A transformada de Laplace

Titulación: Grao en Enxeñaría Civil -- Materia: Matemáticas IIIA transformada de Laplace é un método de gran eficiencia á hora de resolver un certo tipo de ecuacións diferenciais e integrais. Ademais do interese que pode ter desde o punto de vista puramente matemático, constitúe unha ferramenta básica na enxeñaría de control moderna e resulta especialmente útil á hora de resolver determinados problemas de valor inicial con termos non homoxéneos de natureza descontinua ou impulsiva. Este tipo de problemas aparece con relativa frecuencia en enxeñaría, especialmente cando se estudan sistemas mecánicos ou eléctricos. Esta unidade didáctica está deseñada para que, en aproximadamente 10 horas de docencia presencial, o alumnado do Grao en Enxeñaría Civil se introduza no manexo desta ferramenta matemática. Comezaremos presentando un problema relacionado co control da pulverización de tratamentos sobre cultivos. Ese problema, ligado a un oscilador harmónico amortecido, serve de exemplo dun problema asociado a un EDO linear de orde dous que aínda non se sabe resolver, en tanto que o termo non homoxéneo non é unha función continua. Coa intención de poder resolver ese problema (e outros similares) introduciremos a transformada de Laplace,(NOTA: Neste tema preséntase a transformada de Laplace en variable real, pensando en que os alumnos aos que irá dirixida non posúen coñecementos de funcións de variable complexa). Aínda así, debemos sinalar que o emprego da teoría de variable complexa presenta notables vantaxes tanto no desenvolvemento da teoría (permite establecer unha expresión xeral para a transformada inversa), como no das súas aplicacións (noción de impedancia en circuítos eléctricos). Para ver o desenvolvemento da transformada de Laplace en variable complexa o alumno interesado pode consultar, por exemplo, o libro de LePage, 1980.) estableceremos condicións para a súa existencia, estudaremos algunhas das súas propiedades máis importantes e veremos como se pode aplicar para resolver un problema de valor inicial (PVI) asociado a unha EDO linear de orde superior. Finalmente precisaremos como modelar unha función impulso a través dunha delta de Dirac e veremos como a transforma de Laplace permite resolver ese tipo de problemas de natureza impulsiva.Universidade de Santiago de Compostela. Servizo de Normalización Lingüístic

Forecasting variations in profitability and silviculture under climate change of radiata pine plantations through differentiable optimization

Climate change might entail significant alterations in future forest productivity, profitability and management. In this work, we estimated the financial profitability (Soil Expectation Value, SEV) of a set of radiata pine plantations in the northwest of Spain under climate change. We optimized silvicultural interventions using a differentiable approach and projected future productivity using a machine learning model basing on the climatic predictions of 11 Global Climate Models (GCMs) and two Representative Concentration Pathways (RCPs). The forecasted mean SEV for future climate was lower than current SEV (∼22% lower for RCP 4.5 and ∼29% for RCP 6.0, with interest rate = 3%). The dispersion of the future SEV distribution was very high, alternatively forecasting increases and decreases in profitability under climate change depending on the chosen GCM. Silvicultural optimization considering future productivity projections effectively mitigated the potential economic losses due to climate change; however, its ability to perform this mitigation was strongly dependent on interest rates. We conclude that the financial profitability of radiata pine plantations in this region might be significantly reduced under climate change, though further research is necessary for clearing the uncertainties regarding the high dispersion of profitability projectionsS

Equilibrio de Nash para un problema de control multiobjetivo relacionado con la depuraci´on de aguas residuales

En este trabajo, formulamos, estudiamos y resolvemos numéricamente un problema de control multiobjetivo relacionado con la gestión de la depuración de un sistema de aguas residuales. Planteamos el problema de la gestión óptima como la búsqueda de un equilibrio de Nash. Analizamos este problema, probamos un resultado de existencia de un equilibrio de Nash y establecemos un sistema de optimalidad de primer orden que caracteriza los equilibrios de Nash, para lo cual introducimos el estado adjunto. Finalmente, presentamos los resultados numéricos obtenidos en una situación realista planteada en la ría de Vigo

Multi-objective models for the forest harvest scheduling problem in a continuous-time framework

In this study we present several multi-objective models for forest harvest scheduling in forest with single-species, even-aged stands using a continuous formulation. We seek to maximize economic profitability and even-flow of timber harvest volume, both for the first rotation and for the regulated forest. For that, we design new metrics that allow working with continuous decision variables, namely, the harvest time of each stand. Unlike traditional combinatorial formulations, this avoids dividing the planning horizon into periods and simulating alternative management prescriptions before the optimization process. We propose to combine a scalarization technique (weighting method) with a gradient-type algorithm (L-BFGS-B) to obtain the Pareto frontier of the problem, which graphically shows the relationships (trade-offs) between objectives, and helps the decision makers to choose a suitable weighting for each objective. We compare this approach with the widely used in forestry multi-objective evolutionary algorithm NSGA-II. We analyze the model in a Eucalyptus globulus Labill. forest of Galicia (NW Spain). The continuous formulation proves robust in forests with different structures and provides better results than the traditional combinatorial approach. For problem solving, our proposal shows a clear advantage over the evolutionary algorithm in terms of computational time (efficiency), being of the order of 65 times faster for both continuous and discrete formulationsS

Simultaneous optimization of even flow and land and timber value in forest planning: a continuous approach

Background: Forest management planning involves deciding which silvicultural treatment should be applied to each stand and at what time to best meet the objectives established for the forest. For this, many mathematical formulations have been proposed, both within the linear and non-linear programming frameworks, in the latter case generally considering integer variables in a combinatorial manner. We present a novel approach for planning the management of forests comprising single-species, even-aged stands, using a continuous, multi-objective formulation (considering economic and even flow) which can be solved with gradient-type methods. Results: The continuous formulation has proved robust in forest with different structures and different number of stands. The results obtained show a clear advantage of the gradient-type methods over heuristics to solve the problems, both in terms of computational time (efficiency) and in the solution obtained (effectiveness). Their improvement increases drastically with the dimension of the problem (number of stands). Conclusions: It is advisable to rigorously analyze the mathematical properties of the objective functions involved in forest management planning models. The continuous bi-objective model proposed in this paper works with smooth enough functions and can be efficiently solved by using gradient-type techniques. The advantages of the new methodology are summarized as: it does not require to set management prescriptions in advance, it avoids the division of the planning horizon into periods, and it provides better solutions than the traditional combinatorial formulations. Additionally, the graphical display of trade-off information allows an a posteriori articulation of preferences in an intuitive way, therefore being a very interesting tool for the decision-making process in forest planningS

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

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 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-