Francisco J. Fernández

Optimal location of green zones in metropolitan areas to control the urban heat island

In this paper we analyze and numerically solve a problem related to the optimal location of green zones in metropolitan areas in order to mitigate the urban heat island effect. So, we consider a microscale climate model and analyze the problem within the framework of optimal control theory of partial differential equations. Finally we compute its numerical solution using the finite element method, with the help of the interior point algorithm IPOPT, interfaced with the FreeFem++ software package.

Optimal Management of a Bioreactor for Eutrophicated Water Treatment: A Numerical Approach

This paper presents a numerical algorithm for computing the optimal design variables in the management of a bioreactor for the treatment of eutrophicated water: initial and distributed quantities of phytoplankton added along the process, and total duration of the process. This real-world problem is formulated as a state-control constrained optimal control problem, whose numerical resolution is the main aim of this study. After discretizing the control problem, we present a structured algorithm for solving the discrete state systems, computing the corresponding derivatives, and minimizing the objective function. Finally, the good performance of the algorithm is shown by applying it to a realistic example with two pre-reservoirs.

Optimal control of a bioreactor

Abstract This paper is concerned with the analysis of a control problem related to the optimal management of a bioreactor. This real-world problem is formulated as a state-control constrained optimal control problem. We analyze the state system (a complex system of partial differential equations modelling the eutrophication processes for non-smooth velocities), and we prove that the control problem admits, at least, a solution. Finally, we present a detailed derivation of a first order optimality condition – involving a suitable adjoint system – in order to characterize these optimal solutions, and some computational results.

Optimal control of eutrophication processes in a moving domain

Abstract This work deals with an environmental problem related to controlling eutrophication inside a sensitive zone, by means of a regulation of the wastewater discharges in the region. After setting a detailed mathematical formulation of the optimal control problem posed on a free-boundary moving domain, we present several theoretical results on existence-regularity of optimal solutions, and their characterization by a first order optimality system. In the second part of the work a complete numerical algorithm for the resolution of the control problem is proposed, and several numerical examples are also given.

Numerical Optimization of a Bioreactor for the Treatment of Eutrophicated Water

The fundamental idea of a bioreactor consists of holding up eutrophicated water (rich, for instance, in nitrogen) in large tanks where we add a certain quantity of phytoplankton, that we let grow in order to absorb nitrogen and purify water. Its optimal management can be formulated as an optimal control problem with state constraints, where the control can be the quantity of phytoplankton added at each tank or the permanence times, the state variables are the concentrations of the species involved (nutrient, phytoplankton, zooplankton and organic detritus), the objective function to be minimized is the phytoplankton concentration of water leaving the bioreactor, and the state constraints stand for the thresholds imposed on the nitrogen and detritus concentrations in each tank. We recall that this optimal control problem possesses a solution, which can be characterized by a first order optimality condition. After discretizing the control problem, we present a structured algorithm for solving the resulting nonlinear constrained optimization problem. Finally, we also give numerical results for a real-world example.

A Three-Dimensional Eutrophication Model: Analysis and Control

Eutrophication of lakes, reservoirs, streams, and coastal areas is one of the most widespread environmental problems of large water bodies. Eutrophication consists of unnatural enrichment with two plant nutrients: nitrogen and phosphorus. This overnutrification causes undesirable changes in water resources: excessive production of algae, deterioration of water quality and availability, fish kills, health hazards for humans, etc. Controlling the eutrophication is important in order to mitigate and remedy the problem.