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Thematische Bibliographien / Optimal control problems involving partial differential equations / Dissertationen

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Dissertationen zum Thema „Optimal control problems involving partial differential equations“

Autor: Grafiati

Veröffentlicht am 17. September 2021

Zuletzt aktualisiert am 18. Februar 2022

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1

Tsang, Siu Chung. „Preconditioners for linear parabolic optimal control problems“. HKBU Institutional Repository, 2017. https://repository.hkbu.edu.hk/etd_oa/464.

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In this thesis, we consider the computational methods for linear parabolic optimal control problems. We wish to minimize the cost functional while fulfilling the parabolic partial differential equations (PDE) constraint. This type of problems arises in many fields of science and engineering. Since solving such parabolic PDE optimal control problems often lead to a demanding computational cost and time, an effective algorithm is desired. In this research, we focus on the distributed control problems. Three types of cost functional are considered: Target States problems, Tracking problems, and All-time problems. Our major contribution in this research is that we developed a preconditioner for each kind of problems, so our iterative method is accelerated. In chapter 1, we gave a brief introduction to our problems with a literature review. In chapter 2, we demonstrated how to derive the first-order optimality conditions from the parabolic optimal control problems. Afterwards, we showed how to use the shooting method along with the flexible generalized minimal residual to find the solution. In chapter 3, we offered three preconditioners to enhance our shooting method for the problems with symmetric differential operator. Next, in chapter 4, we proposed another three preconditioners to speed up our scheme for the problems with non-symmetric differential operator. Lastly, we have the conclusion and the future development in chapter 5.

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2

Yousept, Irwin. „Optimal control of partial differential equations involving pointwise state constraints: regularization and applications“. Göttingen Cuvillier, 2008. http://d-nb.info/990426513/04.

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3

Flaig, Thomas G. [Verfasser]. „Discretization strategies for optimal control problems with parabolic partial differential equations / Thomas G. Flaig“. München : Verlag Dr. Hut, 2013. http://d-nb.info/103729176X/34.

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4

Trautwein, Christoph [Verfasser], und Peter [Gutachter] Benner. „Optimal control problems constrained by stochastic partial differential equations / Christoph Trautwein ; Gutachter: Peter Benner“. Magdeburg : Universitätsbibliothek Otto-von-Guericke-Universität, 2019. http://d-nb.info/1220034959/34.

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5

Lee, Jangwoon. „Analysis and finite element approximations of stochastic optimal control problems constrained by stochastic elliptic partial differential equations“. [Ames, Iowa : Iowa State University], 2008.

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6

Qi, Meiyu [Verfasser], und Ronald H. W. [Akademischer Betreuer] Hoppe. „Adaptive Mixed Finite Element Approximations of Distributed Optimal Control Problems for Elliptic Partial Differential Equations / Meiyu Qi. Betreuer: Ronald H. W. Hoppe“. Augsburg : Universität Augsburg, 2012. http://d-nb.info/1077700849/34.

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7

Pearson, John W. „Fast iterative solvers for PDE-constrained optimization problems“. Thesis, University of Oxford, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.581405.

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In this thesis, we develop preconditioned iterative methods for the solution of matrix systems arising from PDE-constrained optimization problems. In order to do this, we exploit saddle point theory, as this is the form of the matrix systems we wish to solve. We utilize well-known results on saddle point systems to motivate preconditioners based on effective approximations of the (1,1)-block and Schur complement of the matrices involved. These preconditioners are used in conjunction with suitable iterative solvers, which include MINRES, non-standard Conjugate Gradients, GMRES and BiCG. The solvers we use are selected based on the particular problem and preconditioning strategy employed. We consider the numerical solution of a range of PDE-constrained optimization problems, namely the distributed control, Neumann boundary control and subdomain control of Poisson's equation, convection-diffusion control, Stokes and Navier-Stokes control, the optimal control of the heat equation, and the optimal control of reaction-diffusion problems arising in chemical processes. Each of these problems has a special structure which we make use of when developing our preconditioners, and specific techniques and approximations are required for each problem. In each case, we motivate and derive our preconditioners, obtain eigenvalue bounds for the preconditioners where relevant, and demonstrate the effectiveness of our strategies through numerical experiments. The goal throughout this work is for our iterative solvers to be feasible and reliable, but also robust with respect to the parameters involved in the problems we consider.

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8

Bénézet, Cyril. „Study of numerical methods for partial hedging and switching problems with costs uncertainty“. Thesis, Université de Paris (2019-....), 2019. http://www.theses.fr/2019UNIP7079.

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Nous apportons dans cette thèse quelques contributions à l’étude théorique et numérique de certains problèmes de contrôle stochastique, ainsi que leurs applications aux mathématiques financières et à la gestion des risques financiers. Ces applications portent sur des problématiques de valorisation et de couverture faibles de produits financiers, ainsi que sur des problématiques réglementaires. Nous proposons des méthodes numériques afin de calculer efficacement ces quantités pour lesquelles il n’existe pas de formule explicite. Enfin, nous étudions les équations différentielles stochastiques rétrogrades liées à de nouveaux problèmes de switching, avec incertitude sur les coûts
In this thesis, we give some contributions to the theoretical and numerical study to some stochastic optimal control problems, and their applications to financial mathematics and risk management. These applications are related to weak pricing and hedging of financial products and to regulation issues. We develop numerical methods in order to compute efficiently these quantities, when no closed formulae are available. We also study backward stochastic differential equations linked to some new switching problems, with costs uncertainty

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9

Flaig, Thomas Gerhard [Verfasser], Thomas [Akademischer Betreuer] Apel, Fredi [Akademischer Betreuer] Tröltzsch und Boris [Akademischer Betreuer] Vexler. „Discretization strategies for optimal control problems with parabolic partial differential equations / Thomas Gerhard Flaig. Universität der Bundeswehr München, Fakultät für Bauingenieurwesen und Umweltwissenschaften. Gutachter: Thomas Apel ; Fredi Tröltzsch ; Boris Vexler. Betreuer: Thomas Apel“. Neubiberg : Universitätsbibliothek der Universität der Bundeswehr, 2013. http://d-nb.info/1037118820/34.

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10

Flaig, Thomas G. [Verfasser], Thomas [Akademischer Betreuer] Apel, Fredi [Akademischer Betreuer] Tröltzsch und Boris [Akademischer Betreuer] Vexler. „Discretization strategies for optimal control problems with parabolic partial differential equations / Thomas Gerhard Flaig. Universität der Bundeswehr München, Fakultät für Bauingenieurwesen und Umweltwissenschaften. Gutachter: Thomas Apel ; Fredi Tröltzsch ; Boris Vexler. Betreuer: Thomas Apel“. Neubiberg : Universitätsbibliothek der Universität der Bundeswehr, 2013. http://d-nb.info/1037118820/34.

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11

Trey, Baptiste. „Existence et régularité des formes optimales pour des problèmes d'optimisation spectrale Free boundary regularity for a multiphase shape optimization problem. Communications in Partial Dfferential Equations Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form Existence and regularity of optimal shapes for elliptic operators with drift. Calculus of Variations and Partial Differential Equations“. Thesis, Université Grenoble Alpes, 2020. http://www.theses.fr/2020GRALM019.

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Dans cette thèse, on étudie l'existence et la régularité des formes optimales pour certains problèmes d'optimisation spectrale qui font intervenir un opérateur elliptique avec condition de Dirichlet.On s'intéresse d'abord au problème de la minimisation de la valeur propre principale d'un opérateur avec un terme de transport borné.Que le terme de transport soit fixé ou non, ce problème admet une solution parmi les quasi-ouverts, et si le terme de transport est en outre le gradient d'une fonction Lipschitzienne, alors les solutions sont des ouverts localement de classe C^{1,alpha} en dehors de points exceptionnels.On étudie ensuite en dimension deux la régularité des solutions à un problème d'optimisation à plusieurs phases pour la première valeur propre du Laplacien de Dirichlet.Enfin, on s'intéresse aux ensembles optimaux pour la somme des k premières valeurs propres d'un opérateur elliptique sous forme divergence. On montre que les k premières fonctions propres sur un ensemble optimal sont lipschitziennes de sorte que les ensembles optimaux sont ouverts, et on étudie ensuite la régularité de la frontière des ensembles optimaux
In this thesis, we study the existence and the regularity of optimal shapes for some spectral optimization problems involving an elliptic operator with Dirichlet boundary condition.First of all, we consider the problem of minimizing the principal eigenvalue of an operator with bounded drift under inclusion and volume constraints.Whether the drift is fixed or not, this problem admits solutions among the class of quasi-open sets, and if the drift is furthermore the gradient of a Lipschitz continuous function, then the solutions are open sets and C^{1,alpha}-regular except on a set of exceptional points.Next, we study in dimension two the regularity of the solutions to a multi-phase optimization problem for the first eigenvalue of the Dirichlet Laplacian.Finally, we focus on the optimal sets for the sum of the first k eigenvalues of an operator in divergence form. We prove that the first k eigenfunctions on an optimal set are Lipschitz continuous so that the optimal sets are open sets, and we then study the regularity of the boundary of the optimal sets

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12

Moyano, Gabriel Eduardo. „Modelo de dinámica y control de epidemia de dengue con información a gran escala“. Doctoral thesis, 2016. http://hdl.handle.net/11086/4062.

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Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2016.
El objetivo principal de este trabajo es resolver un problema de control con el fin de minimizar un brote epidémico de fiebre dengue mediante una estrategia de fumigación óptima. El trabajo se divide en dos partes. La primera consiste en plantear un problema de control con el objetivo de minimizar las personas expuesta a la enfermedad y los costos de fumigación, sujeto sobre un modelo de la dinámica espacio-temporal de un brote de fiebre dengue. La segunda parte incluye el desarrollo de un algoritmo eficiente para resolver el problema de control basado en el método del lagrangiano aumentado.
The main objective of this work is to solve a control problem in order to minimize an outbreak of dengue fever through a strategy of optimal spraying. The work is divided into two parts. The first is to pose a control problem in order to minimize people exposed to the disease and fumigation costs, subject to a model of the spatiotemporal dynamics of an outbreak of dengue fever. The second part includes the development of an efficient algorithm to solve the control problem based method increased Lagrangian.

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13

Heese, Harald. „Theory and Numerics for Shape Optimization in Superconductivity“. Doctoral thesis, 2006. http://hdl.handle.net/11858/00-1735-0000-0006-B3F2-0.

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14

Wu, Yen-Lin, und 吳彥霖. „Solutions of Partial Differential Equations Arising from Stochastic Optimal Control Problems and Applications“. Thesis, 2014. http://ndltd.ncl.edu.tw/handle/59434313615487010551.

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博士
國立中央大學
數學系
102
This dissertation is concerned with studying some second order elliptic partial differential equations. We are devote to establishing some qualitative properties of solutions, including existence, uniqueness and structure of solutions to three specific types of nonlinear elliptic equations. In Part 1, we study a gradient constraint equation which is related to a stochastic optimal control problem. We offer the existence and uniqueness of positive radial solutions with certain behavior under weaker conditions on nonlinearity. In Part 2, we consider a semilinear elliptic equation on the hyperbolic space. The asymptotic behavior, existence and uniqueness of positive singular solutions at the origin are proved. In addition, we discuss the structure of solutions of various types via the Pohozaev identity. Finally, in our last chapter, we deal with the Hardy-Sobolev equations and investigate behaviors, existence and uniqueness of solutions for different exponents.

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15

Fischer, Julia [Verfasser]. „Optimal control problems governed by nonlinear partial differential equations and inclusions / vorgelegt von Julia Fischer“. 2010. http://d-nb.info/1004057067/34.

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16

Sardar, Bidhan Chandra. „Study of Optimal Control Problems in a Domain with Rugose Boundary and Homogenization“. Thesis, 2016. http://hdl.handle.net/2005/2883.

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Mathematical theory of partial differential equations (PDEs) is a pretty old classical area with wide range of applications to almost every branch of science and engineering. With the advanced development of functional analysis and operator theory in the last century, it became a topic of analysis. The theory of homogenization of partial differential equations is a relatively new area of research which helps to understand the multi-scale phenomena which has tremendous applications in a variety of physical and engineering models, like in composite materials, porous media, thin structures, rapidly oscillating boundaries and so on. Hence, it has emerged as one of the most interesting and useful subject to study for the last few decades both as a theoretical and applied topic. In this thesis, we study asymptotic analysis (homogenization) of second-order partial differential equations posed on an oscillating domain. We consider a two dimensional oscillating domain (comb shape type) consisting of a fixed bottom region and an oscillatory (rugose) upper region. We introduce optimal control problems for the Laplace equation. There are mainly two types of optimal control problems; namely distributed control and boundary control. For distributed control problems in the oscillating domain, one can apply control on the oscillating part or on the fixed part and similarly for boundary control problem (control on the oscillating boundary or on the fixed part the boundary). We consider all the four cases, namely distributed and boundary controls both on the oscillating part and away from the oscillating part. The present thesis consists of 8 chapters. In Chapter 1, a brief introduction to homogenization and optimal control is given with relevant references. In Chapter 2, we introduce the oscillatory domain and define the basic unfolding operators which will be used throughout the thesis. Summary of the thesis is given in Chapter 3 and future plan in Chapter 8. Our main contribution is contained in Chapters 4-7. In chapters 4 and 5, we study the asymptotic analysis of optimal control problems namely distributed and boundary controls, respectively, where the controls act away from the oscillating part of the domain. We consider both L2 cost functional as well as Dirichlet (gradient type) cost functional. We derive homogenized problem and introduce the limit optimal control problems with appropriate cost functional. Finally, we show convergence of the optimal solution, optimal state and associate adjoint solution. Also convergence of cost-functional. In Chapter 6, we consider the periodic controls on the oscillatory part together with Neumann condition on the oscillating boundary. One of the main contributions is the characterization of the optimal control using unfolding operator. This characterization is new and also will be used to study the limiting analysis of the optimality system. Chapter 7 deals with the boundary optimal control problem, where the control is applied through Neumann boundary condition on the oscillating boundary with a suitable scaling parameter. To characterize the optimal control, we introduce boundary unfolding operators which we consider as a novel approach. This characterization is used in the limiting analysis. In the limit, we obtain two limit problems according to the scaling parameters. In one of the limit optimal control problem, we observe that it contains three controls namely; a distributed control, a boundary control and an interface control.

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17

Delgadino, Matías Gonzalo. „Teoría de control aplicada a tratamientos de quimioterapia“. Bachelor's thesis, 2011. http://hdl.handle.net/11086/53.

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Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2011.
En este trabajo se da una breve introducción y algunas primeras herramientas para la teoría de control y los sistemas de ecuaciones diferenciales con delay. Se utilizan estas herramientas para analizar dos modelos, con miradas diferentes del crecimiento tumoral existentes en la literatura y se propone un nuevo modelo híbrido que contemple no solo la dinámica continua sino también elementos discretos, de forma de buscar un protocolo óptimo de tratamientos de quimioterapia. Para esto, se propone y prueba un teorema donde se caracteriza la derivada del valor final de unavariable con respecto a las duraciones de un sistema con sitches. En la última sección, se encuentra una experiencia numérica mostrando la factibilidad de implementar el teorema anterior al problema propuesto.
Matías Gonzalo Delgadino.

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