Investigation of the spectral properties of a fourth order differential operator with finite transmission conditions

In this paper, the spectral analysis of a singular dissipative fourth order differential operator in lim‐4 case with finite transmission conditions is investigated. For this purpose, the inverse operator with explicit form is used. Finally, with the help of Liv?ic's theorem, it is proved that all root vectors of the fourth order dissipative operator in lim‐4 case with finite transmission conditions are complete in the Hilbert space. Copyright © 2015 John Wiley & Sons, Ltd.     

Spectral analysis of eigenparameter dependent boundary value transmission problems

In this paper, the singular second order differential operators are considered defined on the multi-interval. Some boundary and transmission conditions are imposed on the maximal domain functions with the spectral parameter. After constructing the differential operators associated with the boundary value transmission problems on the suitable Hilbert spaces, it is proved that these operators are the maximal dissipative operators. Finally constructing the model operators which are established with the help of the scattering functions, it is proved that all root vectors of the maximal dissipative operators are complete in the Hilbert spaces.     

On the perturbation determinants of a singular dissipative boundary value problem with finite transmission conditions

In this paper a singular dissipative boundary value problem with finite transmission conditions is investigated. Using Livšic’s theorem, it is proved that the system of all eigen and associated functions of this problem is complete in the Hilbert space.     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.     

一类具一阶奇性核的奇异积分方程

当(?)是复平面C上的光滑封闭曲线,k（z）是在(?)所围成的有界闭区域上连续.在其内部解析的函数时.借助于奇异积分算子的广义逆.讨论了具一阶奇性核的正则型奇异积分方程:　在H类中的求解问题.作为应用,作者给出了当k（z）是一类有理函数时的具体解法,从而统一并推广了 Cauchy核和Hilbert核奇异积分方程的经典结果.     

Finite-time dissipative control for singular discrete-time Markovian jump systems with actuator saturation and partly unknown transition rates

In this work, the finite-time dissipative control problem is considered for singular discrete-time Markovian jumping systems with actuator saturation and partly unknown transition rates. By constructing a proper Lyapunov–Krasonski functional and the method of linear matrix inequalities (LMIs), sufficient conditions that ensure the systems singular stochastic finite-time stability and singular stochastic finite-time dissipative are obtained. Then, the state feedback controllers are designed, and in order to get the optimal values of the dissipative level, the results are extended to LMI convex optimization problems. Finally, numerical examples are given to illustrate the validity of the proposed methods.     

On strongly elliptic singular integral operators with piecewise continuous coefficients

The concept of strongly elliptic operators is one of the main tools for approximating the solution of boundary integral equations by finite element methods (see [4-10]). In the present paper necessary and sufficient conditions for the strong ellipticity of singular integral operators with piecewise continuous matrix coefficients on a closed or open Ljapunov curve are obtained.     

Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operators

In this paper, we investigate the spectral analysis of impulsive quadratic pencil of difference operators. We first present a boundary value problem consisting one interior impulsive point on the whole axis corresponding to the above mentioned operator. After introducing the solutions of impulsive quadratic pencil of difference equation, we obtain the asymptotic equation of the function related to the Wronskian of these solutions to be helpful for further works, then we determine resolvent operator and continuous spectrum. Finally, we provide sufficient conditions guarenteeing finiteness of eigenvalues and spectral singularities by means of uniqueness theorems of analytic functions. The main aim of this paper is demonstrating the impulsive quadratic pencil of difference operator is of finite number of eigenvalues and spectral singularities with finite multiplicities which is an uninvestigated problem proposed in the literature.     

Inverse problem of acoustic scattering for centrally symmetric finite objects in two-dimensional space

Scattering theory for the wave equation in two-dimensional space, perturbed by a finite function of a radial variable, integrable everywhere except, perhaps, the origin of coordinates, is considered from the point of view of the LaxPhillips scheme. The compression operator, related to the corresponding scattering problem, is considered. It is shown that this compression has one-dimensional defect subspaces, and its characteristic operator-function is a meromorphic function, whose zeros and poles coincide, respectively, with the corresponding values of a dissipative operator and its adjoint. The solution of the inverse scattering problem is obtained by reducing it to the inverse problem with two spectra for the singular self-adjoint Sturm-Liouville operator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1649–1657, December, 1990.     

Controllability Results for Second Order Neutral Impulsive Integrodifferential Systems

In this paper, we address the problem of controllability for abstract second order neutral integrodifferential systems with impulsive conditions in Banach spaces. Furthermore, we discuss the controllability of second-order neutral integrodifferential systems in finite dimensional spaces. The results are obtained by using strongly continuous cosine families of operators and the Banach fixed point theorem.     

Banach空间一类奇异脉冲微分方程边值问题的正解

讨论了Banach空间中一类具有奇异性脉冲微分方程的边值问题,利用M(o|¨)nch不动点定理,在与相应的线性算子谱半径有关的条件下,获得了该边值问题正解的存在性.     

Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure

Nonself-adjoint, non-dissipative perturbations of possibly unbounded self-adjoint operators with real purely singular spectrum are considered under an additional assumption that the characteristic function of the operator possesses a scalar multiple. Using a functional model of a nonself-adjoint operator (a generalization of a Sz.-Nagy–Foiaş model for dissipative operators) as a principle tool, spectral properties of such operators are investigated. A class of operators with almost Hermitian spectrum (the latter being a part of the real singular spectrum) is characterized in terms of existence of the so-called weak outer annihilator which generalizes the classical Cayley identity to the case of nonself-adjoint operators in Hilbert space. A similar result is proved in the self-adjoint case, characterizing the condition of absence of the absolutely continuous spectral subspace in terms of the existence of weak outer annihilation. An application to the rank-one nonself-adjoint Friedrichs model is given.     

Variation of the parameters formula and the problem of BIBO for singular measure differential systems with impulse effect

Recently, we first presented and studied singular measure differential systems with impulsive solutions. Some fundamental theory was established. In the present paper, with the help of the Drazin inverse, the variation of the parameters formula of singular measure differential systems is obtained. As an application, the problem of boundary input and boundary output (BIBO) is investigated.     

广义分散控制系统的无穷远固定模

在控制系统的理论与应用研究中,广义系统、分散系统越来越受到人们的重视.不仅在理论上已有不少有意义的结果,而且这些理论研究有明显的实际背景.例如 Leontief的动态投入产出模型,Leslie 的人口增长模型,电力系统等.然而,在一些实际系统中,系统不仅具有广义性的特点,而且从控制角度来看也具有分散性的特点.因此我们把系统     

Singular perturbations in one-dimensional unsteady motion of a real gas : PMM vol. 39, n 6, 1975, pp. 1051–1059

Navier-Stokes equations for one-dimensional motion of gas are reduced to a special dimensionless form convenient for investigations involving a perturbation front. In new variables the transition from limit conditions of motion of an inviscid non-heat-conducting gas to the case of small but finite coefficients of viscosity and thermal conductivity, which is simulated by a perfect gas with singular perturbations induced by the indicated dissipative factors. We establish the inevitability of existence of two regions of singular perturbations, the neighborhood of the perturbation front and that of the point (line, surface) where the investigated motion is generated. The derivation of equations for both boundary layers, which is valid for a fairly general statement of problems of this kind, is presented and conditions of merging with the external (adiabatic) flow are formulated. Examples of computation of motion in boundary layers in problems of piston and point explosion are presented.     

多线性算子在Herz型Hardy空间上的有界性

本文证明了Ｒｎ上奇异积分乘积之有限和的多线性算子是从ＨＫｑ１α１,ｐ１（Ｒｎ）×… ×ＨＫｑｋαｋ,ｐｋ（Ｒｎ）到ＨＫｑα,ｐ（Ｒｎ）有界的,如果它满足由目标空间所确定的直到一定阶的消失矩条件．这些多线性算子所满足的消失矩条件,当αｊ≥０时也是必要的．而且,这里所考虑的奇异积分包括Ｃａｌｄｅｒｏｎ－Ｚｙｇｍｕｎｄ奇异积分及任意阶的分数次积分．     

Determination of coefficients for a dissipative wave equation via boundary measurements

In this paper we consider the inverse problem of recovering the viscosity coefficient in a dissipative wave equation via boundary measurements. We obtain stability estimates by considering all possible measurements implemented on the boundary. We also prove that the viscosity coefficient is uniquely determined by a finite number of measurements on the boundary provided that it belongs to a given finite dimensional vector space.     

Existence of positive solution for second-order nonlinear impulsive singular differential equations of mixed type in Banach spaces

In this paper, by constructing a closed convex set and using the fixed point theory of completely continuous operators, we investigate the existence of positive solutions for an initial value problem of second-order nonlinear impulsive singular integro-differential equations in a Banach space. The method used in this paper is different from that in the literature.     

Singular integral operators in a weighted L2-space

Cauchy singular integral operators are characterized as operators in a weighted L²-space. The integral operator arises from a singular integral equation with variable coefficients. An appropriate weight function associated with the singular integral operator is constructed, and the set of polynomials orthogonal with respect to this weight function is defined. The action of the operator on polynomial sets is studied, and the definition of the operator is extended to a weighted L²-space. In this space, the operator is shown to be bounded, and, in some cases, isometric. Formulas are developed for the composition of the singular integral operator and its one sided inverse.     

Invariant Maximal Positive Subspaces and Polar Decompositions

It is proved that invertible operators on a Krein space which have an invariant maximal uniformly positive subspace and map its orthogonal complement into a nonnegative subspace allow polar decompositions with additional spectral properties. As a corollary, several classes of Krein space operators are shown to allow polar decompositions. An example in a finite dimensional Krein space shows that there exist dissipative operators that do not allow polar decompositions.