Existence of solutions to singular integral equations

Nonnegative solutions are established for singular integral equations of the form y(t) = h(t) + ∫^T₀ k(t, s)f(s, y(s)) ds for t ∈ [0, T]. Here f may be singular at y = 0.     

Lipschitz广义非线性系统观测器设计

研究一类广义非线性系统的观测器设计问题．首先讨论了半正定Lyapunov函数下指数1广义非线性系统稳定及渐近稳定性,然后对一类由线性和Lipschitz非线性项组成的广义非线性系统,给出了渐近稳定观测器存在的条件,并把观测器反馈增益矩阵的设计归结为广义线性系统容许控制以及奇异值计算问题,证明了若容许广义线性系统矩阵的最小奇异值大于系统的Lipschitz常数,容许控制器增益矩阵就是待求的观测器反馈增益矩阵。     

Existence of solutions for a class of nonlinear Volterra singular integral equations

In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space C[0,1] consisting of real functions defined and continuous on the interval [0,1]. The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra-Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.     

A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions

The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions.

     

A numerical approach for solving the high-order linear singular differential-difference equations

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving the high-order linear singular differential-difference equations. With the aid of Bessel polynomials and collocation points, this method converts the singular differential-difference equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives the analytic solutions when the exact solutions are polynomials. Finally, some experiments and their numerical solutions are given; by comparing the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).     

Approximations of finite parts and numerical solutions of singular integral equations

Summary The approximation of the finite parts and the solutions of singular integral equations on a domainQ having several dimensions, often hit against constraints of important computing time. A method very accurate and very rapid of the solution of this two problems is exposed here. This work was supported by the Centre d'Etudes Théoriques de la Détection et des Communications, CETHEDEC (Paris).     

A high-order fast method for computing convolution integral with smooth kernel

In this paper we report on a high-order fast method to numerically calculate convolution integral with smooth non-periodic kernel. This method is based on the Newton-Cotes quadrature rule for the integral approximation and an FFT method for discrete summation. The method can have an arbitrarily high-order accuracy in principle depending on the number of points used in the integral approximation and a computational cost of O(Nlog(N)), where N is the number of grid points. For a three-point Simpson rule approximation, the method has an accuracy of O(h⁴), where h is the size of the computational grid. Applications of the Simpson rule based algorithm to the calculation of a one-dimensional continuous Gauss transform and to the calculation of a two-dimensional electric field from a charged beam are also presented.     

An algorithm based on Q.I. modified splines for singular integral models

In this paper, we propose an algorithm supporting an approximation using quasi-interpolatory (q.i.) splines for the numerical solution of integro-differential equations with Cauchy singular kernel. Some different choices of parameters, defining the numerical model, are analyzed and compared in view of the algorithm efficiency.     

Swarm stability for high-order linear time-invariant singular multi-agent systems

Swarm-stability and swarm-stabilisation problems for high-order linear time-invariant singular multi-agent systems with directed networks are investigated. First, necessary and sufficient conditions for swarm stability and asymptotic swarm stability are proposed, which are independent of the dimensions of Jordan blocks of the Laplacian matrix of the interaction topology. Then, an approach is given to determine the absolute motion as a whole, and it is shown that the absolute motion is completely determined by initial states if the interaction topology is balanced. Furthermore, an approach is presented to determine gain matrices for asymptotic swarm stabilisation. Moreover, leader-following swarm-stability and swarm-stabilisation problems are investigated. Finally, numerical examples are given to demonstrate theoretical results.     

A high-order accurate numerical algorithm for three-dimensional transport prediction

The numerical solution of the three-dimensional pollutant transport equation is obtained with the method of fractional steps; advection is solved by the method of moments and diffusion by cubic splines. Topography and variable mesh spacing are accounted for with coordinate transformations. First estimate wind fields are obtained by interpolation to grid points surrounding specific data locations. Mass consistency is ensured by readjusting the three-dimensional wind field with a Sasaki variational technique. Numerical results agree with results obtained from analytical Gaussian plume relations for ideal conditions. The numerical model is used to simulate the transport of tritium released from the Savannah River Plant on 2 May, 1974. Predicted ground level air concentration 56 km from the release point is within 38% of the experimentally measured value.     

一种适用于多普勒高阶变化率的载波跟踪算法

针对多普勒高阶变化率的载波跟踪问题,分析了现有二阶锁频环辅助三阶锁相环的局限之处,设计了一种三阶锁频环辅助四阶锁相环的载波跟踪算法,从理论上证明了该算法可以对多普勒高阶变化率信号进行无误差跟踪,并根据环路传递函数和环路带宽,推导出了环路所有参数的计算方法,最后通过环路模式切换及环路带宽设置策略,既加快了整个载波跟踪环路的收敛速度,又提高了载波频率跟踪精度,保证了跟踪的稳定性。仿真结果表明,对于多普勒高阶变化率信号,该算法能够有效地完成信号的载波跟踪并正常解调出原始数字信息,因此有较高的应用价值。     

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.     

A recurrence formula for the direct solution of singular integral equations

An iterative method for the solution of singular integral equations is given in this paper by developing a recurrence formula. Discretizing the above formula, by using appropriate quadrature rules, the solution of the singular integral equation is given in an extremely simple form. The number of numerical operations required for such a solution is considerably reduced, when compared to the number of operations required for a classical type of solution. Illustrative examples are given, indicating the efficiency of the method. It is shown that the number of operations in this procedure is only half the number of the operations for a typical numerical method. The convergence of the method is studied in the space of Hölder continuous functions. In the particular case of plane elasticity more efficient bounds are given. In the same case it is proved that the procedure is equivalent to the Schwarz's alternating method and convergence is assured [18].     

迭代张量高阶奇异值分解的图像恢复方法

为了有效地对图像缺失数据进行恢复, 提出一种迭代张量高阶奇异值分解（HOSVD）图像缺失数据恢复方法。该方法首先利用拉格朗日乘子方法将张量核范数目标函数进行子问题分解操作, 简化了求解过程, 然后迭代地采用张量高阶奇异值分解阈值方法进行子问题求解, 最终得到恢复后的图像缺失数据。将矩阵奇异值阈值算法进行扩展而得的HOSVD阈值方法充分利用了图像内部和图像与图像之间的多重约束关系, 大大提高了恢复精度。模拟实验和真实图像实验结果显示该方法具有良好的缺失数据的恢复性能。     

Reduced-order observer-based finite-time control for one-sided Lipschitz nonlinear switched singular systems

The reduced-order observer-based finite-time control problem for one-sided Lipschitz nonlinear switched singular systems is addressed in this paper. First, the design method of the reduced-order observer is given via state transformation. Then, based on the average dwell time (ADT) approach, some new sufficient conditions for regularity, impulse-freeness, have a unique solution and finite-time boundedness (FTB) of the dynamic augmented systems are obtained by exploring the reduced-order observer-based controller. Further, the lower finite-time bound can be obtained by using singular value decomposition method. And the state feedback gain and the observer gain are computed by solving linear matrix inequalities (LMIs). Finally, the validity of the obtained method is illustrated by means of a numerical example and a DC motor system.     

An approximation method for high-order fractional derivatives of algebraically singular functions

The fractional derivative D^qf(s) (0≤s≤1) of a given function f(s) with a positive non-integer q is defined in terms of an indefinite integral. We propose a uniform approximation scheme to D^qf(s) for algebraically singular functions f(s)=s^αg(s) (α>−1) with smooth functions g(s). The present method consists of interpolating g(s) at sample points t_j in [0,1] by a finite sum of the Chebyshev polynomials. We demonstrate that for the non-negative integer m such that m<q<m+1, the use of high-order derivatives g⁽ⁱ⁾(0) and g⁽ⁱ⁾(1) (0≤i≤m) at both ends of [0,1] as well as g(t_j), t_j∈[0,1] in interpolating g(s), is essential to uniformly approximate D^q{s^αg(s)} for 0≤s≤1 when α≥q−m−1. Some numerical examples in the simplest case 1<q<2 are included.     

A regularization algorithm for singular controls of parabolic systems

A practically convenient method of regularization is proposed to solve an optimal control problem for a parabolic equation with a delta function on its right-hand side. The corresponding regularized analogue is proved to converge to a delta function in a negative space. The differential characteristics of the performance criterion are analyzed. An algorithm for finding the optimal control is developed for a regularized problem. Results from a numerical testing of the algorithm are presented. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 86–94, January–February 2006.     

A modified high-order adaptive output feedback control algorithm for linear plants

A modified output feedback control algorithm is proposed for a linear plant subjected to parametric uncertainty; its efficiency is validated and the results of numerical simulations are presented.