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## Archive of Issues

Russia Nizhni Novgorod; Tambov
Year
2021
Volume
31
Issue
2
Pages
265-284
 Section Mathematics Title Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems Author(-s) Sumin V.I.ab, Sumin M.I.ab Affiliations Nizhni Novgorod State Universitya, Tambov State Universityb Abstract We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L_2^m$ , the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations. Keywords convex optimal control, distributed system, functional-operator equation of Volterra type, ill-posedness, iterative regularization, duality, minimizing approximate solution, regularizing operator, Lagrange principle, Pontryagin maximum principle UDC 517.9 MSC 49K20, 39B22, 49N15, 47A52 DOI 10.35634/vm210208 Received 30 December 2020 Language Russian Citation Sumin V.I., Sumin M.I. Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 265-284. References Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal Control, Springer: Boston, 1987. https://doi.org/10.1007/978-1-4615-7551-1 Vasil’ev F.P. Metody optimizatsii (Optimization methods), vols. 1, 2, Moscow: Moscow Center for Continuous Mathematical Education, 2011. Sumin M.I. Regularized parametric Kuhn–Tucker theorem in a Hilbert space, Computational Mathematics and Mathematical Physics, 2011, vol. 51, no. 9, pp. 1489-1509. https://doi.org/10.1134/S0965542511090156 Sumin M.I. On the stable sequential Kuhn-Tucker theorem and its applications, Applied Mathematics, 2012, vol. 3, no. 10A, pp. 1334-1350. https://doi.org/10.4236/am.2012.330190 Sumin M.I. Regularized Lagrange principle and Pontryagin maximum principle in optimal control and in inverse problems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, vol. 25, no. 1, pp. 279-296 (in Russian). https://doi.org/10.21538/0134-4889-2019-25-1-279-296 Sumin M.I. On the regularization of the classical optimality conditions in convex optimal control problems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 2, pp. 252-269 (in Russian). https://doi.org/10.21538/0134-4889-2020-26-2-252-269 Warga J. Optimal control of differential and functional equations, New York; London: Academic Press, 1972. Translated under the title Optimal'noe upravlenie differentsial'nymi i funktsional’nymi uravneniyami, Moscow: Nauka, 1977. Tikhonov A.N., Arsenin V.Ya. Solutions of ill-posed problems, Washington: Winston; New York: Halsted Press, 1977. Sumin M.I. Duality-based regularization in a linear convex mathematical programming problem, Computational Mathematics and Mathematical Physics, 2007, vol. 47, issue 4, pp. 579-600. https://doi.org/10.1134/S0965542507040045 Sumin V.I. Funktsional'nye vol'terrovy uravneniya v teorii optimal'nogo upravleniya raspredelennymi sistemami (Functional Volterra equations in the theory of optimal control of distributed systems), Nizhny Novgorod: Nizhny Novgorod University, 1992. Sumin V.I., Chernov A.V. Operators in spaces of measurable functions: the Volterra property and quasinilpotency, Differential Equations, 1998, vol. 34, no. 10, pp. 1403-1411. https://zbmath.org/?q=an:0958.47014 Gokhberg I.Ts., Krein M.G. Theory and applications of Volterra operators in Hilbert space, Providence: American Mathematical Society, 1970. Sumin V.I. Volterra functional-operator equations in the theory of optimal control of distributed systems, Sov. Math., Dokl., 1989, vol. 39, issue 2, pp. 374-378. https://zbmath.org/?q=an:0695.49006 Sumin V. Volterra functional-operator equations in the theory of optimal control of distributed systems, IFAC-PapersOnLine, 2018, vol. 51, issue 32, pp. 759-764. https://doi.org/10.1016/j.ifacol.2018.11.454 Sumin V.I. Controlled Volterra functional equations and the contraction mapping principle, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, vol. 25, no. 1, pp. 262-278 (in Russian). https://doi.org/10.21538/0134-4889-2019-25-1-262-278 Kuterin F.A., Sumin M.I. The regularized iterative Pontryagin maximum principle in optimal control. II. Optimization of a distributed system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 1, pp. 26-41 (in Russian). https://doi.org/10.20537/vm170103 Ioffe A.D., Tikhomirov V.M. Theory of extremal problems, Amsterdam, New York, Oxford: North-Holland Publishing Company, 1979. Dmitruk A.V. Vypuklyi analiz. Ehlementarnyi vvodnyi kurs (Convex analysis. Elementary introductory course), Moscow: MAKS Press, 2012. Full text