反阻尼一维波动方程的稳定性

文章利用边界控制法研究了一类反阻尼一维波动方程的稳定性. 首先, 通过边界控制的backstepping方法, 引入 包含有两个核函数的积分变换, 将控制系统转化为稳定的目标系统. 核函数个数的增加导致核函数方程更加复杂, 文中 运用了一系列的数学计算技巧求解出核函数, 从而得到反馈控制器; 其次, 运用类似的方法找到变换的逆变换; 最后, 选 择合适范数, 利用变换及逆变换的有界性证明得到闭环系统的稳定性.     

带有大输出时滞的ODE–热方程级联系统的观测器设计

为了估计系统的状态并补偿输出时滞,本文提出一种新颖的观测器设计方法应用在常微分方程(ODE)-热方程级联系统中.不同于传统观测器的设计方法,该方法的核心思想是将大时滞划分为若干段小时滞,再设计一串级联的子观测器逐步地估计系统的状态.这种方法的优势在于对于大时滞仍有效.本文通过backstepping-like方法实现了原系统与目标系统的等价变换,结合Lyapunov-Krasovskii方法和线性矩阵不等式理论,得到了误差系统指数稳定的结果.最后,通过仿真实例验证了级联观测器的有效性.     

具有对称初始数据的二维反应扩散方程的边界镇定

研究了二维圆盘上具有对称初始数据的反应扩散方程的边界控制. 由于初始条件和边界条件关于圆心旋转对称, 系统可以转化为等价的极坐标系下的一维抛物方程. 此时, 极点的奇异性成为了控制器设计中的难点. 本文设计了一系列方程变换, 消除了核函数方程中极点奇异性的影响, 将其转化为修正的Bessel方程, 求出了显式的核函数表达式和精确的边界反馈控制律, 扩展了偏微分方程的backstepping方法. 系统的收敛速度可通过改变控制器中的一个参数来调节. 然后用Lyapunov函数法证明了闭环系统在H1范数下指数稳定, 表明了系统对初值的连续依赖. 最后用数值仿真验证了方法的有效性.     

导出变系数非线性动力学系统拉格朗日函数的两种方法

利用从运动微分方程出发和从第一积分出发导出拉格朗日函数的两种直接方法,构造变系数非线性动力学系统¨x+b(x)x~2+c(x)x=0的拉格朗日函数和c(x)=0特殊情况的拉格朗日函数族.另外,讨论了这种非保守系统广义能量守恒的物理意义.     

基于反步法的耦合分数阶反应扩散系统边界输出反馈控制

针对具有空间依赖耦合系数的分数阶反应扩散系统,利用反步法设计了基于观测器的边界输出反馈控制器,证明了观测增益和控制增益核函数矩阵方程的适定性.针对误差系统和输出反馈的闭环系统,利用分数阶Lyapunov方法分析了系统的Mittag-Leffler稳定性,且利用Wirtinger不等式改进了耦合系统稳定的条件.当系统具有空间依赖的耦合系数时,难以求得控制增益和观测增益核函数的解析解,为此,给出了核函数偏微分方程的数值解方法.数值仿真验证了理论结果.     

改进的小波域耦合偏微分方程图像去噪模型

针对全变分及四阶偏微分方程图像去噪模型的不足,利用小波变换能够聚焦到图像细微变化的优势,提出一种基于小波域的偏微分方程图像去噪算法。并通过对小波的阈值和阈值函数做适当的改进以及利用加权函数将全变分和四阶偏微分方程去噪模型相结合的方法,得到一种改进的小波域耦合偏微分方程图像去噪模型。MATLAB仿真结果表明,该模型和小波软阈值去噪、全变分模型以及四阶偏微分方程图像去噪模型相比,峰值信噪比有明显的提高,而且能够在很好地保留图像的边缘和细节信息的同时提高处理噪声的效率。     

具有线性结构的弹性函数非线性度的新上界

讨论了具有线性结构的弹性函数的两个指标:沃什谱和非线性度,得到了具有线性结构的布尔函数的一些性质.利用沃尔什变换和汉明重量的方法,发现了:如果V是n元布尔函数,f(x)的线性结构,那么得到f(x)的沃尔什变换在为零这一事实,同时得到了一个布尔函数没有k(k≥0)维线性结构的充分条件.最后,利用以上结果推出了具有线性结构的弹性函数的非线性度的上界表达式.     

基于微分方程的退化图像盲复原数学模型构建

经当前方法复原后图像与实际图像之间的吻合度较低,且复原效率低.为此提出基于微分方程的退化图像盲复原数学模型构建方法.通过贝叶斯萎缩方法对阈值选择进行优化,对退化图像做二维离散小波变换,并采用高频系数进行离散Curvelet变换.在此基础上对阈值进行估计,获得最优阈值,实现退化图像的去噪.根据去噪结果在高阶微分方程中引入稀疏测度、卷积滤波器和引导图像特征,通过范数约束获得强边引导图像,在潜在图像中应用稀疏函数和卷积滤波器获得梯度下降方向,得到强边图像,通过强边图像对模糊核进行估计,实现退化图像的盲复原.仿真结果表明,所提方法的复原效率高、复原程度高,说明该方法复原效果较好,应用价值较高.     

PDE与DAE耦合系统求解方法

针对目前Modelica语言只能解决由微分代数方程(DAE)描述的问题,而不能解决由偏微分方程(PDE)表达的问题,提出一种求解PDE与DAE耦合系统的方法.首先采用径向基函数构造近似函数,将未知量场函数的时空变量分开;然后运用配点法对空间变量进行离散,从而将PDE问题转化为DAE问题;最后采用成熟的DAE求解器进行求解,得到场函数在任意时空点的函数值.实例结果表明,该方法在不改变Modelica语法的前提下,能较好地实现PDE与DAE耦合系统的一致求解,且求解精度高、稳定性好、边界条件处理简单.     

数字媒体图像多特征反差增强方法仿真

由于图像在生成的过程中会有相对的噪声产生,而噪声会直接影响该图像中的多特征信息,导致信息表达不完全等情况。基于此提出了数字媒体图像多特征反差增强方法。首先将需要对受噪声污染的图像进行去噪;其次运用偏微分方程(PDE)方法,构建出一种PDE数学模型-TV模型,在对图像去噪的同时,保持原图像的边缘轮廓信息,并且运用Split Bregman算法,并引入辅助变量转换出分段线性函数的泛函极值;在偏微分方程的基础上,运用改进后的反差增强方法,根据直方图的均衡化结果利用累积分布函数的方法,来当做图像灰度的变换函数,以达到合理的调整固定动态范围,且让图像整体视觉效果增强、提高图像中多特征信息的目的。通过仿真结果可以得知,提出的方法可以处理图像中噪声的影响,并且可以将图像中的多特征信息进行有效的反差增强处理,且具有直方图均衡化结果优秀、图像特征增强效率的优点。     

Unsupervised kernel least mean square algorithm for solving ordinary differential equations

In this paper a novel method is introduced based on the use of an unsupervised version of kernel least mean square (KLMS) algorithm for solving ordinary differential equations (ODEs). The algorithm is unsupervised because here no desired signal needs to be determined by user and the output of the model is generated by iterating the algorithm progressively. However, there are several new approaches in literature to solve ODEs but the new approach has more advantages such as simple implementation, fast convergence and also little error. Furthermore, it is also a KLMS with obvious characteristics. In this paper the ability of KLMS is used to estimate the answer of ODE. First a trial solution of ODE is written as a sum of two parts, the first part satisfies the initial condition and the second part is trained using the KLMS algorithm so as the trial solution solves the ODE. The accuracy of the method is illustrated by solving several problems. Also the sensitivity of the convergence is analyzed by changing the step size parameters and kernel functions. Finally, the proposed method is compared with neuro-fuzzy [21] approach.     

Difference kernel iterative method for linear and nonlinear partial differential equations

The purpose of this paper was to propose a new method to solve partial differential equations arising in the field of science and engineering. In this new method, we have reduced the multiple integrals into a single integral and expressed it in terms of a difference kernel. To make the calculation easy and convenient, we have used the Laplace transformation to solve the difference kernel. The method is very simple, easy to understand and calculation minimizing as compared to the Adomian decomposition method and the variational iteration method. Some examples are given to verify the reliability and efficiency of the method.     

Stabilization of a Heat‐ODE System Cascaded at a Boundary Point and an Intermediate Point

This paper considers the stabilization of a heat‐ODE system cascaded at a boundary point and an intermediate point. The stabilizing feedback control law is designed by the backstepping method. Based on a novel transformation, we prove that all the kernel functions in the forward and inverse transformations are of the class C². Moreover, the effectiveness of controller design is shown with a numerical simulation. Finally, we show the coherence between the controllability assumption of the main theorem in this paper and the known one for a special case with λ=0.     

Extended Jacobian elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations

With the aid of computerized symbolic computation, the extended Jacobian elliptic function expansion method and its algorithm are presented by using some relations among ten Jacobian elliptic functions and are very powerful to construct more new exact doubly-periodic solutions of nonlinear differential equations in mathematical physics. The new (2+1)-dimensional complex nonlinear evolution equations is chosen to illustrate our algorithm such that sixteen families of new doubly-periodic solutions are obtained. When the modulus m→1 or 0, these doubly-periodic solutions degenerate as solitonic solutions including bright solitons, dark solitons, new solitons as well as trigonometric function solutions.     

Calculation of internal viscous flows in axisymmetric ducts at moderate to high reynolds numbers

A formal procedure is presented for parabolizing the Navier-Stokes equations for two dimensional internal flows in artitrary ducts with streamline curvature and streamline divergence. This procedure consists of constructing an orthogonal coordinate system from the potential flow solution and parabolizing the corresponding Navier-Stokes equations using the thin channel approximation. Theoretical arguments based on the fundamental existence theorem and numerical examples are used to demonstrate the validity of this procedure. A new method using conformal mapping based on the Schwartz-Christoffel transformation is presented which obtains the solution of the inverse potential flow problem in a direct manner. A numerical solution algorithm based on the two point box scheme is presented for solving the viscous flow equations. This method is shown to be accurate and stable for flows at moderate to high Reynolds numbers over a wide range of conditions.     

On the approximate controllability of semilinear fractional differential systems

Fractional differential equations have wide applications in science and engineering. In this paper, we consider a class of control systems governed by the semilinear fractional differential equations in Hilbert spaces. By using the semigroup theory, the fractional power theory and fixed point strategy, a new set of sufficient conditions are formulated which guarantees the approximate controllability of semilinear fractional differential systems. The results are established under the assumption that the associated linear system is approximately controllable. Further, we extend the result to study the approximate controllability of fractional systems with nonlocal conditions. An example is provided to illustrate the application of the obtained theory.     

改进的自适应多种群DE的机械臂控制方法

根据机械臂关节轴线方向建立了连杆坐标系,利用Denavit-Hartenberg（D-H）法得到连杆坐标系变换矩阵;通过连杆坐标系变换矩阵得到机械臂正运动控制模型;通过正运动模型得到逆运动控制模型,逆运动控制模型是多目标约束优化问题,该模型的最优解既可以保证控制精度,又可以保证各个关节角变动幅度的总代价（定义为旋转副）达到最小。为了求解机械臂逆运动模型,提出了自适应多种群差分演化算法（AMPDE）,多种群策略可以提升个体共享群体信息的能力,自适应变异策略可以提升种群多样性,数值实验表明该算法可以有效求解机械臂逆运动学模型。     

Application of the Haar functions to solution of differential equations

In this paper, it is proposed that Haar functions should be used for solving ordinary differential equations of a time variable in facility. This is because integrated forms of Haar functions of any degree can be illustrated by linear- and linear segment-functions like as triangles. Fortunately, since they are placed where Haar functions are defined in a specified form respectively, these functions are computable by algebraic operations of quasi binary numbers. Therefore, when a given function is approximated in a form of stairsteps on a Haar function system their integration can be termwise executed by shift and add operations of coefficients of the approximation. The use of this system is comparable with an application using the midpoint rule in numerical integration. In this line, nonlinear differential equations can be solved like as linear differential equations.     

Kernels, regularization and differential equations

Many common machine learning methods such as support vector machines or Gaussian process inference make use of positive definite kernels, reproducing kernel Hilbert spaces, Gaussian processes, and regularization operators. In this work these objects are presented in a general, unifying framework and interrelations are highlighted.With this in mind we then show how linear stochastic differential equation models can be incorporated naturally into the kernel framework. And vice versa, many kernel machines can be interpreted in terms of differential equations. We focus especially on ordinary differential equations, also known as dynamical systems, and it is shown that standard kernel inference algorithms are equivalent to Kalman filter methods based on such models.In order not to cloud qualitative insights with heavy mathematical machinery, we restrict ourselves to finite domains, implying that differential equations are treated via their corresponding finite difference equations.