Elements of the mathematical modeling in the control of pollutants emissions

Parra‐Guevara, David; Skiba, Yuri N.

doi:10.1016/s0304-3800(03)00191-1

Cited by 38 publications

(17 citation statements)

References 12 publications

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“…Each point emission source f j (r, t) can be described through its emission rate Q j (t) and emission site r j , that is, f j (r, t) = Q j (t)δ(r − r j ), where δ(r−r j ) is the Dirac delta centered at r j ∈ D. The domain of function f k (r, t) is restricted to a line Γ k ⊂ D in the case of a line source, and to a two-dimensional set A l ⊂ D in the case of an area source f l (r, t). It is important to note that each linearly distributed source, as well as each source distributed over an area, can be approximated by the sum of point sources [16]. However, the formulation of the control problem does not require such transformation, because such details are part of the numerical scheme used to solve the dispersion model (1)- (9).…”

Section: Dispersion Modelmentioning

confidence: 99%

Controlling the Forcing of the Linear Transport Equation to Meet Air Quality Norms at Every Point

Parra‐Guevara

Skiba²,

~{n}a-Maciel

2017

Int. J. of Appl. Math.

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A three-dimensional dispersion air pollution model for point, line or area sources is considered in a limited region. Particular solutions of such model and their respective maximum values are used to pose a quadratic programming problem with the aim to determine optimal emission rates of the sources and meet the standards of air quality at every point in a zone and each instant in an interval of time. The existence and uniqueness of the optimal control problem solution is proved. An efficient algorithm of successive orthogonal projections is used to calculate the optimal solution. Numerical examples obtained in the case of point sources demonstrate the method's ability.

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Section: Dispersion Modelmentioning

confidence: 99%

Controlling the Forcing of the Linear Transport Equation to Meet Air Quality Norms at Every Point

Parra‐Guevara

Skiba²,

~{n}a-Maciel

2017

Int. J. of Appl. Math.

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“…Se asume que la anomalía de la concentración del contaminante  satisface las siguientes ecuaciones (Parra-Guevara y Skiba, 2003).…”

Section: Inestabilidad Del Problema Inverso Simplificadounclassified

Recuperación de la Tasa de Emisión de una Fuente Contaminante: Análisis de la Existencia, la Unicidad y la Estabilidad de las Soluciones

Parra‐Guevara¹,

Skiba²

2016

Inf. tecnol.

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ResumenEn este trabajo se analiza la existencia, la unicidad y la estabilidad de las soluciones en el problema de la recuperación de la tasa de emisión de una fuente puntual no-estacionaria a partir de datos de concentración de contaminantes que contienen una perturbación de pequeña amplitud. Se presentan ejemplos numéricos que muestran la inestabilidad en el problema inverso y permiten identificar el tipo de perturbación en los datos que es más significativa para este problema. La ecuación de balance de materia para un contaminante se utiliza para analizar la existencia de soluciones. A través de las funciones adjuntas y el principio de dualidad para la anomalía de la concentración puntual, se prueba que el problema inverso siempre es inestable cuando el modelo de dispersión tiene asociado un modelo adjunto cuya solución es de clase C 1 . Se analiza la existencia y la unicidad de la tasa de emisión para esta formulación del problema inverso. Se concluye que es indispensable introducir una regularización en el problema inverso que filtre los errores de alta frecuencia en los datos y permita obtener una estimación adecuada de la tasa de emisión. Palabras clave: fuente puntual; tasa de emisión; contaminantes; dispersión; problema inverso; inestabilidad Recovery of the Emission Rate of a Pollution Source: Analysis of the Existence, Uniqueness and Stability of Solutions AbstractIn this work the existence, uniqueness and stability of the solutions in the problem of recovery of the emission rate of a non-steady point source from concentration data of pollutants with a perturbation of small amplitude, are analyzed. Numerical examples that demonstrate the instability of the inverse problem and identify the type of perturbation in the data that is most meaningful to this problem are presented. The mass balance equation for a pollutant is used to analyze the existence of solutions. Through the adjoint functions and the duality principle for the anomaly of the punctual concentration, it is shown that the inverse problem is always unstable when the dispersion model has an associated adjoint model whose solution is of class C 1 . The existence and uniqueness of the emission rate for this formulation of the inverse problem is analyzed. It is concluded that it is indispensable to introduce a regularization in the inverse problem to filter high frequency errors in the data and to obtain an adequate estimate of the emission rate.

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“…In the last years, several works (see, for example, [2][3][4]12,13]) have used mathematical models and optimal control techniques to solve this type of problems. In [12], a wastewater treatment system consisting of several purifying plants discharging into the same domain is considered and, by assuming that every plant is controlled by a unique organization, the management of the treatment system is formulated and studied as an optimal control problem with state and control pointwise constraints.…”

Section: Introductionmentioning

confidence: 99%

Nash equilibrium for a multiobjective control problem related to wastewater management

García-Chan¹,

Muñoz-Sola²,

Vázquez-Méndez³

2009

ESAIM: COCV

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Abstract. This paper is concerned with mathematical modelling in the management of a wastewater treatment system. The problem is formulated as looking for a Nash equilibrium of a multiobjective pointwise control problem of a parabolic equation. Existence of solution is proved and a first order optimality system is obtained. Moreover, a numerical method to solve this system is detailed and numerical results are shown in a realistic situation posed in the estuary of Vigo (Spain).Mathematics Subject Classification. 49J20, 49K20, 90C29, 91B76

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Elements of the mathematical modeling in the control of pollutants emissions

Cited by 38 publications

References 12 publications

Controlling the Forcing of the Linear Transport Equation to Meet Air Quality Norms at Every Point

Controlling the Forcing of the Linear Transport Equation to Meet Air Quality Norms at Every Point

Recuperación de la Tasa de Emisión de una Fuente Contaminante: Análisis de la Existencia, la Unicidad y la Estabilidad de las Soluciones

Nash equilibrium for a multiobjective control problem related to wastewater management

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