On the singular solutions of the Korteweg-de Vries equation

We consider two classes of singular solutions of the KdV equation: singular solutions of the Cauchy problem and singular traveling waves. In both cases, we establish sufficient conditions for their existence.     

Necessary and Sufficient Optimality Conditions for Regular–Singular Stochastic Differential Games with Asymmetric Information

We consider a class of regular–singular stochastic differential games arising in the optimal investment and dividend problem of an insurer under model uncertainty. The information available to the two players is asymmetric partial information and the control variable of each player consists of two components: regular control and singular control. We establish the necessary and sufficient optimality conditions for the saddle point of the zero-sum game. Then, as an application, these conditions are applied to an optimal investment and dividend problem of an insurer under model uncertainty. Furthermore, we generalize our results to the nonzero-sum regular–singular game with asymmetric information, and then the Nash equilibrium point is characterized.     

A problem with a Frankl condition on the boundary of the ellipticity domain for the Gellerstedt equation with a singular coefficient

We study a problem with the Frankl and Bitsadze-Samarskii conditions on the elliptic boundary and on the degeneration line for the Gellerstedt equation with a singular coefficient. We prove the correctness of the stated problem.     

Problem with Frankl and Bitsadze-Samarskii conditions on the degeneration line and on parallel characteristics for the gellerstedt equation with a singular coefficient

We study the well-posedness of a problem for the Gellerstedt equation with a singular coefficient and with the Frankl and Bitsadze-Samarskii conditions on the degeneration line and on parallel characteristics.The uniqueness of the solution of the considered problem is proved with the use of the extremum principle, and the existence of the solution of the problem is justified with the use of the theories of singular integral equations, Wiener-Hopf equations, and Fredholm integral equations.     

Estimates for the Cauchy function of linear singular differential equations and some applications

For higher-order linear singular equations, we find sharp estimates for the Cauchy function and its partial derivatives. We use these estimates to study the properties of solutions of singular differential inequalities and obtain optimal sufficient conditions for the unique solvability of the linear singular Cauchy problem.     

On quadrature and cubature formulas for a class of multiple singular integrals

We consider the problem of conditions for the existence of multiple singular integrals of a certain class at inner and boundary points of a domain. We obtain the quadrature and cubature formulas for calculating multiple singular integrals and present the corresponding estimates for the formulas.     

Simple characterizations of exponential tractability for linear multivariate problems

We study linear multivariate problems defined as the approximation of compact linear multivariate operators over Hilbert spaces. We provide necessary and sufficient conditions on various notions of tractability. These conditions are mainly given in terms of sums of certain functions depending on the singular values of the multivariate problem. In particular, most of these conditions do not require the ordering of these singular values, which in many cases is difficult to achieve.     

Positive solutions for third order multi-point singular boundary value problems

We study a third order singular boundary value problem with multi-point boundary conditions. Sufficient conditions are obtained for the existence of positive solutions of the problem. Recent results in the literature are significantly extended and improved. Our analysis is mainly based on a nonlinear alternative of Leray-Schauder.     

基于开关控制的线性奇异系统的二次状态反馈镇定问题

主要讨论基于开关控制的线性奇异系统的二次状态反馈镇定问题.利用二次反馈镇定的概念,给出了线性奇异系统基于异步开关控制的二次状态反馈镇定问题可解的两个充分条件.进一步,对于带有范数有界的不确定项的奇异线性系统,给出了其可以基于异步开关控制的二次状态反馈鲁棒镇定的可解性条件.     

Riemann problem and singular integral equations with coefficients generated by piecewise constant functions

We study a Riemann boundary value problem with a shift into the interior of the domain. The problem has piecewise constant coefficients that take two values. We find conditions for the existence and uniqueness of a solution of the inhomogeneous problem and formulas for the number of linearly independent solutions of the homogeneous problem. We consider scalar singular integral operators with a shift and matrix characteristic operators whose coefficients are generated by piecewise constant functions and which have automorphic properties. For these operators, we find invertibility conditions.     

On solvability of a two-sided singular control problem

We study a two-sided singular control problem in a general linear diffusion setting and provide a set of conditions under which an optimal control exists uniquely and is of singular control type. Moreover, under these conditions the associated value function can be written in a quasi-explicit form. Furthermore, we investigate comparative static properties of the solution with respect to the volatility and control parameters. Lastly we illustrate the results with two explicit examples.     

有奇异解的无约束最优化问题的一种混合法

本文研究求解含有奇异解的无约束最优化问题算法 .该类问题的一个重要特性是目标函数的Hessian阵可能处处奇异 .我们提出求解该类问题的一种梯度 -正则化牛顿型混合算法 .并在一定的条件下得到了算法的全局收敛性 .而且 ,经一定迭代步后 ,算法还原为正则化 Newton法 .因而 ,算法具有局部二次收敛性 .     

On the absence of local solutions of several evolutionary problems

We consider the existence problem for local (with respect to time) solutions of quasilinear evolutionary partial differential equations and inequalities with singular coefficients and initial conditions. We obtain sufficient conditions for instantaneous blow-up of solutions and show that the results thus obtained cannot be improved in the function class under study.     

Uniqueness in inverse obstacle scattering with conductive boundary conditions

We consider the uniqueness of the inverse obstacle scattering with conductive boundary conditions. This work is based on the original idea of Isakov for transmission boundary conditions, which utilize the solvability of the direct problem, orthogonality relations, approximations to solution of the direct problem and singular solutions. The methodology used is constructive and allows an extension to more general conditions and numerical methods.     

A Note on the p-Laplacian Equation with Singular Coefficients

We consider the solvability of the Dirichlet problem in a unit n-ball for the p-Laplacian equation with singular coefficients which are singular not only as |x| tends to 1 but also as |x| tends to 0. The existence and regularity of positive radial solutions are proved under some conditions related to parameters p, r, λ and q.     

一类四阶奇异边值问题对称正解的最优存在性

本文研究了一类四阶奇异边值问题.通过建立一个特定的锥,利用Leggett-Williams不动点定理,从而在一定的条件下得到一类四阶奇异边值问题对称正解的最优存在性,推广了奇异边值问题对称正解的最优存在性的结果.     

Center conditions and bifurcation of limit cycles at degenerate singular points in a quintic polynomial differential system

The center problem and bifurcation of limit cycles for degenerate singular points are far to be solved in general. In this paper, we study center conditions and bifurcation of limit cycles at the degenerate singular point in a class of quintic polynomial vector field with a small parameter and eight normal parameters. We deduce a recursion formula for singular point quantities at the degenerate singular points in this system and reach with relative ease an expression of the first five quantities at the degenerate singular point. The center conditions for the degenerate singular point of this system are derived. Consequently, we construct a quintic system, which can bifurcates 5 limit cycles in the neighborhood of the degenerate singular point. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of singular point quantities at degenerate singular point is linear and then avoids complex integrating operations.     

Instability of systems with linear delay reducible to singularly perturbed ones

We investigate the stability of linear systems of linear delay differential equations in the case when one of subsystems is singular. We establish sufficient conditions for the instability of solutions to such systems. We solve the considered problem with the help of the Laplace transformation.     

Dirichlet Problem for an Axisymmetric Potential in a Simply Connected Domain of the Meridian Plane

We develop a method for the reduction of the Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane to a Cauchy singular integral equation. In the case where the boundary of the domain is smooth and satisfies certain additional conditions, we regularize the indicated singular integral equation.     

D-admissible hybrid control of a class of singular systems

This paper mainly studies the problem of designing a hybrid state feedback D-admissible controller for a class of linear and nonlinear singular systems. Based on the relationship between singular discrete systems and singular delta operator systems, several necessary and sufficient conditions for a linear singular delta operator system to be D-admissible (i.e. regular, causal and all finite poles lie in a prescribed circular region) with different representations are derived. Then, the existence conditions and explicit expressions of a desirable D-admissible controller are given by means of matrix inequalities and strict linear matrix inequalities, respectively. We further extend the obtained results to singular delta operator systems with Lipschitz nonlinear perturbations, and the design methods of hybrid controller are presented for the nonlinear case as well. Finally, numerical examples as well as simulations are provided to illustrate the effectiveness of the theoretical outcomes obtained in the paper.