In the past three decades, numerous methods have been proposed to transcribe optimal control problems (OCP) into nonlinear programming problems (NLP). In this dissertation work, a unifying weighted residual framework is developed under which most of the existing transcription methods can be derived by judiciously choosing test and trial functions. This greatly simplifies the derivation of optimality conditions and costate estimation results for direct transcription methods. Under the same framework, three new transcription methods are devised which are particularly suitable for implementation in an adaptive refinement setting. The method of Hilbert space projection, the least square method for optimal control and generalized moment method for optimal control are developed and their optimality conditions are derived. It is shown that under a set of equivalence conditions, costates can be estimated from the Lagrange multipliers of the associated NLP for all three methods. Numerical implementation of these methods is described using B-Splines and global interpolating polynomials as approximating functions. It is shown that the existing pseudospectral methods for optimal control can be formulated and analyzed under the proposed weighted residual framework. Performance of Legendre, Gauss and Radau pseudospectral methods is compared with the methods proposed in this research. Based on the variational analysis of first-order optimality conditions for the optimal control problem, an posteriori error estimation procedure is developed. Using these error estimates, an h-adaptive scheme is outlined for the implementation of least square method in an adaptive manner. A time-scaling technique is described to handle problems with discontinuous control or multiple phases. Several real-life examples were solved to show the efficacy of the h-adaptive and time-scaling algorithm.

The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition.

The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition.

The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition. / Science, Faculty of / Mathematics, Department of / Graduate

In this dissertation the solutions of the dynamics and real-time optimal control of magnetic attitude control and formation flying systems are presented. In magnetic attitude control, magnetic actuators for the time-optimal rest-to-rest maneuver with a pseudospectral algorithm are examined. The time-optimal magnetic control is bang-bang and the optimal slew time is about 232.7 seconds. The start time occurs when the maneuver is symmetric about the maximum field strength. For real-time computations, all the tested samples converge to optimal solutions or feasible solutions. We find the average computation time is about 0.45 seconds with the warm start and 19 seconds with the cold start, which is a great potential for real-time computations. Three-axis magnetic attitude stabilization is achieved by using a pseudospectral control law via the receding horizon control for satellites in eccentric low Earth orbits. The solutions from the pseudospectral control law are in excellent agreement with those obtained from the Riccati equation, but the computation speed improves by one order of magnitude. Numerical solutions show state responses quickly tend to the region where the attitude motion is in the steady state. Approximate models are often used for the study of relative motion of formation flying satellites. A modeling error index is introduced for evaluating and comparing the accuracy of various theories of the relative motion of satellites in order to determine the effect of modeling errors on the various theories. The numerical results show the sequence of the index from high to low should be Hill's equation, non- J2, small eccentricity, Gim-Alfriend state transition matrix index, with the unit sphere approach and the Yan-Alfriend nonlinear method having the lowest index and equivalent performance. A higher order state transition matrix is developed using unit sphere approach in the mean elements space. Based on the state transition matrix analytical control laws for formation flying maintenance and reconfiguration are proposed using low-thrust and impulsive scheme. The control laws are easily derived with high accuracy. Numerical solutions show the control law works well in real-time computations.

In this thesis, a general pseudospectral formulation for a class of Sturm-Liouville eigenvalue problems is consructed. It is shown that almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schr&ouml / dinger form may be transformed into a more tractable form. This tractable form will be called here a weighted equation of hypergeometric type with a perturbation (WEHTP) since the non-weighted and unperturbed part of it is known as the equation of hypergeometric type (EHT). It is well known that the EHT has polynomial solutions which form a basis for the Hilbert space of square integrable functions. Pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of WEHTP, and hence the energy eigenvalues of the Schr&ouml / dinger equation. Exemplary computations are performed to support the convergence numerically.