On a Reduction of the Order in a Differential-Algebraic System

Чуйко, С. М.

doi:10.1007/s10958-018-4054-z

Cited by 31 publications

(18 citation statements)

References 10 publications

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“…By analogy with the classication of dierential-algebraic equations [3,6], as well as with the classication of impulse boundary problems for ordinary dierential equations [2,7,8] under condition (2)), in the case of boundedness of matrices A + (k)B(k), A + (k)f (k), we will say that the system of linear dierence-algebraic equations (1) is degenerated. Note that, in contrast to the traditional system of linear dierence equations [1], the solution of the system of linear dierencealgebraic equations (1) under condition (2) depends on an arbitrary bounded vector function ν 0 (k) ∈ R ρ 0 .…”

Section: Nondegenerate Systems Of Linear Dierence-algebraic Equationsmentioning

confidence: 99%

Boundary value problems for systems of non-degenerate difference-algebraic equations

2019

MAMM

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The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N.N. Luzin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of boundary value problems for difference equations, initiated in A.A. Markov, S.N. Bernstein, Ya.S. Besikovich, A.O. Gelfond, S.L. Sobolev, V.S. Ryaben'kii, V.B. Demidovich, A. Halanay, G.I. Marchuk, A.A. Samarskii, Yu.A. Mitropolsky, D.I. Martynyuk, G.M. Vayniko, A.M. Samoilenko, O.A. Boichuk and O.M. Stanzhitsky. Study of nonlinear singularly perturbed boundary value problems for difference equations in partial differences is devoted to the work of V.P. Anosov, L.S. Frank, P.E. Sobolevskii, A.L. Skubachevskii and A. Asheraliev. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A.M. Samoilenko and O.A. Boichuk on linear boundary value problems for difference-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green's operator of the Cauchy problem and the generalized Green operator of a linear boundary value problem for a difference-algebraic equation. The solvability conditions are found in the paper, as well as the construction of a generalized Green operator for the Cauchy problem for a difference-algebraic system. The solvability conditions are found, as well as the construction of a generalized Green operator for a linear Noetherian difference-algebraic boundary value problem. An original classification of critical and noncritical cases for linear difference-algebraic boundary value problems is proposed.

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Section: Nondegenerate Systems Of Linear Dierence-algebraic Equationsmentioning

confidence: 99%

Boundary value problems for systems of non-degenerate difference-algebraic equations

2019

MAMM

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“…Äîñëiäaeó¹ìî çàäà÷ó ïðî ïîáóäîâó ðîçâ'ÿçêiâ z(t) ∈ C 1 [a, b] ëiíiéíîä èôåðåíöiàëüíî-àëãåáðà¨÷íî¨êðàéîâî¨çàäà÷i [1,2,3] A(t)z (t) = B(t)z(t) + f (t), z(•) = α, α ∈ R k ;…”

Section: ëIíiéíi êðàéîâI çàäà÷I äëÿ íåâèðîäAeåíèõ äèôåðåíöIàëüíî-àëãåmentioning

confidence: 99%

“…Òàêèì ÷èíîì, çà óìîâè ρ 0 = 0, ñèñòåìà (3), ðîçâ'ÿçíà âiäíîñíî ïîõiäíî¨, çàëåaeèòü âiä äîâiëüíî¨íåïåðåðâíî¨âåêòîð-ôóíêöi¨ν 0 (t). Ïîçíà÷èìî X 0 (t) íîðìàëüíó ôóíäàìåíòàëüíó ìàòðèöþ X 0 (t) = A + (t)B(t)X 0 (t), X 0 (a) = I n îòðèìàíî¨òðàäèöiéíî¨ñèñòåìè çâè÷àéíèõ äèôåðåíöiàëüíèõ ðiâíÿíü (3). Âiäìiòèìî, ùî íîðìàëüíà ôóíäàìåíòàëüíà ìàòðèöÿ X 0 (t) íåâèðîäaeåíà.…”

Section: ëIíiéíi êðàéîâI çàäà÷I äëÿ íåâèðîäAeåíèõ äèôåðåíöIàëüíî-àëãåunclassified

“…Çà óìîâè (2) ñèñòåìà (3), à îòaeå i ñèñòåìà (1), ìà¹ ðîçâ'ÿçêè âèãëÿäó z(t, c) = X 0 (t)c + K f (s), ν 0 (s) (t), c ∈ R n , äå K f (s), ν 0 (s) (t) := X 0 (t) t a X −1 0 (s) F 0 (s, ν 0 (s)) ds óçàãàëüíåíèé îïåðàòîð Ãðiíà çàäà÷i Êîøi z(a) = 0 äëÿ äèôåðåíöiàëüíî-àëãåáðà¨÷íî¨ñèñòåìè (1). Îñêiëüêè çà óìîâè (2) ñèñòåìà (1) ðîçâ'ÿçíà äëÿ äîâiëüíî¨íåîäíîðiäíîñòi f (t), òî âèïàäîê (2) áóäåìî íàçèâàòè íåâèðîäaeåíèì [3,4,5]. Ïiäñòàâëÿþ÷è çàãàëüíèé ðîçâ'ÿçîê çàäà÷i Êîøi z(a) = c äëÿ äèôåðåíöiàëüíî-àëãåáðà¨÷íîãî ðiâíÿííÿ (1) â êðàéîâó óìîâó (1), ïðèõîäèìî äî ëiíiéíîãî àëãåáðà¨÷íîãî ðiâíÿííÿ, ðîçâ'ÿçíîãî òîäi i òiëüêè òîäi, êîëè [4,5]…”

Section: ëIíiéíi êðàéîâI çàäà÷I äëÿ íåâèðîäAeåíèõ äèôåðåíöIàëüíî-àëãåmentioning

confidence: 99%

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On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case

2019

MAMM

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The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of nonlinear boundary-value problems for ordinary differential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the nonlinear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear oscillations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the nonlinear boundary value problems for the differential algebraic equations, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the nonlinear differential algebraic boundary value problems. In this article we found the conditions of the existence and constructed the iterative scheme for finding the solutions of the weakly nonlinear Noetherian differential-algebraic boundary value problem. The proposed scheme of the research of the nonlinear differential-algebraic boundary value problems in the article can be transferred to the nonlinear matrix differential-algebraic boundary value problems. On the other hand, the proposed scheme of the research of the nonlinear Noetherian differential-algebraic boundary value problems in the critical case in this article can be transferred to the autonomous seminonlinear differential-algebraic boundary value problems.

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“…Как известно [6,7,8], любая (m × n)− матрица A(k) в определенном базисе может быть представлена в виде стандартного разложения…”

Section: стандартное разложение матрицыunclassified

On the regularization of the Cauchy problem for a system of linear difference equations

2018

MAMM

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The article proposes unusual regularization conditions as well as a scheme for finding solutions of the linear Cauchy problem for a system of difference equations in the critical case, significantly using the Moore-Penrose matrix pseudo-inversion technology. The problem posed in the article continues the study of the regularization conditions for linear Noetherian boundary value problems in the critical case given in the monographs by S.G. Krein, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko and A.A. Boichuk. The general case is studied in which a linear bounded operator corresponding to a homogeneous part of a linear Cauchy problem has no inverse. In the article, a generalized Green operator is constructed and the type of a linear perturbation of a regularized linear Cauchy problem for a system of difference equations in the critical case is found. The proposed regularization conditions, as well as the scheme for finding solutions to linear Cauchy problems for a system of difference equations in the critical case, are illustrated in details with examples. In contrast to the earlier articles of the authors, the regularization problem for a linear Cauchy problem for a system of difference equations in the critical case has been resolved constructively, and sufficient conditions has been obtained for the existence of a solution to the regularization problem.

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On a Reduction of the Order in a Differential-Algebraic System

Cited by 31 publications

References 10 publications

Boundary value problems for systems of non-degenerate difference-algebraic equations

Boundary value problems for systems of non-degenerate difference-algebraic equations

On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case

On the regularization of the Cauchy problem for a system of linear difference equations

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