In this dissertation, we apply stochastic approximation (SA) to two different problems addressed respectively in Part I and Part II. / The contribution of Part I is mostly theoretical. We consider the problem of online tracking of moving targets such as a signals, through noisy measurements. In particular, we study a non-stationary environment that is subject to sudden discontinuous changes in the underlying parameters of the system. We assume no a priori knowledge about the parameters nor the change-times. Our approach is based on constant stepsize SA. However, because of the unpredictable discontinuous changes, the choice of stepsize is difficult. Small stepsizes improve precision while large stepsizes allow the SA iterates to react faster to sudden changes. / We first investigate target estimation. Our work appears in [Levy 09]. We propose to combine a small constant stepsize with change-point monitoring, and to reset the process at a value closer to the new target when a change is detected. Because the environment is not stationary, we cannot directly apply the usual limit theorems. We thus give a theoretical characterization and discuss the tradeoff between precision and fast adaptation. We also introduce a new monitoring scheme, the regression-based hypothesis test. / Secondly, we consider an online version of the well-known Q-learning algorithm, which operates directly in its target environment, to optimize a Markov decision process. Online algorithms are challenging because the errors, necessarily made when learning, affect performance. Again, under a switching environment the usual limit theorems are not applicable. We introduce an adaptive stepsize selection algorithm based on weak convergence results for SA. Our algorithm automatically achieves a desirable balance between speed and accuracy. These findings are published in [Levy 06, Costa 09]. / In Part II, we study an applied problem related to the mining industry. Strategic management requires managing large portfolios of investments. Because financial resources are limited, only the projects with the highest net present value (NPV), their measure of economic value, will be funded. To value a mine project we need to consider future uncertainties. The approach commonly taken to value a project is to assume that if funded, the mine will be operated optimally throughout its life. Our final aim is not to provide an exact strategy, but to propose an optimization tool to improve decision-making in complex scenarios. Of all the variables involved, the typically large investments in infrastructure, as well as the uncertainty in commodity price, have the most significant impact on the mine value. We thus adopt a simplified model of the infrastructure and extraction optimization problem, subject to price uncertainty. / Common optimization methods are impractical for realistic size models. Our main contribution is the threshold optimization methodology based on measured valued differentiation (MVD) and SA. We also present another simulation-based method, the particles method [Dallagi 07], for comparison purposes. Both methods are well-adapted for high dimensional problems. We provide numerical results and discuss their characteristics and applicability.

Dans cette thèse j'étudie la géométrie locale des structuresfinslériennes et sous-finslériennes associées à la norme infinien dimension 2 et 3 : géodésiques généralisées courtes, lieu de coupure, lieu conjugué généralisé, lieu de "saut", petites sphères.Pour définir une telle structure au voisinage d'un point $p$ de $mathbb{R}^n$, on se donne une famille de champs de vecteurs $(F_1,dots,F_k)$ et on considère la norme définie sur la distribution$Delta=mbox{vect}{F_1,dots,F_k}$ par $|G|=inf{max{|u_i|} ; | ; G=sum_i u_i F_i} $.En dimension 2, pour $k=2$, si $F_1$ et $F_2$ ne sont pas proportionnels en $p$ alors on obtient une structure finslérienne. Sinon, alors la structure est sous-finslérienne sur une distribution de rang non constant. Nous décrivons les objets géométriques décrits plus haut pour l'ensembles des couples génériques $(F_1,F_2)$.En dimension 3, nous avons étudié la géométrie locale pour les distributions de contact. / In this thesis I study the local geometry of Finslerian and sub-Finslerian structures associated to the maximum norm in dimension 2 and 3 : short generalized geodesics, cut locus, generalized conjugate locus, switching locus, small spheres.To define such a structure in the neighborhood of a point $p$ of $mathbb{R}^n$, we fix a familly of vector fields $(F_1,dots,F_k)$ and consider the norm defined on the distribution $Delta=mbox{vect}{F_1,dots,F_k}$ by $|G|=inf{max{|u_i|} ; | ; G=sum_i u_i F_i} $.In dimension 2, for $k=2$, if $F_1$ and $F_2$ are not proportionnal at $p$ then we obtain a Finslerian structure. If not, the structure is sub-Finslerian on a distribution with non constant rank. We describe the geometric objects for the set of all generic couples $(F_1,F_2)$.In dimension 3, we studied the local geometry for contact distributions.

abstract: Infectious diseases are a leading cause of death worldwide. With the development of drugs, vaccines and antibiotics, it was believed that for the first time in human history diseases would no longer be a major cause of mortality. Newly emerging diseases, re-emerging diseases and the emergence of microorganisms resistant to existing treatment have forced us to re-evaluate our optimistic perspective. In this study, a simple mathematical framework for super-infection is considered in order to explore the transmission dynamics of drug-resistance. Through its theoretical analysis, we identify the conditions necessary for the coexistence between sensitive strains and drug-resistant strains. Farther, in order to investigate the effectiveness of control measures, the model is extended so as to include vaccination and treatment. The impact that these preventive and control measures may have on its disease dynamics is evaluated. Theoretical results being confirmed via numerical simulations. Our theoretical results on two-strain drug-resistance models are applied in the context of Malaria, antimalarial drugs, and the administration of a possible partially effective vaccine. The objective is to develop a monitoring epidemiological framework that help evaluate the impact of antimalarial drugs and partially-effective vaccine in reducing the disease burden at the population level. Optimal control theory is applied in the context of this framework in order to assess the impact of time dependent cost-effective treatment efforts. It is shown that cost-effective combinations of treatment efforts depend on the population size, cost of implementing treatment controls, and the parameters of the model. We use these results to identify optimal control strategies for several scenarios. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2011

In the history of cancer treatment, the immunotherapy is considered to be the most promising treatment approach. The idea behind this breakthrough is to stimulate the patient’s own immune system to recognize the cancer cells and destroy them. In this therapy, the antibodies are known to be powerful medications to activate the immune system in different ways. They circulate throughout the body until they discover a substance that body recognize as alien i.e. antigen and bind to them. Similarly, cancer cells often have molecules on their surface known as tumor-associated antigens. The researchers can design many clones of the antibody that only target a certain antigen type such as one found on tumors or cancer cells. Then, these are used as an effective drug for treating cancer. Thus, the antigen specific antibody interactions play a vital role in cancer immunotherapy. In this study, we propose a dynamic model to represent the population of antigens and antibodies in cancer patients; in particular we focus on the antigen-specific-antibody interactions to elicit an immune response that leads to the death of cancer cells. We formulate a terminal control problem where the schedule and doses of these antibodies are considered as control variables. The objective functional has been formulated as a measure of antigen population at the end of the treatment period. Pontryagin minimum principle (PMP) has been used to obtain the optimal control policies. For illustration, a series of numerical results is presented showing the effectiveness of immune therapy for cancer treatment corresponding to the different scenarios, choices of parameters and treatment periods. The results indicate that the control doses are followed by the emergence of antigen population. This approach would be potentially applicable to determine and prescribe the optimal doses and schedules for cancer patients.

>Magister Scientiae - MSc / Tuberculosis (TB) is currently one of the major public health challenges in South Africa, and in many countries. Mycobacterium tuberculosis is among the leading causes of morbidity and mortality. It is known that tuberculosis is a curable infectious disease. In the case of incomplete treatment, however, the remains of Mycobacterium tuberculosis in the human system often results in the bacterium developing resistance to antibiotics. This leads to relapse and treatment against the resistant bacterium is extremely expensive and difficult. The aim of this work is to present and analyse mathematical models of the population dynamics of tuberculosis for the purpose of studying the effects of efficient treatment versus incomplete treatment. We analyse the spread, asymptotic behavior and possible eradication of the disease, versus persistence of tuberculosis. In particular, we consider inflow of infectives into the population, and we study the effects of screening. A sub-model will be studied to analyse the transmission dynamics of TB in an isolated population. The full model will take care of the inflow of susceptibles as well as inflow of TB infectives into the population. This dissertation enriches the existing literature with contributions in the form of optimal control and stochastic perturbation. We also show how stochastic perturbation can improve the stability of an equilibrium point. Our methods include Lyapunov functions, optimal control and stochastic differential equations. In the stability analysis of the DFE we show how backward bifurcation appears. Various phenomena are illustrated by way of simulations.

abstract: The 2009-10 influenza and the 2014-15 Ebola pandemics brought once again urgency to an old question: What are the limits on prediction and what can be proposed that is useful in the face of an epidemic outbreak? This thesis looks first at the impact that limited access to vaccine stockpiles may have on a single influenza outbreak. The purpose is to highlight the challenges faced by populations embedded in inadequate health systems and to identify and assess ways of ameliorating the impact of resource limitations on public health policy. Age-specific per capita constraint rates play an important role on the dynamics of communicable diseases and, influenza is, of course, no exception. Yet the challenges associated with estimating age-specific contact rates have not been decisively met. And so, this thesis attempts to connect contact theory with age-specific contact data in the context of influenza outbreaks in practical ways. In mathematical epidemiology, proportionate mixing is used as the preferred theoretical mixing structure and so, the frame of discussion of this dissertation follows this specific theoretical framework. The questions that drive this dissertation, in the context of influenza dynamics, proportionate mixing, and control, are: I. What is the role of age-aggregation on the dynamics of a single outbreak? Or simply speaking, does the number and length of the age-classes used to model a population make a significant difference on quantitative predictions? II. What would the age-specific optimal influenza vaccination policies be? Or, what are the age-specific vaccination policies needed to control an outbreak in the presence of limited or unlimited vaccine stockpiles? Intertwined with the above questions are issues of resilience and uncertainty including, whether or not data collected on mixing (by social scientists) can be used effectively to address both questions in the context of influenza and proportionate mixing. The objective is to provide answers to these questions by assessing the role of aggregation (number and length of age classes) and model robustness (does the aggregation scheme selected makes a difference on influenza dynamics and control) via comparisons between purely data-driven model and proportionate mixing models. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2016

The control of complex robotic platforms is a challenging task, especially in designs with high levels of kinematic redundancy. Novel variable impedance actuators (VIAs) have recently demonstrated that, by allowing the ability to simultaneously modulate the output torque and impedance, one can achieve energetically more efficient and safer behaviour. However, this adds further levels of actuation redundancy, making planning and control of such systems even more complicated. VIAs are designed with the ability to mechanically modulate impedance during movement. Recent work from our group, employing the optimal control (OC) formulation to generate impedance policies, has shown the potential benefit of VIAs in tasks requiring energy storage, natural dynamic exploitation and robustness against perturbation. These approaches were, however, restricted to systems with smooth, continuous dynamics, performing tasks over a predefined time horizon. When considering tasks involving multiple phases of movement, including switching dynamics with discrete state transitions (resulting from interactions with the environment), traditional approaches such as independent phase optimisation would result in a potentially suboptimal behaviour. Our work addresses these issues by extending the OC formulation to a multiphase scenario and incorporating temporal optimisation capabilities (for robotic systems with VIAs). Given a predefined switching sequence, the developed methodology computes the optimal torque and impedance profile, alongside the optimal switching times and total movement duration. The resultant solution minimises the control effort by exploiting the actuation redundancy and modulating the natural dynamics of the system to match those of the desired movement. We use a monopod hopper and a brachiation system in numerical simulations and a hardware implementation of the latter to demonstrate the effectiveness and robustness of our approach on a variety of dynamic tasks. The performance of model-based control relies on the accuracy of the dynamics model. This can deteriorate significantly due to elements that cannot be fully captured by analytic dynamics functions and/or due to changes in the dynamics. To circumvent these issues, we improve the performance of the developed framework by incorporating an adaptive learning algorithm. This performs continuous data-driven adjustments to the dynamics model while re-planning optimal policies that reflect this adaptation. The results presented show that the augmented approach is able to handle a range of model discrepancies, in both simulation and hardware experiments using the developed robotic brachiation system.

Le sujet de cette thèse est l'étude de systèmes de production avec impatience. Ces systèmes sont modélisés comme des problèmes d'ordonnancement stochastiques avec des dates d'échéance. Dans la littérature, peu de résultats existent sur le contrôle optimal de ce genre de systèmes. C'est dans ce cadre que s'inscrit cette thèse. Nous considérons un système générique avec une machine, sur laquelle des tâches sont à exécuter. Les durées d'exécution, les dates d'échéance (ou durées d'impatience) et les dates de disponibilité des tâches sont des variables aléatoires. À chaque tâche est associé un poids et l'objectif est de minimiser l'espérance du nombre pondéré de tâches en retard. Dans notre étude, nous utilisons différentes modélisations, rendant compte des différentes contraintes régissant des systèmes réels. Notamment, nous faisons la différence entre l'impatience, le fait d'avoir attendu trop longtemps, et l'abandon, le fait de quitter le système suite à l'impatience. Dans la classe des politiques statiques, nous donnons des ordonnancements optimaux pour des problèmes avec impatience. Dans la classe des politiques dynamiques avec préemption, nous donnons de nouvelles conditions garantissant l'optimalité d'une politique stricte pour des problèmes avec abandon et nous proposons une heuristique plus efficace que celles que l'on trouve dans la littérature. Enfin, nous explorons des variantes et des extensions de ces problèmes, lorsque le système comporte plusieurs machines et lorsque la préemption n'est pas autorisée. / In this thesis, production systems facing abandonments are studied. These problems are modeled as stochastic scheduling problems with due dates. In the literature, few results exist concerning the optimal control of such systems. This thesis aims at providing optimal control policies for systems with impatience. We consider a generic system with a single machine, on which jobs have to be processed. Processing times, due dates (or patience time) and release dates are random variables. A weight is associated to each job and the objective is to minimize the expected weighted number of late jobs. In our study, we use different models, taking into account the specific features of real life problems. For example, we make a difference between impatience, when a customer has been waiting for too long, and abandonment, when a customer leaves the system after getting impatient. In the class of static list scheduling policies, we provide optimal schedules for problems with impatience. In the class of preemptive dynamic policies, we specify conditions under which a strict priority rule is optimal and we give a new heuristic, both extending previous results from the literature. We study variants and extensions of these problems, when several machines are available or when preemption is not authorized.

Made available in DSpace on 2014-10-09T12:50:30Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T13:58:49Z (GMT). No. of bitstreams: 1 00045.pdf: 1150397 bytes, checksum: fd8a86947b37fabf9aa8ff7b1e99d2a9 (MD5) / Dissertacao (Mestrado) / IEA/D / Escola Politecnica, Universidade de Sao Paulo - POLI/USP

L'objectif de cette thèse est de développer des méthodes d'optimisation et de contrôle optimal pour l'amélioration du rendement des cultures en utilisant le modèle de croissance de plantes GreenLab. Les méthodes proposées doivent se placer dans le contexte suivant: (1) la recherche d'amélioration du rendement se fait par des simulations basées sur le modèle structure-fonction GreenLab et (2) les méthodes utilisées sont des algorithmes d'optimisation heuristiques et des techniques de contrôle optimal. L'application de ces méthodes à plusieurs espèces de plantes, allant des plantes agronomiques aux arbres, et avec des différents objectifs a permis d'identifier certaines caractéristiques associées à des plantes ayant un bon rendement. En particulier, les résultats d'optimisation ont révélé la dynamique des relations source-puits au sein de la plante durant sa croissance. Ces résultats peuvent être considérés comme des références pour guider la sélection génétique pour l'amélioration variétale, et également pour l'amélioration des itinéraires culturaux. La perspective à long terme de cette thèse est l'intégration de ses résultats dans des outils d'aide à la décision pour l'agriculture. Pour atteindre les objectifs de cette thèse, nous avons analysé successivement les effets de facteurs endogènes et exogènes (environnementaux) sur la croissance de la plante et sur son rendement. Plus précisément, l'effet des facteurs endogènes a été étudié à conditions environnementales fixées, puis des méthodes de contrôle optimal ont été appliquées sur les variables environnementales, pour un génotype de plante fixé. En conséquence, les problèmes traités dans cette thèse relèvent à la fois de la théorie de l'optimisation et du contrôle optimal. Les principales contributions de cette thèse incluent les points suivants : Des problèmes d'optimisation simple objectif, d'optimisation multi-objectif et d'optimisation sous contrainte ont été formulés et résolus, dans le but de trouver les paramètres endogènes associés à la plante ayant le plus haut rendement, ce qui correspond à la définition d'un idéotype, pour une plante d'espèce donnée. Pour la plupart des problèmes d'optimisation présentés, la méthode la plus appropriée est celle d'un algorithme basé sur une population. Plusieurs algorithmes de ce type ont été comparés et celui ayant les meilleures performances est un algorithme heuristique nommé Particle Swarm Optimization (PSO). Du contrôle optimal a été appliqué pour définir la stratégie d'élagage optimale (application aux feuilles de thé). Come GreenLab peut être formulé comme un système dynamique discret et que la fonction objectif est analytique, une méthode basée sur le gradient, basée sur une approche variationnelle et sur la théorie de Lagrange, a été utilisée. La solution trouvée à été comparée à celle obtenue par la méthode PSO afin de valider cette dernière. Un modèle de dynamique de population d'insectes a été développé à des échelles spatiales et temporelles compatibles avec le modèle GreenLab, afin d'étudier l'interaction plante-insectes. Plus précisément, un écosystème tritrophique a été modélisé, incluant les interactions entre les plantes, les insectes ravageurs et des prédateurs des ravageurs, appelés insectes auxiliaires. un modèle des interaction. L'originalité de ce travail est la rétro-action entre la dynamique de population des insectes et la croissance de la plante, ainsi que la prise en compte de la répartition spatiale des insectes sur chacun des organes de la plantes. Une analyse de sensibilité basée sur la méthode de Morris a été appliquée pour identifier les paramètres les plus ou moins influents sur les sorties d'intérêt. Cela a permis de calculer des stratégies optimales pour l'application des techniques d'éradication des ravageurs. Les paramètres de GreenLab ont été estimés sur environ 400 jeux de données correspondant à 44 génotypes de tomates, à l'aide d'un algorithme des moindres carrés non linéaires généralisés. Considérant l'ensemble des valeurs estimées comme l'espace des paramètres possibles, nous avons calculé les valeurs optimisant le rendement en fruits des tomates. Nous avons analysé les corrélations entre les paramètres estimés ou optimaux et le rendement à l'aide de méthodes statistiques, ce qui a permis d'identifier les plus importants paramètres responsables des différences observées parmi les rendements des différentes plantes. Ainsi, à partir de ces résultat d'optimisation et d'analyse des corrélation, les différences phénotypiques entre différents génotypes ont pu être expliquées d'un point de vue physiologique. / The objective of the thesis is to improve plant yield through optimization and optimal control based on the GreenLab plant growth model. Therefore, the thesis proposed a methodology for investigation of plant yield improvement，whose characteristics are that (1) investigations are all based on the functional-structural plant growth model GreenLab and (2) heuristic optimization algorithm and optimal control techniques are applied to the plant growth model in order to improve plant yield. By applying optimization techniques on different species of plants (crops or trees) and for different kinds of optimization problems, common characteristics that a plant with high yield should possess were obtained. The optimal results in the thesis revealed the source-sink dynamics during the plant growth. The optimization results can be considered as references to guide breeding for ideotype and to improve cultivation modes. The optimization application of GreenLab could thus be possibly used to the agricultural decision support system.To achieve the aims of the thesis, the thesis investigated the effects of endogenous factors and exogenous environmental factors of plant growth on plant yield separately. First, given environmental conditions, the thesis investigated endogenous factors, and then the thesis did optimal control on exogenous environmental factors given plant genotype. Therefore, the problems investigated in the thesis consist of general optimization problems and optimal control problems.The main contributions of the thesis include following issues: According to the species of plants, single optimization problems, multi-objective optimization problems and optimization problems with constraints with respect to plant endogenous factors were formulated and investigated, in order to find the ideotype of plants with high plant yield. A population based algorithm is more suitable for the optimization problems in this thesis. Due to its better performance compared with other heuristic optimization algorithms, all optimization problems were solved by a population-based, heuristic optimization algorithm, namely Particle Swarm Optimization (PSO). Optimal control on the pruning strategy was formulated and investigated in the thesis. As GreenLab can be considered as discrete dynamic system and the objective function of the optimal control problem is analytical, the gradient based method, which is based on the variational approach and Lagrange theory, was used to solve the optimal control problem. Moreover, the optimal solutions were compared with the ones found by PSO, in order to validate the PSO method. The insect population dynamics was modeled mathematically, which was compatible with the plant model GreenLab in terms of spatial and temporal scales, to study the effect of biotic factors on plant growth. The interaction among plants, pests and auxiliaries was implemented, and the ecosystem model, which involves the three tri-trophic components, was thus developed in the thesis. The tri-trophic ecosystem model can simulate the insect population dynamics and the plant growth with consideration of the interaction of insects. Moreover, the tri-trophic ecosystem model considered the partition of individuals in the insect population among plant organs, which is not taken into account in the previous works. A global sensitivity analysis method Morris method was used to analyze the most important parameters and the least influential parameters to model outputs of interest. Through optimization on pest management techniques, the optimal strategies of the application of the pest management techniques were obtained. Estimation of GreenLab parameters with about 400 sets of observation data of 44 tomato genotypes was done in the thesis, by using a generalized non-linear least square algorithm. Taking the estimated parameter values as parameter space, the GreenLab model parameters were optimized, in order to maximize the fruit yield. Through the analysis of the correlation of estimated and optimal parameters with the fruit yield by statistical analysis methods, the most important parameters that result in the difference of fruit yield were found. According to the correlation and optimization results, the phenotypic differences among genotypes were explained from the physiological point of view.