Quantum information is a growing field showing exciting possibilities for computational speed-up and communications. For the successful implementation of quantum computers, high-precision control is required to reach fault-tolerant thresholds. Control of quantum systems pertains to the manipulation of states and their evolution. In order to minimize the effects of the environment on the control operations of the qubits, control pulses should be made time-optimal. In addition, control pulses should be made robust to noise in the system, dispersion in energies and coupling elements, and uncertain parameters. In this thesis, we examine a robust time-optimal gradient ascent technique which is used to develop controls of the motional degrees of freedom for an ensemble of neutral atoms in a one-dimensional optical lattice in the high dispersion regime with shallow trapping potentials. As such, the system is analyzed in the delocalized basis. The system is treated as an ensemble of atoms with a range of possible quasimomenta across the first Brillouin zone. This gives the ensemble of Hamiltonians, indexed by the quasimomenta, a distinct spectra in their motional states and highly inhomogeneous control Hamiltonians. Thus, the optical lattice is seen as a model system for robust control. We find optimized control pulses designed using an ensemble modification of gradient-ascent pulse engineering robust to any range of quasimomentum. We show that it is possible to produce rotation controls with fidelities above 90\% for half of the first Brillouin zone with gate times in the order of several free oscillations. This is possible for a spectrum that shows upwards of 75\% dispersion in the energies of the band structure. We also show that NOT controls for qubit rotations on the entire Brillouin zone fidelities above 99\% were possible for 0.6\% dispersion in energies. The gate times were also in the order of several free oscillations. It is shown that these solutions are palindromic in time due to phase differences in some of the energy couplings when comparing one half of the Brillouin zone to another. We explore the limits of discretized sampling of a continuous ensemble for control.

<p>Methodologies for the analysis and computer simulations of active optimal vibration control of complex elastic structures are considered. The structures, generally represented by a large number of degrees of freedom (DOF), are to be controlled by a comparatively small number of actuators.</p><p>Various techniques presently available to solve the optimal control problems are briefly discussed. A Parametric optimization technique that is versatile enough to solve almost any type of optimization problems is found to give poor accuracy and is time consuming. More promising is the optimality equations approach, which is based on Pontryagins principle. Several new numerical procedures are developed using this approach. Most of the problems in this thesis are analysed in the modal space. Even complex structures can be approximated accurately in the modal space by using only few modes. Different techniques have been first applied to the cases where the number of modes to control was the same as the number of actuators (determined optimal control problems), then to cases in which the number of modes to control is larger than the number of actuators (overdetermined optimal control problems). </p><p>The determined optimal control problems can be solved by applying the Independent Modal Space Control (IMSC) approach. Such an approach is implemented in the Beam Analogy (BA) method that solves the problem numerically by applying the Finite Element Method (FEM). The BA, which uses the ANSYS program, is numerically very efficient. The effects of particular optimization parameters involved in BA are discussed in detail. Unsuccessful attempts have been made to modify this method in order to make it applicable for solving overdetermined or underactuated problems. </p><p>Instead, a new methodology is proposed that uses modified optimality equations. The modifications are due to the extra constraints present in the overdetermined problems. These constraints are handled by time dependent Lagrange multipliers. The modified optimality equations are solved by using symbolic differential operators. The corresponding procedure uses the MAPLE programming, which solves overdetermined problems effectively despite of the high order of differential equations involved.</p><p>The new methodology is also applied to the closed loop control problems, in which constant optimal gains are determined without using Riccatis equations.</p>

Supercavitating vehicles are characterized by substantially reduced hydrodynamic drag with respect to fully wetted underwater vehicles. Drag is localized at the nose of the vehicle, where a cavitator generates a cavity that completely envelops the body. This causes the center of pressure to be always ahead of the center of mass, thus violating a fundamental principle of hydrodynamic stability. This unique loading configuration, the complex and non-linear nature of the interaction forces between vehicle and cavity, and the unsteady behavior of the cavity itself make the control and maneuvering of supercavitating vehicles particularly challenging. This study represents an effort towards the evaluation of optimal trajectories for this class of underwater vehicles, which often need to operate in unsteady regimes and near the boundaries of the flight envelope. Flight trajectories and maneuvering strategies for supercavitating vehicles are here obtained through the solution of an optimal control problem. Given a cost function and general constraints and bounds on states and controls, the solution of the optimal control problem yields the control time histories that maneuver the vehicle according to a desired strategy, together with the associated flight path. The optimal control problem is solved using the direct transcription method, which does not require the derivation of the equations of optimal control and leads to the solution of a discrete parameter optimization problem. Examples of maneuvers and resulting trajectories are given to demonstrate the effectiveness of the proposed methodology and the generality of the formulation.

The major contributions of this thesis are categorized into three areas: (i) magnetic modeling, (ii) optimal performance assessment and (iii) multi-objective design methodology of the linear permanent-magnet (LPM) and linear variable reluctance (LVR) motors for manufacturing automation applications. The target application is to perform repetitive point-to-point positioning tasks on a continuous basis under temperature constraints. Through simplification, the constraint on temperature rise may be replaced by a constraint on average power dissipation, provided that the thermal resistance is constant and known. The basic framework of analysis is first introduced for a class of idealized linear synchronous (LS) motors, where magnetic saturation and spatial harmonics are neglected, to provide clarity and insight. The physics-based force models for the LPM and LVR motors, including spatial harmonics and magnetic saturation as appropriate, are then developed. Due to magnetic linearity, the force model of the LPM motor is derived from the analytical solution of the Poisson Equation. A nonlinear magnetic circuit analysis model is developed for the LVR motor that includes both spatial harmonics and magnetic saturation. The accuracy of both force models are verified by finite element analysis. Applying those force models, the optimal performance assessment of the LPM and LVR motors is explored using the mathematical framework discussed for the idealized LS motors. In particular, the relationship between travel time and travel distance is characterized in terms of average power dissipation. The performance assessment methodologies developed here may be applied to any motor technology used in manufacturing automation applications. The multi-objective design optimization problem is then defined and software for its solution is developed using Monte-Carlo synthesis, the performance assessment tools and dominance-based sorting. Design results for the LPM and LVR motors are then presented. Future research is discussed as the conclusion of the thesis.

The thesis studies determinants and adjustment paths of the people's weight from the view point of rational behavior.It followes the research approach of Becker and Murphy (1988), makes use of the utility function from Levy (2002), and corporates a budget constraint so as to establish an optimal control model for food consumption and weight, and to find out the relationship between them. Negative correlations are found between the steady-state weight and food prices, basal metabolic rate, and time discount rate.Positive correlations are found between the steady-state weight and income, marginal utility of food, and desirable weight. There is a tendancy to guide the actual steady-state weight to a much higher fluctuation margin than that of the desirable weight.In the dynamic analysis, it is also found that, regardless of an increase or decrease of the steady-state weight, both directions of adjustment show that the process of food consumption is always ``overshooting."In other words, when the steady-state weight becomes heavier (lighter), consumers first increase (decrease) their food consumption substantially. And, as the time goes by, there is a gradual decrease (increase) in food consumption owing to the fact that the food consumption is still higher (lower) than what is required for metabolism of the body that makes the weight getting to increase (decrease) till the new equilibrium is arrived.

Despite the immense progress made by molecular biology in cataloging andcharacterizing molecular elements of life and the success in genome sequencing, therehave not been comparable advances in the functional study of complex phenotypes.This is because isolated study of one molecule, or one gene, at a time is not enough byitself to characterize the complex interactions in organism and to explain the functionsthat arise out of these interactions. Mathematical modeling of biological systems isone way to meet the challenge.My research formulates the modeling of gene regulation as a control problem andapplies systems and control theory to the identification, analysis, and optimal controlof genetic regulatory networks. The major contribution of my work includes biologicallyconstrained estimation, dynamical analysis, and optimal control of genetic networks.In addition, parameter estimation of nonlinear models of biological networksis also studied, as a parameter estimation problem of a general nonlinear dynamicalsystem. Results demonstrate the superior predictive power of biologically constrainedstate-space models, and that genetic networks can have differential dynamic propertieswhen subjected to different environmental perturbations. Application of optimalcontrol demonstrates feasibility of regulating gene expression levels. In the difficultproblem of parameter estimation, generalized EM algorithm is deployed, and a set of explicit formula based on extended Kalman filter is derived. Application of themethod to synthetic and real world data shows promising results.

The solution of optimal control problems through the Hamilton-Jacobi-Bellman (HJB) equation offers guaranteed satisfaction of both the necessary and sufficient conditions for optimality. However, finding an exact solution to the HJB equation is a near impossible task for many optimal control problems. This thesis presents an approximation method for solving finite-horizon optimal control problems involving nonlinear dynamical systems. The method uses finite-order approximations of the partial derivatives of the cost-to-go function, and successive higher-order differentiations of the HJB equation. Natural byproducts of the proposed method provide sensitivities of the controls to changes in the initial states, which can be used to approximate the solution to neighboring optimal control problems. For highly nonlinear problems, the method is modified to calculate control sensitivities about a nominal trajectory. In this framework, the method is shown to provide accurate control sensitivities at much lower orders of approximation. Several numerical examples are presented to illustrate both applications of the approximation method.

This research uses the optimal control theory to construct two kinds of dynamic economic systems, which are an economic system without externalities and with externalities. Within each economic system both the centralized economy model and the decentralized economy model are included. The centralized economy (a social planner) model representes a kind of ideal economy, and the goal what the social planner pursues is that the resource allocation of the society satisfies the Pareto Efficiency criteria. On the other hand, the decentralized economy model (consists of a representative producer and a representative consumer) demonstrates the real economy, where economic agents persue their own best interests. While constructing the models, goods market equilibrium, labors market equilibrium, the dynamic accumulation process of capital, and the dynamic accumulation course of pollution are under consideration. Then, I apply optimal control method to get the first order conditions, and compare these f.o.c¡¦.s to verify whether they are unanimous. This paper proves that when externalities of pollution does not exist in the dynamic economic system, the decentralized economy model can achieve the Pareto Efficiency. On the contrary, when externalities of pollution emerges in the dynamic economic system, the decentralized economy model cannot reach Pareto Efficiency. If the externalities of pollution is internalized by the dynamic decentralized economic system economy, it will coincide with Pareto Efficiency. Besides, Pigouvian tax is still an effective policy instrument. Finally, I discuss all dynamic models in this paper to find out whether there exists a long-term and stable steady state. I find stable steady state, saddle-point equilibria, do exist under certain restrictions.

This paper uses the optimal control theory to construct dynamic economic model. The primary purpose of this paper is to discuss how different policies alter the choice problem of the firm and influence the allocation of funds to existing and new research and development activities. I analyze how the fixed-cost regulatory standard and the marginal-cost standard let firm consider externality, in order to protect the consumer of asymmetric information and avoid the problem of adverse selection. The firm maximizes expected profit. At the same time I want to know how the consumer acceptance, mark and audit affect the farmer to purchase the quantity of seed and the allocation of funds . We want to discuss how different price influence the option input path, the option quantity path and the option path . I discuss the different between ultimatum and static model. Finally, I discuss dynamic models in this paper to find out whether there exists a long-term and stable steady state. Saddle-point stability exists under certain restrictions.

<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>