Continuous and discrete-time Successive Backward Sweep (SBS) methods for solving nonlinear optimal control problems involving terminal and control constraints are proposed in this dissertation. They closely resemble the Neighboring Extremals and Differential Dynamic Programming algorithms, which are based on the successive solutions to a series of linear control problems with quadratic performance indices. The SBS methods are relatively insensitive to the initial guesses of the state and control histories, which are not required to satisfy the system dynamics. Hessian modifications are utilized, especially for non-convex problems, to avoid singularities during the backward integration of the gain equations. The SBS method requires the satisfaction of the Jacobi no-conjugate point condition and hence, produces optimal solutions. The standard implementation of the SBS method for continuous-time systems incurs terminal boundary condition errors due to an algorithmic singularity as well as numerical inaccuracies in the computation of the gain matrices. Alternatives for boundary error reduction are proposed, notably the aiming point and the switching between two forms of the sweep expansion formulae. Modification of the sweep formula expands the domain of convergence of the SBS method and allows for a rigorous testing for the existence of conjugate points. Numerical accuracy of the continuous-time formulation of the optimal control problem can be improved with the use of symplectic integrators, which generally are implicit schemes in time. A time-explicit group preserving method based on the Magnus series representation of the state transition is implemented in the SBS setting and is shown to outperform a non-symplectic integrator of the same order. Discrete-time formulations of the optimal control problem, directly accounting for a specific time-stepping method, lead to consistent systems of equations, whose solutions satisfy the boundary conditions of the discretized problem accurately. In this regard, the second-order, implicit mid-point averaging scheme, a symplectic integrator, is adapted for use with the SBS method. The performance of the mid-point averaging scheme is compared with other methods of equal and higher-order non-symplectic schemes to show its advantages. The SBS method is augmented with a homotopy- continuation procedure to isolate and regulate certain nonlinear effects for difficult problems, in order to extend its domain of convergence. The discrete-time SBS method is also extended to solve problems where the controls are approximated to be impulsive and to handle waypoint constraints as well. A variety of highly nonlinear optimal control problems involving orbit transfer, atmospheric reentry, and the restricted three-body problem are treated to demonstrate the performance of the methods developed in this dissertation.

An experimental research was performed to study the effects of surface roughness on the characteristics of separated and reattached turbulent flows in an open channel. A backward facing step was used to induce flow separation. The rough surfaces comprised wire mesh grit-80 and sand grains of average diameter 1.5 mm. In each experiment, the Reynolds number based on the step height and freestream velocity of approach flow was fixed at 3240 and the Reynolds number based on the approach flow depth and freestream velocity was kept constant at 25130. Particle image velocimetry (PIV) technique was used to measure the flow velocity. The results showed that roughness effects on the mean and turbulent quantities are evident only in the recovery region. Moreover, roughness effects on the flow dynamics are dependent on the specific roughness element.

The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic model for assessing the role of age-structure on the disease dynamics is designed. The model undergoes backward bifurcation, a dynamic phenomenon characterized by the co-existence of a stable disease-free and an endemic equilibrium of the model when the associated reproduction number is less than unity. It is shown that adding age-structure to the basic model for malaria transmission does not alter its essential qualitative dynamics. The study is extended to incorporate the use of anti-malaria drugs. Numerical simulations of the extended model suggest that for the case when treatment does not cause drug resistance (and the reproduction number of each of the two strains exceed unity), the model undergoes competitive exclusion. The impact of various effectiveness levels of the treatment strategy is assessed.

This thesis consists of five scientific papers dealing with equations related to the optimal switching problem, mainly backward stochastic differential equations and variational inequalities. Besides the scientific papers, the thesis contains an introduction to the optimal switching problem and a brief outline of possible topics for future research. Paper I concerns systems of variational inequalities with operators of Kolmogorov type. We prove a comparison principle for sub- and supersolutions and prove the existence of a solution as the limit of solutions to iteratively defined interconnected obstacle problems. Furthermore, we use regularity results for a related obstacle problem to prove Hölder continuity of this solution. Paper II deals with systems of variational inequalities in which the operator is of non-local type. By using a maximum principle adapted to this non-local setting we prove a comparison principle for sub- and supersolutions. Existence of a solution is proved using this comparison principle and Perron's method. In Paper III we study backward stochastic differential equations in which the solutions are reflected to stay inside a time-dependent domain. The driving process is of Wiener-Poisson type, allowing for jumps. By a penalization technique we prove existence of a solution when the bounding domain has convex and non-increasing time slices. Uniqueness is proved by an argument based on Ito's formula. Paper IV and Paper V concern optimal switching problems under incomplete information. In Paper IV, we construct an entirely simulation based numerical scheme to calculate the value function of such problems. We prove the convergence of this scheme when the underlying processes fit into the framework of Kalman-Bucy filtering. Paper V contains a deterministic approach to incomplete information optimal switching problems. We study a simplistic setting and show that the problem can be reduced to a full information optimal switching problem. Furthermore, we prove that the value of information is positive and that the value function under incomplete information converges to that under full information when the noise in the observation vanishes.

Semidefinite programming is a powerful modeling tool for a wide range of optimization and feasibility problems. Its prevalent use in practice relies on the fact that a (nearly) optimal solution of a semidefinite program can be obtained efficiently in both theory and practice, provided that the semidefinite program and its dual satisfy the Slater condition. This thesis focuses on the situation where the Slater condition (i.e., the existence of positive definite feasible solutions) does not hold for a given semidefinite program; the failure of the Slater condition often occurs in structured semidefinite programs derived from various applications. In this thesis, we study the use of the facial reduction technique, originally proposed as a theoretical procedure by Borwein and Wolkowicz, as a preprocessing technique for semidefinite programs. Facial reduction can be used either in an algorithmic or a theoretical sense, depending on whether the structure of the semidefinite program is known a priori. The main contribution of this thesis is threefold. First, we study the numerical issues in the implementation of the facial reduction as an algorithm on semidefinite programs, and argue that each step of the facial reduction algorithm is backward stable. Second, we illustrate the theoretical importance of the facial reduction procedure in the topic of sensitivity analysis for semidefinite programs. Finally, we illustrate the use of facial reduction technique on several classes of structured semidefinite programs, in particular the side chain positioning problem in protein folding.

An experimental research was performed to study the effects of surface roughness on the characteristics of separated and reattached turbulent flows in an open channel. A backward facing step was used to induce flow separation. The rough surfaces comprised wire mesh grit-80 and sand grains of average diameter 1.5 mm. In each experiment, the Reynolds number based on the step height and freestream velocity of approach flow was fixed at 3240 and the Reynolds number based on the approach flow depth and freestream velocity was kept constant at 25130. Particle image velocimetry (PIV) technique was used to measure the flow velocity. The results showed that roughness effects on the mean and turbulent quantities are evident only in the recovery region. Moreover, roughness effects on the flow dynamics are dependent on the specific roughness element.

The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic model for assessing the role of age-structure on the disease dynamics is designed. The model undergoes backward bifurcation, a dynamic phenomenon characterized by the co-existence of a stable disease-free and an endemic equilibrium of the model when the associated reproduction number is less than unity. It is shown that adding age-structure to the basic model for malaria transmission does not alter its essential qualitative dynamics. The study is extended to incorporate the use of anti-malaria drugs. Numerical simulations of the extended model suggest that for the case when treatment does not cause drug resistance (and the reproduction number of each of the two strains exceed unity), the model undergoes competitive exclusion. The impact of various effectiveness levels of the treatment strategy is assessed.

This thesis investigates the factors that influence the locational choices of foreign firms investing in China and the factors that influence the level of intermediate goods produced by domestic suppliers in China. It finds that some characteristics of the domestic economy are associated with both, and that foreign enterprises? activities are particularly important in determining the output of domestic suppliers. / PhD Doctorate

Thesis (Ph. D.)--State University of New York at Binghamton, Department of Psychology, 2008. / Includes bibliographical references.

Bien que les modèles hydrologiques soient aujourd'hui utilisés pour la prévision des crues, leur mise en place sur un territoire particulier peut encore poser question. Classiquement, elle nécessite des chronologies observées de pluies et de débits. Cependant, ces données peuvent s'avérer incertaines, insuffisantes voire indisponibles. Cela impact alors l'identification des modèles ainsi que leurs prévisions en les rendant elles-mêmes incertaines et potentiellement inopérantes.Dans cette thèse, nous avons cherché à revisiter ces problématiques liées à la mise en place des modèles, en proposant des approches innovantes étalonnant les modèles sans certaines données pouvant s'avérer trop incertaines ou indisponibles. Ces dernières, habituellement exigées pour l'identification des modèles, sont de plus estimées au cours de l'étalonnage. Deux méthodes applicables quelque-soit la nature du modèle considéré sont proposées. La première est une méthode d'inversion des pluies et des paramètres. Elle permet d'étalonner un modèle hydrologique à partir de chronologies observées de débits et de cumuls de pluies seulement. Cette méthode identifie de plus simultanément les chronologies des pluies au pas de temps horaire. Une application particulière de cette méthode à des modèles déjà établis généralise par ailleurs à des modèles non-analytiques l'approche d'inversion des pluies proposée par Kirchner (2009). La seconde est une méthode de calibration pluie-hauteur d'eau. À partir de chronologies observées de pluies et de hauteurs d'eau seulement, elle permet de mettre en place un modèle hydrologique et d'identifier une courbe de tarage. Outre les analyses originales pouvant être menées sur les chronologies de pluies ou les courbes de tarage estimées par ces méthodes, ces dernières étendent à des contextes plus restreints en données, la possibilité d'identification de modèles pluie-débits et offrent des applications en opérationnel.En parallèle du développement de ces méthodes, cette thèse nous a aussi amené à construire une représentation conceptuelle appelée espace des connaissances. Celle-ci permet d'appréhender de manière commune les questions nouvelles ayant été étudiées ainsi que des approches plus classiques comme la calibration ou la simulation. / Rainfall-runoff models are used for flood forecasting by warning authorities. However their implementation on a particular territory is still a challenge. Indeed, they generally need to be calibrated by using observed rainfall and discharge time series. These data may be subject to errors and uncertainties. They are not always available. Then, the model calibration relevancy may be affected and the forecasts may also be subject to significant uncertainties. This research would like to address such issues related to the rainfall-runoff models calibration, by proposing original methods which may set up a model by using less data than the ``classical'' calibration. The unused data might be either subject to uncertainties or not available. Moreover, these unused data may be estimated by the methods. Two model independant approach were suggested. Both are an heuristic inversion algorithm of rainfall-runoff models. The first method estimates simultaneously hourly rainfall time series and models parameters, by using only observed hourly discharge time series and total areal rainfall of flood events. A specific application of this method to set up models (with fixed parameters), generalises to models which are not invertible analyticaly, the ``hydrology backward'' approach proposed by Kirchner (2009).The second method estimates simultaneously models parameters and a rating-curve, by using only observed hourly rainfall and stage time series. Original analysis may be performed on the rainfall time series and the rating-curve estimated by the methods. Also, they extend the applicability rainfall-runoff models to hydrological context with restricted available data and offer promising operational applications. Yet, this research lead us to build a conceptual framework, denoted knowledge space. This framework unifies not only the original approaches which were proposed, but also some more ``classical'' approaches to hydrology as the calibration and the simulation.