Dissertationen zum Thema „Deterministic optimal control“

Let but de cette thèse est de proposer différents modèles mathématiques de neurones pour l'Optogénétique et d'étudier leur contrôle optimal. Nous définissons d'abord une version contrôlée des modèles déterministes de dimension finie, dits à conductances. Nous étudions un problème de temps minimal pour un système affine mono-entrée dont nous étudions les singulières. Nous appliquons une méthode numérique directe pour observer les trajectoires et contrôles optimaux. Le contrôle optogénétique apparaît comme une nouvelle façon de juger de la capacité des modèles à conductances de reproduire les caractéristiques de la dynamique du potentiel de membrane, observées expérimentalement. Nous définissons ensuite un modèle stochastique en dimension infinie pour prendre en compte le caractère aléatoire des mécanismes des canaux ioniques et la propagation des potentiels d'action. Il s'agit d'un processus de Markov déterministe par morceaux (PDMP) contrôlé, à valeurs dans un espace de Hilbert. Nous définissons une large classe de PDMPs contrôlés en dimension infinie et prouvons le caractère fortement Markovien de ces processus. Nous traitons un problème de contrôle optimal à horizon de temps fini. Nous étudions le processus de décision Markovien (MDP) inclus dans le PDMP et montrons l'équivalence des deux problèmes. Nous donnons des conditions suffisantes pour l'existence de contrôles optimaux pour le MDP, et donc le PDMP. Nous discutons des variantes pour le modèle d'Optogénétique stochastique en dimension infinie. Enfin, nous étudions l'extension du modèle à un espace de Banach réflexif, puis, dans un cas particulier, à un espace de Banach non réflexif
The aim of this thesis is to propose different mathematical neuron models that take into account Optogenetics, and study their optimal control. We first define a controlled version of finite-dimensional, deterministic, conductance based neuron models. We study a minimal time problem for a single-input affine control system and we study its singular extremals. We implement a direct method to observe the optimal trajectories and controls. The optogenetic control appears as a new way to assess the capability of conductance-based models to reproduce the characteristics of the membrane potential dynamics experimentally observed. We then define an infinite-dimensional stochastic model to take into account the stochastic nature of the ion channel mechanisms and the action potential propagation along the axon. It is a controlled piecewise deterministic Markov process (PDMP), taking values in an Hilbert space. We define a large class of infinite-dimensional controlled PDMPs and we prove that these processes are strongly Markovian. We address a finite time optimal control problem. We study the Markov decision process (MDP) embedded in the PDMP. We show the equivalence of the two control problems. We give sufficient conditions for the existence of an optimal control for the MDP, and thus, for the initial PDMP as well. The theoretical framework is large enough to consider several modifications of the infinite-dimensional stochastic optogenetic model. Finally, we study the extension of the model to a reflexive Banach space, and then, on a particular case, to a nonreflexive Banach space

Dans le présent document on aborde trois divers thèmes liés au contrôle et au calcul stochastiques, qui s'appuient sur la notion d'équation différentielle stochastique rétrograde (EDSR) dirigée par une mesure aléatoire. Les trois premiers chapitres de la thèse traitent des problèmes de contrôle optimal pour différentes catégories de processus markoviens non-diffusifs, à horizon fini ou infini. Dans chaque cas, la fonction valeur, qui est l'unique solution d'une équation intégro-différentielle de Hamilton-Jacobi-Bellman (HJB), est représentée comme l'unique solution d'une EDSR appropriée. Dans le premier chapitre, nous contrôlons une classe de processus semi-markoviens à horizon fini; le deuxième chapitre est consacré au contrôle optimal de processus markoviens de saut pur, tandis qu'au troisième chapitre, nous examinons le cas de processus markoviens déterministes par morceaux (PDMPs) à horizon infini. Dans les deuxième et troisième chapitres les équations d'HJB associées au contrôle optimal sont complètement non-linéaires. Cette situation survient lorsque les lois des processus contrôlés ne sont pas absolument continues par rapport à la loi d'un processus donné. Etant donné ce caractère complètement non-linéaire, ces équations ne peuvent pas être représentées par des EDSRs classiques. Dans ce cadre, nous avons obtenu des formules de Feynman-Kac non-linéaires en généralisant la méthode de la randomisation du contrôle introduite par Kharroubi et Pham (2015) pour les diffusions. Ces techniques nous permettent de relier la fonction valeur du problème de contrôle à une EDSR dirigée par une mesure aléatoire, dont une composante de la solution subit une contrainte de signe. En plus, on démontre que la fonction valeur du problème de contrôle originel non dominé coïncide avec la fonction valeur d'un problème de contrôle dominé auxiliaire, exprimé en termes de changements de mesures équivalentes de probabilité. Dans le quatrième chapitre, nous étudions une équation différentielle stochastique rétrograde à horizon fini, dirigée par une mesure aléatoire à valeurs entières sur $R_+ times E$, o`u $E$ est un espace lusinien, avec compensateur de la forme $nu(dt, dx) = dA_t phi_t(dx)$. Le générateur de cette équation satisfait une condition de Lipschitz uniforme par rapport aux inconnues. Dans la littérature, l'existence et unicité pour des EDSRs dans ce cadre ont été établies seulement lorsque $A$ est continu ou déterministe. Nous fournissons un théorème d'existence et d'unicité même lorsque $A$ est un processus prévisible, non décroissant, continu à droite. Ce résultat s’applique par exemple, au cas du contrôle lié aux PDMPs. En effet, quand $mu$ est la mesure de saut d'un PDMP sur un domaine borné, $A$ est prévisible et discontinu. Enfin, dans les deux derniers chapitres de la thèse nous traitons le calcul stochastique pour des processus discontinus généraux. Dans le cinquième chapitre, nous développons le calcul stochastique via régularisations des processus à sauts qui ne sont pas nécessairement des semimartingales. En particulier nous poursuivons l'étude des processus dénommés de Dirichlet faibles, dans le cadre discontinu. Un tel processus $X$ est la somme d'une martingale locale et d'un processus adapté $A$ tel que $[N, A] = 0$, pour toute martingale locale continue $N$. Pour une fonction $u: [0, T] times R rightarrow R$ de classe $C^{0,1}$ (ou parfois moins), on exprime un développement de $u(t, X_t)$, dans l'esprit d'une généralisation du lemme d'Itô, lequel vaut lorsque $u$ est de classe $C^{1,2}$. Le calcul est appliqué dans le sixième chapitre à la théorie des EDSRs dirigées par des mesures aléatoires. Dans de nombreuses situations, lorsque le processus sous-jacent $X$ est une semimartingale spéciale, ou plus généralement, un processus de Dirichlet spécial faible, nous identifions les solutions des EDSRs considérées via le processus $X$ et la solution $u$ d’une EDP intégro-différentielle associée
In the present document we treat three different topics related to stochastic optimal control and stochastic calculus, pivoting on thenotion of backward stochastic differential equation (BSDE) driven by a random measure.After a general introduction, the three first chapters of the thesis deal with optimal control for different classes of non-diffusiveMarkov processes, in finite or infinite horizon. In each case, the value function, which is the unique solution to anintegro-differential Hamilton-Jacobi-Bellman (HJB) equation, is probabilistically represented as the unique solution of asuitable BSDE. In the first chapter we control a class of semi-Markov processes on finite horizon; the second chapter isdevoted to the optimal control of pure jump Markov processes, while in the third chapter we consider the case of controlled piecewisedeterministic Markov processes (PDMPs) on infinite horizon. In the second and third chapters the HJB equations associatedto the optimal control problems are fully nonlinear. Those situations arise when the laws of the controlled processes arenot absolutely continuous with respect to the law of a given, uncontrolled, process. Since the corresponding HJB equationsare fully nonlinear, they cannot be represented by classical BSDEs. In these cases we have obtained nonlinear Feynman-Kacrepresentation formulae by generalizing the control randomization method introduced in Kharroubi and Pham (2015)for classical diffusions. This approach allows us to relate the value function with a BSDE driven by a random measure,whose solution hasa sign constraint on one of its components.Moreover, the value function of the original non-dominated control problem turns out to coincide withthe value function of an auxiliary dominated control problem, expressed in terms of equivalent changes of probability measures.In the fourth chapter we study a backward stochastic differential equation on finite horizon driven by an integer-valued randommeasure $mu$ on $R_+times E$, where $E$ is a Lusin space, with compensator $nu(dt,dx)=dA_t,phi_t(dx)$. The generator of thisequation satisfies a uniform Lipschitz condition with respect to the unknown processes.In the literature, well-posedness results for BSDEs in this general setting have only been established when$A$ is continuous or deterministic. We provide an existence and uniqueness theorem for the general case, i.e.when $A$ is a right-continuous nondecreasing predictable process. Those results are relevant, for example,in the frameworkof control problems related to PDMPs. Indeed, when $mu$ is the jump measure of a PDMP on a bounded domain, then $A$ is predictable and discontinuous.Finally, in the two last chapters of the thesis we deal with stochastic calculus for general discontinuous processes.In the fifth chapter we systematically develop stochastic calculus via regularization in the case of jump processes,and we carry on the investigations of the so-called weak Dirichlet processes in the discontinuous case.Such a process $X$ is the sum of a local martingale and an adapted process $A$ such that $[N,A] = 0$, for any continuouslocal martingale $N$.Given a function $u:[0,T] times R rightarrow R$, which is of class $C^{0,1}$ (or sometimes less), we provide a chain rule typeexpansion for $u(t,X_t)$, which constitutes a generalization of It^o's lemma being valid when $u$ is of class $C^{1,2}$.This calculus is applied in the sixth chapter to the theory of BSDEs driven by random measures.In several situations, when the underlying forward process $X$ is a special semimartingale, or, even more generally,a special weak Dirichlet process,we identify the solutions $(Y,Z,U)$ of the considered BSDEs via the process $X$ and the solution $u$ to an associatedintegro PDE

Les vaccins ont été une grande réussite en matière de santé publique au cours des dernières années. Cependant, le développement de vaccins efficaces contre les maladies infectieuses telles que le VIH ou le virus Ebola reste un défi majeur. Cela peut être attribué à notre manque de connaissances approfondies en immunologie et sur le mode d'action de la mémoire immunitaire. Les modèles mathématiques peuvent aider à comprendre les mécanismes de la réponse immunitaire, à quantifier les processus biologiques sous-jacents et à développer des vaccins fondés sur un rationnel scientifique. Nous présentons un modèle mécaniste de la dynamique de la réponse immunitaire humorale après injection d'un vaccin Ebola basé sur des équations différentielles ordinaires. Les paramètres du modèle sont estimés par maximum de vraisemblance dans une approche populationnelle qui permet de quantifier le processus de la réponse immunitaire et ses facteurs de variabilité. En particulier, le schéma vaccinal n'a d'impact que sur la réponse à court terme, alors que des différences significatives entre des sujets de différentes régions géographiques sont observées à plus long terme. Cela pourrait avoir des implications dans la conception des futurs essais cliniques. Ensuite, nous développons un outil numérique basé sur la programmation dynamique pour optimiser des schémas d'injections répétées. En particulier, nous nous intéressons à des patients infectés par le VIH sous traitement mais incapables de reconstruire leur système immunitaire. Des injections répétées d'un produit immunothérapeutique (IL-7) sont envisagées pour améliorer la santé de ces patients. Le processus est modélisé par un modèle de Markov déterministe par morceaux et des résultats récents de la théorie du contrôle impulsionnel permettent de résoudre le problème numériquement à l'aide d'une suite itérative. Nous montrons dans une preuve de concept que cette méthode peut être appliquée à un certain nombre de pseudo-patients. Dans l'ensemble, ces résultats s'intègrent dans un effort de développer des méthodes sophistiquées pour analyser les données d'essais cliniques afin de répondre à des questions cliniques concrètes
Vaccines have been one of the most successful developments in public health in the last years. However, a major challenge still resides in developing effective vaccines against infectious diseases such as HIV or Ebola virus. This can be attributed to our lack of deep knowledge in immunology and the mode of action of immune memory. Mathematical models can help understanding the mechanisms of the immune response, quantifying the underlying biological processes and eventually developing vaccines based on a solid rationale. First, we present a mechanistic model for the dynamics of the humoral immune response following Ebola vaccine immunizations based on ordinary differential equations. The parameters of the model are estimated by likelihood maximization in a population approach, which allows to quantify the process of the immune response and its factors of variability. In particular, the vaccine regimen is found to impact only the response on a short term, while significant differences between subjects of different geographic locations are found at a longer term. This could have implications in the design of future clinical trials. Then, we develop a numerical tool based on dynamic programming for optimizing schedule of repeated injections. In particular, we focus on HIV-infected patients under treatment but unable to recover their immune system. Repeated injections of an immunotherapeutic product (IL-7) are considered for improving the health of these patients. The process is first by a piecewise deterministic Markov model and recent results of the impulse control theory allow to solve the problem numerically with an iterative sequence. We show in a proof-of-concept that this method can be applied to a number of pseudo-patients. All together, these results are part of an effort to develop sophisticated methods for analyzing data from clinical trials to answer concrete clinical questions

Ce travail de recherche a comme objectif de développer les méthodes d’évaluation des opérations des retenues, nécessaires à la protection contre les crues du bassin versant Vu Gia Thu Bon. La stratégie de contrôle des crues est basée sur un modèle qui associe simulation-optimisation. La fonction objective consiste à minimiser les dégâts totaux d’inondation qui dépend des débits ou des hauteurs d’eau dans les secteurs aval.La méthode proposée comporte trois composants majeurs : (1) la simulation des débits et des niveaux d'eau réalisée par un modèle hydraulique 1D ; (2) la simulation des opérations pour la production hydroélectrique réalisée par un module d'opération de structure ; (3) un modèle d'optimisation (algorithme Shuffled Complex Evolution) destiné à obtenir les règles optimales d’opération pour les retenues.La méthode a été mise en œuvre avec succès pour le système multi-réservoirs dans le bassin versant du Vu Gia Thu Bon, Vietnam. Les performances du modèle d’opération et optimisation pour gestion des crues ont été évaluées la base des crues historiques de 2007, 2009 et 2017. Les résultats obtenus indiquent que les stratégies proposées par le modèle offrent de bien meilleures performances pour la réduction du débit de pointe et sur la diminution du niveau maximal de crue dans les secteurs aval. La méthodologie peut donc être transférée pour la gestion opérationnelle du bassin versant du Vu Gia Thu Bon
The main objective of the current research is to control flood flows and flood levels at various locations at the downstream of the Vu Gia Thu Bon catchment. Due to the characteristics of the system and the targeted optional objectives, a flood control operating strategy has been developed based on coupled simulation-optimization to reduce downstream flood damage of the multi-reservoir system by using spillway gates. The objective function is minimizing the total damages during the flood events that can be expressed as a function of water surface elevations at the inundation zones.The proposed method is based upon combining of three major components: (1) a hydraulic 1D model that allows simulating the flows in the river including the reservoir system, (2) an operation reservoir module adopted for simulation of the multi-reservoir considering physical constraints of the system as well as operation strategies, and (3) an optimization model applied to determine the best set of spillway gates levels, which specify the reservoir release.The method has been successfully implemented for the multi-reservoir system in the Vu Gia Thu Bon catchment. Three flood events in 2007, 2009 and 2017 were selected for demonstration. In order to assess performance of the approach and for comparison purpose, three developed scenarios that are representing operations the reservoir system in the historical, the current rules and the proposed model have been used. The results indicate that the proposed model provides much better performance for all scenarios in terms of reducing the peak flow as well as reducing the maximum water levels at downstream control points compared to the rest scenarios

Breast cancer (BC) has emerged as public health issue in India in last two decades sur-passing cervix cancer. Mortality from breast cancer is 30-40% higher than maternal causes. Breast cancer 5 year survival rates are abysmally low (near 50%). Aforemen-tioned facts singularly trace back to lack of awareness and clinical detection in advanced stage (around 75% cases in stage III). This thesis presents BC epidemiology modelling using two stage deterministic clonal expansion model with time varying parameters. The time varying parameters enable inferences about age specific aggressive cellular growth and mutation rate. However, the model studies only age as the risk factor while other factors stay confounded. We also demonstrate the utility of the parameters to-wards formulating optimal and feasible screening using Clinical Breast Examination (CBE) as the modality for improved mortality reduction for BC in India. The combined approach use carcinogenesis and disease progression models in conjuction. Using car-cinogenesis model, a deterministic clonal expansion model, we derive age specific growth rates for initiated cells, which subsequently progress to fully transformed malignant cells. Scaled up growth rates for initiated cells, when used as proxy to malignant cell growth, provide an estimate of age specific time scales for emergence of screen de-tectable tumors, which we incorporate to formulate adaptive screening policies. The adaptive screening policies are assessed for potential mortality reduction using Markov transition model, i.e. a disease progression approach. In later half of the twentieth century, carcinogenesis, modeled deterministically or stochastically as accumulation of sequential genetic mutations had been demonstrated as successful approach for the description of cancer epidemiology as well experimental data. Knudson’s two hit hypothesis and it’s application to retinoblastoma incidence in children turned out as landmark for validity of the multistage models. The most pop-ular of these models is the Two Stage Clonal Expansion (TSCE) model that posits, car-cinogenesis as a result of two successive hits or events. A first hit initiates the stem cells, transforming it into an intermediate cell. The step is called initiation. The intermediate cell can undergo clonal expansion, and would eventually with a second hit generate a malignant cell, called promotion. The malignant cell could grow up into a full blown tumor or even go extinct. The parameter estimates viz, net growth and mutation rates from these models provide insight into cancer incidence dynamics. We adapted deterministic clonal expansion model to describe breast cancer incidence from different countries using the data from International Agency for Research on Can-cer (IARC). The aim was twofold. To estimate time varying cellular kinetics that could provide qualitative insight on biological features of the underlying population. The non-linear optimization, we propose, seeks to minimize the sum of squared errors (SSRs) for the observed and predicted incidence of breast cancer in piecewise. The model results were tested for the following two cases. 1. Risk factor specific mutation rates. We used risk factor dataset for breast cancer to estimate the ratio of mutation rate with and without the given risk factor. We considered early menarche, parity/age at first birth and presence of first degree relative with breast cancer. 2. Estimation of age specific sojourn time to assess the effect of screening frequency for early detection of breast cancer. We infer an optimal window for annual screen-ing at 38-43 age group. A Markov transition model was with 6 different health states, such as detected and un-detected early and advanced stages, and death was formulated for assessment of adap-tive screening using CBE at population level. The cost effectiveness of the policies for improved mortatlity reduction was reported. The thesis is organized into 6 chapters. Chapter 1 presents the prevailing cancer burden, the cancer control measures, cancer registries, and incidence and mortality trends in India and worldwide using the Inter-national Agency for Research on Cancer publications. An assessment of early detection for breast cancer, survival rates, and suitable screening modality reveals urgent need for population level early detection programs. Chapter 2 presents the details of the data source used. Major was the IARC’s Can-cer in Five Continents (CI5) volumes I-X for three cities in India, Mumbai, Bangalore, Chennai and USA, Finland, and Shanghai. GLOBOCAN 2012 estimates are also dis-cussed and worldwide cancer incidence is compared to draw attention on differences in incidence trends. The Breast Cancer Surveillance Consortium (BCSC) datasets were presented with the risk factor features for BC cases recorded between 2000-2009. The formalism of multistage models, an essential part the of proposed model was presented in brief. Chapter 3 details the proposed Deterministic Clonal Expansion/Two Stage Clonal Ex-pansion model for breast cancer. The assumption and mathematical details are pre-sented. The IARC’s datasets were used for model validation. Since it was known before-hand that all the parameters of the model are not identifiable from the incidence data, we verified the time varying growth rates of stem and initiated cell were correlated. This was expected as these rates are expected to be under influence of growth hormones, es-pecially estrogen. The robustness of the parameter estimation is demonstrated across breast cancer incidence in different countries. Chapter 4 presents the adoption of the validated model for assessment of relative mu-tational rates among different risk factors. Chapter 5 describes a Markov model for natural history of breast cancer to demon-strate the optimal screening policies for the Indian cities. The policies are based on age specific sojourn time estimated from TSCE model, and evaluate the adaptive screening in younger age group (<50 years) in India. Chapter 6 summarizes the conclusion of the thesis work and scope of future work. xiii

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The management of wildlife species as pests involves making choices that determine how much pests control will cost, and what kind of benefits it will deliver. In order to make these choices defensible, the effect courses of action have on how the costs and benefits of pests control accrue should ideally be understood. This study proposes a novel approach to estimate the choice of a wildlife management of an ungulate species in a conservation site (Migliarino-San Rossore-Massaciuccoli Regional Park, Tuscany), combining biological and economical trends. In fact the management of wildlife resources provides contrasting benefits and costs, which ecological or economic approaches alone cannot analyze in their complexity and, at the same time, can only offer a limited insight. The main problem is that, both in protected areas than in country lands (where there are regulated hunting areas), some vertebrate species are considered as pests. In these cases pests are considered as species able to create different kinds of damage to the environment in which they live. The purpose of this work is to adopt an interdisciplinary integration of research expertise from natural sciences, economics and social sciences to manage a fallow deer population in an ex-hunting Estate in Italy, now part of a Regional Park. The aim of this work is to develop a model to achieve a balance constrained by biological and economical variables. Ecological-biological problems regarding environment and wildlife management are usually solved separately by economic tasks. Because bioeconomic control problems are still new objectives of the wildlife management in Italy, this research aims to give an overview of the classical bioeconomic models to introduce a new technique in decisions regarding wildlife species management and eventually harvesting control programs. Bioeconomic models are central to this approach as they combine biological data about population dynamics, sex and age class segregation, habitat use by the biological population, with economic data, deriving by costs for fences to reduce environmental damages and car accidents, costs for harvesting, revenues by venison and trophy, and etc. The primary objective of this work is to produce a bio-economic framework with sufficient structural complexity to analyze the management of this fallow deer population at our local level. This objective could be achieved developing a deterministic biological model that later would be implemented on a bioeconomic one. First, we develop a model in which wildlife managers in a Park seek to balance the revenues by the culling with the costs of the management, as the Italian law restrictions require. In a second step we will try to develop a simulating model imagining our Protected Area in Italy as a Sporting Estate in which the landowner desires to maximize his profit by venison and trophy value in an ecological equilibrium. Later we will use the simulating software Vensim to manipulate our fallow deer population in all the ways and conditions we will, combining the population growth, the size of the cull, and the desired profit. Finally, three different approaches to bioeconomic wildlife management plans are analized to show new possible horizons of the wildlife management activities in Italy: an hunter’s utility maximization problem; a wildlife management maximising the social welfare; a wildlife management maximising meat and trophy value in ecological equilibrium. This work provides techniques to people managing conservation and exploitation of environmental resources to realize the optimal balance between all the variables acting (ecological, economic, social,..).