Approximation schemes for nonlinear neutral optimal control systems

We discuss methods of approximating stable neutral functional differential equations and associated optimal control problems by sequences of optimal control problems for ordinary differential equations. By introducing a class of “mollified” neutral functional differential equations, convergence of the linear interpolating spline and the averaging approximation scheme is proved. A number of numerical examples are included.     

On solvability of functional equations and system of functional equations arising in dynamic programming

The purpose of this paper is to study solvability of two classes of functional equations and a class of system of functional equations arising in dynamic programming of multistage decision processes. By using fixed point theorems, a few existence and uniqueness theorems of solutions and iterative approximation for solving these classes of functional equations are established. Under certain conditions, some existence theorems of coincidence solutions for the class of system of functional equations are shown. Some examples are given to demonstrate the advantage of our results than existing ones in the literature.     

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.     

Approximation properties for solutions to non‐Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non‐Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.     

OSCILLATIONS OF NONLINEAR DELAY DIFFERENTIAL EQUATIONS——GENERAL LINEARIZED OSCILLA TIONS RESULTS

It is shown that for a wide class of the scalar and vector functional equations of the delay and the neutral type a positive solution exists if and only if the linearized approximating equations are non-oscillatory or, equivalently, the characteristic equations of these linear equations have a real root.     

抽象泛函微分方程—整体解的存在性与稳定性

本文在一次近似方程的解具有指数衰减性质的条件下,采用逐步逼近方法得到了原方程整体解的存在唯一性和稳定性结果.     

Stability,prediction-correction,and dynamic programming

It is proven here that a bounded perturbation of the discrete dynamic programming functional equation arising from the Bolza problem yields a bounded change in its solution. This stability property encourages the development of approximation techniques for solving such equations. One such technique, involving the backward solution of an approximate functional equation as a prediction step, followed by a forward reconstruction using true equations as a correction step, is then discussed. Bounds for the errors arising from such an approximation procedure are derived. Successive approximations is suggested, in conclusion, as a means for obtaining improved solutions.     

进化泛函网络多维函数逼近理论及学习算法

提出了一种用于多维函数逼近的进化策略修正泛函网络基函数系数的新算法,并给出了其算法学习过程.利用进化策略的自适应性来确定基函数前的系数,改进了泛函网络的参数通过解方程组来得到这一传统方法.仿真结果表明,这种新的逼近算法简单可行,能够逼近给定的函数到预先给定的精度,具有较快的收敛速度和良好的逼近性能.     

A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria

A weak form of the Hartman-Grobman theorem for retarded functional differential equations around hyperbolic equilibria is presented. Orbits on a center-unstable manifold are compared to orbits on a center-unstable subspace of the linearized equation. The result is applied to obtain a conjugacy between the semidynamical system generated by the functional differential equation and its numerical approximation. A version of the Hartman-Grobman theorem around hyperbolic periodic orbits of functional differential equations is also given.     

W_2~1[a,b]中的最佳逼近泛函及应用

1 引言 线性泛函的逼近问题有着十分广泛的应用背景,本文在具有再生核的W_2~1[a,b]空间中讨论线性泛函L(f)的形如 L_n(f)=sum from i=1 to n(i/1)w_if(x_i) (1)的逼近问题,其中{w_i}_1~n是待定系数,如果存在一组常数{w_i~x}_1~n使 L_n~x=sum from i=1 to n(i/1)w_i~xf(x_i) (2)满足||L—L_n~x||=inf||L—L_n||,则称L_n~x是L的最佳逼近,记 w_i E_n=L—L_n, E_n~3=L—L_n~x,则称E_n~x是最佳逼近误差泛函。 本文在§1中给出L_n~x的表达式及L_(n+1)~x与L_n~x之间的递推公式。并证明L_n~x的收敛性。§3中讨论了上L_n~x(f)在数值积分及常微分方程数值解中的应用,并给出数值算例。     

A method of numerical solution of functional equations

Approximate solutions of some functional equations as the best approximation of the exact solutions in some suitable spaces have been obtained and some applications given.     

A discrete approximation framework for hereditary systems

A discrete approximation framework for initial-value problems involving certain classes of linear functional differential equations (FDE) of the retarded type is constructed. An equivalence between the FDE and abstract evolution equations (AEE) in an appropriately chosen Hilbert space is established. This equivalence is then employed in the development of discrete approximation schemes in which the infinite-dimensional AEE is replaced by a finite-dimensional system of difference equations. Convergence and rates of convergence are demonstrated via the properties of rational functions with operator arguments and both classical and recent results from linear semigroup theory. Two examples of families of approximation schemes which are included in the general framework and which may be implemented directly on high-speed computing machines are developed. A numerical study of examples which illustrates the application and feasibility of the approximation techniques in a variety of problems together with a summary and analysis of the numerical results are also included.     

Discrete Approximations of Controlled Stochastic Systems with Memory: A Survey

This survey article considers discrete approximations of an optimal control problem in which the controlled state equation is described by a general class of stochastic functional differential equations with a bounded memory. Specifically, three different approximation methods, namely (i) semidiscretization scheme; (ii) Markov chain approximation; and (iii) finite difference approximation, are investigated. The convergence results as well as error estimates are established for each of the approximation methods.     

Numerical approximations for a class of differential equations with time- and state-dependent delays

We establish limiting relations between solutions for a large class of functional differential equations with time- and state-dependent delays and solutions of appropriately selected sequences of approximating delay differential equations with piecewise constant arguments. The approximating equations, generated in the above process, lead naturally to discrete difference equations, well suited for computational purposes, and thus provide an approximation framework for simulation studies.     

Spline approximations for linear nonautonomous delay systems

We develop a semi-discrete approximation framework for linear nonautonomous nonhomogeneous functional differential equations of retarded type. The approximation schemes are constructed and convergence results are obtained through the approximation of an associated abstract evolution operator. Computer implementation of the schemes is outlined and a spline-based method included in the framework is constructed. Extensions of the semi-discrete methods to schemes incorporating full discretization and difference equation approximation are also discussed. Numerical results for several examples demonstrating the feasibility of the schemes are presented.     

Doubly Perturbed Neutral Stochastic Functional Equations Driven by Fractional Brownian Motion

In this paper, we study a class of doubly perturbed neutral stochastic functional equations driven by fractional Brownian motion. Under some non-Lipschitz conditions, we will prove the existence and uniqueness of the solution to these equations by providing a semimartingale approximation of a fractional stochastic integration.     

Approximations of solutions to neutral functional differential equations with nonlocal history conditions

This work is concerned with a class of neutral functional differential equations with nonlocal history conditions in a Hilbert space. The approximation of solution to a class of such problems is studied. Moreover, the convergence of Faedo-Galerkin approximation of solution is shown. For illustration, an example is worked out.     

On the approximation of eigenvalues associated with functional differential equations

In this paper an approximation method based upon spline functions is developed for the eigenvalue problem associated with functional differential equations. Convergence results are established and the rate of convergence is investigated. Numerical results for cubic and quintic spline based methods are given. The paper concludes with a brief discussion of other possible approximation methods.     

Superconvergence of functional approximation methods for integral equations

In this work, a functional approximation method for calculating the linear functional of the solution of second-kind Fredholm integral equations is developed. When the method is applied to the collocation method or to the multi-projection method, it generates approximations which exhibit superconvergence.     

Stationary Schr?dinger equation in nonrelativistic quantum mechanics and the functional integral

We formulate a method for representing solutions of homogeneous second-order equations in the form of a functional integral or path integral. As an example, we derive solutions of second-order equations with constant coefficients and a linear potential. The method can be used to find general solutions of the stationary Schr?dinger equation. We show how to find the spectrum and eigenfunctions of the quantum oscillator equation. We obtain a solution of the stationary Schr?dinger equation in the semiclassical approximation, without a singularity at the turning point. In that approximation, we find the coefficient of transmission through a potential barrier. We obtain a representation for the elastic potential scattering amplitude in the form of a functional integral.