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## Archive of Issues

Russia Nizhni Novgorod
Year
2015
Volume
25
Issue
4
Pages
483-491
 Section Mathematics Title On singular controls of a maximum principle for the problem of the Goursat-Darboux system optimization Author(-s) Lisachenko I.V.ab, Sumin V.I.b Affiliations Nizhni Novgorod State Technical Universitya, Nizhni Novgorod State Universityb Abstract The paper deals with the terminal optimization problem connected with the Goursat-Darboux control system. The right-hand side of the differential equation is a full nonlinear Caratheodory function. We consider the case in which solutions of the Goursat-Darboux system necessarily belong to a class of functions with $p$-integrable (for some $p>1$) mixed derivatives. In our case a choice of this class is defined by boundary functions. We study singular controls in the sense of the pointwise maximum principle that are controls for which this principle is strong degenerate, i.e., degenerate together with second-order optimality conditions. It is shown that for strong degeneration of the pointwise maximum principle it is sufficient that right-hand side with respect to state derivatives is affine and these derivatives and control are separated additively. Necessary optimality conditions of the singular controls are given for this case. These conditions generalize similar necessary optimality conditions which were obtained for more smooth right-hand sides in the case of solutions with bounded mixed derivatives. Keywords nonlinear Goursat-Darboux system, solutions having summable mixed derivatives, terminal optimization problem, maximum principle, singular controls UDC 517.95 MSC 49K20 DOI 10.20537/vm150405 Received 2 October 2015 Language Russian Citation Lisachenko I.V., Sumin V.I. On singular controls of a maximum principle for the problem of the Goursat-Darboux system optimization, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 4, pp. 483-491. References Vasil'ev O.V., Srochko V.A., Terletskii V.A. Metody optimizatsii i ikh prilozheniya. Chast' 2. Optimal'noe upravlenie (Optimization methods and their applications. Part 2. Optimal Control), Novosibirsk: Nauka, 1990, 151 p. Sumin V.I. Strong degeneration of singular controls in distributed optimization problems, Sov. Math., Dokl., 1992, vol. 44, no. 2, pp. 453-458. Sumin V.I. On singular controls in the sence of the pointwise maximum principle in distributed optimization problems, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 3, pp. 70-80 (in Russian). Mansimov K.B., Mardanov M.J. Kachestvennaya teoriya optimal'nogo upravleniya sistemami Gursa-Darbu (Qualitative theory of optimal control connected with Goursat-Darboux system), Baku: Elm, 2010, 360 p. Gabasov R., Kirillova F.M., Mansimov K.B. Necessary second-order optimality conditions for systems with distributed parameters, Akad. Nauk Beloruss. SSR, Inst. Mat., Prepr., Minsk, 1982, no. 31, 32 p. (in Russian). Mansimov K.B. Singular controls in control problems of distributed parameter systems, J. Math. Sci., New York, 2008, vol. 148, no. 3, pp. 331-381. Sumin V.I. Optimization of control Volterra systems, Cand. Sci. (Phys.-Math.) Dissertation, Gor'kii, 1975, 158 p. Lisachenko I.V., Sumin V.I. Nonlinear Goursat-Darboux control problem: conditions for the preservation of global solvability, Differential Equations, 2011, vol. 47, no. 6, pp. 858-870. Lisachenko I.V., Sumin V.I. The maximum principle for terminal optimization problem connected with Goursat-Darboux system in the class of functions having summable mixed derivatives, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 2, pp. 52-67 (in Russian). Lisachenko I.V., Sumin V.I. On singular controls in the sense of the maximum principle for terminal optimization problem connected with Goursat-Darboux system, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2012, no. 1 (39), pp. 80-81 (in Russian). Lisachenko I.V., Sumin V.I. About singular controls of maximum principle for terminal optimization problem connected with Goursat-Darboux system, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 2015, vol. 20, no. 5, pp. 1264-1274 (in Russian). Ioffe A.D., Tikhomirov V.M. Teoriya ekstremal'nykh zadach (Theory of extremal problems), Moscow: Nauka, 1974, 479 p. Full text