Periodic optimal control for biomass productivity maximization in a photobioreactor using natural light

We address the question of optimization of the microalgal biomass long term productivity in the framework of production in photobioreactors under the influence of day/night cycles. For that, we propose a simple bioreactor model accounting for light attenuation in the reactor due to biomass density and obtain the control law that optimizes productivity over a single day through the application of Pontryagin's maximum principle, with the dilution rate being the main control. An important constraint on the obtained solution is that the biomass in the reactor should be at the same level at the beginning and at the end of the day so that the same control can be applied everyday and optimizes some form of long term productivity. Several scenarios are possible depending on the microalgae's strain parameters and the maximal admissible value of the dilution rate: bang-bang or bang-singular-bang control or, if the growth rate of the algae is very strong in the presence of light, constant maximal dilution. A bifurcation diagram is presented to illustrate for which values of the parameters these different behaviors occur. Finally, a simple sub-optimal bang-bang strategy is proposed that numerically achieves productivity levels that almost match those of the optimal strategy

Stability analysis of a wastewater treatment plant with saturated control

International audienceThis paper presents a saturated proportional controller that achieves depollution of wastewater in a continuous anaerobic digester. This goal is reached by defining a region of the state-space where the depollution is achieved and forcing attractivity and invariance of this region. The control variable is the dilution rate and the controlled variable is a linear combination (S λ) of the substrates con-centrations, that could be the Chemical Oxygen Demand (COD) or the Biological Oxygen Demand (BOD), depending on the value of λ. No measurement of the substrates concentrations in the input flow is required; the only necessary measurement is S λ

Polytopic Lyapunov functions for persistence analysis of competing species

We show that stability of the equilibrium of a family of interconnected scalar systems can be proved by using a sum of monotonic ${\mathcal C}^0$ functions as Lyapunov function. We prove this result in the general framework of nonlinear systems and then in the special case of Kolmogorov systems. As an application, it is then used to show that intra-specific competition can explain coexistence of several species in a chemostat where they compete for a single substrate. This invalidates the Competitive Exclusion Principle, that states that in the classical case (without this intra-specific competition), it is indeed known that only one of the species will survive

Computation of time-optimal switchings for linear systems with complex poles

International audienceThe minimum-time bounded control of linear systems is generically bang-bang and the number of switchings does not exceed the dimension of the system if the eigenvalues of the system matrix are real or if the initial condition is sufficiently close to the target. This paper extends the method of Grognard & Sepulchre (2001) for computing the switching times of time-optimal controllers to linear systems with complex poles and demonstrates its application on MPC schemes

Influence of Intrapredatory Interferences on Impulsive Biological Control Efficiency

International audienceIn this paper, a model is proposed for the biological control of a pest by its natural predator. The model incorporates a qualitative description of intrapredatory interference whereby predator density decreases the per capita predation efficiency and generalises the classical Beddington-DeAngelis formulation. A pair of coupled ordinary differential equations are used, augmented by a discrete component to depict the periodic release of a fixed number of predators into the system. This number is defined in terms of the rate of predator release and the frequency at which the releases are to be carried out. This formulation allows us to compare different biological control strategies in terms of release size and frequency that involve the same overall number of predators. The stability properties of the zero-pest solution are analysed. We obtain an upper bound on the interference strength (the biological condition) and a minimal bound on the predator release rate (the managerial condition) required to eradicate a pest population. We demonstrate that increasing the frequency of releases reduces this minimal rate and also increases the rate of convergence of the system to the zero-pest solution for a given release rate. Additionally, we show that other conclusions are to be expected if the interferences between predators have weaker or stronger effects than the generalised Beddington-DeAngelis formulation proposed in this paper

Two models of interfering predators in impulsive biological control

International audienceIn this paper, we study the effects of Beddington-DeAngelis interference and squabbling, respectively, on the minimal rate of predator release required to drive a pest population to zero. A two-dimensional system of coupled ordinary differential equations is considered, augmented by an impulsive component depicting the periodic release of predators into the system. This periodic release takes place independently of the detection of the pests in the field. We establish the existence of a pest-free solution driven by the periodic releases, and express the global stability conditions for this solution in terms of the minimal predator rate required to bring an outbreak of pests to nil. In particular, we show that with the interference effects, the minimal rate will only guarantee eradication if the releases are carried out frequently enough. When Beddington-DeAngelis behaviour is considered, an additional constraint for the existence itself of a successful release rate is that the pest growth rate should be less than the predation pressure, the latter explicitly formulated in terms of the predation function and the interference parameters

The Effects of Partial Crop Harvest on Biological Pest Control

In this paper, the effects of periodic partial harvesting of a continuously grown crop on augmentative biological control are analyzed. Partial harvesting can remove a proportion of both pests and biological control agents, so its influence on the control efficiency cannot be a priori neglected. An impulsive model consisting of a general predator-prey model in ODE, augmented by a discrete component to depict releases of biological control agents and the periodic partial harvesting is used. The periods are taken as integer multiples of each other. A stability condition for pest eradication is expressed as the minimal value of the budget per unit time to spend on predators. We consider the partial harvesting period to be fixed by both the plant's physiology and market forces so that the only manipulated variable is the release period. It is shown that varying the release period with respect to the harvest period influences the minimal budget value when the former is carried out more often than the latter and has no effect when releases take place as often as or less frequently than the partial harvests

Modelling plant compensatory effects in plant-insect dynamics

International audienceModelling plant-pest interactions is not an obvious task since the involved processes are numerous and complex. We propose a minimal model based on trophic relations and the concept of plant compensation capacity. We only consider three main components in our system: the plant foliar biomass, the compensation capacity, and the pest population. We prove that there exist two threshold parameters, N1 and N2, and show that the system admits different equilibria, which are locally asymptotically stable or unstable, depending on the value of the previous threshold parameters. Finally, we summarize our theoretical results in a bifurcation diagram that allows to discuss possible control strategies to lower the impacts of the pest or even to obtain a better biomass yield