Master of Arts / Department of Economics / Steven P. Cassou / This report shows the mathematics behind the solution to continuous time optimization problems. It shows how to specify the Hamiltonian function, how to use the Hamiltonian to obtain the optimal conditions for a typical economic optimal control problem and applies these techniques to several optimal control problems commonly encountered in macroeconomics. An appendix shows how to set up the optimal conditions for the case in which the state and co-state variables are both vectors. A second appendix shows how to approach the control situation for a system of optimal control problems where the co-state variable for the first sub-optimal control problem is the state variable for the second sub-optimal control problem.

Pure mathematics is often classified as continuous or discrete, that is into topology and combinatorics. Classical topology is the study of spaces in the small, modern topology or homology theory is the study of their large scale structure. The latter and its applications to General Systems Theory and implications on computer programming are the subject of our investigations. A general homology theory includes boundary and adjoint operators defined over a graded category. Singular homology theory describes the structure of high dimensional Simplicial complexes, and is the basis of Kron's tearing of electrical networks. De ~ham Cohomology Theory describes the structure of exterior differential forms used to ~nalyse distributed fields in high dimensional spaces. Likewise optimal control ~roblems can be described by abstract homology theories. Ideas from tensor theory are ~sed to identify the homological structure of Leontief's economic model as a real ~xample of an optimal control system. The common property of each of the above ~ystems is that of optimisation or equivalently the mapping of an error to zero. The ~~iterion may be a metric in space, or energy in an electrical or mechanical network ~~ system, or an abstract cost function in state space or money in an economic system ~~d is always the product of a covariant and a contravariant variable. ~e axiomatic nature of General Homology Theory depends on the definition of an ~~missable category, be it group, ring or module structure. Similarly real systems ~~e analysed in terms of mutually recursive algebras, vector, matrix or polynomial. ~~rther the group morphisms or mode operators are defined recursively. An orthogonal ~~mputer language, Algo182, is proposed which is capable of manipulating the objects ~~scribed by homological systems theory, thus alleviating the tedium and insecurity t~curred in iDtplementing computer programs to analyse engineering systems.

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.

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.

Recent developments in quantum information processing have presented new and interesting ways to perform advanced algorithms and improve signal to noise ratios. Examples of these include optimal control theory pulse generation algorithms and the usage of high Q-factor resonators. However, these developments are blocked by current spectrometer designs. This thesis details the design and testing of a new spectrometer with sufficient accuracy, bandwidth, and control to implement these advances. The proposed solution is to use an FPGA-based development board together with custom computer software. This gives access to high-speed analogue inputs and outputs, as well as digital output pins. The spectrometer is then used in two X-band electron spin resonance experiments, showing how the advantages of the system allow for superior results to that possible with the previous equipment. In addition, the setup is used in a Nitrogen Vacancy (NV) system where a rabi experiment is performed.

Falls from heights (FFH) are one of the leading causes of fatalities in skilled labor divisions such as construction, mining, agriculture/forestry, and manufacturing. Previous research has established that localized muscle fatigue (LMF) increases center of mass (COM)- and center of pressure (COP)-based measures of quiet stance. This is important because these increases have been linked to elevated risk of falls, and workers in the construction industry frequently engage in fatiguing activities while working at heights. In addition, the rate of fatality due to an occupational fall increases exponentially with age. Improved methods of fall prevention may be obtained through increased understanding of factors that have a deleterious effect on balance and postural control such as LMF and aging. An initial study was conducted to investigate the effects of LMF and aging on balance recovery from a postural perturbation without stepping. Sagittal plane postural perturbations were administered to young and older groups of participants before and after exercises to fatigue the lumbar extensors or ankle plantar flexors. Measures of balance recovery were based on the COM and COP trajectories and the maximum perturbation that could be withstood without stepping. Balance recovery measures were consistent with an LMF-induced decrement to recover from perturbations without stepping. Aging was also associated with an impaired ability to recover from the perturbations. The second study in the series investigated the effects of aging and LMF on the neural control of upright stance during small postural perturbations. Small magnitude postural perturbations were administered to young and older groups before and after fatiguing exercises. A single degree of freedom (DOF) model of the human body was developed that accurately simulated the experimentally collected kinematics during recovery from the perturbations. The model was controlled by invariant feedback gains that operated on the time-delayed kinematics. Feedback gains and time-delay were optimized for each participant, and a novel delay margin analysis was performed to assess system robustness toward instability. Results indicated that older individuals had a longer "effective" time-delay and exhibited greater reliance on afferent velocity information. No changes in feedback controller gains, time-delay, or delay margins were found with LMF in either age group. The final study investigated the use of a nonlinear controller to simulate responses to large magnitude postural perturbations. A three DOF model of the human body was developed and controlled with the state-dependent Riccati equation (SDRE). Parameters of the SDRE were optimized to fit the experimentally recorded kinematics. Unlike other forms of nonlinear control, the SDRE provides meaningful parameters for interpretation in the system identification. The SDRE approach was successful at stabilizing the dynamical system; however, accurate results were not obtained. Reasons for these errors are discussed, and an alternative formulation to the time-delayed optimal control problem using Roesser state space equations is presented. / Ph. D.

<div>Previous studies have shown that implementing trajectory optimization can reduce state estimations errors. These navigation based path planning problems are often diffcult to solve being computationally burdensome and exhibiting other numerical issues, so former studies have often used lower-delity methods or lacked explanatory power.</div><div><br></div><div><div>This work utilizes indirect optimization methods, particularly optimal control theory, to obtain high-quality solutions minimizing state estimation errors approximated by a continuous-time extended Kalman lter. Indirect methods are well-suited to this because necessary conditions of optimality are found prior to discretization and numerical computation. They are also highly parallelizable enabling application to increasingly larger problems.</div></div><div><br></div><div><div>A simple one dimensional problem shows some potential obstacles to solving problems of this type including regions of the trajectory where the control is unimportant. Indirect trajectory optimization is applied to a more complex scenario to minimize location estimation errors of a single cart traveling in a 2-D plane to a goal location and measuring range from a xed beacon. This resulted in a 96% reduction of the location error variance when compared to the minimum time solution. The single cart problem also highlights the importance of the matrix that encodes the linearization of the vehicle's measurement with respect to state. It is shown in this case that the vehicle roughly attempts to maximize the magnitude of its elements. Additionally, the cart problem further illustrates problematic regions of a design space where the objective is not signicantly affected by the trajectory.</div></div><div><br></div><div><div>An aircraft descent problem demonstrates the applicability of these methods to aerospace problems. In this case, estimation error variance is reduced 28.6% relative to the maximum terminal energy trajectory. Results are shown from two formulations of this problem, one with control constraints and one with control energy cost, to show the benets and disadvantages of the two methods. Furthermore, the ability to perform trade studies on vehicle and trajectory parameters is shown with this problem by solving for dierent terminal velocities and different initial locations.</div></div>

The scheduling and Control Group at the University of South Australia has been studying the optimal control of trains for many years, and has developed in-cab devices that help drivers stay on time and minimise energy use. In this thesis, we re-examine the optimal control theory for the train control problem. In particular, we study the optimal control around steep sections of track. To calculate an optimal driving strategy we need a realistic model of train performance. In particular, we need to know a coefficient of rolling resistance and a coefficient of aerodynamic drag. In practice, these coefficients are different for every train and difficult to predict. In the thesis, we study the use of mathematical filters to estimate model parameters from observations of actual train performance.