The application of techniques from nonlinear analysis to materials science has seen great developments in the recent years and it has really been a driving force for substantial mathematical research in the area of partial differential equations and the multi-dimensional calculus of variations. This thesis has been motivated by two recent and remarkable experimental observations of H. Seiner in shape-memory alloys which we attempt to interpret mathematically. Much of the work is original and has given rise to deep problems in the calculus of variations. Firstly, we study the formation of non-classical austenite-martensite interfaces. Ball & Carstensen (1997, 1999) theoretically investigated the possibility of the occurrence of such interfaces and studied the cubic-to-tetragonal case extensively. In this thesis, we present an analysis of non-classical austenite-martensite interfaces recently observed by Seiner et al.~in a single crystal of a CuAlNi shape-memory alloy, undergoing a cubic-to-orthorhombic transition. We show that these can be described by the general nonlinear elasticity model and we make some predictions regarding the admissible volume fractions of the martensitic variants involved, as well as the habit plane normals. Interestingly, in the above experimental observations, the interface between the austenite and the martensitic configuration is never exactly planar, but rather slightly curved, resulting from the pattern of martensite not being exactly homogeneous. However, it is not clear how one can reconstruct the inhomogeneous configuration as a stress-free microstructure and, instead, a theoretical approach is followed. In this approach, a general method is provided for the construction of a compatible curved austenite-martensite interface and, by exploiting the structure of quasiconvex hulls, the existence of curved interfaces is shown in two and three dimensions. As far as the author is aware of, this is the first construction of such a curved austenite-martensite interface. Secondly, we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy consisting of a single variant of stabilized 2H martensite. The nucleation process is induced by localized heating and it is observed that, regardless of where the localized heating is applied, the nucleation points are always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant throughout the specimen, provided this is suitably cut. The proofs of quasiconvexity are based on a rigidity argument and are specific to the change of symmetry in the phase transformation. To the best of the author's knowledge, quasiconvexity conditions at edges and corners have not been considered before.

This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C^1,α-Hölder continuity, for some 0 < α < 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential A-harmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energy-minimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a sub-group of residual symmetries.

In this work we introduce a novel queueing model with two classes of users in which, instead of accessing a resource, users wait in the system to match with a candidate from the other class. The users are selective and the matchings occur probabilistically. This new model is useful for analysing the traffic in web portals that match people who provide a service with people who demand the same service, e.g. employment portals, matrimonial and dating sites and rental portals. We first provide a Markov chain model for these systems and derive the probability distribution of the number of matches up to some finite time given the number of arrivals. We then prove that if no control mechanism is employed these systems are unstable for any set of parameters. We suggest four different classes of control policies to assure stability and conduct analysis on performance measures under the control policies. Contrary to the intuition that the rejection rate should decrease as the users become more likely to be matched, we show that for certain control policies the rejection rate is insensitive to the matching probability. Even more surprisingly, we show that for reasonable policies the rejection rate may be an increasing function of the matching probability. We also prove insensitivity results related to the average queue lengths and waiting times. Further, to gain more insight into the behaviour of probabilistic matching systems, we propose approximation methods based on fluid and diffusion limits using different scalings. We analyse the basic properties of these approximations and show that some performance measures are insensitive to the matching probability agreeing with the results found by the exact analysis. Finally we study the optimal control and revenue management for the systems with the objective of profit maximization. We formulate mathematical models for both unobservable and observable systems. For an unobservable system we suggest a deterministic optimal control, while for an observable system we develop an optimal myopic state dependent pricing.

Approved for public release, distribution is unlimited / problem, with engine thrust as the only possible perturbation. The optimal control problems are solved using the general purpose dynamic optimization software, DIDO. The dynamical model together with the fuel optimal control problem is validated by simulating several well known orbit transfers. By replicating the single satellite model, this thesis shows that a multi-satellite model which optimizes all vehicles concurrently can be easily built. The specific scenario under study involves the injection of multiple satellites from a common launch vehicle; however, the methods and model are applicable to spacecraft formation problems as well. / Major, United States Air Force

Cancer is a generic term for a large group of diseases where cells of the body lose their normal mechanisms for growth so that they grow in an uncontrolled way. One of the most common treatments of cancer is chemotherapy that aims to kill abnormal proliferating cells; however normal cells and other organs of the patients are also adversely affected. In practice, it's often difficult to maintain optimum chemotherapy doses that can maximise the abnormal cell killing as well as reducing side effects. The most chemotherapy drugs used in cancer treatment are toxic agents and usually have narrow therapeutic indices, dose levels in which these drugs significantly kill the cancerous cells are close to the levels which sometime cause harmful toxic side effects. To make the chemotherapeutic treatment effective, optimum drug scheduling is required to balance between the beneficial and toxic side effects of the cancer drugs. Conventional clinical methods very often fail to find drug doses that balance between these two due to their inherent conflicting nature. In this investigation, mathematical models for cancer chemotherapy are used to predict the number of tumour cells and control the tumour growth during treatment. A feedback control method is used so as to maintain certain level of drug concentrations at the tumour sites. Multi-objective Genetic Algorithm (MOGA) is then employed to find suitable solutions where drug resistances and drug concentrations are incorporated with cancer cell killing and toxic effects as design objectives. Several constraints and specific goal values were set for different design objectives in the optimisation process and a wide range of acceptable solutions were obtained trading off among different conflicting objectives. Abstract v In order to develop a multi-objective optimal control model, this study used proportional, integral and derivative (PID) and I-PD (modified PID with Integrator used as series) controllers based on Martin's growth model for optimum drug concentration to treat cancer. To the best of our knowledge, this is the first PID/I-PD based optimal chemotherapy control model used to investigate the cancer treatment. It has been observed that some solutions can reduce the cancer cells up to nearly 100% with much lower side effects and drug resistance during the whole period of treatment. The proposed strategy has been extended for more drugs and more design constraints and objectives.

This monograph addresses control and filtering problems for systems with sudden changes in their behavior and whose changes are detected and estimated by an imperfect detector. More precisely it considers continuous-timeMarkov Jump Linear Systems (MJLS) where the current mode of operation is estimated by a detector. This detector is assumed to be imperfect in the sense that it is possible that the detected mode of operation diverges from the real mode of operation. Furthermore the probabilities for these detections are considered to be known. It is assumed that the detector has its own dynamic, which means that the detected mode of information can change independently from the real mode of operation. The novelty of this approach lies in how uncertainties are modeled. A Hidden Markov Model (HMM) is used to model the uncertainties introduced by the detector. For these systems the following problems are addressed: i) Stochastic Stabilizability in mean-square sense, ii) H2 control, iii) H? control and iv) the H? filtering problem. Solutions based on Linear Matrix Inequalities (LMI) are developed for each of these problems. In case of the H2 control problem, the solutionminimizes an upper bound for the H2 norm of the closed-loop control system. For the H? control problem a solution is presented that minimizes an upper bound for the H? norm of the closed-loop control system. In the case of the H? filtering, the solution presented minimizes the H? norm of a system representing the estimation error. The solutions for the control problems are illustrated using a numerical example modeling a simple two-tank process. / Esta monografia aborda problemas de controle e filtragem em sistemas com saltos espontâneos que alteram seu comportamento e cujas mudanças são detectadas e estimadas por um detector imperfeito. Mais precisamente, consideramos sistemas lineares cujos saltos podem ser modelados usando um processo markoviano (Markov Jump Linear Systems) e cujo modo de operação corrente é estimado por um detector. O detector é considerado imperfeito tendo em vista a possibilidade de divergência entre o modo real de operação e o modo de operação detectado. Ademais, as probabilidades das deteccões são consideradas conhecidas. Assumimos que o detector possui uma dinâmica própria, o que significa que o modo de operação detectado pode mudar independentemente do modo real de operação. A novidade dessa abordagem está na modelagem das incertezas. Um processo oculto de Markov (HMM) é usado para modelar as incertezas introduzidas pelo detector. Para esses sistemas, os seguintes problemas são abordados: i) estabilidade quadrática ii) controle H2, iii) controle H? e iv) o problema da filtragem H?. Soluções baseadas em Desigualdades de Matriciais Lineares (LMI) são desenvolvidas para cada um desses problemas. No caso do problema de controle H2, a solução minimiza um limite superior para a norma H2 do sistema de controle em malha fechada. Para o problema H? -controle é apresentada uma solução que minimiza um limite superior para a norma H? do sistema de controle em malha fechada. No caso da filtragem H?, a solução apresentada minimiza a norma H? de um sistema que representa o erro de estimativa. As soluções para os problemas de controle são ilustradas usando um exemplo numérico que modela um processo simples de dois tanques.

Neste trabalho procuramos determinar o controle ótimo para problemas de custo médio a longo prazo (CMLP) de sistemas lineares com saltos markovianos (SLSMs) com observação parcial dos estados da cadeia de Markov, e, para isso, implementamos métodos computacionais heurísticos como algoritmos evolutivos de primeira geração - algoritmo genético (AG) básico - e os algoritmos UMDA(Univariate Marginal Distribution Algorithm) e BOA(Bayesian Optimization Algorithm), de segunda geração. Utilizamos um algoritmo variacional para comparar com os métodos implementados e medir a qualidade de suas soluções. Desenvolvemos uma abordagem de transição de níveis de observação (ATNO), partindo de um problema de observação completa e migrando através de problemas parcialmente observados. Cada um dos métodos mencionados acima foi implementado também no contexto da ATNO. Para realizar uma análise estatística sobre o desempenho dos métodos computacionais, utilizamos um gerador de SLSMs com importantes características da teoria de controle como: estabilidade, estabilizabilidade, observabilidade, controlabilidade e detetabilidade. Por fim, apresentamos alguns resultados sobre o CMLP com controles estabilizantes e resultados parciais a respeito da unicidade de solução / In this work we are interested in the optimal control for the long run average cost (LRAC) problem for linear systems with Markov jump parameters (LSMJP), using heuristic methods like first generation evolutionary algorithms - genetic algorithm (GA) - and second generation algorithms including UMDA (Univariate Marginal Distribution Algorithm) and BOA (Bayesian Optimization Algorithm). We have developed a scheme that employs different problems with intermediate levels of observation of the Markov chain, starting with complete observation and shifting to the partial observation problem. The aforementioned methods have been implemented using this scheme. Moreover, in order to compare the methods, we use an algorithm for generating a number of LSMJP and we present a basic statistical analysis of the results. Finally, we present some results on the LRAC with stabilizing control and some partial results on the uniqueness of the solution

A importância do planejamento de sistemas hidrotérmicos é largamente reconhecida. A questão constitui-se em um complicado problema de otimização que envolve aspectos não-lineares e estocásticos, e que se torna mais complexo quando várias usinas são consideradas em conjunto, em virtude da ocorrência de interações. Apesar de se conhecerem formalizações teóricas do problema e da solução, a complexidade acarreta sérios entraves computacionais e muitas vezes consideram-se apenas cenários simplificados, como o determinístico ou estocástico com um modelo de reservatório equivalente. Este trabalho tem como objetivo conciliar a facilidade numérica de problemas determinísticos com a consideração indireta da natureza estocástica do problema. Com esta finalidade, propõe-se o uso de um funcional de custo que reflete o risco de vertimento a cada período. Outra linha estudada neste trabalho foi a de rastreamento de alvo utilizando a técnica de controle Linear Quadrático (LQ). A eficácia das estratégias propostas é avaliada através de diversos casos de estudo, incluindo comparação com o planejamento determinístico usual / The scheduling of hydrothermal systems is an important and challenging problem. In fact, the problem is complex since it involves the optimization of a cost functional related to a nonlinear and stochastic system. Although mathematical formulations of the problem are available, there are computational issues (as the course of dimensionality in dynamical programming) that make difficult to obtain the solution. In this situation, some simplifyied equivalent models or deterministic formulations were considered in the literature, which are simpler to deal with. The aim of this work is to obtain a solution thet presents the ease of computaion of deterministic models and, simultaneously, it reflects the stochastic nature of the problem, at some extension. The strategy consists of taking into account a cost functional that is related to the risk of spillage at each period. Another strategy considered here is the linear quadratic control with a target. Several case estudies are presented, which employ different models for the stochastic inflow, allwing us to evaluate the effectiveness of the strategies and to compare them with the usual deterministic control

A depressão metabólica em aves e mamíferos, dada a alta demanda energética destes animais, se apresenta, geralmente, como resposta às condições de escassez de alimentos e baixas temperaturas. Desta forma, este projeto busca explorar, no campo teórico, como o sistema de termorregulação poderia atuar no sentido de maximizar as reservas energéticas minimizando os gastos metabólicos (depressão metabólica). Para tanto, fazemos uso de teorias da engenharia de controle que propiciam ferramental teórico para analisar como se dariam essas minimizações, ou seja, como o sistema nervoso atuaria estabelecendo um controle (set-point hipotalâmico) que minimizasse estes gastos à medida que se desse o processo de termorregulação. Neste contexto, propomos um modelo básico de termorregulação que leva em conta temperatura corpórea, taxa metabólica e temperatura ambiente, no qual o set-point atua como um controle. Mostramos como este modelo de regulação térmica propicia, devido à sua configuração, significativa redução dos distúrbios causados por variações da temperatura ambiente. Através da teoria de controle ótimo, mostramos como o set-point hipotalâmico pode surgir como resultado da minimização de um funcional relacionado ao custo com a termorregulação. Além disso, fez-se uma análise de como a temperatura ambiente pode definir diferentes situações em termos de vantagens da depressão metabólica como mecanismo de minimização de gasto energético. Para este tipo de análise, propomos um índice de razão entre o custo metabólico constante e o obtido sob atuação do controlador durante o período em que se dá o processo. Após um período em depressão metabólica, os indivíduos devem voltar a sua condição de eutermia, e, em situações de baixa temperatura, o custo deste retorno pode suplantar as vantagens para um dado indivíduo. Assim, são analisadas as influencias da massa corpórea, onde se observa aumento do custo em decorrência da entrada em depressão metabólica por parte dos indivíduos de maior massa. Tal aumento de custo é acentuado nas situações de menor temperatura ambiente. Finalmente, uma análise relativa ao tempo para retorno à condição de eutermia é apresentada, sendo que os resultados vão ao encontro das evidencias atuais sobre a flexibilidade estratégica de muitos hibernantes. / Metabolic depression of mammals and birds, animals of high metabolic demands, normally emerges as a response to food shortage and low ambient temperature. The main goal of this research is to explore, in a theoretical perspective, how the thermoregulatory system could extend the energy reserves of these endotherms decreasing metabolic costs under those environmental conditions. To approach the problem, we propose the use of control engineering theories to analyze the way the this minimization could occur, in other words, how the nervous system would act establishing a control (hypothalamic set-point) to minimize those costs during the thermoregulatory process. In this context, we propose a basic thermoregulation model that takes into account body temperature, metabolic rate and environmental temperature, and in which the set-point acts as a control. We show how this model can significantly reduce disturbances generated by ambient temperature. Using optimal control theory, we show how the hypothalamic set-point can emerge as a result of a minimization process of a functional related to thermoregulation costs. Also, how ambient temperature can define different metabolic profiles is explored, in terms of metabolic depression and the necessary return to euthermic conditions. To quantify this analysis we propose an index, based on the ratio between a constant metabolic cost and the metabolic cost defined by the controller. After a period in metabolic depression individuals should return to their euthermic condition, and, in situations of low environmental temperature, it is shown that the cost to return can be larger than the advantages. In this way, analyzing body mass influences we observed increased metabolic depression cost in larger individuals. This cost is even higher under lower environmental temperature. Finally, the cost related to the time elapsed, until the euthermic state is reached again, is considered. These last results are in accordance with current conception about the flexibility in hibernation process.

Este trabalho trata das aplicações da Teoria de Controle Ótimo ao problema de otimização da Política Monetária. Esse problema consiste em minimizar uma função que representa o custo da inflação para a sociedade, por meio de manipulações na variável de controle, que é a taxa de juros da economia. Serão considerados dois modelos para a dinâmica macroeconômica: um keynesiano e um novo-keynesiano. O problema de minimização sujeito à dinâmica keynesiana pode ser resolvido por meio dos conceitos tradicionais de controle ótimo, como o LQR e LQG. Por outro lado, o modelo novokeynesiano possui uma dinâmica mais complexa e não-recursiva, impossibilitando a aplicação direta dos métodos de programação estocástica. Assim, o problema de minimização sujeito à essa dinâmica requer métodos mais complexos, como o método do Lagrangiano ou o método do ponto de sela recursivo. É apresentada a solução analítica para o problema de controle ótimo em cada tipo de dinâmica. Em seguida, o problema de estimação de parâmetros é abordado. Métodos como o OLS e o GMM são empregados para estimar os parâmetros do modelo. Também são realizadas simulações para determinar numericamente as políticas de controle ótimo em alguns cenários. Por fim, a política monetária ótima é determinada para o período entre 2008 e 2009 e comparada com a política monetária adotada pelo governo. / This article discusses the applications of the Optimal Control Theory to the Monetary Policy optimization problem. This problem consists in minimizing a function that represents the inflation cost to society, through manipulation on the control variable, which is the interest rate of the economy. It will be considered two models for macroeconomic dynamics: a Keynesian and a new-Keynesian model. The minimization problem subject to Keynesian dynamics can be solved through traditional optimal control tools, such as LQR and LQG. On the other hand, the second model has a more complex and non-recursive dynamic, precluding the direct application of stochastic programming methods. Thus, the minimization problem restricted to this dynamic requires more complex methods, like the Lagrangian or the recursive saddlepoint method. It is presented the analytical solution to the optimal control problem for each type of dynamics. Then, the parameter estimation problem is addressed. Methods such as OLS and GMM are used to estimate the model parameters. Simulations are also carried out to determine numerically the optimal control policies in some scenarios. Finally, the optimal monetary policy is determined for the period between 2008 and 2009 and compared with the monetary policy adopted by the government.