Linear Noetherian boundary value problem for a matrix difference-algebraic Lyapunov equation
Abstract
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. Standzhitsky. 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. 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 linear
boundary value problems for the differential-algebraic boundary value problem for a matrix Lyapunov equation, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear differential-algebraic boundary value problem for a matrix Lyapunov equation.
In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a matrix Lyapunov equation. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a matrix Lyapunov equation in the critical case in this article can be transferred to the seminonlinear differential-algebraic boundary value problem for a matrix Lyapunov equation.
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References
А. А. Boichuk, S. М. Chuiko, Ya. V. Kalinichenko. Linear Noetherian boundary value problem for the matrix difference equation, Ukrainian math. journal, – 2020. – 3. 72. – P. 386-402. DOI: 10.37863/umzh.v72i3.637.
А. А. Boichuk. Boundary value problems for systems of difference equations, Ukrainian mathematical journal, – 1997. – 6. 49. – P. 832-835.
S. M. Chuiko. The Green’s operator of a generalized matrix linear differential-algebraic boundary value problem, Siberian Mathematical Journal. – 2015. – 4. 56. – P. 752-760. DOI: 10.17377/smzh.2015.56.417.
J. R. Magnus. L-structured matrices and linear matrix equations, Linear algebra and its appl., – 1983. – 14. – P. 67-88. DOI: https://doi.org/10.1080/03081088308817543
S. M. Chuiko, E. V. Chuiko, Ya. V. Kalinichenko. Boundary value problems for systems of linear difference-agebraic equations, Nonlinear oscillations. – 2019. – 4. 22. – P. 560-573.
A. A. Boichuk, A. M. Samoilenko. Generalized inverse operators and Fredholm boundary-value problems (2-th edition). – Berlin; Boston: De Gruyter, 2016, 298 p. DOI: https://doi.org/10.1515/9783110378443
V. I. Korobov, M. O. Bebiya. Stabilization of one class of nonlinear systems, uncontrollable in the first approximation, Rep. NAS Ukraine. – 2014. – 2. – P. 20-25.
A. N. Tikhonov, V. Ya. Arsenin. Methods for solving ill-posed problems. 1986. Nauka, М., 288 p.
S. G. Krein. Linear equations in a Banach space. 1971. Nauka, М., 104 p.
S. M. Chuiko, Ya. V. Kalinichenko. On the regularization of the Cauchy problem for a system of linear difference equations, Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, – 2018. – 88. – P. 27-34. DOI: https://doi.org/10.26565/2221-5646-2018-88-03
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