This thesis presents a numerical approach based on generalized fractional-order Chebyshev wavelets for solving fractional-order optimal control problems. The exact value of the Riemann– Liouville fractional integral operator of the generalized fractional-order Chebyshev wavelets is computed by applying the regularized beta function. We apply the given wavelets, the exact formula, and the collocation method to transform the studied problem into a new optimization problem. The convergence analysis of the proposed method is provided. The present method is extended for solving fractional-order, distributed-order, and variable-order optimal control problems. Illustrative examples are considered to show the advantage of this method in comparison with the existing methods in the literature.

<p> This work focuses on solving the general optimal control problems with smart-learning-enabled and theory-supported optimal control (SET-OC) approaches. The proposed SET-OC includes two main directions. Firstly, according to the basic idea of the direct method, the smart-learning-enabled iterative optimization algorithm (SEIOA) is proposed for solving discrete optimal control problems. Via discretization and reformulation, the optimal control problem is converted into a general quadratically constrained quadratic programming (QCQP) problem. Then, the SEIOA is applied to solving QCQPs. To be specific, first, a structure-exploiting decomposition scheme is introduced to reduce the complexity of the original problem. Next, an iterative search, combined with an intersection-cutting plane, is developed to achieve global convergence. Furthermore, considering the implicit relationship between the algorithmic parameters and the convergence rate of the iterative search, deep learning is applied to design the algorithmic parameters from an appropriate amount of training data to improve convergence property. To demonstrate the effectiveness and improved computational performance of the proposed SEIOA, the developed algorithms have been implemented in extensive real-world application problems, including unmanned aerial vehicle path planning problems and general QCQP problems. According to the theoretical analysis of global convergence and the simulation results, the efficiency, robustness, and improved convergence rate of the optimization framework compared to the state-of-the-art optimization methods for solving general QCQP problems are analyzed and verified. Secondly, the onboard learning-based optimal control method (L-OCM) is proposed to solve the optimal control problems. Supported by the optimal control theory, the necessary conditions of optimality for optimal control of the optimal control problem can be derived, which leads to two two-point-boundary-value-problems (TPBVPs). Then, critical parameters are identified to approximate the complete solutions of the TPBVPs. To find the implicit relationship between the initial states and these critical parameters, deep neural networks are constructed to learn the values of these critical parameters in real-time with training data obtained from the offline solutions.  To demonstrate the effectiveness and improved computational performance of the proposed L-OCM approaches, the developed algorithms have been implemented in extensive real-world application problems, including two-dimensional human-Mars entry, powered-descent, landing guidance problems, and fuel-optimal powered descent guidance (PDG) problems. In addition, considering there is no thorough analysis of the properties of the optimal control profile for PDG when considering the state constraints, a rigid theoretical analysis of the fuel-optimal PDG problem with state constraints is further provided. According to the theoretical analysis and simulation results, the optimality, robustness, and real-time performance of the proposed L-OCM are analyzed and verified, which indicates the potential for onboard implementation. </p>

Diese Dissertation besteht aus drei unabhängigen Artikeln in dem Forschungsfeld der Informationsökonomik. Ein wiederkehrendes Motiv in allen drei Artikeln ist die ambivalente Rolle von privater Information. In Kontrast zur klassischen Entscheidungstheorie, in der mehr Informationen Individuen niemals schlechter stellt, analysiere ich drei verschiedene Umgebungen, in denen mehr Konsumenteninformation die Konsumentenrente verringern kann. / This dissertation comprises three independent chapters in the field of information economics. The recurrent theme of all three chapters is the ambiguous role of information: While in standard decision theory additional information enables individuals to weakly increase utility through making better choices, I analyze three di erent environments in which more information to consumers may actually be detrimental to consumer utility.

Orientador: Valeriano Antunes de Oliveira / Resumo: Neste trabalho, primeiramente, serão apresentados problemas de otimização nos quais a função objetivo é de múltiplas variáveis e de valor intervalar e as restrições de desigualdade são dadas por funcionais clássicos, isto é, de valor real. Serão dadas as condições de otimalidade usando a E−diferenciabilidade e, depois, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade usando a gH−diferenciabilidade total são do tipo KKT e as suficientes são do tipo de convexidade generalizada. Em seguida, serão estabelecidos problemas de controle ótimo nos quais a funçãao objetivo também é com valor intervalar de múltiplas variáveis e as restrições estão na forma de desigualdades e igualdades clássicas. Serão fornecidas as condições de otimalidade usando o conceito de Lipschitz para funções intervalares de várias variáveis e, logo, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade, usando a gH−diferenciabilidade total, estão na forma do célebre Princípio do Máximo de Pontryagin, mas desta vez na versão intervalar. / Abstract: In this work, firstly, it will be presented optimization problems in which the objective function is interval−valued of multiple variables and the inequality constraints are given by classical functionals, that is, real−valued ones. It will be given the optimality conditions using the E−differentiability and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability are of KKT−type and the sufficient ones are of generalized convexity type. Next, it will be established optimal control problems in which the objective function is also interval−valued of multiple variables and the constraints are in the form of classical inequalities and equalities. It will be furnished the optimality conditions using the Lipschitz concept for interval−valued functions of several variables and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability is in the form of the celebrated local Pontryagin Maximum Principle, but this time in the intervalar version. / Doutor

[pt] Esquemas semi-Lagrangeanos usados para a aproximação do princípio da programação dinâmica são baseados em uma discretização temporal reconstruída no espaço de estado. O uso de uma malha estruturada torna essa abordagem inviável para problemas de alta dimensão devido à maldição da dimensionalidade. Nesta tese, apresentamos uma nova abordagem para problemas de controle ótimo de horizonte infinito onde a função valor é calculada usando Funções de Base Radial (RBFs) pelo método de aproximação de mínimos quadrados móveis de Shepard em malhas irregulares. Propomos um novo método para gerar uma malha irregular guiada pela dinâmica e uma rotina de otimizada para selecionar o parâmetro responsável pelo formato nas RBFs. Esta malha ajudará a localizar o problema e aproximar o princípio da programação dinâmica em alta dimensão. As estimativas de erro para a função valor também são fornecidas. Testes numéricos para problemas de alta dimensão mostrarão a eficácia do método proposto. Além do controle ótimo de EDPs clássicas mostramos como o método também pode ser aplicado ao controle de equações não-locais. Também fornecemos um exemplo analisando a convergência numérica de uma equação não-local controlada para o modelo contínuo. / [en] Semi-Lagrangian schemes for the approximation of the dynamic programming principle are based on a time discretization projected on a state-space grid. The use of a structured grid makes this approach not feasible for highdimensional problems due to the curse of dimensionality. In this thesis, we present a new approach for infinite horizon optimal control problems where the value function is computed using Radial Basis Functions (RBF) by the Shepard s moving least squares approximation method on scattered grids. We propose a new method to generate a scattered mesh driven by the dynamics and an optimal routine to select the shape parameter in the RBF. This mesh will help to localize the problem and approximate the dynamic programming principle in high dimension. Error estimates for the value function are also provided. Numerical tests for high dimensional problems will show the effectiveness of the proposed method. In addition to the optimal control of classical PDEs, we show how the method can also be applied to the control of nonlocal equations. We also provide an example analyzing the numerical convergence of a nonlocal controlled equation towards the continuous model.

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Cette thèse a pour but de développer des approches déterministes et heuristiques pour résoudre certaines classes des problèmes d'optimisation en télécommunication et la mobilité d'un réseau de transport : problèmes de routage, problèmes de covoiturage, problèmes de contrôle de l'alimentation dans un réseau sans fil, problèmes d'équilibrage du spectre dans les réseaux DSL. Il s'agit des problèmes d'optimisation non convexe de très grande taille. Nos approches sont basées sur la programmation DC&DCA, méthode de décomposition proximale et la méthode d'étiquetage des graphes. Grâce aux techniques de formulation/reformulation et de pénalité exacte, nous avons établi des programmes DC équivalents en vue de leur résolution par DCA. Selon la structure de ces problèmes, on peut fournir des décompositions DC appropriées ou de bons points initiaux de DCA. Nos méthodes ont été programmées sous MATLAB, C/C++. Ils montrent la performance de nos algorithmes par rapport à des méthodes existantes. / In this thesis, we focus on developing deterministic and heuristic approaches for solving some classes of optimization problems in Telecommunication and Mobility & Transport domain: Routing problems, Car pooloing problems, Power control problems in wireless network, Optimal spectrum balancing problems in DSL networks. They are large-scale nonconvex optimization problems. Our methodologies are focus on DC programming and DCA, Proximal decomposition method and Labeling method in graph theory. They are well-known as powerful tools in optimization. The considered problems were reformulated using the DC formulation/reformulation and exact penalty techniques and the DCA was used to obtain the solution. Also, depending on the structure of considered problems, we can provide appropriate DE decompositions or good initial points for DCA. All these proposed methods have been implemented with MATLAB, C/C++ to confirm the practical aspects and enhance our research works.