk | Light and Photosynthesis in Aquatic Ecosystems[END_REF];

ξ(X) = kX + K bg , (4)
with k > 0 the specific light extinction coefficient of the microalgae specie and K bg ≥ 0 the background turbidity that summarizes the light absorption and difussion due to all non-microalgae components i.e. suspended solids and dissolved colored material. Typical values of the coefficient k for several microalgae species are given in Table 1. To use expression (4), the density of the culture must be low enough so that most of the photons are diffused at most once. For multi diffusion regimes, which characterize industrial reactors, 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 cul-620 ture[END_REF]. Various empirical expressions have therefore been developed to macroscopically account for multi scattering (see table 6 in [START_REF] Béchet | Modeling the effects of light and temperature on algae growth: State of the art and critical assessment for productivity pre-625 diction during outdoor cultivation[END_REF]). For example, in [START_REF] Yun | Kinetic modeling of the lightdependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF] the following expression is proposed

ξ(X) = k 1 X k 2 + X ,
We note that for ξ = 0 (transparent culture), expression ( 5) is reduced to

μ(•) = µ(I in ). ( 6
)
This is consistent with the fact that all microalgae will perceive the same light intensity. To study the case ξ > 0, let us define A Ω (z) as the area of the cross section of Ω at depth z ∈ [0, L]. The following lemma gives a useful expression for determining the AGR in different PBRs.

Lemma 1. If ξ > 0, then μ(•) = 1 ξV (Ω) Iin Iout(ξL,Iin) µ(I) I A Ω 1 ξ ln I in I dI. (7)
Proof. By doing the change of variables I = I(ξ, I in , z) we rewrite expression (5);

μ(•) = 1 ξV (Ω) (x,y)∈Ω0 Iin I(ξ,Iin,h(x,y)) µ(I) I dIdxdy.
After changing the order of integration in the latter expression we obtain Flat-plate PBR. In the case of a flat-plate PBR (i.e. h(x, y) = L > 0 for all (x, y) ∈ Ω 0 ), A Ω (z) is constant and V (Ω) = LA Ω (z). By using Eq.( 7), it is straightforward to verify that

μ(•) = 1 ξV (Ω) Iin Iout(ξL,Iin) (x,
μ(•) = 1 ξL Iin Iout(ξL,Iin) µ(I) I dI. (9)
This shows that in flat-plate PBRs, the AGR is 170 completely determined by ξL and I in .

Optical depth. The product ξL will be denoted by θ. This variable is usually known as optical depth.

The optical depth reflects the actual amount of light energy absorbed by the culture medium. Indeed, θ can be determined from

θ = ln(I in /I out (θ, I in )). (10)
Thus, in view of Eq.( 9), in a flat-plate PBR, the AGR is determined by the light intensities at the top and at the bottom of the PBR.

Triangular horizontal cylinder. Consider now that the PBR is a triangular horizontal cylinder (see

Fig. 2A) i.e. h(x, y) = -m 1 y + L for y ∈ [0, L/m 1 ] and h(x, y) = m 2 y + L for y ∈ [-L/m 2 , 0] with m 1 , m 2 > 0. So we have that A Ω (z) = H 1 m1 + 1 m2 (z + L) and V (Ω) = HL 2 (1/m 1 + 1/m 2 )/2
with H the length of the cylinder. Thus, Lemma 1 gives

μ(•) = - 2 θ 2 Iin Iout(θ,Iin) µ(I) I ln I in I dI + 2 θ Iin Iout(θ,Iin) µ(I) I dI.
(11) This expression shows that the family of all the triangular PBRs have the same AGR, which is determined by the incident light intensity and the optical depth (or the light intensity at the lowest point according to Eq.( 10)). Expression [START_REF] Martínez | Maximizing microalgae productivity by shading outdoor cultures[END_REF] is clearly different from expression [START_REF] Mairet | Adaptive control for optimizing microalgae production[END_REF]. This shows that the form of the reactor must be taken into account for determining the AGR.

Shape of a PBR. We say that two PBRs have equivalent shape if one of them can be obtained by scaling the other one. Mathematically, a PBR occupying a region Ω has equivalent shape of a PBR occupying a region Ω if there exist positive numbers s x ,s y , and s z such that

Ω = {(s x x, s y y, s z z) ; (x, y, z) ∈ Ω}.
If s x = s y = s z we speak of an uniform scaling, otherwise of a non-uniform scaling. The following theorem shows that by fixing the shape of the PBR, we obtain the same AGR and that is determined by the incident light intensity and the optical depth.

Theorem 2. Two PBRs with equivalent shape lead to the same AGR, which is completely determined by the incident light intensity and the optical depth.

Proof. First we note that

V (Ω) = L 0 A Ω (z)dz = Iin Iout(θ,Iin) 1 I A Ω z = 1 ξ ln Iin I dI.
Then, we can write

μ(•) = Iin Iout(ξL,Iin) µ(I)f (I; ξ, I in , Ω)dI, (12)
with f defined by

f (I; ξ, I in , Ω) := 1 I A Ω 1 ξ ln Iin I Iin Iout(ξL,Iin) 1 I A Ω 1 ξ ln Iin I dI .
(13) Now, consider another PBR of depth L > L occupying a region Ω obtained by scaling Ω. That is, there are s x , s y > 0 such that

Ω := {(s x x, s y y, (L /L)z); (x, y, z) ∈ Ω}. Assume now that the extinction coefficient ξ in this PBR satisfies ξL = ξ L . If we note that A Ω (z) = s x s y A Ω ((L/L )z), it is straightforward to verify that f (I; ξ , I in , Ω ) = f (I; ξ, I in , Ω). ( 14
)
Eq.( 14) indicates that the value of f is determined 200 by I in , the product ξL, and shape of the PBR. This holds of course for the AGR, since I out depends only on I in and ξL.

Horizontal semicircular cylinder. As a last example of a flat light-exposed surface PBR, consider a semicircular horizontal cylinder (see Fig. 2B) i.e. h(x, y) = L 2 -y 2 . So we have that

A(z) = 2 √ L 2 -z 2 and Lemma 1 gives μ(•) = 4 πθ Iin Iout(θ,Iin) µ(I) I 1 - 1 θ 2 ln 2 I in I dI.
(15) This expression for the AGR is clearly different from that of Eq.( 9). According to Theorem 2, this 205 expression is also valid for semi-elliptical horizontal cylinders.

Cylindrical PBR radially enlightened. The cylindrical PBR, evenly illuminated around, is also commonly used. The region Ω occupied by the PBR corresponds to a cylinder of radius R, which is radially and evenly illuminated over the sides (not over the bases). From [START_REF] Evers | A model for light-limited continuous cultures: Growth, shading, and maintenance[END_REF], the path length of light to a point at distance r from the center as a function of the angle φ (see Fig. 3) equals

s(r, φ) = r cos φ + R 2 -r 2 sin 2 φ. ( 16
)
By using the law of Lambert-Beer and accounting for the light flux coming from all the directions (φ moving between 0 and 2π), the light intensity at a distance r from the center is given by

I(r) = 2π 0
I in e -ξs(r,φ) dφ.

By using cylindrical coordinates, the AGR can be written as

μ(•) := 1 πR 2 2π 0 R 0 µ(I(r))rdrdφ (18)
I(r) is increasing as a function r (see Proof of Theorem 3). Thus, the darkest zone of the PBR is 210 in the center (r = 0) and the most illuminated zone is on the sides (r = R). Based on the distance between these two zones, we define the optical depth as θ := ξR. We have the following version of Theorem 2 for the cylindrical PBR. Theorem 3. The AGR defined in Eq.( 18) is completely determined by the optical depth and the incident light intensity.

Before proving Theorem 3, note that the optical depth can be obtained explicitly from

θ = ln 2π I in I(0) . ( 19
)
This shows that Theorem 3 could be stated in terms of I in and I(0).

I(0) I(R) µ(I)H(I; ξ, I in , R)dI, (21)
with

H(I; ξ, I in , R) = ϕ(I; ξ, I in , R) R 2 f (ϕ(I; ξ, R); ξ, I in , R)
.

By evaluating I(0) and I(R), it is easy to verify that they are determined by ξR and I in . It remains to prove the same for H. For this purpose, consider another PBR of radius R and assume that the extinction coefficient ξ in this PBR satisfies 235 ξR = ξ R . As in Theorem 2 we have to prove that H(I; ξ, I in , R) = H(I; ξ , I in , R ), but this follows rapidly after noting that ϕ(I; ξ , I in , R ) = ϕ(I; ξ, I in , R)R /R.

3. Properties of the AGR in flat-plate PBRs.

240

Definition of the function g. In a flat-plate PBR the AGR is given by Eq.( 9). If I in > 0, by substituting [START_REF] Mairet | The photoinhibistat: Operating 565 microalgae culture under photoinhibition for strain selection[END_REF] in [START_REF] Mairet | Adaptive control for optimizing microalgae production[END_REF] we have

μ(•) = g(I in , I out (θ, I in )), (22)
with

g(I in , I out ) = 1 ln(I in /I out ) Iin Iout µ(I) I dI. (23)
Eq.( 23) gives an explicit expression for the AGR as a function of the light intensities at the top and at the bottom of the PBR. The function g was first defined in [START_REF] Gerla | Photoinhibition 555 and the assembly of light-limited phytoplankton communities[END_REF] and is useful for studying the properties of the AGR. This function highlights the fundamental link between AGR and the extreme values of light intensity in the PBR.

Assumptions over µ. µ corresponds to the carbon gain rate i.e. µ = p -m, with p the carbon uptake rate and m the specific carbon loss rate that is assumed to be constant. In the literature we find different models for p, for example the classical model of Steele [START_REF] Steele | Environmental control of photosynthesis 635 in the sea[END_REF] p

(I) = p max I I * e 1-I I * , (24)
with p max the maximum value of p, or Haldane-type models 2 like the one presented in [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] p(I) = p max I

I + pmax α I I * -1 2 , ( 25
)
where α is the initial slope of p. Table 2 shows the kinetic parameters of model ( 25) for three different microalgae species. Fig. 4 shows p given in [START_REF] Yun | Kinetic modeling of the lightdependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF] with kinetic parameters from In what follows of this section, we will describe the AGR when p is given by [START_REF] Yun | Kinetic modeling of the lightdependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF] or [START_REF] Béchet | Modeling the effects of light and temperature on algae growth: State of the art and critical assessment for productivity pre-625 diction during outdoor cultivation[END_REF]. However, the theory of turbid cultures is derived for any net growth rate satisfying the following assumptions. 25)) for Chlorella vulgaris with kinetic parameters from Table 2, experimental data from [START_REF] Yeh | Effect of light supply and carbon source on cell growth and cellular composition of a newly isolated microalga chlorella vulgaris esp-31[END_REF], and graphical description of σ.

Assumption 5 states that for high light intensities (I → ∞) phytoplankton grows similarly 265 as under absence of light (I = 0). Indeed, when p is given by 25 we have µ(0) = lim I→∞ µ(I) = -m.

When p is given by Eq.( 25) or Eq.( 24) it is possible to obtain explicit expressions for the AGR (see 270

Appendix B).

Symmetry of p. We note that for any I > 0 different from I * , there exists another light intensity, that will be denoted σ(I), such that

p(I) = p(σ(I)). (26)
A graphic representation of σ is shown in Fig. 4. It seems clear to define σ(I * ) = I * which ensures the continuity of σ. As we will see below, σ is relevant when describing the AGR. In the case that p is given by Eq.( 25) we obtain a simple expression for σ(I)

σ(I) = I * 2 I . (27)
Critical optical depth. Consider that the incident light intensity I in is fixed and lower than or equal to I * . According to Lemma A1, the AGR (or the function g) increases as the light intensity at the bottom I out increases. Consequently, the maximum value of the AGR is obtained when I out equals I in . Recalling Eq.( 1), this equality is only possible if θ = 0 (i.e. the medium is transparent). Consider now that I in is fixed at a value higher than I * . According to Lemma A1, there exists γ(I in ) < I * such that the AGR is maximal when I out = γ(I in ), or, in terms of the optical depth, when θ = ln(I in /γ(I in )).

Following these ideas we define the critical optical depth, denoted by θ(I in ), as

θ(I in ) := ln Iin γ(Iin) if I in > I * , 0 if I in ≤ I * . ( 28
)
The critical optical depth (associated to a critical value if biomass) reflects the value of the optical depth for which the AGR is maximal. Fig. 5A shows how the AGR varies with the optical depth for different incident light intensities. When I in = I * (or I in < I * ), the AGR decreases with θ. When I in > I * , the AGR increases with θ until reaching its maximum value at θ = θ(I in ) and then decreases. As θ tends to be too high, the AGR approaches the value of the specific growth rate in absence of light (lim θ→∞ μ(•) = µ(0)). Proposition A2 states in detail this behavior. We note that θ increases with I in , which is true in general as stated in Proposition A4. When p is given by Eq.( 25), according to Proposition A4, we have that for any

I in > I * ln I in I * < θ(I in ) < 2 ln I in I * . (29)
This inequality shows that θ increases logarithmically. Fig. 6 shows θ(I in ) as a function of I in for three different microalgae species with kinetic parameters from Table 2.

Critical incident light. Fig. 5B shows the AGR as a function of the incident light for different values of the optical depth. According to Lemma A2, there is an incident light intensity I(θ), that we call critical incident light intensity, at which the AGR is maximal. When θ = 0, the AGR coincides with the specific growth rate. As θ increases, the form of the AGR becomes wider and reaches its maximum at a higher incident light intensity. At low incident light intensity (i.e. for I in < I(θ)), light is a limiting factor in the sense that increasing light enhances the growth rate. This notion is not straightforward since some cells (in surface) can be photoinhibited. At high light (and especially for low θ), light is globally inhibiting.

According to Lemma A2, the critical incident light intensity is such that µ is the same at the top and at the bottom of the PBR (see Eq. ( 40)). In terms of σ this can be written as Thus, if p is given by Eq.( 25), then

σ(I(θ)) = I(θ)e -θ . (30)
I(θ) = I * e θ/2 . ( 31
)
If p is given by Eq.( 24), we cannot determine an explicit expression for σ, but we can determine directly from Eq.(40) (Appendix A) the form of the critical incident light intensity

I(θ) = I * θ 1 -e -θ ,
These expressions show that the critical incident light increases with the optical depth. This is true in the general case according to Proposition A5. 2.

sume that ξ is given by ( 4). In that case the optical depth is given by θ(X) = (kX + K bg )L.

From the previous section, we know that the AGR is maximal if θ(X) equals the critical optical depth θ. Thus, if the optical depth in absence of microalgae θ(0) = K bg L is lower than θ, there is a microalgae population density X = θ-K bg L kL maximizing the AGR. This shows the existence of an Allee effect, i.e. the AGR is maximal at an intermediate population density. This is illustrated in Fig. [START_REF] Gerla | Photoinhibition 555 and the assembly of light-limited phytoplankton communities[END_REF] that shows the AGR as a function of X for different values of K bg . Now, if θ(0) = K bg L ≥ θ, any increase in the microalgae population will move the optical depth further away from its optimal value. In consequence, the AGR rate is maximal when X = 0. In that case there is no Allee effect.From Proposition A4, θ increases with I in , thus, high values of I in and low values of K bg L (low optical depth associated to the background turbidity) favor the presence of an Allee effect.

Experimental evidence of a positive optimal population density is for example described in [START_REF] Qiang | Physiological characteristics of spirulina platensis (cyanobacteria) cultured at ultrahigh cell densities 1[END_REF]. The optimal density is 2.5 g/L for a culture of Spirulina platensis in a glass column photobioreactor illuminated with an incident light intensity of 2000 µ mol m -2 s -1 . Incident light intensity as a limiting factor. By a limiting factor we understand a factor such that the AGR increases with any increase of it. Fig. 5B shows that when θ = 10, the incident light intensity is a limiting factor on the range 0 -2000 µ mol m -2 s -1 . This does not mean that microalgae does not suffer from photoinhibition, in fact, they do near the surface. It only means that the AGR increases. To determine if in outdoor cultures illuminated with natural light the light is a limiting factor, let us assume that I max is the maximal incident light intensity (at midday) that the culture can receive. Thus, if I max ≤ I(θ), then light is a limiting factor during all the day. If p is given by Eq.( 25), recalling Eq.( 31), this is equivalent to θ ≥ 2 ln

I max I * . ( 32
)
Thus, if I max = 2000 µ mol m -2 s -1 , and I * = 90 µ mol m -2 s -1 , condition [START_REF] Qiang | Productivity and photosynthetic efficiency ofspirulina platensis as affected by light intensity, algal density and rate of mixing in a flat plate photobioreactor[END_REF] says that for cultures with θ ≥ 6.1, the light is a limiting factor. In terms of microalgae concentration, if ξ is given by Eq.( 4), Eq.( 32) can be put as

LX ≥ 2 ln I max I * -LK bg /k. ( 33
)
If K bg = 0 m -1 and k = 0.2 gC m -2 , the previous condition says that for cultures with an areal 325 microalgal concentration higher than 30.5 gC m -2 light is a limiting factor.

AGR in different geometries. In section 3, we described some properties of the AGR in vertical cylindrical (or flat-plate) PBRs illuminated over one base. Even if we did not provide any formal demonstration for these different cases, these properties are expected to be valid for PBRs with a different shape. For sake of simplicity, we will refer to the different PBRs as flat-plate (Eq.( 9)), triangular (Eq.( 11)), semicircular (Eq.15)), and cylindrical PBR (Eq.( 18)). Fig. 8A shows the AGR for different PBRs as a function of the optical depth. We can see that for low values of θ the AGR is higher in the flat-plate PBR while for high values of θ it is higher in the triangular PBR. Fig. 8B shows the AGR for the different PBRs as a function of the incident light intensity. We can see that for high values of I in the AGR is higher in the flat-plate PBR while for low values of I in it is higher in the cylindrical PBR. These differences can be justified by the distribution of the microalgal culture with respect to the light gradient.

Productivity. In order to compare the productivity of the different PBRs presented in this paper, we benchmark them considering that they all have the same volume V and the same irradiated culture area A. The depth of the flat-plate, the triangular, and the semicircular PBRs are V /A, 2V /A, and 4 π V /A respectively. For the cylindrical PBR the radius is 2V /A. B = XV /A corresponds to the biomass concentration per unit of irradiated area. Thus, the biomass productivity per unit of irradiated area is

P := B μ(I in , θ). (34)
Writing the AGR as a function of the optical depth and the incident light intensity is justified by Theorems 2 and 3. Let us assume that the extinction coefficient is given by Eq. ( 4). Then, in the flatplate PBR, the optical depth is given by

θ = (kX + K bg )V /A = kB + K bg V /A. (35)
Consequently, P depends on V /A but not on the values of V and A separately. The same is valid for the other PBRs. In the case that K bg = 0, the productivity P is independent of V /A. Figure 9 shows the productivity as a function of B. We note that when I in > I * the highest productivity is reached in the rectangular PBR. However, when A. I in = 500 µ mol m -2 s -1 is fixed. B. θ = 3 is fixed. We consider p given by [START_REF] Yun | Kinetic modeling of the lightdependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF] with the kinetic parameters of C. vulgaris given in Table 2 and m = 0.1 d -1 .

I in < I * the highest productivity is reached in the column PBR. The microalgae population density that yields maximal productivity is known as optimal population (or cell) density (OCD). The 360 OCD has been studied experimentally in many works [START_REF] Qiang | Physiological characteristics of spirulina platensis (cyanobacteria) cultured at ultrahigh cell densities 1[END_REF][START_REF] Qiang | Productivity and photosynthetic efficiency ofspirulina platensis as affected by light intensity, algal density and rate of mixing in a flat plate photobioreactor[END_REF][START_REF] Hu | Ultrahigh-cell-density culture of a marine green alga chlorococcum littorale in a flat-plate photobioreactor[END_REF][START_REF] Qiang | Combined effects of light intensity, light-path and culture density on output rate of spirulina platensis (cyanobacteria)[END_REF].

In practice, background turbidity is not zero. It can even be very high if algae are used in wastewa-365 ter treatment. In that case, the productivity varies with V /A (or with the light path-length). Figure 10 shows the maximal productivity for different values of V /A. As shown experimentally in [START_REF] Qiang | Combined effects of light intensity, light-path and culture density on output rate of spirulina platensis (cyanobacteria)[END_REF], the productivity decreases as the light path-length A. I in = 1000 µ mol m -2 s -1 B. I in = 50 µ mol m -2 s -1 . We consider p given by [START_REF] Yun | Kinetic modeling of the lightdependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF] with the kinetic parameters of C. vulgaris given in Table 2. The other parameters are taken to be k = 0.2 m 2 gC -1 , m = 0.1 d -1 , and K bg = 0.

increases. Note that the optical depth depends on K bg V /A and not on K bg and V /A separately. Thus, the operational parameters determining the productivity are the incident light intensityI in and the dimensionless parameter K bg V /A describing the background optical depth.

Conclusions

In this paper we have described the AGR of microalgae in perfectly mixed PBRs. We showed that, given a form of the reactor, the AGR is completely determined by the incident light intensity and the optical depth. This result is due to the Lambert-Beer law and to the fact that the extinction coefficient is independent of the location in the PBR. From a mathematical point of view, p is given by ( 25) with kinetic parameters of C. vulgaris given in Table 2. Other parameters are taken to be

I in = 1000 µ mol m -2 s, k = 0.2 m 2 gC -1 , m = 0.1 d -1 , K bg = 10 m -1 .
very general, in particular, we did not assume that the specific growth rate µ is differentiable as usually.

390

In the case of flat-plate PBRs illuminated from above, we studied in details the properties of the AGR. We showed the existence of values of the incident light intensity (critical light intensity) and the optical depth (critical optical depth) 395 maximizing the AGR. We also studied how these values vary.

These results are important for understanding how different environmental factors can affect the growth rate. In particular, they are useful for determining conditions for the occurrence 400 of Allee effect or conditions such that the incident light intensity is a limiting factor.

Appendix A

These Appendix present the lemmas and propositions used in the two previous sections. We assume that assumptions 4 and 5 hold. The first lemma gives in detail the properties of the function g defined in [START_REF] Fernández | A model for light distribution and average solar irradiance inside outdoor tubular photobioreactors for the microalgal mass cul-620 ture[END_REF]. We note that Eq.( 23) does not define g when I in or I out are equal to zero. However, from the definition of the AGR (Eq.( 5)), it is natural to define g(0, I out ) = g(I in , 0) = µ(0).

In the reality I out is never higher that I in . However, we study the behavior of g for any value of I out . This is necessary for proving Lemma A3.

Lemma A1. (Properties of g) Let g : R + ×R + -→ R defined by Eq.( 23) and Eq.(36). • if I in > I * , then γ(I in ) ∈ (σ(I in ), I * ),

• if I in = I * , then γ(I in ) = I * , and

• if I in < I * , then γ(I in ) ∈ (I * , σ(I in )).
Proof. The proof of the part a) follows directly from the definition of g.

For the part b), following [START_REF] Hsu | Dynamics of phytoplankton communities under photoinhibition[END_REF], we can easily determine that lim Iout→Iin g(I in , I out ) = µ(I in ) and lim Iout→0 g(I in , I out ) = µ(0). Thus c) is proved.

For the part c), let I in > 0 be given. By using [START_REF] Fernández | A model for light distribution and average solar irradiance inside outdoor tubular photobioreactors for the microalgal mass cul-620 ture[END_REF], we can determine the partial derivative of g with respect to I out ; Fig. 11 shows graphically the part c) of Lemma A1. The following proposition states the main properties of the AGR (see Eq.( 9) or Eq.( 22)) as a function of the optical depth θ and the incident light intensity I in .

implies that for any interval J ⊂ [0, I in ], if g(I in , •) is strictly increasing (decreasing)in J, then μ(•, I in ) is strictly decreasing (increasing) in J -1 (as a function of θ), where J -1 = {θ ≥ 0 ; I out (θ, I in ) ∈ J}. If I in ≤ I * , then θ(I in ) = 0, hence we have to prove that μ(•, I in ) is strictly decreasing on R + . From Lemma A1 we have that g(I in , •) is strictly increasing in J = R + , then μ(•, I in ) is strictly decreasing in J

-1 = R + . If I in > I * , then θ(I in ) = ln Iin γ(Iin) > 0. From Lemma 465 A1 we have that g(I in , •) is strictly increasing in J = [0, γ(I in )], then μ(•, I in ) is strictly decreasing in J -1 = [0, θ(I in )].
In the same way we prove that μ(•, I in ) is strictly increasing in [ θ(I in ), ∞).

470

For the part c), first we note that for any θ > 0,

∂ μ(θ,Iin) ∂Iin = 1 θIin h(I in ) with h(I) = µ(I) -µ(Ie -θ
). Since µ is strictly increasing on (0, I * ], we have that h(I) > 0 for all I ∈ (0, I * ].

(41)

In the same way, since µ is strictly decreasing on [I * , ∞), we have that

h(I) < 0 for all I ∈ [I * e θ , ∞). (42)
From ( 41) and (42), we conclude that there exists I(θ) ∈ J := (I * , I * e θ ) such that h(I(θ)) = 0. Now let I 1 , I 2 ∈ J such that

I 1 < I 2 . It is not difficult to note that h(I 2 ) -h(I 1 ) = µ(I 2 ) -µ(I 1 ) <0 + µ(I 1 e -θ ) -µ(I 2 e -θ ) <0 < 0,
from where h is strictly decreasing on J. Com-475 bining this result with the inequalities (41) and (42), we conclude that h is positive on (0, I(θ)) and negative on (I(θ), ∞). Part c) follows directly from this result.

480

For the part d), if θ → ∞, then I out (θ, I in ) → 0. Thus, μ(θ, I in ) → g(0, I in ) = µ(0).

The following lemma states how γ (defined in Lemma A1) varies with I in . This lemma is necessary for proving Proposition A4.

485

Lemma A3. γ is continuous and strictly decreasing as a function of I in on [I * , ∞).

Proof. Let I in , I in > I * such that I in < I in . For any I out ∈ [γ(I in ), I * ), the function g(I out , •) is strictly decreasing on [γ(I out ), ∞). Since I out < I * , we have that γ(I out ) ∈ (I * , σ(I out )). We also note that I out > γ(I in ) > σ(I in ), therefore σ(I out ) < I in . The last inequality implies that the function g(I out , •) is strictly decreasing on [I in , ∞). By symmetry of g, we conclude that the function g(•, I out ) is strictly decreasing on [I in , ∞). Thus, we have that g(I in , I out ) > g(I in , I out ), for all I out ∈ [γ(I in ), I * ). This shows that g(I in , •) cannot intersect µ on [γ(I in ), I * ), therefore γ(I in ) < γ(I in ). Consequently, γ is strictly decreasing on (I * , ∞).

490

To prove that γ is continuous, it is enough to prove that γ is a bijection. Since γ is strictly decreasing, it is enough to prove that for any y ∈ (0, I * ) there exist I y ∈ (I * , ∞) such that g(I y , y) = µ(y). The following proposition states some properties of the critical optical depth (defined in [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF]). This implies the inequality (43). For the part b), from Lemma A3 and the definition of θ (see Eq. 28), we conclude that θ is strictly increasing on [I * , ∞). By taking the limit when I in → ∞ in (43), we conclude that lim Iin→∞ θ(I in ) = ∞. For the continuity, we take the limit when I in → I * + in Eq.(43), we obtain lim Iin→I * + θ(I in ) = 0 = θ(I * ), from where θ is continuous in I * . The continuity over all R + follows directly from the definition of θ and Lemma

515

A3.

The following proposition gives some properties of the critical light intensity Proposition A5. (Properties of the critical light intensity) I is strictly increasing and continuous. This equation shows that as θ increases the fraction on the left increases. Since σ is strictly decreasing, this is only possible if I(θ) increases. Thus, I is strictly increasing as a function of θ. From Eq.( 6), we also note that I is a bijection, indeed, it is inverse function I -1 (I) : [I * , ∞) -→ [0, ∞) is given by

I -1 (I) = ln I σ(I) . ( 44
)
Since I is strictly increasing and bijective, we conclude that is continuous.

Appendix B

Assume that p is given by Eq. [START_REF] Yun | Kinetic modeling of the lightdependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF]. Assume that µ is given by Eq. [START_REF] Béchet | Modeling the effects of light and temperature on algae growth: State of the art and critical assessment for productivity pre-625 diction during outdoor cultivation[END_REF]

Figure 1 :

 1 Figure 1: Region Ω occupied by the culture in a Cartesian coordinate system. I in is the incident light intensity, Ω 0 ⊂ R 2 is the illuminated flat-surface, Ω is the volume occupied by the culture, h(x, y) is the height of the culture at different positions (x, y) ∈ Ω 0 , L is the maximum depth of the culture, and Iout is the light intensity through the lowest part of the culture.

  I) corresponds to the projection onto the plane x -y of the set formed by all the points of Ω at which the light intensity equals I. From Lambert-165 Beer law, the latter is the intersection of Ω and the plane parallel to Ω 0 that intersects z = 1 ξ ln Iin I . Thus (x,y)∈Ω0(I) dxdy = A Ω 1 ξ ln Iin I and we conclude the proof.

Figure 2 :

 2 Figure 2: Schematic representation of a cross section through a cylindrical horizontal PBR. A. Triangular cylinder. The base measures L(1/m 1 + 1/m 2 ) and the height L. B. Semicircular cylinder of radius L.

Figure 3 :

 3 Figure 3: Schematic representation of one-half of a cross section through a cylindrical chemostat vessel (s, light path; R, cylinder radius; r, distance from the center; φ, angle of light path with line through the center.

Proof. (of Theorem 3 )

 3 Let us denote by f the first derivative of I(r) i.e. f (r; ξ, I in , R) = -ξI in 2π 0 e -ξs(r,φ;R) ∂ ∂r s(r, φ; R)dφ (20) Note that we explicitly indicate the dependence on the radius of the functions f and s . It is straightforward to verify that I (r) > 0 for any r ∈ [0, R). Thus, f is a strictly increasing function with respect to r. Consequently, 225 f (r; ξ, I in , R) > f (0; ξ, I in , R) = 0. Thus, I is strictly increasing. The latter implies that I(r) : [0, R] -→ [I(R), I(0)] has an inverse function ϕ(I; ξ, I in , R) : [I(R), I(0)] -→ [0, R]. 230 By integrating Eq.(18) with respect to φ and doing the change of variables I = I(r), the AGR can be written as μ(•) = 2

Assumption 4 .Figure 4 :

 44 Figure4: Carbon uptake rate (see Eq.(25)) for Chlorella vulgaris with kinetic parameters from Table2, experimental data from[START_REF] Yeh | Effect of light supply and carbon source on cell growth and cellular composition of a newly isolated microalga chlorella vulgaris esp-31[END_REF], and graphical description of σ.

Figure 5 :

 5 Figure 5: A. Plot of the AGR as a function of the optical depth for three different values of the incident light. B. Plot of the AGR as a function of the incident light intensity for three different values of the optical depth. We consider p given by (25) with the kinetic parameters of C. vulgaris given in Table 2 and m = 0.1 d -1 .

295 4 .

 4 DiscussionAGR as a function of the microalgae population density. Let us consider a flat-plate PBR, and as-

Figure 6 :

 6 Figure 6: Critical optical depth θ as a function of the incident light I in for three different microalgae species. Kinetic parameters are taken from Table2.

Figure 7 :

 7 Figure 7: Plot of the AGR as a function of X. We consider p given by (25) with the kinetic parameters of C. vulgaris given in Table 2. The other parameters are taken to be I in = 500 µ mol m -2 s, k = 0.2 m 2 gC -1 , L = 0.4 m, and m = 0.1 d -1 .

Figure 8 :

 8 Figure 8: Plots of the AGR for different shapes of the PBR.A. I in = 500 µ mol m -2 s -1 is fixed. B. θ = 3 is fixed. We consider p given by[START_REF] Yun | Kinetic modeling of the lightdependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF] with the kinetic parameters of C. vulgaris given in Table2and m = 0.1 d -1 .

Figure 9 :

 9 Figure 9: Productivity as a function of the biomass concentration per illuminated surface.A. I in = 1000 µ mol m -2 s -1 B. I in = 50 µ mol m -2 s -1 . We consider p given by[START_REF] Yun | Kinetic modeling of the lightdependent photosynthetic activity of the green microalga chlorella vulgaris[END_REF] with the kinetic parameters of C. vulgaris given in Table2. The other parameters are taken to be k = 0.2 m 2 gC -1 , m = 0.1 d -1 , and K bg = 0.

  385our assumptions over the specific growth rate are

Figure 10 :

 10 Figure10: Maximal productivity as a function of V /A. p is given by (25) with kinetic parameters of C. vulgaris given in Table2. Other parameters are taken to beI in = 1000 µ mol m -2 s, k = 0.2 m 2 gC -1 , m = 0.1 d -1 , K bg = 10 m -1 .

  a) (Symmetry) g(I in , I out ) = g(I out , I in ) for allI in , I out ≥ 0. b) (Partial continuity) g(I in , •) is continuous on R + for all I in ∈ R + .c) (Unimodal function with respect to I out ) For any I in > 0, there exists a unique γ(I in ) > 0 satisfyingg(I in , γ(I in )) = µ(γ(I in )). (37)It holds that g(I in , •) is strictly increasing on [0, γ(I in )] and strictly decreasing on [γ(I in ), ∞), moreover,

  ∂g(I in , I out ) ∂I out = 1 I out [ln(I in ) -ln(I out )] 2 p(I out ), (38) with p(I out ) = Iin Iout µ(I)-µ(Iout) I dI. Since I out [ln(I in ) -ln(I out )] 2 > 0 for any I out ∈ (0, I in ), the sign of ∂g(Iin,Iout) ∂Iout is determined by the sign of p(I out ).Let us assume that I in > I * and letI out ∈ [0, σ(I in )].Since µ is strictly increasing on [0, σ(I in )], we have that µ(I) > µ(I out ) for all I ∈ [I out , σ(I in )). This implies that p(I out ) > 0. Since I out was chosen arbitrarily we conclude that p(I out ) > 0 for all I out ∈ [0, σ(I in )]. In the same way we can prove that p(I out ) < 0 for allI out ∈ [I * , ∞)By the intermediate value theorem there exists I ∈ (σ(I in ), I * ) such that p(I ) = 0. Suppose there exists another I ∈ (σ(I in ), I * ) such that p(I ) = 0. By using elemental properties of integrals, it can be shown that the equality p(I ) = p(I ) implies Iin I µ(I ) -µ(I ) Suppose that I > I . Since µ is strictly increasing on [0, I * ], we have that µ(I) ≥ µ(I ) for all I ∈ [I , I ]. Thus the left side in (39) is negative, while the right side is positive. This contradiction shows that I is the unique root of p. Hence p is positive 435 on [0, I ) and negative on (I , ∞). Consequently, g(I in , •) is strictly increasing on [0, I ) and strictly decreasing on (I , ∞). By taking γ(I in ) = I we conclude the proof. The case I in ≤ I * follows the same arguments.

  440

Figure 11 :

 11 Figure 11: Plot of specific growth rate µ (continuous line) and the function g(I in , •) (dotted line) as functions of Iout. a) I in > I * ; the function µ has a peak at the optimal irradiance I * , while the function g(I in , •) has a peak at γ(I in ), where intersects µ. According to Lemma A1, σ(I in ) < γ(I in ) < I * . Both functions reach the same value at I in . b) I in ≤ I * ; both functions µ and g(I in , •) are strictly increasing reaching the same value when Iout = I in .

  For this purpose, we define the function G :[I * , ∞) -→ R such that G(I in ) = ln(I in /y)[g(I in , y) -µ(y)] It is clear that G(I * ) > 0 (see proof Lemma A1 c)). Let M (I) = µ(I)-µ(y)= µ(0) -µ(y) < 0, we conclude that lim Iin→∞ G(I in ) = -∞. Thus, since G is 495 continuous, there is I y > I * such that G(I y ) = 0.From Lemma A1 c), we have that θ(I in ) < γ(I in ) < I * .By taking the limit when I → I * + in the later inequality, we conclude that γ(I in ) → I * as I in → I * + . Therefore γ is continuous on [I * , ∞].

  500

  Proposition A4. (Properties of the critical optical depth.) a) For anyI in ∈ [I * , ∞) it holds ln I in I * < θ(I in ) < ln I in σ(I in ) . (43) b) θ is continuous on R + , strictly increasing on 505 [I * , ∞), and lim Iin→∞ θ(I in ) = ∞. Proof. From Lemma A1 part c), we have that for any I in > I * , σ(I in ) < γ(I in ) < I * .

520Proof.

  From Eq.(40) we obtain thatI(θ) σ(I(θ)) = e θ .

Let ∆ = 4 µ max αI * - 1 , then 525 •

 1525 if ∆ = 0, μ(θ, I in ) = I in -I out θ 4p max I * (I * + I in )(I * + I out ) -m • if ∆ > 0, μ(θ, I in ) = 2pmax/ √ ∆ θ [arctan(κ(I in )) -arctan(κ(I out ))] -m,with κ(I) = 2pmax ln (Iin-I * +ξ2)(Iout-I * +ξ1) (Iin-I * +ξ1)(Iout-I * +ξ2) -m, where ξ 1 = I * 2 α 2pmax

  Then, μ(θ, I in ) = pmax θ e 1-I out I * e I in -I out I * -1 -m

Table 1 :

 1 with k 1 and k 2 empirical parameters. Values of the specific light attenuation coefficient for different microalgae species[START_REF] Béchet | Modeling the effects of light and temperature on algae growth: State of the art and critical assessment for productivity pre-625 diction during outdoor cultivation[END_REF]

	Extinction		Algae
	coefficient (m 2 gC -1 )
	0.214		Monodus subterraneu
	0.175		Spirulina platensis
	0.150		Porphyridium cruentum
	0.200		Chlorella pyrenoidosa
	AGR definition. The specific growth rate of mi-
	croalgae, denoted by µ, represents the net growth
	potential of the population. If biomass is defined
	in terms of cell density, µ is the cell division rate,
	minus the mortality rate. If X is a mass of algal
	carbon, then µ is the rate of CO 2 fixation, minus
	the respiration rate. In section 3 (paragraph As-
	sumptions over µ) some explicit expressions of µ
	as a function of the light intensity I perceived by
	microalgae are presented. In this section, we only
	assume that I is the single factor that limits algae
	growth i.e µ : R + -→ R is a function of I. Let
	V (Ω) be the volume of the culture. Following [6],
	we compute the average growth rate (AGR) in the
	PBR, denoted by μ, by integrating the local specific
	growth rates over all the culture
	μ(•) :=	1 V (Ω) Ω	µ(I(ξ, I in , z))dxdydz.

Table 2

 2

					Unit
	p max	1.63	2	1.32	d -1
	I *	87.2	275	119	µmol m -2 s -1
	α	0.027 0.05 0.086 µmol -1 m 2 s d -1

for C.vulgaris. Parm. C.V. C.P. S.C.

Table 2 :

 2

Kinetic parameters of p in (25) for different microalgae species; C.V.: Chlorella vulgaris at 25 • C [29], C.P.: Chlorella pyrenoidosa at optimal temperature [28], S.C.: Scenedesmus crassus at 25 • C [30].

This is necessary to ensure the existence of the maximum of h on Ω 0 and define L at the end of this paragraph.

$ This work was supported by the CONICYT doctoral grant (Carlos Martínez), and by the Phycover (ANR-14-CE04-0011) and IPL Algae in silico (INRIA) projects.