For a second-order parabolic equation with degenerations, we construct the solution of the problem of optimal control for systems described by the first boundary-value problem with internal and startup controls. The coefficients of parabolic equation have power singularities of any order in time and space variables on a certain set of points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. M. Isaryuk and I. D. Pukal’skii, “The boundary-value problems for parabolic equations with a nonlocal condition and degenerations,” Ukr. Mat. Visnyk, 11, No. 4, 480–496 (2014); English translation: J. Math. Sci., 207, No. 1, 26–38 (2015); https://doi.org/10.1007/s10958-015-2352-2.
I. M. Isaryuk and I. D. Pukal’s’kyi, “Nonlocal problem with oblique derivative and the optimization problem for parabolic equations,” Nauk. Visn. Chernivtsi Univ., Mat., Issue 528, 62–69 (2010).
I. M. Isaryuk and I. D. Pukal’s’kyi, “Nonlocal parabolic problem with degeneration,” Ukr. Mat. Zh., 66, No. 2, 208–215 (2014); English translation: Ukr. Math. J., 66, No. 2, 232–241 (2014); https://doi.org/10.1007/s11253-014-0925-8.
e, Linear and Quasilinear Equations of Parabolic Type, Naukа, Moscow (1967); English translation: Transl. Math. Monogr., Vol. 23, AMS, Providence, RI (1968.
E. M. Landis, Second-Order Elliptic and Parabolic Equations [in Russian], Nauka, Moscow (1971).
M. I. Matiichuk, Parabolic and Elliptic Boundary-Value Problems with Singularities [in Ukrainian], Prut, Chernivtsi (2003).
B. Yo. Ptashnyk, V. S. Il’kiv, I. Ya. Kmit’, and V. M. Polishchuk, Nonlocal Boundary-Value Problems for Partial Differential Equations [in Ukrainian], Naukova Dumka, Kyiv (2002).
I. D. Pukal’s’kyi, Boundary-Value Problems for Nonuniformly Parabolic and Elliptic Equations with Degenerations and Singularities [in Ukrainian], Ruta, Chernivtsi (2008).
I. D. Pukal’s’kyi, “A parabolic boundary-value problem and a problem of optimal control,” Mat. Metody Fiz.-Mekh. Polya, 52, No. 4, 34–41 (2009); English translation: J. Math. Sci., 174, No. 2, 159–168 (2011); https://doi.org/10.1007/s10958-011-0287-9.
I. D. Pukal’s’kyi and I. M. Isaryuk, “Nonlocal parabolic boundary-value problems with singularities,” Mat. Metody Fiz.-Mekh. Polya, 56, No. 4, 54–66 (2013); English translation: J. Math. Sci., 208, No. 3, 327–343 (2015); https://doi.org/10.1007/s10958-015-2449-7.
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs (1964); Russian translation: Mir, Moscow (1968).
S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type (Ser. Operator Theory: Adv. and Appl., Vol. 152.), Birkhäuser, Basel (2004); https://doi.org/10.1007/978-3-0348-7844-9.
H. Lange and H. Teismann, “Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state,” Math. Meth. Appl. Sci., 30, No. 13, 1483–1505 (2007); https://doi.org/10.1002/mma.849.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 63, No. 2, pp. 17–28, April–June, 2020.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Isariuk, І.М., Pukal’s’kyi, І.D. Internal and Startup Controls of the Solutions of Boundary-Value Problem for Parabolic Equations with Degenerations. J Math Sci 272, 14–28 (2023). https://doi.org/10.1007/s10958-023-06396-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06396-z