Oscillation of nonlinear partial difference equations with delays

In this paper, we first derive a discrete Gaussian formula and then apply the formula to two classes of nonlinear (parabolic and hyperbolic) partial difference equations with delays to obtain sufficient conditions under which every solution of the two classes of nonlinear partial difference equations with delays is oscillatory.     

面向偏微分方程的连续反演控制算法综述

连续反演控制算法是一种面向偏微分方程(partial differential equations,PDEs)模型控制对象,配合边界控制方式的分布参数系统(distributed parameter systems,DPSs)控制算法.该算法基于Volterra映射运算,思路较为新颖,具有鲁棒性、逆最优性,便于获得显式的精确控制律和闭环系统的精确解,并能结合观测器、自适应控制等领域已取得的成果拓展应用范围.本文概述了连续反演算法的基本原理和设计过程,总结了该算法在抛物线偏微分模型、双曲线偏微分模型、复合偏微分模型、非线性偏微分模型等各方面的最新进展,最后归纳了该算法的主要特点,并探讨了未来研究的发展方向.     

Convolutional solutions of partial difference equations

A significant part of the theory of one-dimensional linear shift-invariant systems is based on the concept of weighting function (or impulse response): the output is the convolution of the weighting function with the input. This paper introduces the concept of linear translation-invariant systems and uses this notion in studying impulse response, z-transforms, and transfer functions for multidimensional systems.     

Oscillation criteria of certain nonlinear partial difference equations

This paper is concerned with the nonlinear partial difference equation with continuous variables

,where a, σ_i, τ_i are positive numbers, h_i(x, y, u) ε C(R⁺ × R⁺ × R, R), uh_i(x, y, u) > 0 for u ≠ 0, h_i is nondecreasing in u, i = 1, …, m. Some oscillation criteria of this equation are obtained.     

Accelerating technique for implicit finite difference schemes for two dimensional parabolic partial differential equations

In this paper we define a new accurate fast implicit method for the finite difference solution of the two dimensional parabolic partial differential equations with first level condition, which may be obtained by any other method. The stability region is discussed. The suggested method is considered as an accelerating technique for the implicit finite difference scheme, which is used to find the first level condition. The obtained results are compared with some famous finite difference schemes and it is in satisfactory agreement with the exact solution.     

The oscillation of partial difference equations with continuous arguments

For the partial difference equations

and

we shall obtain sufficient conditions for the oscillation of all solutions of these equations.     

Stability of difference Volterra equations: Direct Liapunov method and numerical procedure

A procedure to construct Liapunov functionals for discrete Volterra equations is proposed. Using this procedure stability conditions are derived for general Volterra difference equations. Some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for Volterra integro-differential equations are presented.     

A concise proof for existence and uniqueness of solutions of linear parabolic PDEs in the context of optimal control

We present a concise proof for existence and uniqueness of solutions of linear parabolic PDEs. It is based on an analysis of the corresponding differential operator and its adjoint in appropriate spaces and simple enough to be presented in the context of an introductory lecture on optimal control of PDEs. Our approach also clarifies some aspects in the structure of first order optimality conditions as illustrated in an example.     

非线性抛物型偏差分系统迭代学习控制

本文研究了一类时空离散的非线性抛物型偏差分系统的迭代学习控制问题.首先,针对系统含有不确定系数与非线性特点,设计了开环P型迭代学习控制算法;然后,建立了输出跟踪误差沿迭代轴收敛的充分条件,并利用离散Gronwall不等式、λ范数以及压缩映射原理,详细给出了收敛性分析证明.最后通过仿真实例说明了算法的有效性.     

Stability and convergence of difference schemes for multi-dimensional parabolic equations with variable coefficients and mixed derivatives

ABSTRACT

We present second-order difference schemes for a class of parabolic problems with variable coefficients and mixed derivatives. The solvability, stability and convergence of the schemes are rigorously analysed by the discrete energy method. Using the Richardson extrapolation technique, the fourth-order accurate numerical approximations both in time and space are obtained. It is noted that the Richardson extrapolation algorithms can preserve stability of the original difference scheme. Finally, numerical examples are carried out to verify the theoretical results.     

Asymptotic behavior of certain Riccati difference equations

Matrix Riccati difference equations are investigated on the infinite index set. Under natural assumptions an existence and uniqueness theorem is proven. The existence of the asymptotic expansion of the solution and computability of its coefficients are shown, provided the coefficients of the equation have such an expansion.     

Structural stability of linear difference equations in Hilbert space

In this note, we prove that linear difference equations in Hilbert space are structurally stable if and only if they have an exponential dichotomy.     

On sparse lu factorization procedures for the solution of parabolic differential equations in three space dimensions

The numerical solution of partial differential equations in 3 dimensions by finite difference methods leads to the problem of solving large order sparse structured linear systems.

In this paper, a factorization procedure in algorithmic form is derived yielding direct and iterative methods of solution of some interesting boundary value problems in physics and engineering.     

A convex penalty for switching control of partial differential equations

A convex penalty for promoting switching controls for partial differential equations is introduced; such controls consist of an arbitrary number of components of which at most one should be simultaneously active. Using a Moreau–Yosida approximation, a family of approximating problems is obtained that is amenable to solution by a semismooth Newton method. The efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples.     

On the asymptotic stability of coupled delay differential and continuous time difference equations

A two-step Liapunov-Krasovskii methodology for checking the asymptotic stability of nonlinear coupled delay differential and continuous time difference equations is proposed here. The feasibility of such methodology is shown by means of Liapunov-Krasovskii functionals with nonconstant kernels in the integrals, for instance discretized Liapunov-Krasovskii ones. An illustrative example taken from the literature, showing the effectiveness of the proposed method, is reported.     

Non-oscillation of solutions of difference equations of third order

In this paper, sufficient conditions in terms of coefficient functions are obtained for non-oscillation of all solutions of a class of linear homogeneous third order difference equations of the form

     

基于伪谱展开的抛物型系统镇定与边界观测研究(英文)

模型截断设计方法在无限维系统控制中得到广泛的应用,但是其自身存在信息丢失的缺陷而限制了控制器对高频扰动抑制的性能.本文研究具有内部和Neumann边界控制的抛物型系统,其中系统采用边界测量.内部控制采用比例反馈形式,其中反馈增益核由Sturm-Liouville系统稳定性分析来待定;类似地,边界反馈的设计也采用待定反馈增益核的方式,最终对描述系统稳定性的Sturm-Liouville系统采用伪谱方法进行求解.数字仿真结果表明了该方法的有效性.     

Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences

We consider linear difference equations with polynomial coefficients over C and their solutions in the form of sequences indexed by the integers (sequential solutions). We investigate the C-linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to Z of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to Z of entire solutions and that the dimension of this space is equal to the order of the original equation.We also consider d-dimensional (d≥1) hypergeometric sequences, i.e., sequential and subanalytic solutions of consistent systems of first-order difference equations for a single unknown function. We show that the dimension of the space of subanalytic solutions is always at most 1, and that this dimension may be equal to 0 for some systems (although the dimension of the space of all sequential solutions is always positive).Subanalytic solutions have applications in computer algebra. We show that some implementations of certain well-known summation algorithms in existing computer algebra systems work correctly when the input sequence is a subanalytic solution of an equation or a system, but can give incorrect results for some sequential solutions.     

Oscillations and nonoscillations of half-linear difference equations generated by deviating arguments

For the half-linear difference equation

where α > 0, we shall offer sufficient conditions for the oscillation of all solutions, as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions. Several examples which dwell upon the importance of our results are also included.     

推广形式传输控制协议流量控制方程的方差稳定性

给定计算机网络中的传输控制协议（transmission control protocol,TCP）流量控制算法,如何确定其稳定域,是网络设计中的一个重要问题.由于网络上控制算法受大量随机因素影响,这相当于对一个由随机微分/差分方程描述的控制系统进行稳定性分析.目前已有研究大多直接对系统方程取期望,转为讨论期望的稳定性,而简单忽略受控TCP流的随机震荡.本文意在指出这种随机震荡给稳定性带来的不可忽视的影响.本文以TCP/RED（含早期随机检测的TCP流）系统为例,首先,从系统的随机微分方程出发,通过在平衡点处线性化,将系统化为含加乘混合噪声的多维线性时不变系统.然后,给出了分别对应时间连续与离散情况的推广的TCP流量控制方程,即含多噪声源的一次时不变随机微分/差分方程组.接着,对此推广形式,推导了其协方差矩阵所满足的矩阵方程,并在此基础上,得到了协方差矩阵极限渐近稳定的充要条件以及此极限的计算公式.在工程设计中,此条件可以作为系统稳定与否的一个替代判据,方差极限公式可用来估计系统的运动范围.最后,将一般公式应用到具体例子上,展示了考虑方差稳定性后系统稳定域的变化.进一步,仿照确定性系统中的处理方法,本文结论还可推广到非线性系统及时变系统.