@article{KarlNeitzelWachsmuth2020, author = {Karl, Veronika and Neitzel, Ira and Wachsmuth, Daniel}, title = {A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems}, series = {Computational Optimization and Applications}, volume = {77}, journal = {Computational Optimization and Applications}, issn = {0926-6003}, doi = {10.1007/s10589-020-00223-w}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232811}, pages = {7831-869}, year = {2020}, abstract = {In this paper we apply an augmented Lagrange method to a class of semilinear ellip-tic optimal control problems with pointwise state constraints. We show strong con-vergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.}, language = {en} } @phdthesis{Karl2020, author = {Karl, Veronika}, title = {Augmented Lagrangian Methods for State Constrained Optimal Control Problems}, doi = {10.25972/OPUS-21384}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-213846}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {This thesis is concerned with the solution of control and state constrained optimal control problems, which are governed by elliptic partial differential equations. Problems of this type are challenging since they suffer from the low regularity of the multiplier corresponding to the state constraint. Applying an augmented Lagrangian method we overcome these difficulties by working with multiplier approximations in \$L^2(\Omega)\$. For each problem class, we introduce the solution algorithm, carry out a thoroughly convergence analysis and illustrate our theoretical findings with numerical examples. The thesis is divided into two parts. The first part focuses on classical PDE constrained optimal control problems. We start by studying linear-quadratic objective functionals, which include the standard tracking type term and an additional regularization term as well as the case, where the regularization term is replaced by an \$L^1(\Omega)\$-norm term, which makes the problem ill-posed. We deepen our study of the augmented Lagrangian algorithm by examining the more complicated class of optimal control problems that are governed by a semilinear partial differential equation. The second part investigates the broader class of multi-player control problems. While the examination of jointly convex generalized Nash equilibrium problems (GNEP) is a simple extension of the linear elliptic optimal control case, the complexity is increased significantly for pure GNEPs. The existence of solutions of jointly convex GNEPs is well-studied. However, solution algorithms may suffer from non-uniqueness of solutions. Therefore, the last part of this thesis is devoted to the analysis of the uniqueness of normalized equilibria.}, subject = {Optimale Kontrolle}, language = {en} } @article{RoyBorziHabbal2017, author = {Roy, S. and Borz{\`i}, A. and Habbal, A.}, title = {Pedestrian motion modelled by Fokker-Planck Nash games}, series = {Royal Society Open Science}, volume = {4}, journal = {Royal Society Open Science}, number = {9}, doi = {10.1098/rsos.170648}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-170395}, pages = {170648}, year = {2017}, abstract = {A new approach to modelling pedestrians' avoidance dynamics based on a Fokker-Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to the output of real experiments (conducted with humans) for four test cases.}, language = {en} } @phdthesis{Poerner2018, author = {P{\"o}rner, Frank}, title = {Regularization Methods for Ill-Posed Optimal Control Problems}, edition = {1. Auflage}, publisher = {W{\"u}rzburg University Press}, address = {W{\"u}rzburg}, isbn = {978-3-95826-086-3 (Print)}, doi = {10.25972/WUP-978-3-95826-087-0}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-163153}, school = {W{\"u}rzburg University Press}, pages = {xiii, 166}, year = {2018}, abstract = {This thesis deals with the construction and analysis of solution methods for a class of ill-posed optimal control problems involving elliptic partial differential equations as well as inequality constraints for the control and state variables. The objective functional is of tracking type, without any additional \(L^2\)-regularization terms. This makes the problem ill-posed and numerically challenging. We split this thesis in two parts. The first part deals with linear elliptic partial differential equations. In this case, the resulting solution operator of the partial differential equation is linear, making the objective functional linear-quadratic. To cope with additional control constraints we introduce and analyse an iterative regularization method based on Bregman distances. This method reduces to the proximal point method for a specific choice of the regularization functional. It turns out that this is an efficient method for the solution of ill-posed optimal control problems. We derive regularization error estimates under a regularity assumption which is a combination of a source condition and a structural assumption on the active sets. If additional state constraints are present we combine an augmented Lagrange approach with a Tikhonov regularization scheme to solve this problem. The second part deals with non-linear elliptic partial differential equations. This significantly increases the complexity of the optimal control as the associated solution operator of the partial differential equation is now non-linear. In order to regularize and solve this problem we apply a Tikhonov regularization method and analyse this problem with the help of a suitable second order condition. Regularization error estimates are again derived under a regularity assumption. These results are then extended to a sparsity promoting objective functional.}, subject = {Optimale Steuerung}, language = {en} } @phdthesis{Sprengel2017, author = {Sprengel, Martin}, title = {A Theoretical and Numerical Analysis of a Kohn-Sham Equation and Related Control Problems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-153545}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2017}, abstract = {In this work, multi-particle quantum optimal control problems are studied in the framework of time-dependent density functional theory (TDDFT). Quantum control problems are of great importance in both fundamental research and application of atomic and molecular systems. Typical applications are laser induced chemical reactions, nuclear magnetic resonance experiments, and quantum computing. Theoretically, the problem of how to describe a non-relativistic system of multiple particles is solved by the Schr{\"o}dinger equation (SE). However, due to the exponential increase in numerical complexity with the number of particles, it is impossible to directly solve the Schr{\"o}dinger equation for large systems of interest. An efficient and successful approach to overcome this difficulty is the framework of TDDFT and the use of the time-dependent Kohn-Sham (TDKS) equations therein. This is done by replacing the multi-particle SE with a set of nonlinear single-particle Schr{\"o}dinger equations that are coupled through an additional potential. Despite the fact that TDDFT is widely used for physical and quantum chemical calculation and software packages for its use are readily available, its mathematical foundation is still under active development and even fundamental issues remain unproven today. The main purpose of this thesis is to provide a consistent and rigorous setting for the TDKS equations and of the related optimal control problems. In the first part of the thesis, the framework of density functional theory (DFT) and TDDFT are introduced. This includes a detailed presentation of the different functional sets forming DFT. Furthermore, the known equivalence of the TDKS system to the original SE problem is further discussed. To implement the TDDFT framework for multi-particle computations, the TDKS equations provide one of the most successful approaches nowadays. However, only few mathematical results concerning these equations are available and these results do not cover all issues that arise in the formulation of optimal control problems governed by the TDKS model. It is the purpose of the second part of this thesis to address these issues such as higher regularity of TDKS solutions and the case of weaker requirements on external (control) potentials that are instrumental for the formulation of well-posed TDKS control problems. For this purpose, in this work, existence and uniqueness of TDKS solutions are investigated in the Galerkin framework and using energy estimates for the nonlinear TDKS equations. In the third part of this thesis, optimal control problems governed by the TDKS model are formulated and investigated. For this purpose, relevant cost functionals that model the purpose of the control are discussed. Henceforth, TDKS control problems result from the requirement of optimising the given cost functionals subject to the differential constraint given by the TDKS equations. The analysis of these problems is novel and represents one of the main contributions of the present thesis. In particular, existence of minimizers is proved and their characterization by TDKS optimality systems is discussed in detail. To this end, Fr{\´e}chet differentiability of the TDKS model and of the cost functionals is addressed considering \(H^1\) cost of the control. This part is concluded by deriving the reduced gradient in the \(L^2\) and \(H^1\) inner product. While the \(L^2\) optimization is widespread in the literature, the choice of the \(H^1\) gradient is motivated in this work by theoretical consideration and by resulting numerical advantages. The last part of the thesis is devoted to the numerical approximation of the TDKS optimality systems and to their solution by gradient-based optimization techniques. For the former purpose, Strang time-splitting pseudo-spectral schemes are discussed including a review of some recent theoretical estimates for these schemes and a numerical validation of these estimates. For the latter purpose, nonlinear (projected) conjugate gradient methods are implemented and are used to validate the theoretical analysis of this thesis with results of numerical experiments with different cost functional settings.}, subject = {Optimale Kontrolle}, language = {en} } @article{SchindeleBorzi2016, author = {Schindele, Andreas and Borz{\`i}, Alfio}, title = {Proximal Methods for Elliptic Optimal Control Problems with Sparsity Cost Functional}, series = {Applied Mathematics}, volume = {7}, journal = {Applied Mathematics}, number = {9}, doi = {10.4236/am.2016.79086}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-145850}, pages = {967-992}, year = {2016}, abstract = {First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.}, language = {en} } @phdthesis{Merger2016, author = {Merger, Juri}, title = {Optimal Control and Function Identification in Biological Processes}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-138900}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2016}, abstract = {Mathematical modelling, simulation, and optimisation are core methodologies for future developments in engineering, natural, and life sciences. This work aims at applying these mathematical techniques in the field of biological processes with a focus on the wine fermentation process that is chosen as a representative model. In the literature, basic models for the wine fermentation process consist of a system of ordinary differential equations. They model the evolution of the yeast population number as well as the concentrations of assimilable nitrogen, sugar, and ethanol. In this thesis, the concentration of molecular oxygen is also included in order to model the change of the metabolism of the yeast from an aerobic to an anaerobic one. Further, a more sophisticated toxicity function is used. It provides simulation results that match experimental measurements better than a linear toxicity model. Moreover, a further equation for the temperature plays a crucial role in this work as it opens a way to influence the fermentation process in a desired way by changing the temperature of the system via a cooling mechanism. From the view of the wine industry, it is necessary to cope with large scale fermentation vessels, where spatial inhomogeneities of concentrations and temperature are likely to arise. Therefore, a system of reaction-diffusion equations is formulated in this work, which acts as an approximation for a model including computationally very expensive fluid dynamics. In addition to the modelling issues, an optimal control problem for the proposed reaction-diffusion fermentation model with temperature boundary control is presented and analysed. Variational methods are used to prove the existence of unique weak solutions to this non-linear problem. In this framework, it is possible to exploit the Hilbert space structure of state and control spaces to prove the existence of optimal controls. Additionally, first-order necessary optimality conditions are presented. They characterise controls that minimise an objective functional with the purpose to minimise the final sugar concentration. A numerical experiment shows that the final concentration of sugar can be reduced by a suitably chosen temperature control. The second part of this thesis deals with the identification of an unknown function that participates in a dynamical model. For models with ordinary differential equations, where parts of the dynamic cannot be deduced due to the complexity of the underlying phenomena, a minimisation problem is formulated. By minimising the deviations of simulation results and measurements the best possible function from a trial function space is found. The analysis of this function identification problem covers the proof of the differentiability of the function-to-state operator, the existence of minimisers, and the sensitivity analysis by means of the data-to-function mapping. Moreover, the presented function identification method is extended to stochastic differential equations. Here, the objective functional consists of the difference of measured values and the statistical expected value of the stochastic process solving the stochastic differential equation. Using a Fokker-Planck equation that governs the probability density function of the process, the probabilistic problem of simulating a stochastic process is cast to a deterministic partial differential equation. Proofs of unique solvability of the forward equation, the existence of minimisers, and first-order necessary optimality conditions are presented. The application of the function identification framework to the wine fermentation model aims at finding the shape of the toxicity function and is carried out for the deterministic as well as the stochastic case.}, subject = {Optimale Kontrolle}, language = {en} } @phdthesis{Wurst2015, author = {Wurst, Jan-Eric}, title = {Hp-Finite Elements for PDE-Constrained Optimization}, publisher = {W{\"u}rzburg University Press}, address = {W{\"u}rzburg}, isbn = {978-3-95826-024-5 (print)}, doi = {10.25972/WUP-978-3-95826-025-2}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-115027}, school = {W{\"u}rzburg University Press}, pages = {188}, year = {2015}, abstract = {Diese Arbeit behandelt die hp-Finite Elemente Methode (FEM) f{\"u}r linear quadratische Optimal-steuerungsprobleme. Dabei soll ein Zielfunktional, welches die Entfernung zu einem angestrebten Zustand und hohe Steuerungskosten (als Regularisierung) bestraft, unter der Nebenbedingung einer elliptischen partiellen Differentialgleichung minimiert werden. Bei der Anwesenheit von Steuerungsbeschr{\"a}nkungen k{\"o}nnen die notwendigen Bedingungen erster Ordnung, die typischerweise f{\"u}r numerische L{\"o}sungsverfahren genutzt werden, als halbglatte Projektionsformel formuliert werden. Folglich sind optimale L{\"o}sungen oftmals auch nicht-glatt. Die Technik der hp-Diskretisierung ber{\"u}cksichtigt diese Tatsache und approximiert raue Funktionen auf feinen Gittern, w{\"a}hrend Elemente h{\"o}herer Ordnung auf Gebieten verwendet werden, auf denen die L{\"o}sung glatt ist. Die erste Leistung dieser Arbeit ist die erfolgreiche Anwendung der hp-FEM auf zwei verwandte Problemklassen: Neumann- und Interface-Steuerungsprobleme. Diese werden zun{\"a}chst mit entsprechenden a-priori Verfeinerungsstrategien gel{\"o}st, mit der randkonzentrierten (bc) FEM oder interface konzentrierten (ic) FEM. Diese Strategien generieren Gitter, die stark in Richtung des Randes beziehungsweise des Interfaces verfeinert werden. Um f{\"u}r beide Techniken eine algebraische Reduktion des Approximationsfehlers zu beweisen, wird eine elementweise interpolierende Funktion konstruiert. Außerdem werden die lokale und globale Regularit{\"a}t von L{\"o}sungen behandelt, weil sie entscheidend f{\"u}r die Konvergenzgeschwindigkeit ist. Da die bc- und ic- FEM kleine Polynomgrade f{\"u}r Elemente verwenden, die den Rand beziehungsweise das Interface ber{\"u}hren, k{\"o}nnen eine neue L2- und L∞-Fehlerabsch{\"a}tzung hergeleitet werden. Letztere bildet die Grundlage f{\"u}r eine a-priori Strategie zum Aufdatieren des Regularisierungsparameters im Zielfunktional, um Probleme mit bang-bang Charakter zu l{\"o}sen. Zudem wird die herk{\"o}mmliche hp-Idee, die daraus besteht das Gitter geometrisch in Richtung der Ecken des Gebiets abzustufen, auf die L{\"o}sung von Optimalsteuerungsproblemen {\"u}bertragen (vc-FEM). Es gelingt, Regularit{\"a}t in abz{\"a}hlbar normierten R{\"a}umen f{\"u}r die Variablen des gekoppelten Optimalit{\"a}tssystems zu zeigen. Hieraus resultiert die exponentielle Konvergenz im Bezug auf die Anzahl der Freiheitsgrade. Die zweite Leistung dieser Arbeit ist die Entwicklung einer v{\"o}llig adaptiven hp-Innere-Punkte-Methode, die Probleme mit verteilter oder Neumann Steuerung l{\"o}sen kann. Das zugrundeliegende Barriereproblem besitzt ein nichtlineares Optimilit{\"a}tssystem, das eine numerische Herausforderung beinhaltet: die stabile Berechnung von Integralen {\"u}ber Funktionen mit m{\"o}glichen Singularit{\"a}ten in Elementen h{\"o}herer Ordnung. Dieses Problem wird dadurch gel{\"o}st, dass die Steuerung an den Integrationspunkten {\"u}berwacht wird. Die Zul{\"a}ssigkeit an diesen Punkten wird durch einen Gl{\"a}ttungsschritt garantiert. In dieser Arbeit werden sowohl die Konvergenz eines Innere-Punkte-Verfahrens mit Gl{\"a}ttungsschritt als auch a-posteriori Schranken f{\"u}r den Diskretisierungsfehler gezeigt. Dies f{\"u}hrt zu einem adaptiven L{\"o}sungsalgorithmus, dessen Gitterverfeinerung auf der Entwicklung der L{\"o}sung in eine Legendre Reihe basiert. Hierbei dient das Abklingverhalten der Koeffizienten als Glattheitsindikator und wird f{\"u}r die Entscheidung zwischen h- und p-Verfeinerung herangezogen.}, subject = {Finite-Elemente-Methode}, language = {en} }