On the degree of the polynomial defining a planar algebraic curves of constant width

In this paper, we consider a family of closed planar algebraic curves $\mathcal{C}$ which are given in parametrization form via a trigonometric polynomial $p$. When $\mathcal{C}$ is the boundary of a compact convex set, the polynomial $p$ represents the support function of this set. Our aim is to examine properties of the degree of the defining polynomial of this family of curves in terms of the degree of $p$. Thanks to the theory of elimination, we compute the total degree and the partial degrees of this polynomial, and we solve in addition a question raised by Rabinowitz in \cite{Rabi} on the lowest degree polynomial whose graph is a non-circular curve of constant width. Computations of partial degrees of the defining polynomial of algebraic surfaces of constant width are also provided in the same way.Comment: 13 page

Optimization of the separation of two species in a chemostat

International audienceIn this work, we study a two species chemostat model with one limiting substrate, and our aim is to optimize the selection of the species of interest. More precisely, the objective is to find an optimal feeding strategy in order to reach in minimal time a target where the concentration of the first species is significantly larger than the concentration of the other one. Thanks to Pontryagin maximum principle, we introduce a singular feeding strategy which allows to reach the target, and we prove that the feedback control provided by this strategy is optimal whenever initial conditions are chosen in the invariant attractive manifold of the system. The optimal synthesis of the problem in presence of more than one singular arc and for initial conditions outside this set is also investigated

Second order analysis for strong solutions in the optimal control of parabolic equations

International audienceIn this paper we provide a second order analysis for strong solutions in the optimal control of parabolic equations. We consider the case of box constraints on the control and final integral constraints on the state. In contrast to sufficient conditions assuring quadratic growth in the weak sense, i.e. when the cost increases at least quadratically for admissible controls uniformly near to the nominal one (see e.g. [16, 26]), our main result provides a sufficient condition for quadratic growth of the cost for admissible controls whose associated states are uniformly near to the state of the nominal one. As a consequence of our results, for qualified problems with a strictly convex and quadratic Hamiltonian, we prove that both notions of quadratic growth coincide

Weak and strong minima : from calculus of variation toward PDE optimization

This note summarizes some recent advances on the theory of optimality conditions for PDE optimization. We focus our attention on the concept of strong minima for optimal control problems governed by semi-linear elliptic and parabolic equations. Whereas in the field of calculus of variations this notion has been deeply investigated, the study of strong solutions for optimal control problems of partial differential equations (PDEs) has been addressed recently. We first revisit some well-known results coming from the calculus of variations that will highlight the subsequent results. We then present a characterization of strong minima satisfying quadratic growth for optimal control problems of semi-linear elliptic and parabolic equations and we end by describing some current investigations

Objets convexes de largeur constante (en 2D) ou d'épaisseur constante (en 3D) : du neuf avec du vieux

International audienceLes objets convexes de largeur constante (dans le plan) ou d'épaisseur constante (dans l'espace) ont fait l'objet d'une attention soutenue de la part des mathématiciens du XIXe comme du XXe siècle, y compris par les plus célèbres d'entre eux (H. Minkowski, H. Lebesgue, W. Blaschke, A. Hurwitz, etc.). Malgré tous les efforts déployés et le nombre de résultats obtenus, certains problèmes posés depuis longtemps à propos de ces objets convexes restent encore ouverts. Les techniques modernes comme celles issues du calcul variationnel ou du contrôle optimal ont néanmoins permis soit de retrouver d'une nouvelle manière des résultats déjà démontrés, soit d'en améliorer significativement certains autres. Dans cet article, qui se veut de synthèse et à but essentiellement pédagogique, nous passons en revue les propriétés et caractérisations essentielles, plutôt de type " variationnel ", des corps convexes de largeur constante (en 2D) ou d'épaisseur constante (en 3D), en insistant sur les différences fondamentales en 2D ou 3D ; ce faisant, nous arrivons sur le front de la recherche récente sur les problèmes restés ouverts, en particulier la conjecture sur le corps convexe de l'espace d'épaisseur constante donnée et de volume minimal

Minimal time problem for a fed-batch bioreactor with a non admissible singular arc

International audienceIn this paper, we consider an optimal control problem for a system describing a fed-batch bioreactor with one species and one substrate. Our aim is to find an optimal feedback control in order to steer the system to a given target in minimal time. The growth function is of Haldane type implying the existence of a singular arc. Unlike many studies on the minimal time problem governed by an affine system w.r.t. the control with one input, we assume that the singular arc is non-necessary controllable. This brings interesting issues in terms of optimal synthesis. Thanks to the Pontryagin Maximum Principle, we provide the optimal synthesis of the problem, It turns out that singular extremal trajectories are no longer optimal on a subset of the singular arc

Improvement of performances of continuous biological water treatment with periodic solutions

We study periodic solutions of the chemostat model under an integral constraint, either on the flow rate (Pb. 1) or on the substrate concentration (Pb. 2). We give conditions on the growth kinetics for which it is possible to improve the averaged water quality (Pb. 1) or the total quantity of treated water (Pb. 2) over a given time period, compared to steady-state. When this is possible, we characterize optimal periodic solutions and show a duality between the two optimization problems. The results are illustrated on four types of growth kinetics, given by Monod, Haldane, Hill and Contois functions

Optimal periodic control for scalar dynamics under integral constraint on the input

International audienceThis paper studies a periodic optimal control problem governed by a one-dimensional system, linear with respect to the control u, under an integral constraint on u. We give conditions for which the value of the cost function at steady state with a constant control \bar u can be improved by considering periodic control u with average value equal to \bar u. This leads to the so-called "over-yielding" met in several applications. With the use of the Pontryagin Maximum Principle, we provide the optimal synthesis of periodic strategies under the integral constraint. The results are illustrated on a single population model in order to study the effect of periodic inputs on the utility of the stock of resource

Minimal time control of fed-batch bioreactor with product inhibition

International audienceThis paper is devoted to the minimal time control problem for fed-batch bioreactors, in presence of an inhibitory product, which is released by the biomass proportionally to its growth. We first consider a growth rate with substrate saturation and product inhibition, and we prove that the optimal strategy is fill and wait (bang-bang). We then investigate the case of the Jin growth rate which takes into account substrate and product inhibition. For this type of growth function, we can prove the existence of singular arc paths defining singular strategies. Several configurations are addressed depending on the parameter set. For each case, we provide an optimal feedback control of the problem (of type bang-bang or bang-singular-bang). These results are obtained gathering the initial system into a planar one by using conservation laws. Thanks to Pontryagin maximum principle, Green's theorem, and properties of the switching function, we obtain the optimal synthesis. A methodology is also proposed in order to implement the optimal feeding strategies

Analysis of a periodic optimal control problem connected to microalgae anaerobic digestion

International audienceIn this work, we study the coupling of a culture of microalgae limited by light and an anaerobic digester in a two-tank bioreactor. The model for the reactor combines a periodic day-night light for the culture of microalgae and a classical chemostat model for the digester. We first prove the existence and attraction of periodic solutions of this problem for a 1 day period. Then, we study the optimal control problem of optimizing the production of methane in the digester during a certain timeframe, the control on the system being the dilution rate (the input flow of microalgae in the digester). We apply Pontryagin's Maximum Principle in order to characterize optimal controls, including the computation of singular controls. We present numerical simulations by direct and indirect methods for different light models and compare the optimal 1-day periodic solution to the optimal strategy over larger timeframes. Finally, we also investigate the dependence of the optimal cost with respect to the volume ratio of the two tanks