Abstract
In this paper, we examine the unique solvability (in the sense of strong solutions) of the Cauchy problem for a linear inhomogeneous equation in a Banach space solved with respect to the Caputo fractional derivative. We assume that the operator acting on the unknown function in the right-hand side of the equation generates an analytic resolving operator family for the corresponding homogeneous equation. We obtain a representation of a strong solution of the Cauchy problem and examine the solvability of optimal control problems with a convex, lower semicontinuous, lower bounded, coercive functional for the equation considered. The general results obtained are used to prove the existence of an optimal control in problems with specific functionals. Abstract results obtained for a control system described by an equation in a Banach space are illustrated by examples of optimal control problems for a fractional equation whose special cases are the subdiffusion equation and the diffusion wave equation.
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References
E. G. Bajlekova, Fractional evolution equations in Banach spaces, Ph.D. thesis, Eindhoven (2001).
P. Clément, H. J. A. M. Hoijmans, S. Angenent, C. J. van Duijn, and B. de Pagier, One-Parameter Semigroups, North-Holland, Amsterdam (1987).
V. E. Fedorov, D. M. Gordievskikh, and M. V. Plekhanova, “Equations in Banach spaces with a degenerate operator under the fractional derivative,” Differ. Uravn., 51, No. 10, 1367–1375 (2015).
V. E. Fedorov, E. A. Romanova, and A. Debbouche, “Analytic in a sector resolving families of operators for degenerate evolution equations of a fractional order,” Sib. Zh. Chist. Prikl. Mat., 16, No. 2, 93–107 (2016).
A. V. Fursikov, Optimal Control for Distributed Systems. Theory and Applications [in Russian], Nauchnaya Kniga, Novosibirsk (1999).
V. A. Kostin, “On the Solomyak–Yosida theorem for analytic semigroups,” Algebra Anal., 11, No. 1, 118–140 (1999).
M. V. Plekhanova, “Optimal control problems for linear degenerate fractional equations,” Itogi Nauki Tekhn. Sovr. Mat. Prilozh., 149, 72–83 (2018).
M. V. Plekhanova, “Strong solutions to nonlinear degenerate fractional order evolution equations,” J. Math. Sci., 230, No. 1, 146–158 (2018).
J. Prüss, Evolutionary Integral Equations and Applications, Springer, Basel (1993).
E. A. Romanova and V. E. Fedorov, “Resolving operators of a linear degenerate evolution equation with Caputo derivative. The sectorial case,” Mat. Zametki SVFU, 23, No. 4 (92), 58–72 (2016).
M. Z. Solomyak, “Application of the theory of semigroups to the study of differential equations in Banach spaces,” Dokl. Akad. Nauk SSSR, 122, No. 5, 766–769 (1958).
K. Yosida, Functional Analysis, Springer-Verlag, Berlin–Göttingen–Heidelberg (1965).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 167, Proceedings of the IV International Scientific Conference “Actual Problems of Applied Mathematics,” Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part III, 2019.
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Plekhanova, M.V. Strong Solution and Optimal Control Problems for a Class of Fractional Linear Equations. J Math Sci 260, 315–324 (2022). https://doi.org/10.1007/s10958-022-05696-0
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DOI: https://doi.org/10.1007/s10958-022-05696-0
Keywords and phrases
- Caputo fractional derivative
- fractional evolution equation
- resolving family of operators analytic in a sector
- optimal control problem
- subdiffusion equation
- diffusion wave equation