We construct and substantiate an approximate control in the form of feedback for the problem of approximate bounded synthesis with semidefinite quality criterion for a parabolic equation containing a nonlinear term that depends regularly on a small parameter.
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J.-L. Lions, Contrô le Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris (1968).
V. I. Ivanenko and V. S. Mel’nik, Variational Methods in Problems of Control for Systems with Distributed Parameters [in Russian], Naukova Dumka, Kiev (1988).
A. I. Egorov, Optimal Control by Thermal and Diffusion Processes [in Russian], Nauka, Moscow (1978).
T. K. Sirazetdinov, Optimization of Systems with Distributed Parameters [in Russian], Nauka, Moscow (1977).
V. E. Kapustyan, “Optimal control by parabolic equations with aftereffect,” in: A. A. Ligun, A. A. Kapustyan, V. E. Kapustyan, and Yu. I. Volkov (editors), Special Problems of the Theory of Approximations and Optimal Control by Distributed Systems [in Russian], Vyshcha Shkola, Kiev (1990), pp. 75–154.
A. I. Egorov and T. F. Mikhailova, “Synthesis of optimal control by a heat process with bounded control. Part 1,” Avtomatika, No. 3, 57–61 (1990).
A. I. Egorov and T. F. Mikhailova, “Synthesis of optimal control by a heat process with bounded control. Part 2,” Avtomatika, No. 4, 31–39 (1990).
B. N. Bublik and A. I. Nevidomskii, “Synthesis of optimal concentrated control for the heat equation,” Model. Sist. Obrab. Inform., Issue 1, 78–87 (1982).
B. M. Bublyk and A. I. Nevidoms’kyi, “An optimal regulator for the heat equation with variable weight coefficients in a functional,” Dopov. Akad. Nauk Ukr. RSR, Ser. A, Fiz.-Mat. Tekhn. Nauky, No. 10, 56–58 (1985).
V. E. Kapustyan, “Optimal stabilization of solutions of a parabolic boundary-value problem by bounded concentrated control,” Probl. Upravl. Inform., No. 6, 58–67 (1999).
V. E. Belozerov and V. E. Kapustyan, Geometric Methods of Modal Control [in Russian], Naukova Dumka, Kiev (1999).
M. R. Rakhimov, “On some quadratic regulator problems for linear hyperbolic equations,” Differents. Uravn., 23, No. 1, 143–154 (1987).
M. R. Rakhimov, “On the application of the method of dynamical programming to quadratic regulator problems for linear hyperbolic equations,” Dokl. Akad. Nauk SSSR, 292, No. 3, 553–558 (1987).
O.V. Kapustyan, V. S. Mel’nik, J. Valero, and V.V. Yasinsky, Global Attractors of Multi-Valued Dynamical Systems and Evolution Equations without Uniqueness, Naukova Dumka, Kyiv (2008).
A. V. Sukretna and O. A. Kapustyan, “Approximate averaged synthesis of the problem of optimal control for a parabolic equation,” Ukr. Mat. Zh., 56, No. 10, 1384–1394 (2004).
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Translated from Neliniini Kolyvannya, Vol. 12, No. 3, pp. 291–298, July–September, 2009.
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Kapustyan, O.V., Kapustyan, O.A. & Sukretna, A.V. Approximate bounded synthesis for one weakly nonlinear boundary-value problem. Nonlinear Oscill 12, 297–304 (2009). https://doi.org/10.1007/s11072-010-0078-0
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DOI: https://doi.org/10.1007/s11072-010-0078-0