Controlling eutrophication by means of water recirculation: an optimal control perspective

In this work, the artificial recirculation of water is presented and analyzed, from the perspective of the optimal control of partial differential equations, as a tool to prevent eutrophication effects in large waterbodies. A novel formulation of the environmental problem, based on the coupling of nonlinear models for hydrodynamics, water temperature and concentrations of the different species involved in the eutrophication processes, is introduced. After a complete and rigorous analysis of the existence of optimal solutions, a full numerical algorithm for their computation is proposed. Finally, some numerical results for a realistic scenario are shown, in order to prove the efficiency of our approach.Xunta de Galicia | Ref. ED431C 2019/02Agencia Estatal de Investigación | Ref. MTM2016-75140-

Mathematical analysis and numerical resolution of a heat transfer problem arising in water recirculation

This work is devoted to the analysis and resolution of a well-posed mathematical model for several processes involved in the artificial circulation of water in a large waterbody. This novel formulation couples the convective heat transfer equation with the modified Navier–Stokes system following a Smagorinsky turbulence model, completed with a suitable set of mixed, nonhomogeneous boundary conditions of diffusive, convective and radiative type. We prove several theoretical results related to existence of solution, and propose a full algorithm for its computation, illustrated with some realistic numerical examples.Ministerio de Economía y Competitividad | Ref. MTM2015-65570-PXunta de Galicia | Ref. ED431D 2017/1

On existence and uniqueness of solution for a hydrodynamic problem related to water artificial circulation in a lake

In this work we introduce a well-posed mathematical model for the processes involved in the artificial circulation of water, in order to avoid eutrophication phenomena, for instance, in a lake. This novel and general formulation is based on the modified Navier–Stokes equations following the Smagorinsky model of turbulence, and presenting a suitable nonhomogeneous Dirichlet boundary condition. For the analytical study of the problem, we prove several theoretical results related to existence, uniqueness and smoothness for the solution of this recirculation model.Xunta de Galicia | Ref. ED431C 2018/50Ministerio de Economía y Competitividad | Ref. MTM2015-65570-

Algal cultivation for bioenergy production: first mathematical modelling results in raceways

The most used algae cultivation systems are the open-channel raceway ponds for their low maintenance and energy costs. Raceways allow algal cultivation using wastewater, where algae mass can be employed as source for bioenergy production. One of the main external factors influencing algal productivity is the velocity of the liquid inside the pond, that can be easily controlled by the position and/or rotational speed of the turning paddle wheel, and by the height of water. In this work we introduce a novel methodology to automate the optimization of the design of raceway ponds based on techniques of optimal control of partial differential equations. So, we formulate the problem as a control problem where the state system is given by the coupled nonlinear equations for hydrodynamics and algae/nitrogen/phosphorus concentrations, and the objective function to be maximized represents the global concentration of algae at final time. We present here a detailed, rigorous mathematical formulation of the optimal control problem, we propose a numerical algorithm for its resolution, and we show some preliminary computational results related to the numerical modelling of the problem

Optimizing the design of an estuarine water quality monitoring network by optimal control techniques

In this work, we propose a novel methodology in order to automatically optimize the location of the sampling points for a water quality monitoring network in an estuary, in such a way that any unknown pollution source can be identified (both in intensity and location) from the data supplied by those sampling points. In the central part of the article, after a rigorous mathematical formulation of the environmental problem, the full details of its numerical implementation are given. Finally, we present and analyze the results when applying the above proposed technique to study a real case in Ría of Vigo (northwestern Spain).Agencia Estatal de Investigación | Ref. TED2021-129324B-I00Universidade de Vigo/CISU

Optimal management of an urban road network with an environmental perspective

Within the framework of numerical simulation and optimal control of partial differential equations, in this work we deal with the mathematical modelling and optimal management of urban road networks. In particular, we are interested in finding the optimal management of the network intersections in order to reduce traffic congestion and atmospheric pollution. So, we consider two different multi-objective control problems (the former from a cooperative viewpoint, the latter within a hierarchical paradigm), propose a complete numerical algorithm to solve them, and, finally, present several numerical tests for a realistic case posed in the Guadalajara Metropolitan Area (Mexico), where the possibilities of our methodology are shown.Sistema Nacional de Investigadores | Ref. SNI-52768Programa para el Desarrollo Profesional Docente | Ref. PRODEP/103.5/16/8066Ministerio de Economía y Competitividad | Ref. MTM2015-65570-

Bilevel optimal control of urban traffic-related air pollution by means of Stackelberg strategies

Air contamination and road congestion are two major problems in modern cities. Both are closely related and present the same source: traffic flow. To deal with these problems, governments impose traffic restrictions preventing the entry of vehicles into sensitive areas, with the final goal of decreasing pollution levels. Unfortunately, these restrictions force drivers to look for alternative routes that usually generate traffic congestions, resulting in longer travel times and higher levels of contamination. In this work, blending computational modelling and optimal control of partial differential equations, we formulate and analyse a bilevel optimal control problem with air pollution and drivers’ travel time as objectives and look for optimal solutions in the sense of Stackelberg. In this setting, the leader (local government) implements traffic restrictions meanwhile the follower (drivers set) acts choosing travel preferences against leader constraints. We discretize the problem and propose a numerical algorithm to solve it, combining genetic-elitist algorithms and interior-point methods. Finally, computational results for a realistic case posed in the Guadalajara Metropolitan Area (Mexico) are shown.Sistema Nacional de Investigadores | Ref. SNI-52768Programa para el Desarrollo Profesional Docente | Ref. PRODEP/103.5/16/8066Xunta de Galicia | Ref. ED431C 2018/50

Designing an ecologically optimized road corridor surrounding restricted urban areas: a mathematical methodology

The use of optimization techniques for the optimal design of roads and railways has increased in recent years. The environmental impact of a layout is usually given in terms of the land use where it runs (avoiding some ecologically protected areas), without taking into account air pollution (in these or other sensitive areas) due to vehicular traffic on the road. This work addresses this issue and proposes an automatic method for obtaining a specific corridor (optimal in terms of air pollution), where the economically optimized road must be designed in a later stage. Combining a 1D traffic simulation model with a 2D air pollution model, and using classical techniques for optimal control of partial differential equations, the problem is formulated and solved in the framework of Mixed Integer Nonlinear Programming. The usefulness of this approach is shown in a real case study posed in a region that suffers from serious episodes of environmental pollution, the Guadalajara Metropolitan Area (México)Xunta de Galicia | Ref. ED431C 2018/50Sistema Nacional de Investigadores | Ref. SNI-52768Programa para el Desarrollo Profesional Docente | Ref. PRODEP/103.5/16/806

Optimal location of exit doors for efficient evacuation of crowds at gathering places

This work deals with the optimal design for the location of the exit doors at meeting places (such as sports centers, public squares, street markets, transport stations, etc.) to guarantee a safer emergency evacuation in events of a sporting, social, entertainment or religious type. This problem is stated as an optimal control problem of nonlinear partial differential equations, where the state system is a reformulation of the Hughes model (coupling the eikonal equation for a density-weighted walking velocity of pedestrians and the continuity equation for conservation of the pedestrian density), the control is the location of the exit doors at the domain boundary (subject to several geometric constraints), and the cost function is related to the evacuation rate. We provide a full numerical algorithm for solving the problem (a finite element technique for the discretization and a gradient-free procedure for the optimization), and show several numerical results for a realistic case.Ministerio de Ciencia e Innovación | Ref. TED2021-129324B-I00Sistema Nacional de Investigadores, México | Ref. SNI-52768Programa para el Desarrollo Profesional Docente (México) | Ref. PRODEP/103.5/16/8066CONACyT | Ref. 21755

Towards a more efficient evacuation of crowds by means of an optimal location of exit doors

In this work we present a new strategy, employing optimal control techniques of partial differential equations, to automate the optimization of locations for a given number of exit doors at gathering places, so that the evacuation of crowds takes place in a safer and faster way. Once given a detailed mathematical formulation of the problem, in order to solve the constrained optimal control problem numerically, we propose its full discretization, with a space semi-discretization via the finite element method over a family of triangular meshes of the domain under study, and a time semi-discretization via the Euler algorithm. Finally, for the resulting discretized minimization problem, we try its optimization by means of a derivative-free algorithm. Numerical examples, corresponding to different scenarios for a real-world study case posed on “Plaza de la Liberacion” (Guadalajara, Mexico), are presented and discussed to assess the effectiveness of our approach