Au cours de cette thèse, le problème général de la conception de méthodes couplées de contrôle et d'estimation pour des systèmes dynamiques non linéaires a été étudié. La cible principale était la navigation par corrélation de terrain (TAN en anglais), où le problème était de guider et d’estimer la position 3D d’un drone survolant une zone connue. Dans cette application, on suppose que les seules données disponibles sont la vitesse du système, une mesure de la différence entre l'altitude absolue du drone et l'altitude du sol survolé et une carte du sol. La TAN est un bon exemple d'application non linéaire dans laquelle le principe de séparation ne peut pas être appliqué. En réalité, la qualité des observations dépend du contrôle et plus précisément de la zone survolée par le drone. Par conséquent, il existe un besoin de méthodes couplées d'estimation et de contrôle. Il est à noter que le problème d'estimation créé par TAN est en soi difficile à analyser et à résoudre. En particulier, les sujets suivants ont été traités:• Conception d'observateur non linéaire et commande en retour de sortie pour la TAN avec des cartes au terrain analytiquesdans un cadre déterministe à temps continu.• La modélisation conjointe du filtrage optimal non linéaire et du contrôle optimal stochastique en temps discretavec des informations imparfaites.• la conception de schémas de contrôle prédictif stochastique duaux associés à un filtre particulaire et leur implémentation numérique pour la TAN. / During this PhD, the general problem of designing coupled control and estimation methods for nonlinear dynamical systems has been investigated. The main target application was terrain-aided navigation (TAN), where the problem is to guide and estimate the 3D position of a drone flying over a known area. In this application, it is assumed that the only available data are the speed of the system, a measurement of the difference between the absolute altitude of the drone and the altitude of the ground flied over and a map of the ground. TAN is a good example of a nonlinear application where the separation principle cannot be applied. Actually, the quality of the observations depends on the control and more precisely on the area that is flied over by the drone. Therefore, there is a need for coupled estimation and control methods. It is to be noted that the estimation problem created by TAN is in itself difficult to analyse and solve. In particular, the following topics have been treated:• Nonlinear observer design and outputfeedback control for TAN with analytical ground mapsin a deterministic continuous-time framework.• The joint modelling of nonlinear optimal filtering and discrete-time stochastic optimal controlwith imperfect information.• The design of output-feedback Explicit dual stochastic MPC schemes coupled with a particlefilter and their numerical implementation to TAN.

Multi-period portfolio optimization (MPO) has gained a lot of interest in modern portfolio theory due to its consideration for inter-temporal trading e effects, especially market impacts and transactions costs, and for its subtle reliability on return predictability. However, because of the heavy computational demand, portfolio policies based on this approach have been sparsely explored. In that regard, a tractable MPO framework proposed by N. Gârleanu & L. H. Pedersen has been investigated. Using the stochastic control framework, the authors provided a closed form expression of the optimal policy. Moreover, they used a specific, yet flexible return predictability model. Excess returns were expressed using a linear factor model, and the predicting factors were modeled as mean reverting processes. Finally, transactions costs and market impacts were incorporated in the problem formulation as a quadratic function. The elaborated methodology considered that the market returns dynamics are governed by fast and slow mean reverting factors, and that the market transactions costs are not necessarily quadratic. By controlling the exposure to the market returns predicting factors, the aim was to uncover the importance of the mean reversion speeds in the performance of the constructed trading strategies, under realistic market costs. Additionally, for the sake of comparison, trading strategies based on a single-period mean variance optimization were considered. The findings suggest an overall superiority in performance for the studied MPO approach even when the market costs are not quadratic. This was accompanied with evidence of better usability of the factors' mean reversion speed, especially fast reverting factors, and robustness in adapting to transactions costs. / Portföljoptimering över era perioder (MPO) har fått stort intresse inom modern portföljteori. Skälet till detta är att MPO tar hänsyn till inter-temporala handelseffekter, särskilt marknadseffekter och transaktionskostnader, plus dess tillförlitlighet på avkastningsförutsägbarhet. På grund av det stora beräkningsbehovet har dock portföljpolitiken baserad på denna metod inte undersökts mycket. I det avseendet, har en underskriven MPO ramverk som föreslagits av N.Gârleanu L. H. Pedersen undersökts. Med hjälp av stokastiska kontrollramen tillhandahöll författarna formuläret för sluten form av den optimala politiken. Dessutom använde de en specifik, men ändå flexibel returförutsägbarhetsmodell. Överskjutande avkastning uttrycktes med hjälp av en linjärfaktormodell och de förutsägande faktorerna modellerades som genomsnittligaåterföringsprocesser. Slutligen inkorporerades transaktionskostnader och marknadseffekter i problemformuleringen som en kvadratisk funktion. Den utarbetade metodiken ansåg att marknadens avkastningsdynamik styrs av snabba och långsammaåterhämtningsfaktorer, och att kostnaderna för marknadstransaktioner inte nödvändigtvis är kvadratiska. Genom att reglera exponeringen mot marknaden återspeglar förutsägande faktorer, var målet att avslöja vikten av de genomsnittliga omkastningshastigheterna i utförandet av de konstruerade handelsstrategierna, under realistiska marknadskostnader. Dessutom, för jämförelses skull, övervägdes handelsstrategier baserade på en enstaka genomsnittlig variansoptimering. Resultaten tyder på en överlägsen överlägsenhet i prestanda för det studerade MPO-tillvägagångssättet, även när marknadsutgifterna inte är kvadratiska. Detta åtföljdes av bevis för bättre användbarhet av faktorernas genomsnittliga återgångshastighet, särskilt snabba återställningsfaktorer och robusthet vid anpassning till transaktionskostnader

Markov jump linear systems find application in many areas including economics, fault-tolerant control, and networked control. Despite significant attention paid to Markov jump linear systems in the literature, few authors have investigated Markov jump linear systems with time-inhomogeneous Markov chains (Markov chains with time-varying transition probabilities), and even fewer authors have considered time-inhomogeneous Markov chains with a priori unknown transition probabilities. This dissertation provides a formal stability and disturbance attenuation analysis for a Markov jump linear system where the underlying Markov chain is characterized by an a priori unknown sequence of transition probability matrices that assumes one of finitely-many values at each time instant. Necessary and sufficient conditions for uniform stochastic stability and uniform stochastic disturbance attenuation are reported. In both cases, conditions are expressed as a set of finite-dimensional linear matrix inequalities (LMIs) that can be solved efficiently. These finite-dimensional LMI analysis results lead to nonconservative LMI formulations for optimal controller synthesis with respect to disturbance attenuation. As a special case, the analysis also applies to a Markov jump linear system with known transition probabilities that vary in a finite set. / Ph. D.

This thesis presents several problems based on papers written jointly by the author and Dr. Christian-Oliver Ewald. Firstly, the author extends the model presented by Fershtman and Nitzan (1991), which studies a deterministic differential public good game. Two types of volatility are considered. In the first case the volatility of the diffusion term is dependent on the current level of public good, while in the second case the volatility is dependent on the current rate of public good provision by the agents. The result in the latter case is qualitatively different from the first one. These results are discussed in detail, along with numerical examples. Secondly, two existing lines of research in game theoretic studies of fisheries are combined and extended. The first line of research is the inclusion of the aspect of predation and the consideration of multi-species fisheries within classical game theoretic fishery models. The second line of research includes continuous time and uncertainty. This thesis considers a two species fishery game and compares the results of this with several cases. Thirdly, a model of a fishery is developed in which the dynamic of the unharvested fish population is given by the stochastic logistic growth equation and it is assumed that the fishery harvests the fish population following a constant effort strategy. Explicit formulas for optimal fishing effort are derived in problems considered and the effects of uncertainty, risk aversion and mean reversion speed on fishing efforts are investigated. Fourthly, a Dixit and Pindyck type irreversible investment problem in continuous time is solved, using the assumption that the project value follows a Cox-Ingersoll- Ross process. This solution differs from the two classical cases of geometric Brownian motion and geometric mean reversion and these differences are examined. The aim is to find the optimal stopping time, which can be applied to the problem of extracting resources.

Le contexte de nos travaux de thèse est l'intégration de l'énergie éolienne sur les réseaux insulaires. Ces travaux sont soutenus par EDF SEI, l'opérateur électrique des îles françaises. Nous étudions un système éolien-stockage où un système de stockage d'énergie doit aider un producteur éolien à tenir, vis-à-vis du réseau, un engagement de production pris un jour à l'avance. Dans ce contexte, nous proposons une démarche pour l'optimisation du dimensionnement et du contrôle du système de stockage (gestion d'énergie). Comme les erreurs de prévision J+1 de production éolienne sont fortement incertaines, la gestion d'énergie du stockage est un problème d'optimisation stochastique (contrôle optimal stochastique). Pour le résoudre, nous étudions tout d'abord la modélisation des composants du système (modélisation énergétique du stockage par batterie Li-ion ou Sodium-Soufre) ainsi que des entrées (modélisation temporelle stochastique des entrées incertaines). Nous discutons également de la modélisation du vieillissement du stockage, sous une forme adaptée à l'optimisation de la gestion. Ces modèles nous permettent d'optimiser la gestion de l'énergie par la méthode de la programmation dynamique stochastique (SDP). Nous discutons à la fois de l'algorithme et de ses résultats, en particulier de l'effet de la forme des pénalisations sur la loi de gestion. Nous présentons également l'application de la SDP sur des problèmes complémentaires de gestion d'énergie (lissage de la production d'un houlogénérateur, limitation des rampes de production éolienne). Cette étude de l'optimisation de la gestion permet d'aborder l'optimisation du dimensionnement (choix de la capacité énergétique). Des simulations temporelles stochastiques mettent en évidence le fort impact de la structure temporelle (autocorrélation) des erreurs de prévision sur le besoin en capacité de stockage pour atteindre un niveau de performance donné. La prise en compte de paramètres de coût permet ensuite l'optimisation du dimensionnement d'un point de vue économique, en considérant les coûts de l'investissement, des pertes ainsi que du vieillissement. Nous étudions également le dimensionnement du stockage lorsque la pénalisation des écarts à l'engagement comporte un seuil de tolérance. Nous terminons ce manuscrit en abordant la question structurelle de l'interaction entre l'optimisation du dimensionnement et celle du contrôle d'un système de stockage, car ces deux problèmes d'optimisation sont couplés. / The context of this PhD thesis is the integration of wind power into the electricity grid of small islands. This work is supported by EDF SEI, the system operator for French islands. We study a wind-storage system where an energy storage is meant to help a wind farm operator fulfill a day-ahead production commitment to the grid. Within this context, we propose an approach for the optimization of the sizing and the control of the energy storage system (energy management). Because day-ahead wind power forecast errors are a major source of uncertainty, the energy management of the storage is a stochastic optimization problem (stochastic optimal control). To solve this problem, we first study the modeling of the components of the system. This include energy-based models of the storage system, with a focus on Lithium-ion and Sodium-Sulfur battery technologies. We then model the system inputs and in particular the stochastic time series like day-ahead forecast errors. We also discuss the modeling of storage aging, using a formulation which is adapted to the control optimization. Assembling all these models enables us to optimize the energy management of the storage system using the stochastic dynamic programming (SDP) method. We introduce the SDP algorithms and present our optimization results, with a special interest for the effect of the shape of the penalty function on the energy control law. We also present additional energy management applications with SDP (mitigation of wind power ramps and smoothing of ocean wave power). Having optimized the storage energy management, we address the optimization of the storage sizing (choice of the rated energy). Stochastic time series simulations show that the temporal structure (autocorrelation) of wind power forecast errors have a major impact on the need for storage capacity to reach a given performance level. Then we combine simulation results with cost parameters, including investment, losses and aging costs, to build a economic cost function for sizing. We also study storage sizing when the penalization of commitment deviations includes a tolerance threshold. We finish this manuscript with a structural study of the interaction between the optimizations of the sizing and the control of an energy storage system, because these two optimization problems are coupled.

We have examinined the problem of constructing efficient strategies for continuous-time dynamic asset allocation. In order to obtain efficient investment strategies; a stochastic optimal control approach was applied to find optimal transaction control. Two mathematical problems are formulized and studied: Model I; a dynamic programming approach that maximizes an isoelastic functional with respect to given underlying portfolio dynamics and Model II; a more sophisticated approach where a time-inconsistent state dependent mean-variance functional is considered. In contrast to the optimal controls for Model I, which are obtained by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation; the efficient strategies for Model II are constructed by attaining subgame perfect Nash equilibrium controls that satisfy the extended HJB equation, introduced by Björk et al. in [1]. Furthermore; comprehensive execution algorithms where designed with help from the generated results and several simulations are performed. The results reveal that optimality is obtained for Model I by holding a fix portfolio balance throughout the whole investment period and Model II suggests a continuous liquidation of the risky holdings as time evolves. A clear advantage of using Model II is concluded as it is far more efficient and actually takes time-inconsistency into consideration. / Vi har undersökt problemet som uppstår vid konstruktion av effektiva strategier för tidskontinuerlig dynamisk tillgångsallokering. Tillvägagångsättet för konstruktionen av strategierna har baserats på stokastisk optimal styrteori där optimal transaktionsstyrning beräknas. Två matematiska problem formulerades och betraktades: Modell I, en metod där dynamisk programmering används för att maximera en isoelastisk funktional med avseende på given underliggande portföljdynamik. Modell II, en mer sofistikerad metod som tar i beaktning en tidsinkonsistent och tillståndsberoende avvägning mellan förväntad avkastning och varians. Till skillnad från de optimala styrvariablerna för Modell I som satisfierar Hamilton-Jacobi-Bellmans (HJB) partiella differentialekvation, konstrueras de effektiva strategierna för Modell II genom att erhålla subgame perfekt Nashjämvikt. Dessa satisfierar den utökade HJB ekvationen som introduceras av Björk et al. i [1]. Vidare har övergripande exekveringsalgoritmer skapats med hjälp av resultaten och ett flertal simuleringar har producerats. Resultaten avslöjar att optimalitet för Modell I erhålls genom att hålla en fix portföljbalans mellan de riskfria och riskfyllda tillgångarna, genom hela investeringsperioden. Medan för Modell II föreslås en kontinuerlig likvidering av de riskfyllda tillgångarna i takt med, men inte proportionerligt mot, tidens gång. Slutsatsen är att det finns en tydlig fördel med användandet av Modell II eftersom att resultaten påvisar en påtagligt högre grad av effektivitet samt att modellen faktiskt tar hänsyn till tidsinkonsistens.