Criterion of the solvability of matrix equations of the Lyapunov type

Бойчук, А. А.; Krivosheya, S. A.

doi:10.1007/bf02513089

Cited by 26 publications

(10 citation statements)

References 3 publications

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“…Вообще говоря, предполагаем α β δ γ. Условия разрешимости и структура решения линейной дифференциальной системы (3) были приведены в монографии [Беллман, 1969]. Конструктивные условия разрешимости и структура периодического решения линейной дифференциальной системы (3) при условии α = β получены в статье [Boichuk, Krivosheya, 2001] с использованием обобщенного обращения матриц и операторов, описанного в статье [Boichuk, Krivosheya, 1998]. Таким образом, задача о построении решений матричного дифференциального уравнения (1), подчиненного краевому условию (2), является обобщением периодической задачи для матричного уравнения Риккати [Boichuk, Krivosheya, 2001], нетеровых краевых задач для систем обыкновенных дифференциальных уравнений [Boichuk, Samoilenko, 2004;Boichuk, 1988;Бойчук, 1990], а также задачи Коши для матричного уравнения Бернулли [Деревенский, 2008, ч.…”

Section: постановка задачиunclassified

Nonlinear boudary value problem in the case of parametric resonance

Chujko¹,

Nesmelova²,

Sysoev³

2015

CRM

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Найдены необходимые и достаточные условия существования решений нелинейной матричной краевой задачи для системы обыкновенных дифференциальных уравнений в случае параметрического резонанса. Построена сходящаяся итерационная схема для нахождения приближений к решению нелинейной матричной краевой задачи для системы обыкновенных дифференциальных уравнений в случае параметрического резонанса. В качестве примера применения построенной итерационной схемы найдены приближения к решениями периодической краевой задачи для уравнения типа Риккати с параметрическим возмущением. Для контроля точности найденных приближений к решениям периодической краевой задачи для уравнения типа Риккати использованы невязки этих приближений. Ключевые слова: нелинейная нетерова краевая задача, матричные дифференциальные уравнения, обобщенный оператор Грина, параметрический резонанс

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Section: постановка задачиunclassified

Nonlinear boudary value problem in the case of parametric resonance

Chujko¹,

Nesmelova²,

Sysoev³

2015

CRM

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“…A lot of applied problems [1], [3], [4] come down to solving systems of algebraic equations. Therefore, it is dicult to overestimate the role of methods for nding solutions of such systems in applied mathematics.…”

Section: Introductionmentioning

confidence: 99%

Solving Systems of Second-Order Matrix Equations

Nedashkovska

Kukharska

2022

VAM

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In view of the growing role of unmanned aerial vehicles in the military sphere and the increasing automation of enterprises, the problem of improving control quality is urgent. Matrix equations and systems of matrix equations are widely used in some applied disciplines, in particular, in optimization problems of control systems. However, there is no universal approach to solving all problems of this class: only methods of solving the most popular matrix equations, such as the Sylvester, Riccati, and Lyapunov equations, have been developed. There is an opportunity to explore this topic in more detail in the papers of Beavers A.N., Denman E.D., Boichuk A.A., Krivosheya S.A. and Kramer K. In one of our previous articles, a method for solving systems of algebraic equations over the eld of real numbers was proposed. In this paper, the previously considered approach is generalized, and an approximate solution scheme for systems of polynomial matrix equations of the second degree with many unknowns is presented. A recurrent formula for calculating the approximate solution of systems in the form of a continued matrix fraction is also given. The convergence of the proposed method is investigated based on Vorpitsky's sucient condition. The results of numerical experiments that conrm the validity of theoretical calculations and the eectiveness of the proposed scheme for the approximate solution of matrix equations are presented.

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“…Numerous examples of various problems that can be studied by operator methods of perturbation theory can be found in the monograph [6]. The present paper uses the theory of generalized inverse and Moore-Penrose pseudoinverse operators [9] - [13] to construct perturbation theory for the Lyapunov type equation [14], [15] in resonance cases [16]. We investigate bifurcation conditions of solutions of boundary value problems in the Hilbert space when considering problem has the set of solutions not for any right hand side of the equation.…”

Section: Introductionmentioning

confidence: 99%

“…3. Under condition (16) or (17), the strong generalized solutions, classical solutions, and quasi-solutions of the boundary value problem (14), (15) have the form…”

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confidence: 99%

Exponential Dichotomy and Bifurcation Conditions of Solutions of the Hamiltonian Operators Boundary Value Problems in the Hilbert Space

Oleksandr¹

2018

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Criterion of the solvability of matrix equations of the Lyapunov type

Cited by 26 publications

References 3 publications

Nonlinear boudary value problem in the case of parametric resonance

Nonlinear boudary value problem in the case of parametric resonance

Solving Systems of Second-Order Matrix Equations

Exponential Dichotomy and Bifurcation Conditions of Solutions of the Hamiltonian Operators Boundary Value Problems in the Hilbert Space

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