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Mathematical theory of homogenization of partial differential equations is relatively a new area of research (30-40 years or so) though the physical and engineering applications were well known. It has tremendous applications in various branches of engineering and science like : material science ,porous media, study of vibrations of thin structures, composite materials to name a few. There are at present various methods to study homogenization problems (basically asymptotic analysis) and there is a vast amount of literature in various directions. Homogenization arise in problems with oscillatory coefficients, domain with large number of perforations, domain with rough boundary and so on. The latter one has applications in fluid flow which is categorized as oscillating boundaries. In fact ,in this thesis, we consider domains with oscillating boundaries. We plan to study to homogenization of certain optimal control problems with oscillating boundaries. This thesis contains 6 chapters including an introductory Chapter 1 and future proposal Chapter 6. Our main contribution contained in chapters 2-5. The oscillatory domain under consideration is a 3-dimensional cuboid (for simplicity) with a large number of pillars of length O(1) attached on one side, but with a small cross sectional area of order ε2 .As ε0, this gives a geometrical domain with oscillating boundary. We also consider 2-dimensional oscillatory domain which is a cross section of the above 3-dimensional domain. In chapters 2 and 3, we consider the optimal control problem described by the Δ operator with two types of cost functionals, namely L2-cost functional and Dirichlet cost functional. We consider both distributed and boundary controls. The limit analysis was carried by considering the associated optimality system in which the adjoint states are introduced. But the main contribution in all the different cases(L2 and Dirichlet cost functionals, distributed and boundary controls) is the derivation of error estimates what is known as correctors in homogenization literature. Though there is a basic test function, one need to introduce different test functions to obtain correctors. Introducing correctors in homogenization is an important aspect of study which is indeed useful in the analysis, but important in numerical study as well. The setup is the same in Chapter 4 as well. But here we consider Stokes’ Problem and study asymptotic analysis as well as corrector results. We obtain corrector results for velocity and pressure terms and also for its adjoint velocity and adjoint pressure. In Chapter 5, we consider a time dependent Kirchhoff-Love equation with the same domain with oscillating boundaries with a distributed control. The state equation is a fourth order hyperbolic type equation with associated L2-cost functional. We do not have corrector results in this chapter, but the limit cost functional is different and new. In the earlier chapters the limit cost functional were of the same type.

<p>Current indirect optimal control problem (IOCP) solvers, like beluga or PINs, use symbolic math to derive the necessary conditions to solve the IOCP. This limits the capability of IOCP solvers by only admitting symbolically representable functions. The purpose of this thesis is to present a framework that extends those solvers to derive the necessary conditions of an IOCP with fully numeric methods. With fully numeric methods, additional types of functions, including conditional logic functions and look-up tables can now be easily used in the IOCP solver.</p> <p><br></p> <p>This aim was achieved by implementing algorithmic differentiation (AD) as a method to derive the IOCP necessary conditions into a new solver called Giuseppe. The Brachistochrone problem was derived analytically and compared Giuseppe to validate the automatic derivation of necessary conditions. Two additional problems are compared and extended using this new solver. The first problem, the maximum cross-range problem, demonstrates a trajectory can be optimized indirectly while utilizing a conditional density function that switches as a function of height according to the 1976 U.S. atmosphere model. The second problem, the minimum time to climb problem, demonstrates a trajectory can be optimized indirectly while utilizing 6 interpolated look up tables for lift, drag, thrust, and atmospheric conditions. The AD method yields the exact same precision as the symbolic methods, rather than introducing numeric error as finite difference derivatives would with the benefit of admitting conditional switching functions and look-up tables. </p>

Simulation modeling is important for resource management and operational strategy within the industry. Optimation AB specializes in modeling and simulation of complex systems using Dymola, but also offers solutions for decision support by solving simplified optimal control problems (OCPs). Since simulation models can be exported as functional mock-up units (FMUs), interfacing the underlying equations, this thesis explores the use of FMUs to formulate and solve OCPs in Python, proposing a workflow based on the softwares CasADi, Rockit and IPOPT. Test cases of increasing complexity, including a cogeneration plant OCP, were employed to evaluate the workflow. Promising results were obtained for simplified models, though scaling, initial guesses and solver settings require further consideration. Collocation demonstrated the fastest convergence time and overall robustness. It could be concluded that integrating FMUs into OCPs is feasible, although complex models require modifications. This suggest that creating simplified component libraries in Dymola, tailored for optimization, could improve method implementation and re-usability.

Orientador: Edvaldo Assunção / Resumo: Neste trabalho estudamos problemas de controle ótimo intervalar e intervalar fuzzy. Em particular, propomos problemas de controle ótimo via teoria de incerteza generalizada e teoria dos conjuntos fuzzy. Dentre os vários tipos de incerteza generalizada utilizamos apenas a intervalar. Embora as abordagens do processo de solução dos problemas de controle ótimo intervalar e intervalar fuzzy sejam similares, as premissas iniciais para o uso e identificação de aplicação delas em problemas práticos são distintas assim como é distinto o processo de tomada de decisão. Assim, propomos inicialmente o problema de controle ótimo intervalar em tempo discreto. A primeira proposta de solução para o problema de controle ótimo intervalar em tempo discreto é construída usando a aritmética intervalar restrita de níveis simples juntamente com a técnica de programação dinâmica. As respostas do problema de controle ótimo intervalar contêm as possibilidades de soluções viáveis, e para implementar uma solução viável para o usuário final usamos a solução que minimiza o arrependimento máximo nos exemplos numéricos. A segunda proposta de solução para o problema de controle ótimo intervalar em tempo discreto é realizada com a aritmética intervalar restrita uma vez que essa aritmética intervalar é mais geral do que a aritmética intervalar restrita de níveis simples pois não considera os intervalos envolvidos nas operações variando de forma dependente. Exemplos numéricos também foram construídos e ilustram... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: In this work we study the interval optimal control problem and fuzzy interval optimal control problem. In particular, we propose optimal control problems via theory of generalized uncertainty and fuzzy set theory. Among the various types of generalized uncertainty we use only the interval uncertainty. Although the approaches to solve the interval optimal control problem and fuzzy interval optimal control problem are similar, the input data for problems with generalized uncertainty and flexibility are distinct as is distinct the decision-making process. Thus, we initially propose the discrete-time interval optimal control problem. The first solution method to solve the discrete-time interval optimal control problem is constructed using single-level constrained interval arithmetic coupled with a dynamic programming technique. The optimal interval solution contains the real-valued optimal solutions, and to implement a feasible solution to the user we use the minimax regret criterion in numerical examples. The second solution method to solve the discrete-time interval optimal control problem is done with the constrained interval arithmetic since this interval arithmetic is more general than the single-level constrained interval arithmetic because it does not have its intervals varying of dependent form in interval operations. Numerical examples have also been constructed and illustrate the method of solution. Finally, we study the discrete-time fuzzy interval optimal control prob... (Complete abstract click electronic access below) / Doutor

In this thesis, we study homogenization of optimal control problems in various oscillatory domains. Specifically, we consider four types of domains given in Figure 1 below. Figure 1: Oscillating Domains The thesis is organized into six chapters. Chapter 1 provides an introduction to our work and the rest of the thesis. The main contributions of the thesis are contained in Chapters 2-5. Chapter 6 presents the conclusions of the thesis and possible further directions. A brief description of our work (Chapters 2-5) follows: Chapter 2: Asymptotic behaviour of a fourth order boundary optimal control problem with Dirichlet boundary data posed on an oscillating domain as in Figure 1(A) is analyzed. We use the unfolding operator to study the asymptotic behavior of this problem. Chapter 3: Homogenization of a time dependent interior optimal control problem on a branched structure domain as in Figure 1(B) is studied. Here we pose control on the oscillating interior part of the domain. The analysis is carried out by appropriately defined unfolding operators suitable for this domain. The optimal control is characterized using various unfolding operators defined at each branch of every level. Chapter 4: A new unfolding operator is developed for a general oscillating domain as in Figure 1(C). Homogenization of a non-linear elliptic problem is studied using this new un-folding operator. Using this idea, homogenization of an optimal control problem on a circular oscillating domain as in Figure 1(D) is analyzed. Chapter 5: Homogenization of a non-linear optimal control problem posed on a smooth oscillating domain as in Figure 1(C) is studied using the unfolding operator.

The work selects a specific issue from the numerical analysis of optimal control problems. We investigate a linear-quadratic optimal control problem based on a partial differential equation on 3-dimensional non-convex domains. Based on efficient solution methods for the partial differential equation an algorithm known from control theory is applied. Now the main objectives are to prove that there is no degradation in efficiency and to verify the result by numerical experiments. We describe a solution method which has second order convergence, although the intermediate control approximations are piecewise constant functions. This superconvergence property is gained from a special projection operator which generates a piecewise constant approximation that has a supercloseness property, from a sufficiently graded mesh which compensates the singularities introduced by the non-convex domain, and from a discretization condition which eliminates some pathological cases. Both isotropic and anisotropic discretizations are investigated and similar superconvergence properties are proven. A model problem is presented and important results from the regularity theory of solutions to partial differential equation in non-convex domains have been collected in the first chapters. Then a collection of statements from the finite element analysis and corresponding numerical solution strategies is given. Here we show newly developed tools regarding error estimates and projections into finite element spaces. These tools are necessary to achieve the main results. Known fundamental statements from control theory are applied to the given model problems and certain conditions on the discretization are defined. Then we describe the implementation used to solve the model problems and present all computed results.

The primary goal of this thesis is to study finite element based a priori and a posteriori error estimates of optimal control problems of various kinds governed by linear elliptic PDEs (partial differential equations) of second and fourth orders. This thesis studies interior and boundary control (Neumann and Dirichlet) problems. The initial chapter is introductory in nature. Some preliminary and fundamental results of finite element methods and optimal control problems which play key roles for the subsequent analysis are reviewed in this chapter. This is followed by a brief literature survey of the finite element based numerical analysis of PDE constrained optimal control problems. We conclude the chapter with a discussion on the outline of the thesis. An abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed in the second chapter. The analysis establishes the best approximation result from a priori analysis point of view and delivers a reliable and efficient a posteriori error estimator. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. Subsequently, the applications of p p - interior penalty methods for a boundary control problem as well as a distributed control problem governed by the bi-harmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. In the third chapter, an alternative energy space based approach is proposed for the Dirichlet boundary control problem and then a finite element based numerical method is designed and analyzed for its numerical approximation. A priori error estimates of optimal order in the energy norm and the m norm are derived. Moreover, a reliable and efficient a posteriori error estimator is derived with the help an auxiliary problem. An energy space based Dirichlet boundary control problem governed by bi-harmonic equation is investigated and subsequently a l y - interior penalty method is proposed and analyzed for it in the fourth chapter. An optimal order a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution has minimum regularity. Further an optimal order l l norm error estimate is derived. The fifth chapter studies a super convergence result for the optimal control of an interior control problem with Dirichlet cost functional and governed by second order linear elliptic PDE. An optimal order a priori error estimate is derived and subsequently a super convergence result for the optimal control is derived. A residual based reliable and efficient error estimators are derived in a posteriori error control for the optimal control. Numerical experiments illustrate the theoretical results at the end of every chapter. We conclude the thesis stating the possible extensions which can be made of the results presented in the thesis with some more problems of future interest in this direction.

Submitted by DANIELLA PORTO null (danielinha.dani@gmail.com) on 2016-03-24T18:05:56Z No. of bitstreams: 1 TESE Daniella Porto.pdf: 1058349 bytes, checksum: ed5227eb69daeb674962db0bf4513f1f (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-03-24T19:42:30Z (GMT) No. of bitstreams: 1 porto_d_dr_sjrp.pdf: 1058349 bytes, checksum: ed5227eb69daeb674962db0bf4513f1f (MD5) / Made available in DSpace on 2016-03-24T19:42:30Z (GMT). No. of bitstreams: 1 porto_d_dr_sjrp.pdf: 1058349 bytes, checksum: ed5227eb69daeb674962db0bf4513f1f (MD5) Previous issue date: 2016-03-15 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Consideramos dois tipos de problemas de controle ótimo: a) Problemas de controle impulsivo e b) problemas de controle ótimo sobre domínios estratificados. Organizamos o trabalho em duas partes distintas. A primeira parte é dedicada ao estudo de um problema de controle impulsivo onde a técnica de reparametrização usual do problema impulsivo é usada para obter um problema regular. Então nós damos resultados de aproximações consistentes via discretização de Euler em que uma sequência de problemas aproximados é obtida com a propriedade que se existe uma subsequência de processos que são ótimos para o correspondente problema discreto que converge para algum processo limite, então o último é ótimo para o problema reparametrizado original. A partir da solução ótima reparametrizada somos capazes de fornecer a solução do problema impulsivo original. A segunda parte considera o problema de tempo mínimo definido sobre domínios estratificados. Definimos o problema e estabelecemos desigualdades de Hamilton Jacobi. Então, damos algumas motivações via Lei de Snell e o problema do Elvis e finalmente fornecemos condições de otimalidade necessárias e suficientes. / We consider two types of optimal control problems: a) Impulsive control problems and b) optimal control problems in stratified domains. So we organize this work in two distinct parts. The first part is dedicated to the study of an impulsive optimal control problem where the usual reparametrization technique of the impulsive problem is used to obtain a regular problem. Then we provide consistent approximation results via Euler discretization in which a sequence of related approximated problems is obtained with the property that if there is a subsequence of processes which are optimal for the corresponding discrete problems which converge to some limit process, then the latter is optimal to the original reparametrized problem. From the reparametrized optimal solution we are able to provide the solution to the original impulsive problem. The second part is regarding the minimal time problem defined on stratified domains. We sate the problem and establish Hamilton-Jacobi inequalities. Then we give some motivation via Snell's law and the Elvis problem and finally we provide necessary and sufficient conditions of optimality. / FAPESP: 2011/14121-9

The work selects a specific issue from the numerical analysis of optimal control problems. We investigate a linear-quadratic optimal control problem based on a partial differential equation on 3-dimensional non-convex domains. Based on efficient solution methods for the partial differential equation an algorithm known from control theory is applied. Now the main objectives are to prove that there is no degradation in efficiency and to verify the result by numerical experiments. We describe a solution method which has second order convergence, although the intermediate control approximations are piecewise constant functions. This superconvergence property is gained from a special projection operator which generates a piecewise constant approximation that has a supercloseness property, from a sufficiently graded mesh which compensates the singularities introduced by the non-convex domain, and from a discretization condition which eliminates some pathological cases. Both isotropic and anisotropic discretizations are investigated and similar superconvergence properties are proven. A model problem is presented and important results from the regularity theory of solutions to partial differential equation in non-convex domains have been collected in the first chapters. Then a collection of statements from the finite element analysis and corresponding numerical solution strategies is given. Here we show newly developed tools regarding error estimates and projections into finite element spaces. These tools are necessary to achieve the main results. Known fundamental statements from control theory are applied to the given model problems and certain conditions on the discretization are defined. Then we describe the implementation used to solve the model problems and present all computed results.