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## Archive of Issues

Russia Nizhni Novgorod
Year
2017
Volume
27
Issue
2
Pages
162-177
 Section Mathematics Title Regularization of the Pontryagin maximum principle in the problem of optimal boundary control for a parabolic equation with state constraints in Lebesgue spaces Author(-s) Gorshkov A.A.a, Sumin M.I.a Affiliations Nizhni Novgorod State Universitya Abstract A convex optimal control problem is considered for a parabolic equation with a strictly uniformly convex cost functional, with boundary control and distributed pointwise state constraints of equality and inequality type. The images of the operators that define pointwise state constraints are embedded into the Lebesgue space of integrable with $s$-th degree functions for $s\in(1,2)$. In turn, the boundary control belongs to Lebesgue space with summability index $r\in (2,+\infty)$. The main results of this work in the considered optimal control problem with pointwise state constraints are the two stable, with respect to perturbation of input data, sequential or, in other words, regularized principles: Lagrange principle in nondifferential form and Pontryagin maximum principle. Keywords optimal boundary control, parabolic equation, sequential optimization, dual regularization, stability, pointwise state constraint in the Lebesgue space, Lagrange principle, Pontryagin's maximum principle UDC 517.97 MSC 47A52 DOI 10.20537/vm170202 Received 10 November 2016 Language Russian Citation Gorshkov A.A., Sumin M.I. Regularization of the Pontryagin maximum principle in the problem of optimal boundary control for a parabolic equation with state constraints in Lebesgue spaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 2, pp. 162-177. References Vasil’ev F.P. Metody optimizatsii (Optimization methods), vols. 1, 2, Moscow: Moscow Center for Continuous Mathematical Education, 2011, 620 p., 432 p. Sumin M.I. Stable sequential convex programming in a Hilbert space and its application for solving unstable problems, Comput. Math. Math. Phys., 2014, vol. 54, issue 1, pp. 22-44. DOI: 10.1134/S0965542514010138 Sumin M.I. A regularized gradient dual method for the inverse problem of a final observation for a parabolic equation, Comput. Math. Math. Phys., 2004, vol. 44, issue 11, pp. 1903-1921. Sumin M.I. Duality-based regularization in a linear convex mathematical programming problem, Comput. Math. Math. Phys., 2007, vol. 47, issue 4, pp. 579-600. DOI: 10.1134/S0965542507040045 Sumin M.I. Nekorrektnye zadachi i metody ikh resheniya. Materialy k lektsiyam dlya studentov starshikh kursov: Uchebnoe posobie (Ill-posed problems and their solutions. Materials for lectures for senior students: Textbook), Nizhnii Novgorod: Lobachevsky State University of Nizhnii Novgorod, 2009, 289 p. Sumin M.I. Regularized parametric Kuhn-Tucker theorem in a Hilbert space, Comput. Math. Math. Phys., 2011, vol. 51, issue 9, pp. 1489-1509. DOI: 10.1134/S0965542511090156 Sumin M.I. On the stable sequential Kuhn-Tucker theorem and its applications, Applied Mathematics, 2012, vol. 3, issue 10, pp. 1334-1350. DOI: 10.4236/am.2012.330190 Raymond J.-P., Zidani H. Pontryagin's principle for state-constrained control problems governed by parabolic equations with unbounded controls, SIAM J. Control Optim., 1998, vol. 36, issue 6, pp. 1853-1879. DOI: 10.1137/S0363012996302470 Casas E., Raymond J.-P., Zidani H. Pontryagin's principle for local solutions of control problems with mixed control-state constraints, SIAM J. Control Optim., 2000, vol. 39, issue 4, pp. 1182-1203. DOI: 10.1137/S0363012998345627 Sumin M.I. Suboptimal control of a semilinear elliptic equation with a phase constraint and a boundary control, Differential Equations, 2001, vol. 37, issue 2, pp. 281-300. DOI: 10.1023/A:1019226011838 Sumin M.I. Parametric dual regularization for an optimal control problem with pointwise state constraints, Comput. Math. Math. Phys., 2009, vol. 49, issue 12, pp. 1987-2005. DOI: 10.1134/S096554250912001X Sumin M.I. Regularized sequential Pontryagin maximum principle in the convex optimal control with pointwise state constraints, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2012, issue 1 (39), pp. 130-133 (in Russian). Sumin M.I. Stable sequential Pontryagin maximum principle in optimal control problem with state constraints, XII Vserossiiskoe soveshchanie po problemam upravleniya (VSPU–2014): Trudy (Proc. XII All-Russia Conf. on Control Problems (RCCP–2014)), Moscow: Inst. of Control Problems, 2014, pp. 796-808 (in Russian). Sumin M.I. Stable sequential Pontryagin maximum principle in optimal control for distributed systems, Dinamika sistem i protsessy upravleniya: Trudy Mezhdunarodnoi konferentsii (System dynamic and control processes: Proceedings of Int. Conf. Dedicated to the 90th Anniversary of Academician N.N. Krasovskii), Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, 2015, pp. 301-308 (in Russian). Sumin M.I. Subdifferentiability of value functions and regularization of Pontryagin maximum principle in optimal control for distributed systems, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 2015, vol. 20, issue 5, pp. 1461-1477 (in Russian). Sumin M.I. On the stable sequential Lagrange principle in the convex programming and its applications for solving unstable problems, Trudy Inst. Mat. Mekh. Ural Otd. Ross. Akad. Nauk, 2013, vol. 19, no. 4, pp. 231-240 (in Russian). Warga J. Optimal control of differential and functional equations, New York: Academic Press, 1972, 531 p. Translated under the title Optimal'noe upravlenie differentsial'nymi i funktsional'nymi uravneniyami, Moscow: Nauka, 1977, 624 p. Sumin M.I. Dual regularization and Pontryagin's maximum principle in a problem of optimal boundary control for a parabolic equation with nondifferentiable functionals, Proc. Steklov Inst. Math., 2011, vol. 275, suppl. 1, pp. 161-177. DOI: 10.1134/S0081543811090124 Gorshkov A.A. On dual regularization in convex programming in uniformly convex space, Vestn. Nizhegorod. Univ. N.I. Lobachevskogo, 2013, no. 3 (1), pp. 172-180 (in Russian). Gorshkov A.A., Sumin M.I. The stable Lagrange principle in sequential form for the problem of convex programming in uniformly convex space and its applications, Russian Mathematics, 2015, vol. 59, issue 1, pp. 11-23. DOI: 10.3103/S1066369X15010028 Gorshkov A.A. Regularized Pontryagin maximum principle in optimal control for a parabolic equation with phase constraints in Lebesgue spaces, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 2015, vol. 20, issue 5, pp. 1104-1110 (in Russian). Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Linear and quasilinear equations of parabolic type, Providence, R.I.: AMS, 1968, 648 p. Vladimirov A.A., Nesterov Yu.E., Chekanov Yu.N. On uniformly convex functionals, Mosc. Univ. Comput. Math. Cybern., 1978, no. 3, pp. 10-21. Ekeland I., Temam R. Convex analysis and variational problems, SIAM, 1999, 402 p. Aubin J.-P., Ekeland I. Applied nonlinear analysis, New York: John Wiley and Sons, 1988, 584 p. Mordukhovich B.S. Variational analysis and generalized differentiation. I: Basic Theory, Berlin: Springer, 2006, 595 p. Full text