We propose an approach to the construction of solutions and quasisolutions of a boundary-value problem for the Lyapunov equation in a Banach space. If the necessary and sufficient conditions for the solvability of this boundary-value problem are satisfied, then the corresponding solutions of the problem are constructed by using the generalized inverse operator. As an example, we consider the problem in the space of bounded sequences with countably dimensional matrices.
Similar content being viewed by others
References
S. G. Krein, Linear Equations in Banach Spaces [in Russian], Nauka, Moscow (1971).
S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1967).
M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).
E. V. Panasenko and O. O. Pokutnyi, “Boundary-value problems for differential equations in a Banach space with unbounded operator in the linear part,” Nelin. Kolyv., 16, No. 4, 518–526 (2013); English translation: J. Math. Sci., 203, No. 3, 366–374 (2014).
O. A. Boichuk and S. A. Krivosheya, “Criterion for the solvability of matrix equations of the Lyapunov type,” Ukr. Mat. Zh., 50, No. 8, 1021–1026 (1998); English translation: Ukr. Math. J., 50, No. 8, 1162–1169 (1998).
A. A. Boichuk and S. A. Krivosheya, “A critical periodic boundary-value problem for a matrix Riccati equation,” Different. Equat., 37, No. 4, 464–471 (2001).
S. M. Chuiko, “On the solution of matrix Lyapunov equations,” Visn. Kharkiv. Univ., Ser. Mat. Prikl. Mat. Mekh., No. 1120, 85–94 (2014).
A. E. Bryson, Jr., and Yu-Ch. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Blaisdell Publ. Co., London (1969).
A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Generalized Inverse Operators and Noetherian Boundary-Value Problems [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1995).
A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht (2004).
O. O. Pokutnyi, “Generalized inverse operator in Fr´echet, Banach, and Hilbert spaces,” Visn. Kyiv. Nats. Univ., Ser. Fiz.-Mat. Nauk., No. 4, 158–161 (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 19, No. 2, pp. 240–246, April–June, 2016.
Rights and permissions
About this article
Cite this article
Panasenko, E.V., Pokutnyi, O.O. Boundary-Value Problems for the Lyapunov Equation in Banach Spaces. J Math Sci 223, 298–304 (2017). https://doi.org/10.1007/s10958-017-3356-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-017-3356-x