Linear Two-point Boundary Value Problems Research Articles

Congestion Control for Nonlinear TD-SCDMA Discrete Networks Based on TCP/IP

A successive approximation approach (SAA) is developed to obtain a new congestion controller for the nonlinear TD-SCDMA network control systems based on TCP/IP. By using the successive approximation approach, the original optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term that is the limit of the solution sequence of the adjoint vector differential equations. By using the finite-time iteration of nonlinear compensation term of optimal solution sequence, we can obtain a suboptimal control law for TD-SCDMA network control systems based on TCP/IP.

Optimal L ∞ estimates for Galerkin methods for nonlinear singular two-point boundary value problems

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

An iterative method for suboptimal control of linear time-delayed systems

This article presents a new approach for solving the Optimal Control Problem (OCP) of linear time-delay systems with a quadratic cost functional. The proposed method can also be used for designing optimal control time-delay systems with disturbance. In this study, the Variational Iteration Method (VIM) is employed to convert the original Time-Delay Optimal Control Problem (TDOCP) into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Finally, Illustrative examples are included to demonstrate the validity and applicability of the technique.

Wavelet multilevel augmentation method for linear boundary value problems

This work presents a new approach to numerically solve the general linear two-point boundary value problems with Dirichlet boundary conditions. Multilevel bases from the anti-derivatives of the Daubechies wavelets are constructed in conjunction with the augmentation method. The accuracy of numerical solutions can be improved by increasing the number of basis levels, but the computational cost also increases drastically. The multilevel augmentation method can be applied to reduce the computational time by splitting the coefficient matrix into smaller submatrices. Then the unknown coefficients in the higher level can be solved separately. The convergent rate of this method is $2^{s}$ , where $1 \leq s \leq p+1$ , when the anti-derivatives of the Daubechies wavelets order p are applied. Some numerical examples are also presented to confirm our theoretical results.

Necessary and Sufficient Conditions for the Solvability of Linear Boundary-Value Problems for the Fredholm Integrodifferential Equations

We propose a method for the investigation and solution of linear boundary-value problems for the Fredholm integrodifferential equations based on the partition of the interval and introduction of additional parameters. Every partition of the interval is associated with a homogeneous Fredholm integral equation of the second kind. The definition of regular partitions is presented. It is shown that the set of regular partitions is nonempty. A criterion for the solvability of the analyzed problem is established and an algorithm for finding its solutions is constructed. Integrodifferential equations are often encountered in applications as mathematical models of various processes in natural sciences. In the monograph [1], their role in the study of hereditary processes is indicated and a survey of the papers devoted to the boundary-value problems for the Volterra and Fredholm integrodifferential equations is presented. The solvability of various problems for the Fredholm integrodifferential equations and approximate methods for finding their solutions are studied by many authors [2–11]. Linear boundary-value problems for Fredholm integrodifferential equations with degenerate kernels are considered in the monograph [12]. Necessary and sufficient conditions for the solvability of boundary-value problems are established and the algorithms for finding the solutions of these problems are constructed. In the present paper, we consider a linear two-point boundary-value problem

A Fourth-Order Collocation Scheme for Two-Point Interface Boundary Value Problems

A fourth-order accurate orthogonal spline collocation scheme is formulated to approximate linear two-point boundary value problems with interface conditions. The coefficients of the differential operator may have jump discontinuities at the interface point, a nodal point of the scheme. Existence and uniqueness of the numerical solution are proved. Optimal order error estimates in the maximum norm are obtained, and a superconvergence property of the numerical solution in the maximal nodal norm is proved. Numerical results are presented confirming the theoretical estimates.

Augmented Lagrangian Method for One Dimensional Optimal Control Problems Governed by Delay Differential Equation

In this research, numerical solutions of continuous optimal control problems governed by linear damping evolution with delay and real coefficients are presented. The necessary conditions obtained from the knowledge of calculus of variation for optimal control problem constrained by delay differential equation is a linear two-point boundary value problem involving both delay and advance terms. Clearly, this coupling that exists between the state variable and the control variable is not amenable to analytical solution hence a direct numerical approach is adopted. We propose an augmented discretized continuous algorithm via quadratic programming, which is capable of handling optimal control problems constrained by delay differential equations. The discretization of the problem using trapezoidal rule (a one step second order numerical scheme) and Crank-Nicholson with quadratic formulation amenable to quadratic programming technique for solution of the optimal control problems are considered. A control operator (penalized matrix), through the augmented Lagrangian method, is constructed. Important properties of the operator as regards sequential quadratic programming techniques for determining the optimal point are shown.

Criteria for the unique solvability of a linear two-point boundary value problem for systems of integro-differential equations

For a linear boundary value problem for a Fredholm integro-differential equation, we obtain necessary and sufficient conditions for the unique solvability in terms of a matrix Qνm (h) formed on the basis of the matrices of boundary conditions, the differential part, the integral term, and a partition with increment h > 0 of the interval on which the problem is defined.

An algorithm for solving a linear two-point boundary value problem for an integrodifferential equation

family of algorithms for solving a linear two-point boundary value problem is constructed in terms of the data of the integrodifferential equation and the boundary condition involved. The convergence conditions for the algorithms are established, and necessary and sufficient conditions for the well-posedness of the problem are found.

Parametric septic splines approach for the solution of linear sixth-order two-point boundary value problems

Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a class of numerical methods for the solution of linear sixth-order two-point boundary value problems. Convergence analysis of the methods is discussed through standard procedure. It is shown that the present methods give approximations, which are better than those produced by other splines and domain decomposition methods. Three numerical examples are given to illustrate the practical usefulness of the new approach.

Variable Structure Disturbance Rejection Control for Nonlinear Uncertain Systems with State and Control Delays via Optimal Sliding Mode Surface Approach

The paper considers the problem of variable structure control for nonlinear systems with uncertainty and time delays under persistent disturbance by using the optimal sliding mode surface approach. Through functional transformation, the original time-delay system is transformed into a delay-free one. The approximating sequence method is applied to solve the nonlinear optimal sliding mode surface problem which is reduced to a linear two-point boundary value problem of approximating sequences. The optimal sliding mode surface is obtained from the convergent solutions by solving a Riccati equation, a Sylvester equation, and the state and adjoint vector differential equations of approximating sequences. Then, the variable structure disturbance rejection control is presented by adopting an exponential trending law, where the state and control memory terms are designed to compensate the state and control delays, a feedforward control term is designed to reject the disturbance, and an adjoint compensator is designed to compensate the effects generated by the nonlinearity and the uncertainty. Furthermore, an observer is constructed to make the feedforward term physically realizable, and thus the dynamical observer-based dynamical variable structure disturbance rejection control law is produced. Finally, simulations are demonstrated to verify the effectiveness of the presented controller and the simplicity of the proposed approach.

Optimal Ascent Guidance for Air-Breathing Launch Vehicle Based on Optimal Trajectory Correction

An optimal guidance algorithm for air-breathing launch vehicle is proposed based on optimal trajectory correction. The optimal trajectory correction problem is a nonlinear optimal feedback control problem with state inequality constraints which results in a nonlinear and nondifferentiable two-point boundary value problem (TPBVP). It is difficult to solve TPBVP on-board. To reduce the on-board calculation cost, the proposed guidance algorithm corrects the reference trajectory in every guidance cycle to satisfy the optimality condition of the optimal feedback control problem. By linearizing the optimality condition, the linear TPBVP is obtained for the optimal trajectory correction. The solution of the linear TPBVP is obtained by solving linear equations through the Simpson rule. Considering the solution of the linear TPBVP as the searching direction for the correction values, the updating step size is generated by linear search. Smooth approximation is applied to the inequality constraints for the nondifferentiable Hamiltonian. The sufficient condition for the global convergence of the algorithm is given in this paper. Finally, simulation results show the effectiveness of the proposed algorithm.

Assessment of He's homotopy perturbation method for optimal control of linear time-delay systems

This paper renders an optimal regulator for a system with state time-delay and quadratic performance index. First, by means of Maximum Principle and necessary optimal conditions, a continues-time two-point boundary-value problem (TPBVP) including both time-delay and time-advance is derived. Then, using Homotopy Perturbation Method (HPM), the obtained TPBVP is transformed into a sequence of linear time-invariant TPBVP without any time-delay or timeadvance. Solving the resulted linear TPBVP sequence in a recursive manner, concludes that the optimal control law in the form of rapid convergent series. At the end, an iterative algorithm with low complexity in computation and fast convergence rate is proposed to achieve an accurate enough suboptimal control law. It's worth noting that simulation based on application of a harmonic oscillator is conducted to demonstrate efficiency of this method.

Approximate Solutions of Singular Two-Point BVPs Using Legendre Operational Matrix of Differentiation

Exact and approximate analytical solutions of linear and nonlinear singular two-point boundary value problems (BVPs) are obtained for the first time by the Legendre operational matrix of differentiation. Different from other numerical techniques, shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The accuracy of the technique is demonstrated through several linear and nonlinear test examples.

New Spectral Second Kind Chebyshev Wavelets Algorithm for Solving Linear and Nonlinear Second-Order Differential Equations Involving Singular and Bratu Type Equations

A new spectral algorithm based on shifted second kind Chebyshev wavelets operational matrices of derivatives is introduced and used for solving linear and nonlinear second-order two-point boundary value problems. The main idea for obtaining spectral numerical solutions for these equations is essentially developed by reducing the linear or nonlinear equations with their initial and/or boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. Convergence analysis and some efficient specific illustrative examples including singular and Bratu type equations are considered to demonstrate the validity and the applicability of the method. Numerical results obtained are compared favorably with the analytical known solutions.

Analysis of a moving collocation method for one-dimensional partial differential equations

A moving collocation method has been shown to be very efficient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two robust codes MOVCOL and MOVCOL4. In this paper, the relations between the method and the traditional collocation and finite volume methods are investigated. It is shown that the moving collocation method inherits desirable properties of both methods: the ease of implementation and high-order convergence of the traditional collocation method and the mass conservation of the finite volume method. Convergence of the method in the maximum norm is proven for general linear two-point boundary value problems. Numerical results are given to demonstrate the convergence order of the method.

A Successive Approximation Approach of Nonlinear Optimal Control with R-Rank Persistent Disturbances

A new feedforward and feedback optimal control law for a class of nonlinear systems with persistent disturbances is presented in this paper. By using a successive approximation approach (SAA), the original nonlinear optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The optimal control law obtained consists of analytical linear feedforward and feedback terms and a nonlinear compensation term which is the limit of the solution sequence for the adjoint vector differential equations. By using the finite-step iteration of nonlinear compensation sequence, we can obtain a feedforward and feedback suboptimal control law. A disturbance observer is designed to get a physically realizable controller. Simulation examples show the effectiveness of the approach.

A numerical method for solving boundary value problems for fractional differential equations

A numerical scheme, based on the Haar wavelet operational matrices of integration for solving linear two-point and multi-point boundary value problems for fractional differential equations is presented. The operational matrices are utilized to reduce the fractional differential equation to system of algebraic equations. Numerical examples are provided to demonstrate the accuracy and efficiency and simplicity of the method.

Cubic Trigonometric B-Spline Applied to Linear Two-Point Boundary Value Problems of Order Two

Cubic Trigonometric B-Spline Applied to Linear Two-Point Boundary Value Problems of Order Two

Criteria for the well-posedness of a linear two-point boundary value problem for systems of integro-differential equations

We consider a linear two-point boundary value problem for systems of integro-differential equations. By using the parametrization method and an approximation of the integro-differential equation by a loaded differential equation, we establish coefficient tests for the well-posedness of the considered problem and suggest an algorithm for finding the solution.