By using the generalized inversion theory of operators, we establish a criterion of solvability and the general form of solutions of the operator equations with control that are not everywhere solvable and of linear boundary-value problems for these operators in Banach spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. N. Krasovskii, Theory of Control of Motion [in Russian], Nauka, Moscow (1968).
R. E. Kalman, R. L. Falb, and M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York (1969).
N. N. Danilov, A Course of Mathematical Economics [in Russian], Sib. Otdelen. Ross. Akad. Nauk, Novosibirsk (2002).
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Fizmatlit, Moscow (2005).
D. E. Kirk, Optimal Control Theory: An Introduction, Dover Publ., New York (2004).
O. A. Kapustyan and O. K. Mazur, “Solvability of the problem of optimal control with minimal energy for one parabolic boundaryvalue problem with nonlocal boundary conditions,” Zh. Obchysl. Prykl. Mat., 120, No. 3, 6–10 (2015).
V. I. Zubov, “Construction of program motions in linear controlled systems,” Differents. Uravn., 6, No. 4, 632–633 (1970).
A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Normally Solvable Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (2019).
O. A. Boichuk, E. S. Voitushenko, and L. M. Shehda, “A Noetherian impulsive control problem,” Nel. Kolyv., 19, No. 3, 362–366 (2016); English translation: J. Math. Sci., 226, No. 3, 254–259 (2017).
I. A. Bondar, “Control conditions for integrodifferential equations with degenerate kernel that are not always solvable and boundaryvalue problems for these equations,” Bukovyn. Mat. Zh., 4, No. 1-2, 13–17 (2016).
I. Ts. Gokhberg and N. Ya. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Operators [in Russian], Shtiintsa, Kishinev (1973).
M. M. Popov, “Complementable spaces and some problems of the contemporary geometry of Banach spaces,” Mat. S’ohodni’07, Issue 13, 78–116 (2007).
S. G. Krein, Linear Equations in Banach Spaces [in Russian], Nauka, Moscow (1971).
V. F. Zhuravlev, “Solvability criterion and representation of solutions of n-normal and d-normal linear operator equations in a Banach space,” Ukr. Mat., Zh., 62, No. 2, 167–182 (2010); English translation: Ukr. Math. J., 62, No. 2, 186–202 (2010).
V. F. Zhuravlev, N. P. Fomin, and P. N. Zabrodskiy, “Conditions of solvability and representation of the solutions of equations with operator matrices,” Ukr. Mat. Zh., 71, No 4, 471–485 (2019); English translation: Ukr. Math. J., 71, No 4, 537–553 (2019).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 5, pp. 602–616, May, 2021. Ukrainian DOI: 10.37863/umzh.v73i5.6537.
Rights and permissions
About this article
Cite this article
Boichuk, O.A., Zhuravliov, V.P. Boundary-Value Problems with Control for Operator Equations in Banach Spaces. Ukr Math J 73, 701–717 (2021). https://doi.org/10.1007/s11253-021-01954-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-021-01954-7