Abstract
We prove the unique solvability of a nonlinear controlled functional operator equation in a Banach ideal space. We also establish sufficient conditions for the global solvability of all controls from a pointwise bounded set, provided that some majorant equation for the given family of these controls is globally solvable. We give examples of controlled boundary value problems reducible to the considered equation.
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References
V. I. Sumin, “Functional-Operator Volterra Equations in Theory of Optimal Control with Distributed Parameters,” Sov. Phys. Dokl. 305(5), 1056–1059 (1989).
V. I. Sumin, “The Features ofGradientMethods for Distributed Optimal Control Problems,” Zhurn. Vychisl. Matem. i Matem. Fiz. 30(1), 3–21 (1990).
V. I. Sumin, Functional Volterra Equations in the Optimal Control Theory for Distributed Systems (Nizhegorodsk. Gos. Univ., N. Novgorod, 1992), Vol. 1 [in Russian].
V. I. Sumin, “On Functional Volterra Equations,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 9, 67–77 (1995) [Russian Mathematics (Iz. VUZ) 39 (9), 65–75 (1995)].
V. I. Sumin, “The Controlled Functional Volterra Equations in Lebesgue Spaces,” Vestn. Nizhegorodsk. Univ. im N. I. Lobachevskogo. Ser. Mat. Model. i Optim. Uprav. (NNGU, N. Novgorod, 1998), No. 2, 138–151.
V. I. Sumin and A. V. Chernov, “On Sufficient Stability Conditions for the Existence of Global Solutions of Volterra Operator Equations,” Vestn. Nizhegorodsk. Univ. im N. I. Lobachevskogo. Ser. Mat. Model. i Optim. Uprav. (Nizhegorodsk. Gos. Univ., N. Novgorod, 2003), No. 1, 39–49.
V. I. Sumin and A. V. Chernov, Volterra Operator Equations in Banach Spaces: Stability of Existence of Global Solutions, Available from VINITI, No. 1198-V00 (Nizhni Novgorod, 2000).
L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1984) [in Russian].
V. I. Sumin and A. V. Chernov, “Operators in Spaces of Measurable Functions: the Volterra Property and Quasinilpotency,” Differents. Uravneniya 34(10), 1402–1411 (1998).
B. Sh. Mordukhovich, Approximation Methods in Problems of Optimization and Control (Nauka, Moscow, 1988) [in Russian].
V. S. Pugachev, Lectures on Functional Analysis (Mosk. Aviat. Inst., Moscow, 1996) [in Russian].
O. A. Ladyzhenskaya, The Mixed Problem for a Hyperbolic Equation (GITTL, Moscow, 1953) [in Russian].
V.M. Fedorov, A Course in Functional Analysis (Lan’, St. Petersburg, 2005) [in Russian].
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Original Russian Text © A.V. Chernov, 2011, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2011, No. 3, pp. 95–107.
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Chernov, A.V. A majorant criterion for the total preservation of global solvability of controlled functional operator equation. Russ Math. 55, 85–95 (2011). https://doi.org/10.3103/S1066369X11030108
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DOI: https://doi.org/10.3103/S1066369X11030108