Mathematical theory of homogenization of partial differential equations is relatively a new area of research (30-40 years or so) though the physical and engineering applications were well known. It has tremendous applications in various branches of engineering and science like : material science ,porous media, study of vibrations of thin structures, composite materials to name a few. There are at present various methods to study homogenization problems (basically asymptotic analysis) and there is a vast amount of literature in various directions. Homogenization arise in problems with oscillatory coefficients, domain with large number of perforations, domain with rough boundary and so on. The latter one has applications in fluid flow which is categorized as oscillating boundaries. In fact ,in this thesis, we consider domains with oscillating boundaries. We plan to study to homogenization of certain optimal control problems with oscillating boundaries. This thesis contains 6 chapters including an introductory Chapter 1 and future proposal Chapter 6. Our main contribution contained in chapters 2-5. The oscillatory domain under consideration is a 3-dimensional cuboid (for simplicity) with a large number of pillars of length O(1) attached on one side, but with a small cross sectional area of order ε2 .As ε0, this gives a geometrical domain with oscillating boundary. We also consider 2-dimensional oscillatory domain which is a cross section of the above 3-dimensional domain. In chapters 2 and 3, we consider the optimal control problem described by the Δ operator with two types of cost functionals, namely L2-cost functional and Dirichlet cost functional. We consider both distributed and boundary controls. The limit analysis was carried by considering the associated optimality system in which the adjoint states are introduced. But the main contribution in all the different cases(L2 and Dirichlet cost functionals, distributed and boundary controls) is the derivation of error estimates what is known as correctors in homogenization literature. Though there is a basic test function, one need to introduce different test functions to obtain correctors. Introducing correctors in homogenization is an important aspect of study which is indeed useful in the analysis, but important in numerical study as well. The setup is the same in Chapter 4 as well. But here we consider Stokes’ Problem and study asymptotic analysis as well as corrector results. We obtain corrector results for velocity and pressure terms and also for its adjoint velocity and adjoint pressure. In Chapter 5, we consider a time dependent Kirchhoff-Love equation with the same domain with oscillating boundaries with a distributed control. The state equation is a fourth order hyperbolic type equation with associated L2-cost functional. We do not have corrector results in this chapter, but the limit cost functional is different and new. In the earlier chapters the limit cost functional were of the same type.

This thesis studies the optimal control of vehicular systems, focusing on the solution of minimum-lap-time problems for a Formula 1 car. The basic optimal control theory is summarised as an infinite-dimensional extension of optimisation theory. The relevant numerical techniques for optimisation and integral approximation are compared in view of the application to vehicle systems. The classical brachistochrone problem is revisited from an optimal control perspective, with two vehicle-relevant generalisations. Closed-form solutions are derived for both the optimal trajectory and transit time. Problems involving a steerable disc rolling on the interior surface of a hemisphere are studied. For three-dimensional problems of this type, which involve rolling bodies and nonholonomic constraints, numerical solutions are used. The identification of 3D race track models from measured GPS data is treated as a problem in the differential geometry of curves and surfaces. Curvilinear coordinates are adopted to facilitate optimal control solutions. The track is specified in terms of three displacement-dependent curvatures and two edge variables. The differential model is smoothed using numerical optimal control techniques. The Barcelona track is considered as an illustrative example. The minimum-lap-time problem for a Formula 1 car on a flat track is solved using direct transcription. The driven line and multiple car setup parameters are optimised simultaneously. It is shown that significant lap-time reductions can be obtained from track-specific setup parameter optimisation. Reduced computing times are achieved using a combination of analytical derivatives, model non-dimensionalisation and problem scaling. The optimal control of the car on a 3D track is studied; the results are compared with flat-track solutions. Contemporary kinetic energy-recovery systems are studied and compared with future hybrid kinetic-thermal energy-recovery systems. It is demonstrated that these systems can produce contemporary lap time using approximately two-thirds of the fuel required by present-day vehicles.

Heavy duty powertrains are complex systems with components from various domains, different response times during transient operations and different efficient operating ranges. To ensure efficient transient operation of a powertrain, e.g. with low fuel consumption or short transient duration, it is important to come up with proper control strategies. In this dissertation, optimal control theory is used to calculate and analyze efficient heavy duty powertrain controls during transient operations in different applications. This is enabled by first developing control ready models, usable for multi-phase optimal control problem formulations, and then using numerical optimal control methods to calculate the optimal transients. Optimal control analysis of a wheel loader operating in a repetitive loading cycle is the first studied application. Increasing fuel efficiency or reducing the operation time in such repetitive loading cycles sums up to large savings over longer periods of time. Load lifting and vehicle traction consume almost all of the power produced by a diesel engine during wheel loader operation. Physical models are developed for these subsystems where the dynamics are described by differential equations. The model parameters are tuned and fuel consumption estimation is validated against measured values from real wheel loader operation. The sensitivity of wheel loader trajectory with respect to constrains such as the angle at which the wheel loader reaches the unloading position is also analyzed. A time and fuel optimal trajectory map is calculated for various unloading positions. Moreover, the importance of simultaneous optimization of wheel loader trajectory and the component transients is shown via a side to side comparison between measured fuel consumption and trajectories versus optimal control results. In another application, optimal control is used to calculate efficient gear shift controls for a heavy duty Automatic Transmission system. A modeling and optimal control framework is developed for a nine speed automatic transmission. Solving optimal control problems using the developed model, time and jerk efficient transient for simultaneous disengagement of off-going and engagement of in-coming shift actuators are obtained and the results are analyzed. Optimal controls of a diesel-electric powertrain during a gear shift in an Automated Manual Transmission system are calculated and analyzed in another application of optimal control. The powertrain model is extended by including driveline backlash angle as an extra state in the system. This is enabled by implementation of smoothing techniques in order to describe backlash dynamics as a single continuous function during all gear shift phases. Optimal controls are also calculated for a diesel-electric powertrain corresponding to a hybrid bus during a tip-in maneuver. It is shown that for optimal control analysis of complex powertrain systems, minimizing only one property such as time pushes the system transients into extreme operating conditions far from what is achievable in real applications. Multi-objective optimal control problem formulations are suggested in order to obtain a compromise between various objectives when analyzing such complex powertrain systems.

Self-assembly is a process of non-intrusive transformation of a system from a disordered to an ordered state. For engineering purposes, self-assembly of microscopic objects can benefit significantly from macroscopic guidance and control. This dissertation is concerned with controlling self-assembly in binary monolayers of electrically charged particles that follow basic laws of statistical mechanics. First, a simple macroscopic model is used to determine an optimal thermal control for self-assembly. The model assumes that a single rate-controlling mechanism is responsible for the formation of spatially ordered structures and that its rate follows an Arrhenius form. The model parameters are obtained using molecular dynamics simulations. The optimal control is derived in an analytical form using classical optimization methods. Two major lessons were learned from that work: (i) isothermal control was almost as effective as optimal time-dependent thermal control, and (ii) neither electrostatic interactions nor thermal control were particularly effective in eliminating voids formed during self-assembly. Accordingly, at the next stage, the focus is on temperature-pressure control under isothermal-isobaric conditions. In identifying optimal temperature and pressure conditions, several assumptions, that allow one to relate the optimal conditions to the phase diagram, are proposed. Instead of verifying the individual assumptions, the entire approach is verified using molecular dynamics simulations. It is estimated that under optimal isothermal-isobaric conditions the rate of self-assembly is about five time faster than that under optimal temperature control conditions. It is argued that the proposed approach of relating optimal conditions to the phase diagram is applicable to other systems. Further, the work reveals numerous and useful parallels between self-assembly and crystal physics, which are important to exploit for developing robust engineering self-assembly processes. / text

Localization and motion control (navigation) are two major tasks for a successful mobile robot navigation. The motion controller determines the appropriate action for the robot’s actuator based on its current state in an operating environment. A robot recognizes its environment through some sensors and executes physical actions through actuation mechanisms. However, sensory information is noisy and hence actions generated based on this information may be non-deterministic. Therefore, a mobile robot provides actions to its actuators with a certain degree of uncertainty. Moreover, when no prior knowledge of the environment is available, the problem becomes even more difficult, as the robot has to build a map of its surroundings as it moves to determine the position. Skilled navigation of a differential drive mobile robot (DDMR) requires solving these tasks in conjunction, since they are inter-dependent. Having resolved these tasks, mobile robots can be employed in many contexts in indoor and outdoor environments such as delivering payloads in a dynamic environment, building safety, security, building measurement, research, and driving on highways. This dissertation exploits the use of the emerging Radio Frequency IDentification (RFID) technology for the design and implementation of cost-effective and modular control techniques for navigating a mobile robot in an indoor environment. A successful realization of this process has been addressed with three separate navigation modules. The first module is devoted to the development of an indoor navigation system with a customized RFID reader. This navigation system is mainly pioneered by mounting a multiple antenna RFID reader on the robot and placing the RFID tags in three dimensional workspace, where the tags’ orthogonal position on the ground define the desired positions that the robot is supposed to reach. The robot generates control actions based on the information provided by the RFID reader for it to navigate those pre-defined points. On the contrary, the second and third navigation modules employ custom-made RFID tags (instead of the RFID reader) which are attached at different locations in the navigation environment (on the ceiling of an indoor office, or on posts, for instance). The robot’s controller generates appropriate control actions for it’s actuators based on the information provided by the RFID tags in order to reach target positions or to track pre-defined trajectory in the environment. All three navigation modules were shown to have the ability to guide a mobile robot in a highly reverberant environment with variant degrees of accuracy.

Many engineering problems are naturally posed as optimal control problems. It may involve moving between two points in the fastest possible way, or to put a satellite into orbit with minimum energy consumption. Many optimal control problems are too difficult to be solved analytically and therefore require the use of numerical methods. The numerical methods that are the most widespread are the so-called direct methods. However, there is one major drawback with these. If the problem is non-convex, the solution is not guaranteed globally optimal, that is, the absolute best, instead it is guaranteed locally optimal, that is the best in its vicinity. To compensate for this, the problem should be solved several times, under different conditions, in order to investigate whether the solution is a good candidate for the global optimum. CasADi is a software specifically designed for dynamic optimization. It has gained wide spread in recent years because it provides all the necessary building blocks for dynamic optimization. This has given individual engineers and scientists the ability to independently formulate and solve all sorts of optimal control problems. However, this requires good theoretical knowledge of the necessary numerical methods. The advantage of a toolbox, which solves general optimal control problems, is that the underlying numerical methods have been tested and shown to function on optimal control problems with known solutions. This means that the user does not need exhaustive knowledge of the numerical methods involved, but can focus on formulating and solving optimal control problems. The main contribution of this thesis is an optimal control toolbox for MATLAB based on CasADi. The toolbox does not require expert knowledge of the numerical methods, but provides an alternative lower level abstraction that allows for more complex problem formulations. The toolbox implements two direct methods, direct multiple shooting and direct collocation. This allows a problem formulation with many degrees of freedom. The most important property of the toolbox is that the discretization can be changed, without the problem formulation needing to be altered. This way the user can easily change the conditions for his/her problem. The thesis describes how the two implemented direct methods work, and the design choices made. It also describes what remains to test and evaluate, and the problems that have been used as a reference during the development process.

In this thesis, we present a novel approach to solving optimization problems that are defined on agent-based models (ABM). The approach utilizes concepts in genetic programming (GP) and is demonstrated here using an optimization problem on the Sugarscape ABM, a prototype ABM that includes spatial heterogeneity, accumulation of agent resources, and agents with different attributes. The optimization problem seeks a strategy for taxation of agent resources which maximizes total taxes collected while minimizing impact on the agents over a finite time. We demonstrate how our GP approach yields better taxation policies when compared to simple flat taxes and provide reasons why GP-generated taxes perform well. We also look at ways to improve the performance of the GP optimization method. / McAnulty College and Graduate School of Liberal Arts; / Computational Mathematics / MS; / Thesis;

The mean-field game theory is the study of strategic decision making in very large populations of weakly interacting individuals. Mean-field games have been an active area of research in the last decade due to its increased significance in many scientific fields. The foundations of mean-field theory go back to the theory of statistical and quantum physics. One may describe mean-field games as a type of stochastic differential game for which the interaction between the players is of mean-field type, i.e the players are coupled via their empirical measure. It was proposed by Larsy and Lions and independently by Huang, Malhame, and Caines. Since then, the mean-field games have become a rapidly growing area of research and has been studied by many researchers. However, most of these studies were dedicated to diffusion-type games. The main purpose of this thesis is to extend the theory of mean-field games to jump case in both discrete and continuous state space. Jump processes are a very important tool in many areas of applications. Specifically, when modeling abrupt events appearing in real life. For instance, financial modeling (option pricing and risk management), networks (electricity and Banks) and statistics (for modeling and analyzing spatial data). The thesis consists of two papers and one technical report which will be submitted soon: In the first publication, we study the mean-field game in a finite state space where the dynamics of the indistinguishable agents is governed by a controlled continuous time Markov chain. We have studied the control problem for a representative agent in the linear quadratic setting. A dynamic programming approach has been used to drive the Hamilton Jacobi Bellman equation, consequently, the optimal strategy has been achieved. The main result is to show that the individual optimal strategies for the mean-field game system represent 1/N-Nash equilibrium for the approximating system of N agents. As a second article, we generalize the previous results to agents driven by a non-linear pure jump Markov processes in Euclidean space. Mathematically, this means working with linear operators in Banach spaces adapted to the integro-differential operators of jump type and with non-linear partial differential equations instead of working with linear transformations in Euclidean spaces as in the first work. As a by-product, a generalization for the Koopman operator has been presented. In this setting, we studied the control problem in a more general sense, i.e. the cost function is not necessarily of linear quadratic form. We showed that the resulting unique optimal control is of Lipschitz type. Furthermore, a fixed point argument is presented in order to construct the approximate Nash Equilibrium. In addition, we show that the rate of convergence will be of special order as a result of utilizing a non-linear pure jump Markov process. In a third paper, we develop our approach to treat a more realistic case from a modelling perspective. In this step, we assume that all players are subject to an additional common noise of Brownian type. We especially study the well-posedness and the regularity for a jump version of the stochastic kinetic equation. Finally, we show that the solution of the master equation, which is a type of second order partial differential equation in the space of probability measures, provides an approximate Nash Equilibrium. This paper, unfortunately, has not been completely finished and it is still in preprint form. Hence, we have decided not to enclose it in the thesis. However, an outlook about the paper will be included.

Doctor of Philosophy / Department of Industrial & Manufacturing Systems Engineering / Chih-Hang Wu / The topic of infectious disease epidemics has recently attracted substantial attentions in research communities and it has been shown that the changes of human behaviors have significant impacts on the dynamics of disease transmission. However, the study and understanding of human reactions into spread of infectious disease are still in the very beginning phase and how human behaviors change during the spread of infectious disease has not been systematically investigated. Moreover, the study of human behaviors includes not only various enforced measures by public authorities such as school closure, quarantine, vaccination, etc, but also the spontaneous self-protective actions which are triggered by risk perception and fear of diseases. Hence, the goal of this research is to study the impacts of human behaviors to the epidemic from these two perspectives: spontaneous behavioral changes and public intervention strategies. For the sake of studying spontaneous changes of human behaviors, this research first time applied evolutionary spatial game into the study of human reactions to the spread of infectious disease. This method integrated contact structures and epidemics information into the individuals’ decision processes, by adding two different types of information into the payoff functions: the local information and global information. The new method would not only advance the field of game theory, but also the field of epidemiology. In addition, this method was also applied to a classic compartmental dynamic system which is a widely used model for studying the disease transmission. With extensive numerical studies, the results first proved the consistency of two models for the sake of validating the effectiveness of the spatial evolutionary game. Then the impacts of changes of human behaviors to the dynamics of disease transmission and how information impacts human behaviors were discussed temporally and spatially. In addition to the spontaneous behavioral changes, the corresponding intervention strategies by policy-makers played the key role in process of mitigating the spread of infectious disease. For the purpose of minimizing the total lost, including the social costs and number of infected individuals, the intervention strategies should be optimized. Sensitivity analysis, stability analysis, bifurcation analysis, and optimal control methods are possible tools to understand the effects of different combination of intervention strategies or even find an appropriate policy to mitigate the disease transmission. One zoonotic disease, named Zoonotic Visceral Leishmaniasis (ZVL), was studied by adopting different methods and assumptions. Particularly, a special case, backward bifurcation, was discussed for the transmission of ZVL. Last but not least, the methodology and modeling framework used in this dissertation can be expanded to other disease situations and intervention applications, and have a broad impact to the research area related to mathematical modeling, epidemiology, decision-making processes, and industrial engineering. The further studies can combine the changes of human behaviors and intervention strategies by policy-makers so as to seek an optimal information dissemination to minimize the social costs and the number of infected individuals. If successful, this research should aid policy-makers by improving communication between them and the public, by directing educational efforts, and by predicting public response to infectious diseases and new risk management strategies (regulations, vaccination, quarantine, etc.).

Master of Science / Department of Electrical and Computer Engineering / Balasubramaniam Natarajan / Noel Schulz / With increasing costs for fuel there is a growing interest in improving fuel efficiency and performance of military vehicles by employing (1) hybrid drive train architecture; (2) reliable vehicle power system structure, and (3) effective power management strategies of multiple power sources (engine, battery and ultracapacitor) and vehicle electrical loads. However, current ruled-based power management strategies that focus primarily on traction fail to meet the rapidly increasing requirements of military vehicles, including: (1) better fuel economy; (2) the ability to support pulsed power weapon loads; (3) maintaining battery SOC for power offloading applications, and (4) the ability to perform load scheduling of vehicle non-traction electrical loads to save energy. In this thesis, we propose an optimal control based algorithm in conjunction with a rule-based control strategy to optimally manage three power sources (engine, battery and pulsed power supply module) and an effective power management solution for vehicle non-traction electrical loads such that: (1) all traction, non-traction and pulsed power needs are met; (2) power drawn from the engine for specific mission is minimized; (3) a certain desired battery SOC is guaranteed for offloading power, and (4) the ability to perform load scheduling based on different mission requirements. The proposed approach is validated using simulation of a mission specific profile and is compared with two other popular control strategies. The improvements in power efficiency, desired SOC level and ability to perform optimal load scheduling are demonstrated.