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We present a novel, high-order numerical method to solve viscous Burger's equation with smooth initial and boundary data. The proposed method combines Cole-Hopf transformation with well conditioned integral reformulations to reduce the problem into either a single easy-to-solve integral equation with no constraints, or an integral equation provided by a single integral boundary condition. Fully exponential convergence rates are established in both spatial and temporal directions by embracing a full Gegenbauer collocation scheme based on Gegenbauer-Gauss (GG) mesh grids using apt Gegenbauer parameter values and the latest technology of barycentric Gegenbauer differentiation and integration matrices. The global collocation matrices of the reduced algebraic linear systems were derived allowing for direct linear system solvers to be used. Rigorous error and convergence analyses are presented in addition to two easy-to-implement pseudocodes of the proposed computational algorithms. We further show three numerical tests to support the theoretical investigations and demonstrate the superior accuracy of the method even when the viscosity paramter $\nu \to 0$, in the absence of any adaptive strategies typically required for adaptive refinements.

In this dissertation, I have developed some computational methods aimed at solving nonlinear optimal control problems. The bulk of the effort has involved the development of some efficient numerical
methods for solving boundary value problems, integral equations, and integro-differential equations. Chapters 1 and 2 provide an introduction to the necessary tools drawn from the calculus of variations, special function theory and optimization. In Chapters 3 and 5, Gegenbauer-based quadrature rules and boundary value problem solvers are
developed and shown to be highly effective (corresponding to two papers published in J. Comp. Appl. Math.) An important contribution is Chapter 4, which contains a negative result – namely that adding the
Gegenbauer parameter (alpha) as an additional degree of freedom to be optimized in the “outer loop” can lead to incorrect results (which are difficult to detect). Chapter 6 returns to the issue of optimal control and reformulates the differential constraint problem in integral form to avoid the ill-conditioning associated with spectral differentiation
matrices. The theory is nicely put together and the numerical evidence supports the notion that this approach is superior to classical methods in many regards. Chapter 7 considers a similar class of problems but with higher order differential constraints. These are treated in a very stable manner by setting the unknown to be the highest order derivative of the state vector, resulting in a well conditioned integral reformulation of the problem. A discussion of future work is included in Chapter 8.

The analytic solutions of simple optimal control problems may be found using the classical tools such as the calculus of variations, dynamic programming, or the minimum principle. However, in practice, a closed form expression of the optimal control is difficult or even impossible to determine for general nonlinear optimal control problems. Therefore such intricate optimal control problems must be solved numerically. The numerical solution of optimal control problems has been the subject of a significant amount of study since the last century; yet determining the optimal control within high precision remains very challenging in many optimal control applications. The classes of direct orthogonal collocation methods and direct pseudospectral methods are some of the most elegant numerical methods for solving nonlinear optimal control problems nowadays. These methods offer several advantages over many other popular discretization methods in the literature. The key idea of these methods is to transcribe the infinite-dimensional continuous-time optimal control problem into a finite-dimensional nonlinear programming problem. These methods are based on spectral collocation methods, which have been extensively applied and actively used in many areas. Many polynomials approximations and various discretization points have been introduced and studied in the literature for the solution of optimal control problems using control and/or state parameterizations. The commonly used basis polynomials in direct orthogonal collocation methods and direct pseudospectral methods are the Chebyshev and Legendre polynomials, and the collocation points are typically chosen to be of the Gauss or Gauss-Lobatto type of points. The integral operation in the cost functional of an optimal control problem is usually approximated by the well-known Gauss quadrature rules. The differentiation operations are frequently calculated by multiplying a constant differentiation matrix known as the spectral differentiation matrix by the matrix of the function values at a certain discretization/collocation nodes. Thus, the cost functional, the dynamics, and the constraints of the optimal control problem are approximated by a set of algebraic equations. Unfortunately, there are two salient limitations associated with the applications of typical direct orthogonal collocation methods and direct pseudospectral methods: (i) The spectral differentiation matrix, especially those of higher-orders, are widely known to be ill-conditioned; therefore, the numerical computations may be very sensitive to round-off errors. In fact, for a higher-order spectral differentiation matrix, the ill-conditioning becomes very extreme to the extent that the development of efficient preconditioners is a necessity. (ii) The popular spectral differentiation matrix employed frequently in the literature of direct orthogonal collocation methods and direct pseudospectral methods is a square and dense matrix. Therefore, to determine approximations of higher-orders, one usually has to increase the number of collocation points in a direct pseudospectral method, which in turn increases the number of constraints and the dimensionality of the resulting nonlinear programming problem. Also increasing the number of collocation points in a direct orthogonal collocation method increases the number of constraints of the reduced nonlinear programming problem. Eventually, the increase in the size of the spectral differentiation matrix leads to larger nonlinear programming problems, which may be computationally expensive to solve and time-consuming. The research goals of this dissertation are to furnish an efficient, accurate, rapid and robust optimal control solver, and to produce a significantly small-scale nonlinear programming problem using considerably few collocation points. To this end, we introduce a direct optimization method based on a novel Gegenbauer collocation integration scheme which draws upon the power of the well-developed nonlinear programming techniques and computer codes, and the well-conditioning of the numerical integration operators. This modern technique adopts two principle elements to achieve the research goals: (i) The discretization of the optimal control problem is carried out within the framework of a complete integration environment to take full advantage of the well-conditioned numerical integral operators. (ii) The integral operations included in the components of the optimal control problem are approximated through a novel optimal numerical quadrature in a certain optimality measure. The introduced numerical quadrature outperforms classical spectral quadratures in accuracy, and can be established efficiently through the Hadamard multiplication of a constant rectangular spectral integration matrix by the vector of the integrand function values at some optimal Gegenbauer-Gauss interpolation nodes, which usually differ from the employed integration/collocation nodes. The work presented in this dissertation shows clearly that the rectangular form of the developed numerical integration matrix is substantial for the achievement of very precise solutions without affecting the size of the reduced nonlinear programming problem. Chapter 1 is an introductory chapter highlighting the strengths and the weaknesses of various solution methods for optimal control problems, and provides the motivation for the present work. The chapter concludes with a general framework for using Gegenbauer expansions to solve optimal control problems and an overview for the remainder of the dissertation. Chapter 2 presents some preliminary mathematical background and basic concepts relevant to the solution of optimal control problems. In particular, the chapter introduces some key concepts of the calculus of variations, optimal control theory, direct optimization methods, Gegenbauer polynomials, Gegenbauer collocation, in addition to some other essential topics. Chapter 3 presents a published article in Journal of Computational and Applied Mathematics titled “Optimal Gegenbauer quadrature over arbitrary integration nodes.” In this chapter, we introduce a novel optimal Gegenbauer quadrature to efficiently approximate definite integrations numerically. The novel numerical scheme introduces the idea of exploiting the strengths of the Chebyshev, Legendre, and Gegenbauer polynomials through a unified approach, and using a unique numerical quadrature. In particular, the numerical scheme developed employs the Gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms. For a large-scale number of expansion terms, the numerical quadrature has the advantage of converging to the optimal Chebyshev and Legendre quadratures in the $L^{infty}$-norm and $L^2$-norm, respectively. The developed Gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes. Moreover, exact integrations are obtained for polynomials of any arbitrary degree $n$ if the number of columns in the developed Gegenbauer integration matrix is greater than or equal to $n$. The error formula for the Gegenbauer quadrature is derived. Moreover, a study on the error bounds and the convergence rate shows that the optimal Gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of Gegenbauer expansion terms. Two efficient computational algorithms are presented for optimally constructing the Gegenbauer quadrature, and to ideally maintain the robustness and the rapid convergence of the discrete approximations. We illustrate the high-order approximations of the optimal Gegenbauer quadrature through extensive numerical experiments including comparisons with conventional Chebyshev, Legendre, and Gegenbauer polynomial expansion methods. The present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods. Chapter 4 presents a published article in Advances in Computational Mathematics titled “On the optimization of Gegenbauer operational matrix of integration.” The chapter is focused on the intriguing question of “which value of the Gegenbauer parameter $alpha$ is optimal for a Gegenbauer integration matrix to best approximate the solution of various dynamical systems and optimal control problems?” The chapter highlights those methods presented in the literature which recast the aforementioned problems into unconstrained/constrained optimization problems, and then add the Gegenbauer parameter $alpha$ associated with the Gegenbauer polynomials as an extra unknown variable to be optimized. The theoretical arguments presented in this chapter prove that this naive policy is invalid since it violates the discrete Gegenbauer orthonormality relation, and may in turn produce false optimization problems analogs to the original problems with poor solution approximations. Chapter 5 presents a published article in Journal of Computational and Applied Mathematics titled “Solving boundary value problems, integral, and integro-differential equations using Gegenbauer integration matrices.” The chapter resolves the issues raised in the previous chapter through the introduction of a hybrid Gegenbauer collocation integration method for solving various dynamical systems such as boundary value problems, integral and integro-differential equations. The proposed method recasts the original problems into their integral formulations, which are then discretized into linear systems of algebraic equations using a hybridization of the Gegenbauer integration matrices developed in Chapter 3. The resulting linear systems are generally well-conditioned and can be easily solved using standard linear system solvers. A study on the error bounds of the proposed method is presented, and the spectral convergence is proven for two-point boundary-value problems. Comparisons with other competitive methods in the recent literature are included. The proposed method results in an efficient algorithm, and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems. The developed numerical scheme provides a viable alternative to other solution methods when high-order approximations are required using only a relatively small number of solution nodes. Chapter 6 presents a published article in The Proceedings of 2012 Australian Control Conference, AUCC 2012, titled “Solving optimal control problems using a Gegenbauer transcription method.” The chapter presents a novel direct orthogonal collocation method using Gegenbauer-Gauss collocation for solving continuous-time optimal control problems with nonlinear dynamics, state and control constraints, where the admissible controls are continuous functions. The framework of the novel method involves the mapping of the time domain onto the interval $[0, 1]$, and transforming the dynamical system given as a system of ordinary differential equations into its integral formulation through direct integration. In this manner, the proposed Gegenbauer transcription method unifies the process of the discretization of the dynamics and the integral cost function. The state and the control variables are then fully parameterized using Gegenbauer expansion series with some unknown Gegenbauer spectral coefficients. The proposed Gegenbauer transcription method recasts the performance index, the reduced dynamical system, and the constraints into systems of algebraic equations using the optimal Gegenbauer quadrature introduced in Chapter 3. Finally, the Gegenbauer transcription method transcribes the infinite-dimensional optimal control problem into a finite-dimensional nonlinear programming problem, which can be solved in the spectral space; thus approximating the state and the control variables along the entire time horizon. The high precision and the spectral convergence of the discrete solutions are verified through two optimal control test problems with nonlinear dynamics and some inequality constraints. In particular, we investigate the application of the proposed method for finding the best path in 2D of an unmanned aerial vehicle moving in a stationary risk environment. Moreover, we compare the performance of the proposed Gegenbauer transcription method with another classical variational technique to demonstrate the efficiency and the accuracy of the proposed method. Chapter 7 presents a published article in Journal of Computational and Applied Mathematics titled “Fast, accurate, and small-scale direct trajectory optimization using a Gegenbauer transcription method.” This chapter extends the Gegenbauer transcription method introduced in the preceding chapter to deal further with continuous-time optimal control problems including different orders time derivatives of the states by solving the continuous-time optimal control problem directly for the control $u(t)$ and the highest-order time derivative $x^{(N)}(t), N in mathbb{Z}^+$. The state vector and its derivatives up to the $(N-1)$th-order derivative can then be stably recovered by successive integration. Moreover, we present our solution method for solving linear quadratic regulator problems as we aim to cover a wider collection of continuous-time optimal control problems with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature. The advantages of the proposed direct Gegenbauer transcription method over other traditional discretization methods are shown through four well-studied optimal control test examples. The present work is a major breakthrough in the area of computational optimal control theory as it delivers significantly accurate solutions using considerably small numbers of collocation points, states and controls expansions terms. Moreover, the Gegenbauer transcription method produces very small-scale nonlinear programming problems, which can be solved very quickly using modern nonlinear programming software. The Gegenbauer collocation integration scheme adopted in this dissertation allows for the solution of continuous-time optimal control problems governed by various types of dynamical systems; thus encompassing a wider collection of problems than standard optimal control solvers. Moreover, the method is simple and very suitable for digital computations. Chapter 8 presents some concluding remarks on the works developed in this dissertation including some suggestions for future research.

We present a novel, high-order, efficient, and exponentially convergent shifted Gegenbauer integral pseudo-spectral method (SGIPSM) to solve numerically Lane-Emden equations provided with some mixed Neumann and Robin boundary conditions. The framework of the proposed method includes: (i) recasting the problem into its integral formulation, (ii) collocating the latter at the shifted flipped-Gegenbauer-Gauss-Radau (SFGGR) points, and (iii) replacing the integrals with accurate and well-conditioned numerical quadratures constructed via SFGGR-based shifted Gegenbauer integration matrices. The integral formulation is eventually discretized into linear/nonlinear system of equations that can be solved easily using standard direct system solvers. The implementation of the proposed method is further illustrated through four efficient computational algorithms. The theoretical study is enriched with rigorous error, convergence, and stability analyses of the SGIPSM. The paper highlights some interesting new findings pertaining to
``the apt choice of Gegenbauer collocation set of points'' that could largely influence the proper use of Gegenbauer polynomials as basis polynomials for polynomial interpolation and collocation. Five numerical test examples are presented to verify the effectiveness, accuracy, exponential convergence, and numerical stability of the proposed method. The numerical simulations are associated with extensive numerical comparisons with other rival methods in the literature to demonstrate further the power of the proposed method. The SGIPSM is broadly applicable and represents a strong addition to common numerical methods for solving linear/nonlinear differential equations when high-order approximations are required using a relatively small number of collocation points.

The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.

We present a high-order shifted Gegenbauer pseudospectral method (SGPM) to
solve numerically the second-order one-dimensional hyperbolic telegraph
equation provided with some initial and Dirichlet boundary conditions. The
framework of the numerical scheme involves the recast of the problem into its
integral formulation followed by its discretization into a system of
well-conditioned linear algebraic equations. The integral operators are
numerically approximated using some novel shifted Gegenbauer operational
matrices of integration. We derive the error formula of the associated
numerical quadratures. We also present a method to optimize the constructed
operational matrix of integration by minimizing the associated quadrature error
in some optimality sense. We study the error bounds and convergence of the
optimal shifted Gegenbauer operational matrix of integration. Moreover, we
construct the relation between the operational matrices of integration of the
shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive
the global collocation matrix of the SGPM, and construct an efficient
computational algorithm for the solution of the collocation equations. We
present a study on the computational cost of the developed computational
algorithm, and a rigorous convergence and error analysis of the introduced
method. Four numerical test examples have been carried out in order to verify
the effectiveness, the accuracy, and the exponential convergence of the method.
The SGPM is a robust technique, which can be extended to solve a wide range of
problems arising in numerous applications.

This paper reports a novel direct Gegenbauer (ultraspherical) transcription method (GTM) for solving continuous-time optimal control (OC) problems (CTOCPs) with linear/nonlinear dynamics and path constraints. In (Elgindy et al. 2012) [1], we presented a GTM for solving nonlinear CTOCPs directly for the state and the control variables, and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment. This article extends the GTM to deal further with problems including higher-order time derivatives of the states by solving the CTOCP directly for the control u(t)u(t) and the highest-order time derivative x(N)(t),N∈Z+x(N)(t),N∈Z+. The state vector and its derivatives up to the (N−1)(N−1)th-order derivative can then be stably recovered by successive integration. Moreover, we present our solution method for solving linear–quadratic regulator (LQR) problems as we aim to cover a wider collection of CTOCPs with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature. The proposed numerical scheme fully parameterizes the state and the control variables using Gegenbauer expansion series. For problems with various order time derivatives of the state variables arising in the cost function, dynamical system, or path/terminal constraints, the GTM seeks to fully parameterize the control variables and the highest-order time derivatives of the state variables. The time horizon is mapped onto the closed interval [0,1][0,1]. The dynamical system characterized by differential equations is transformed into its integral formulation through direct integration. The resulting problem on the finite interval is then transcribed into a nonlinear programming (NLP) problem through collocation at the Gegenbauer–Gauss (GG) points. The integral operations are approximated by optimal Gegenbauer quadratures in a certain optimality sense. The reduced NLP problem is solved in the Gegenbauer spectral space, and the state and the control variables are approximated on the entire finite horizon. The proposed method achieves discrete solutions exhibiting exponential convergence using relatively small-scale number of collocation points. The advantages of the proposed direct GTM over other traditional discretization methods are shown through four well-studied OC test examples. The present work is a major breakthrough in the area of computational OC theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points, states and controls expansion terms. Moreover, the GTM produces very small-scale NLP problems, which can be solved very quickly using the modern NLP software.

The theory of Gegenbauer (ultraspherical) polynomial approximation has received considerable attention in recent decades. In particular, the Gegenbauer polynomials have been applied extensively in the resolution of the Gibbs phenomenon, construction of numerical quadratures, solution of ordinary and partial differential equations, integral and integro-differential equations, optimal control problems, etc. To achieve better solution approximations, some methods presented in the literature apply the Gegenbauer operational matrix of integration for approximating the integral operations, and recast many of the aforementioned problems into unconstrained/constrained optimization problems. The Gegenbauer parameter α associated with the Gegenbauer polynomials is then added as an extra unknown variable to be optimized in the resulting optimization problem as an attempt to optimize its value rather than choosing a random value. This issue is addressed in this article as we prove theoretically that it is invalid. In particular, we provide a solid mathematical proof demonstrating that optimizing the Gegenbauer operational matrix of integration for the solution of various mathematical problems by recasting them into equivalent optimization problems with α added as an extra optimization variable violates the discrete Gegenbauer orthonormality relation, and may in turn produce false solution approximations.

In this paper we describe a novel direct optimization method using Gegenbauer-Gauss (GG) collocation for solving continuous-time optimal control (OC) problems (CTOCPs) with nonlinear dynamics, state and control constraints. The time domain is mapped onto the interval [0; 1], and the dynamical system formulated as a system of ordinary differential equations is transformed into its integral formulation through direct integration. The state and the control variables are fully parameterized using Gegenbauer expansion series with some unknown Gegenbauer spectral coefficients. The proposed Gegenbauer transcription method (GTM) then recasts the performance index, the reduced dynamical system, and the constraints into systems of algebraic equations using optimal Gegenbauer quadratures. Finally, the GTM transcribes the infinite-dimensional OC problem into a parameter nonlinear programming (NLP) problem which can be solved in the spectral space; thus approximating the state and the control variables along the entire time horizon. The high precision and the spectral convergence of the discrete solutions are verified through two OC test problems with nonlinear dynamics and some inequality constraints. The present GTM offers many useful properties and a viable alternative over the available direct optimization methods.