On solvability of linear matrix boundary-value problem

S. M. Chuiko

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2018), pp. 86-97.

Résumé

We find conditions of solvability and a construction of generalized Green's operator of linear matrix boundary-value problem; at that, we propose an operator which reduces a linear matrix equation to traditional linear Noetherian boundary-value problem. To solve a linear matrix system we use an operator which reduces a linear matrix equation to a linear algebraic equation with rectangular matrix.

Détail

Keywords: Green operator, Noetherian boundary-value problem, matrix differential equation.

Bibliographie

[1] Azbelev N. V., Maksimov N. P., Rakhmatullina L. F., Vvedenie v teoriyu funktsionalno-differentsialnykh uravnenii, Nauka, M., 1991

[2] Boichuk A. A., Samoilenko A. M., Generalized inverse operators and Fredholm boundary-value problems, 2nd Ed., De Gruyter, Berlin–Boston, 2016 | MR | Zbl

[3] Bellman R., Vvedenie v teoriyu matrits, Nauka, M., 1969 | MR

[4] Boichuk A. A., Krivosheya S. A., “Criterion of the solvability of matrix equations of the Lyapunov type”, Ukr. Math. J., 50:8 (1998), 1162–1169 | DOI | MR | Zbl

[5] Chuiko S. M., “A generalized matrix differential-algebraic equation”, J. Math. Sci. (N. Y.), 210:1 (2015), 9–21 | DOI | MR | Zbl

[6] Campbell S. L., Singular systems of differential equations, Pitman Advanced Publishing Program, San Francisco–London–Melbourne, 1980 | MR | Zbl

[7] Boyarintsev Yu. E., Chistyakov V. F., Algebro-differentsialnye sistemy. Metody resheniya i issledovaniya, Nauka, Novosibirsk, 1998 | MR

[8] Chuiko S. M., “Green's operator of a generalized matrix linear differential-algebraic boundary value problem”, Sib. Math. J., 56:4 (2015), 752–760 | DOI | MR | Zbl

[9] Boichuk A. A., Krivosheya S. A., “A critical periodic boundary value problem for a matrix Riccati equations”, Differential Equat., 37:4 (2001), 464–471 | DOI | MR | Zbl

[10] Chuiko S. M., “O reshenii lineinykh matrichnykh uravnenii”, Nauchn. vestn. Kharkovsk. un-ta im. V. N. Karazina. Ser. matem., prikladnaya matem., mekhan., 81 (2015), 28–34

[11] Voevodin V. V., Kuznetsov Yu. A., Matritsy i vychisleniya, Nauka, M., 1984 | MR

[12] Samoilenko A. M., Shkil M. I., Yakovets V. P., Liniini sistemi diferentsialnikh rivnyan z virodzhennyam, Vischa shkola, Kiiv, 2000

[13] Boichuk A. A., Pokutnyi A. A., Chistyakov V. F., “Application of perturbation theory to the solvability analysis of differential algebraic equations”, Comput. Math. and Math. Physics, 53:6 (2013), 777–788 | DOI | MR | Zbl

[14] Chuiko S. M., “Lineinye neterovy kraevye zadachi dlya differentsialno-algebraicheskikh sistem”, Komp. issledov. i modelirovanie, 5:5 (2013), 769–783

[15] Laptinskii V. N., Makovetskii I. I., “K konstruktivnomu analizu dvukhtochechnoi kraevoi zadachi dlya nelineinogo uravneniya Lyapunova”, Differents. uravneniya, 41:7 (2005), 994–996 | MR | Zbl

[16] Bondarev A. N., Laptinskii V. N., “Multipoint boundary value problem for the Lyapunov equation in the case of strong degeneration of the boundary conditions”, Differ. Equ., 47 (2011), 778–786 | DOI | MR | Zbl

[17] Chuiko S. M., “Lineinaya kraevaya zadacha dlya matrichnogo differentsialnogo uravneniya”, O seminare po kachestvennoi teorii differentsialnykh uravnenii v Moskovskom universitete, Differents. uravneniya, 52:11 (2016), 1578–1579

[18] Chuiko S. M., “Operator Grina lineinoi neterovoi kraevoi zadachi dlya matrichnogo differentsialnogo uravneniya”, Dinamicheskie sistemy, 4(32):1–2 (2014), 101–107 | Zbl