In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function J J in the sense of strong solutions. This means that the function J J grows quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin’s principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of the Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions.

In this paper, we study the optimization problem of maximizing biogas production at the steady state in a two-stage anaerobic digestion model, which was initially proposed in [4]. Nominal operating points, consisting in steady states where the involved microorganisms coexist, are usually referred to as desired operational conditions, in particular for maximizing biogas production. Nevertheless, we prove that under some conditions related to input substrate concentrations and microorganism growth functions, the optimal steady state can be the extinction of one of the two species. We provide some numerical examples of this situation.

International audienceWe address the problem of finding an optimal feedback control for feeding a fed-batch bioreactor with one species and one substrate from a given initial condition to a given target value in a minimal amount of time. Recently, the optimal synthesis (optimal feeding strategy) has been obtained in systems in which the microorganisms involved are represented by increasing growth functions or growth functions with one maxima, with either Monod or Haldane functions, respectively (widely used in bioprocesses modeling). In the present work, we allow impulsive controls corresponding to instantaneous dilutions, and we assume that the growth function of the microorganism present in the process has exactly two local maxima. This problem has been considered from a numerical point of view in [15] without impulsive controls. In this article, we introduce two singular arc feeding strategies, and we define explicit regions of initial conditions in which the optimal strategy is either the first singular arc strategy or the second strategy

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This 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 controlū can be improved by considering periodic control u with average value equal toū. 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.

<p style='text-indent:20px;'>In this paper, we consider a resource-consumer model taking into account a linear coupling between species (with constant rate). The corresponding operator is proportional to a discretization of the Laplacian in such a way that the resulting dynamical system can be viewed as a regular perturbation of the classical chemostat system. We prove the existence of a unique locally asymptotically stable steady-state for every value of the transfer-rate and every value of the dilution rate not exceeding a critical value. In addition, we give an expansion of the steady-state in terms of the transfer-rate and we prove a uniform persistence property of the dynamics related to each species. Finally, we show that this equilibrium is globally asymptotically stable for every value of the transfer-rate provided that the dilution rate is with small enough values.</p>

Abstract. Curves which can be rotated freely in an n-gon (that is, an regular polygon with n sides) so that they always remain in contact with every side of the n-gon are called rotors. Using optimal control theory, we prove that the rotor with minimal area consists of a finite union of arcs of circles. Moreover, the radii of these arcs are exactly the distances of the diagonals of the n-gon from the parallel sides. Finally, using the extension of Noether's theorem to optimal control (as performed in [D. F. M. Torres, WSEAS Trans. Math., 3 (2004), pp. 620-624]), we show that a minimizer is necessarily a regular rotor, which proves a conjecture formulated in 1957 by Goldberg (see [M. Golberg, Amer. Math. Monthly, 64 (1957), pp. 71-78]).

In 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.