<p>In racing, it is useful to analyze vehicle performance and driving strategies to achieve the best result possible in competitions. This is often done by simulations and test driving.</p><p>In this thesis optimal control is used to examine how a racing car should be driven to minimize the lap time. This is achieved by calculating the optimal racing line at various tracks. The tracks can have arbitrary layout and consist of corners with non-constant radius. The road can have variable width. A four wheel vehicle model with lateral and longitudinal weight transfer is used.</p><p>To increase the performance of the optimization algorithm, a set of additional techniques are used. The most important one is to divide tracks into smaller overlapping segments and find the optimal line for each segment independently. This turned out to be useful when the track is long.</p><p>The optimal racing line is found for various tracks and cars. The solutions have several similarities to real driving techniques. The result is presented as driving instructions in Racer, a car simulator.</p>

Cette thèse, intitulée méthodes particulaires en commande optimale stochastique s'intéresse aux problèmes d'optimisation dans l'incertain et a leur résolution. Le terme particulaire renvoie au fait que nous considèrons des méthodes basées sur une approche de type Monte-Carlo, contrairement aux méthodes par programmation dynamiques stochastiques qui utilisent une discrétisation faite a priori.<br />La résolution des problèmes d'optimisation stochastique nécessite deux étapes : une étape d'approximation et une étape d'optimisation. Les deux premiers chapitres de ce manuscrit seront consacrées a la partie optimisation. Nous traiterons dans les chapitres qui suivront de l'approximation des problèmes d'optimisation dans l'incertain. Nous commencerons, dans ce manuscrit, (chapitre I) par présenter les problèmes qui seront abordés ; nous nous attarderons surtout sur la représentation de la structure d'information d'un probléme d'optimisation stochastique. Deux principales représentations se dégagent : une représentation algébrique et une représentation fonctionnelle. A partir de la nature de cette structure d'information, nous ferons la typologie des problémes d'optimisation stochastique : boucle ouverte, boucle fermée, information statique ou information dynamique. Le deuxième chapitre (chapitre II) traitera des conditions d'optimalité pour les problèmes de commande optimale stochastique : à partir des représentations algébriques ou fonctionnelles de l'information, nous présenterons des conditions d'optimalité du type Karush-Kuhn-Tucker. Les conditions présentées dans le chapitre II comportent presque invariablement des opérateurs d'espérance conditionnelle. La résolution de ces problèmes impose alors d'approximer ces opérateurs. Nous commencerons dans le chapitre III par motiver notre approche avant de passer à une revue de la littérature des problèmes d'estimation de densité, densité conditionnelle et espérance conditionnelle. Dans le chapitre IV, nous présentons la méthode des élements finis particulaires qui consiste en l'approximation de la structure d'information par une restriction du feedback à une classe donnée a priori de fonctions de base. Différents résultats de convergence et d'erreur asymptotique seront donné. L'avant dernier chapitre (chapitre V) présentera un algorithme chaotique de gradient pour la résolution de problémes d'optimisation stochastique en boucle fermée. Un résultat de convergence, de vitesse ainsi qu'une application numérique seront donnés. Nous nous intéresserons dans le dernier chapitre (chapitre VI) aux aspects numérique de la résolution des problèmes de commande optimale stochastique à partir des difféerentes méthodes présentes dans les chapitres précedents. Nous présenterons diffèrents algorithmes et heuristiques pour résoudre un problème de gestion de production d'un barrage hydro-électrique.

Thèse de doctorat : Mathématiques : Lyon, École normale supérieure (sciences) : 2005. / Bibliogr. p. 257-263.

We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton-Jacobi-Bellman equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. By analyzing the properties of the free boundary problem, we provide an explicit characterization of model parameters for which the value function is finite. Furthermore, we prove that the value function, as well as the slopes of the lines demarcating the no-trading region, can be expanded as a series of integer powers of [lambda superscript 1/3]. The coefficients of arbitrary order in this expansion can be computed. / text

A traffic network can be disrupted by work zones and incidents. Calculating diversion rate is a core issue for estimating demand changes, which is needed to select a suitable work zone configuration and work schedule. An urban network can provide multiple alternative routes, so traffic assignment is the best tool to analyze diversion rates on network level and the local level. Compared with the results from static traffic assignment, dynamic traffic assignment predicts a higher network diversion rate in the morning peak period and off-peak period, a lower local diversion rate in the morning peak period. Additionally, travelers may benefit from knowing real-time traffic condition to avoid the traffic incident areas. Deploying variable message signs (VMSs) is one possible solution. One key issue is optimizing locations of VMSs. A planning model is created to solve the problem. The objective is minimize total system travel time. The link transmission model is used to evaluate the performance of the network, and bounded rational behavior is used to represent drivers' response to VMSs. A self-adapting genetic algorithm (GA) is formulated to solve the problem. This model selects the best locations to provide VMSs, typically places are that allow travelers to switch to alternative routes. Results show that adding more VMSs beyond a certain threshold level does not further reduce travel time. / text

The object of this thesis is the study of some new financial models. The common feature is that they all involve optimal decisions. Some of the decisions take the form of a control and we enter the theory of stochastic optimal control and of Hamilton-Jacobi-Bellman (HJB) equations. Other decisions are "binary" and we deal with the theory of optimal stopping and free boundary problems. Throughout the thesis we will prefer a heuristic and intuitive approach to a too technical one which could hide the underlying ideas. In the first part we introduce the reader to option pricing, HJB equations and free boundary problems, and we review briefly the use of these mathematical tools in finance. The second part of the thesis deals with passport options. The pricing of these exotic options involves stochastic optimal control and free boundary problems. Finally, in the last part we study the end-of-the-year bonus for traders: how to optimally reward a trader?

Vehicle behaviour near the limit of adhesion is studied using linear optimal . control techniques and relatively simple vehicle models. Both time-invariant and time-varying approaches are used. Controllability is applied as a post-processing tool to analyse the resultant vehicle behaviour. First, a 4WS controller is developed using a linear time-invariant method, with a reference model control structure. Two handling objectives are defined, which are thought to provide predictable dynamics. Advantages of using a reference model control are clearly shown. With a developed control structure, it is shown that the prescribed target dynamics is achieved, provided tyre forces are available. It is also found that the controller is robust to small changes in the various vehicle parameter values. As a next step, time-varying modelling approach was used in order to better represent the vehicle operating conditions through the various dynamic range, including the limit of adhesion. An iterative vehicle path optimisation problem is formulated using a linear time-varying control approach. The validity of the optimisation method is studied against the steady-state simulation result at the limit of adhesion. It is shown that the method is capable of finding a trajectory in the vicinity of the friction limit, where the front tyres are used fully whilst retaining some margin at the rears. However, a couple of Issues are discovered. First, due to the quadratic nature of the road geometry cost function, the trajectory could get locked if the vehicle runs very close to the edge of the road. Hence, the . optimisation needs to be formulated such that the level of "optimality" on the trajectory remains consistent throughout the manoeuvre at each iteration. Secondly, it is found that inappropriate control demands are produced if the system matrix becomes poorly conditioned near the limit. This results in optimisation failure. In order to understand the mechanism of this failure, controllability of linear timevarying system was analysed and its properties were discussed in detail. First, the calculation methods of the controllability gramian matrix are investigated and some practical limitations are found. The gramian matrix is then used to define an open loop control sequence. It is found that the damping of the system has a significant influence on the control strategy. Subsequently, "the moving controllability window of a fixed time period" is found to provide the most relevant information of changing dynamics through the time. The study showed that the failure of the optimisation in the vicinity of the friction limit was indeed due to lack of controllability and the optimisation method itself was functioning correctly. The vehicle path optimisation problem is then extended to include longitudinal dynamics, enabling simulation of more general manoeuvres. The single corner simulation showed that the optimisation converges to an "out-in-out" path, with iterative solution improving continuously in a first order manner. Simulations with various controller settings showed that the strategy is reasonably robust provided that the changes in parameter settings are kept within a reasonable magnitude. It is also confirmed that the optimisation is able to drive a vehicle close to the limit under different types of operations required, i.e. braking, cornering and acceleration. The study was then performed with slightly more complex road geometry in order to investigate if the· optimisation is capable of prioritising certain· part of the manoeuvre in order to achieve better overall result. Unfortunately, this problem could not be solved successfully. The optimisation concentrated on the latter part of the manoeuvre as it had higher sensitivity to the final cost. This resulted in clearly sub-optimal overall performance. Finally, relatively simple study is conducted to investigate the correlation between various vehicle settings and optimisation results. Using the path optimisation problem formulation, iris found that the more oversteer vehicles are able to achieve better· result with more margin left in rear tyre force capacity. The handling objective functions used for the 4 WS controller is also calculated for the resultant trajectories. It is found that the neutral steer cost had a strong correlation, whereas the linearity cost showed no noticeable correlation. The controllability analysis was applied on the various vehicle settings using step steer simulation. It showed that more understeering vehicle retains higher controllability throughout the dynamics range. It is also found that higher inertia gives better controllability near the limit, however, it gives less controllability at more moderate operating conditions.

This report is mainly based on the paper "Optimal multiple stopping and valuation of swing options" by R. Carmona and N. Touzi (1). Here the authors model and solve optimal stopping problems with more than one exercise time. The existence of optimal stopping times is firstly proved and they then construct the value function of American put options with multiple exercises in the case of the Black-Scholes model, characterizing the exercise boundaries of the perpetual case. Finally, they extend the analysis to the swing contracts with infinitely many exercise rights. In this report, we concentrate on explaining their rigorous mathematical analysis in detail, especially for the valuation of the perpetual American put options with single exercise and two exercise rights, and the characteristics of the exercise boundaries of the multiple stopping case. These results are presented as theorems in Chapter 2 and Chapter 3. / text

Optimal large eddy simulation (OLES) is an approach to LES sub-grid modeling that requires multi-point correlation data as input. Until now, this has been obtained by analyzing DNS statistics. In the finite-volume OLES formulation studied here, under the assumption of small-scale homogeneity and isotropy, these correlations can be theoretically determined from Kolmogorov inertial-range theory, small-scale isotropy, along with the quasi-normal approximation. These models are expressed as generalized quadratic and linear finite volume operators that represent the convective momentum flux. These finite volume operators have been analyzed to determine their characteristics as numerical approximation operators and as models of small-scale effects. In addition, the dependence of the model operators on the anisotropy of the grid and on the size of the stencils is analyzed to develop idealized general operators that can be used on general grids. The finite volume turbulence operators developed here will be applicable in a wide range of LES problems. / text

Logistic mixed effects models are used to determine whether optimal codons associate with two specific properties of the expressed protein: solvent accessibility, aggregation propensity, or evolutionary conservation. Both random components and fixed structures in the models are decided by following certain selection procedures. More models are also developed by considering different factor combinations using the same selection procedure. The results show that evolutionary conservation is the most important factor for predicting for the optimal codon usage for most amino acids; aggregation propensity is also an important factor, and solvent accessibility is the least important factor for most amino acids.The results of this analysis are consistent with the previous literature, provide more straightforward way to study the research question and also more information for the insight relationships. / text