Построение решения задачи управляемости для линейных интегро-дифференциальных уравнений с ограничениями
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Ключевые слова:
линейные интегро-дифференциальные уравнения, фазовые и интегральные ограничения, оптимизационная задача, минимизирующие последовательностиАннотация
Предлагается метод решения задачи управляемости для процессов описываемых линейными интегро-дифференциальными уравнениями с краевыми условиями при наличии фазовых и интегральных ограничений с учетом ограничений на значения управления. Путем введения вспомогательных управляющих функций исходная краевая задача погружается в краевую задачу линейного дифференциального уравнения. Определяется множество всех управлений, каждый элемент которого переводит траекторию линейной системы из любой начальной точки в любое желаемое конечное состояние. Такой подход позволяет получить равносильные тождества и свести решения исходной краевой задачи к начальной задаче оптимального управления. Строятся минимизирующие последовательности предельные точки которых являются решениями задачи управляемости для линейных интегро-дифференциальных уравнений с ограничениями. Конструктивность предлагаемого метода показана на примере.
Библиографические ссылки
[1] Aisagaliev S. A., «Upravlyaemost nekotoroy sistemyi differentsialnyih uravneniy» [Controllability of a differential-equation system], Differentsialnyie uravneniya (1991) : 1476–1486.
[2] Aisagaliev S. A., «Obschee reshenie odnogo klassa integralnyih uravneniy [The general solution of a class of integral equations]», Matematicheskiy zhurnal (2005) : 17–34.
[3] Aisagaliev S. A. Teoriya upravlyaemosti dinamicheskih sistem [Controllability theory of dynamical systems] (Kazakh universiteti, 2014), 158.
[4] Aisagaliev S. A. Konstruktivnaya teoriya kraevyih zadach obyiknovennyih differentsialnyih uravneniy [Constructive theory of boundary value problems for ordinary differential equations](Almaty: Kazakh universiteti, 2015), 207.
[5] Aisagaliev S. A., Aisagalieva S. S., «Suschestvovanie resheniya zadachi upravlyaemosti dlya lineynyih integro- differentsialnyih uravneniy s ogranicheniyami [The existence of a solution of the controllability problem for linear integral and differential equations with restrictions]», Vestnik KazNU, ser. meh., mat., inf. (2017) : 3-17.
[6] Aisagaliev S. A., Belogurov A. P., «Upravlyaemost i byistrodeystvie protsessa, opisyivaemogo parabolicheskim uravneniem s ogranichennyim upravleniem [Controllability and speed of the process described by a parabolic equation with bounded control]», Sibirskiy matematicheskiy zhurnal (2012) : 20–36.
[7] Aisagaliev S. A., Kabidoldanova A. A. «Optimalnoe upravlenie lineynyimi sistemami s lineynyim kriteriem kachestva i ogranicheniyami [On the optimal control of linear systems with linear performance criterion and constraints]», Differentsialnyie uravneniya (2012) : 826–836.
[8] Aisagaliev S. A., Kalimoldaev M. N., «Konstruktivnyiy metod resheniya kraevoy zadachi dlya obyiknovennyih differentsialnyih uravneniy [Constructive method for solving a boundary value problem for ordinary differential equations]», Differentsialnyie uravneniya (2015) : 147–160.
[9] Ananevskiy I. M., Anahin N. V., Ovseevich A. I., «Sintez ogranichennogo upravleniya lineynyimi dinamicheskimi sistemami s pomoschyu obschey funktsii Lyapunova [Synthesis of limited control for linear dynamical systems with the using of the general Lyapunov function]», Dokladyi RAN (2010) : S. 319-323.
[10] Bellman R. Matematicheskie metodyi v meditsine [Mathematical Methods in Medicine] (M.: Mir, 1987), 282.
[11] Byikov Ya.V., O nekotoryih zadachah teorii integro-differentsialnyih uravneniy [On some problems of the theory of integro-differential equations] (Frunze: Ilim, 1957), 312.
[12] Emelyanov S. V., Krischenko A. P., «Stabilizatsiya neregulyarnyih sistem [Stabilization of irregular systems]», Differentsialnyie uravneniya (2012) : 1516-1524.
[13] Gabasov R., Kirillov F. M. Kachestvennaya teoriya optimalnyih protsessov [Qualitative theory of optimal processes] (M.: Nauka, 1971), 421.
[14] Glushkov V.M., Ivanov V.V., Yanenko V.M. Modelirovanie razvivayuschihsya sistem [Modeling of developing systems] (M.: Nauka, 1983), 285.
[15] Imanaliev M.I. Metodyi resheniya nelineynyih obratnyih zadach i ih prilozheniya [Methods for solving nonlinear inverse problems and their applications] (Frunze: Ilim, 1977), 302.
[16] Imanaliev M.I. Obobschennyie resheniya integralnyih uravneniy pervogo roda [Generalized solutions of integral equations of the first kind] (Frunze: Ilim, 1981), 265.
[17] Kalman R. E., «Ob obschey teorii sistem upravleniya [On the General Theory of Control Systems]». Trudyi 4 Kongressa Mezhdunarodnaya federatsii po avtomaticheskomu upravleniyu, AN SSSR, 1961.
[18] Korovin S. K., Kapalin I. V., Fomichev V. V., «Minimalnyie stabilizatoryi dlya lineynyih dinamicheskih sistem [Minimum stabilizers for linear dynamic systems]». Dokladyi RAN, 2011.
[19] Krasnov M.L. Integralnyie uravneniya [Integral equations] (M.: Nauka, 1975), 304.
[20] Krasovskiy N. N. Teoriya upravleniya dvizheniem [Theory of motion control] (M.: Nauka, 1968), 475.
[21] Lakshmikantham V., Rao M.R. Theory of integro-differential equations (London, 1995), 402.
[22] Li E. V., Markus L. Osnovyi teorii optimalnogo upravleniya [Fundamentals of Optimal Control Theory] (M.: Nauka, 1972), 575.
[23] Nikolis G., Prigozhin I. Samoorganizatsiya v neravnovesnyih sistemah [Self-organization in nonequilibrium systems] (M.: Mir, 1979), 245.
[24] Romanovskiy Yu.M., Stepanova N.V., Chernavskiy D.S. Matematicheskaya biofizika [Mathematical Biophysics] (M.: Nauka, 1984), 290.
[25] Rubin A.B. Termodinamika biologicheskih protsessov [Thermodynamics of biological processes] (Izd-vo MGU, 1984), 352.
[26] Semenov Yu. M., «O polnoy upravlyaemosti lineynyih neavtonomnyih sistem [On the complete controllability of linear nonautonomous systems]», Differentsialnyie uravneniya (2012) : 1263-1277.
[27] Varga Dzh. Optimalnoe upravlenie differentsialnyimi i funktsionalnyimi uravneniyami [Optimal control of differential and functional equations] (M.: Nauka, 1977), 585.
[28] Volterra V. Matematicheskaya teoriya borbyi za suschestvovaniya [The mathematical theory of the struggle for existence] (M.: Nauka, 1976), 320.
[29] Zubov V. I. Lektsii po teorii upravleniya [Lectures on the control theory] (M.: Nauka, 1975), 502.
[2] Aisagaliev S. A., «Obschee reshenie odnogo klassa integralnyih uravneniy [The general solution of a class of integral equations]», Matematicheskiy zhurnal (2005) : 17–34.
[3] Aisagaliev S. A. Teoriya upravlyaemosti dinamicheskih sistem [Controllability theory of dynamical systems] (Kazakh universiteti, 2014), 158.
[4] Aisagaliev S. A. Konstruktivnaya teoriya kraevyih zadach obyiknovennyih differentsialnyih uravneniy [Constructive theory of boundary value problems for ordinary differential equations](Almaty: Kazakh universiteti, 2015), 207.
[5] Aisagaliev S. A., Aisagalieva S. S., «Suschestvovanie resheniya zadachi upravlyaemosti dlya lineynyih integro- differentsialnyih uravneniy s ogranicheniyami [The existence of a solution of the controllability problem for linear integral and differential equations with restrictions]», Vestnik KazNU, ser. meh., mat., inf. (2017) : 3-17.
[6] Aisagaliev S. A., Belogurov A. P., «Upravlyaemost i byistrodeystvie protsessa, opisyivaemogo parabolicheskim uravneniem s ogranichennyim upravleniem [Controllability and speed of the process described by a parabolic equation with bounded control]», Sibirskiy matematicheskiy zhurnal (2012) : 20–36.
[7] Aisagaliev S. A., Kabidoldanova A. A. «Optimalnoe upravlenie lineynyimi sistemami s lineynyim kriteriem kachestva i ogranicheniyami [On the optimal control of linear systems with linear performance criterion and constraints]», Differentsialnyie uravneniya (2012) : 826–836.
[8] Aisagaliev S. A., Kalimoldaev M. N., «Konstruktivnyiy metod resheniya kraevoy zadachi dlya obyiknovennyih differentsialnyih uravneniy [Constructive method for solving a boundary value problem for ordinary differential equations]», Differentsialnyie uravneniya (2015) : 147–160.
[9] Ananevskiy I. M., Anahin N. V., Ovseevich A. I., «Sintez ogranichennogo upravleniya lineynyimi dinamicheskimi sistemami s pomoschyu obschey funktsii Lyapunova [Synthesis of limited control for linear dynamical systems with the using of the general Lyapunov function]», Dokladyi RAN (2010) : S. 319-323.
[10] Bellman R. Matematicheskie metodyi v meditsine [Mathematical Methods in Medicine] (M.: Mir, 1987), 282.
[11] Byikov Ya.V., O nekotoryih zadachah teorii integro-differentsialnyih uravneniy [On some problems of the theory of integro-differential equations] (Frunze: Ilim, 1957), 312.
[12] Emelyanov S. V., Krischenko A. P., «Stabilizatsiya neregulyarnyih sistem [Stabilization of irregular systems]», Differentsialnyie uravneniya (2012) : 1516-1524.
[13] Gabasov R., Kirillov F. M. Kachestvennaya teoriya optimalnyih protsessov [Qualitative theory of optimal processes] (M.: Nauka, 1971), 421.
[14] Glushkov V.M., Ivanov V.V., Yanenko V.M. Modelirovanie razvivayuschihsya sistem [Modeling of developing systems] (M.: Nauka, 1983), 285.
[15] Imanaliev M.I. Metodyi resheniya nelineynyih obratnyih zadach i ih prilozheniya [Methods for solving nonlinear inverse problems and their applications] (Frunze: Ilim, 1977), 302.
[16] Imanaliev M.I. Obobschennyie resheniya integralnyih uravneniy pervogo roda [Generalized solutions of integral equations of the first kind] (Frunze: Ilim, 1981), 265.
[17] Kalman R. E., «Ob obschey teorii sistem upravleniya [On the General Theory of Control Systems]». Trudyi 4 Kongressa Mezhdunarodnaya federatsii po avtomaticheskomu upravleniyu, AN SSSR, 1961.
[18] Korovin S. K., Kapalin I. V., Fomichev V. V., «Minimalnyie stabilizatoryi dlya lineynyih dinamicheskih sistem [Minimum stabilizers for linear dynamic systems]». Dokladyi RAN, 2011.
[19] Krasnov M.L. Integralnyie uravneniya [Integral equations] (M.: Nauka, 1975), 304.
[20] Krasovskiy N. N. Teoriya upravleniya dvizheniem [Theory of motion control] (M.: Nauka, 1968), 475.
[21] Lakshmikantham V., Rao M.R. Theory of integro-differential equations (London, 1995), 402.
[22] Li E. V., Markus L. Osnovyi teorii optimalnogo upravleniya [Fundamentals of Optimal Control Theory] (M.: Nauka, 1972), 575.
[23] Nikolis G., Prigozhin I. Samoorganizatsiya v neravnovesnyih sistemah [Self-organization in nonequilibrium systems] (M.: Mir, 1979), 245.
[24] Romanovskiy Yu.M., Stepanova N.V., Chernavskiy D.S. Matematicheskaya biofizika [Mathematical Biophysics] (M.: Nauka, 1984), 290.
[25] Rubin A.B. Termodinamika biologicheskih protsessov [Thermodynamics of biological processes] (Izd-vo MGU, 1984), 352.
[26] Semenov Yu. M., «O polnoy upravlyaemosti lineynyih neavtonomnyih sistem [On the complete controllability of linear nonautonomous systems]», Differentsialnyie uravneniya (2012) : 1263-1277.
[27] Varga Dzh. Optimalnoe upravlenie differentsialnyimi i funktsionalnyimi uravneniyami [Optimal control of differential and functional equations] (M.: Nauka, 1977), 585.
[28] Volterra V. Matematicheskaya teoriya borbyi za suschestvovaniya [The mathematical theory of the struggle for existence] (M.: Nauka, 1976), 320.
[29] Zubov V. I. Lektsii po teorii upravleniya [Lectures on the control theory] (M.: Nauka, 1975), 502.
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Как цитировать
Aisagaliev S. А., & Aisagalieva, S. S. (2018). Построение решения задачи управляемости для линейных интегро-дифференциальных уравнений с ограничениями. Вестник КазНУ. Серия математика, механика, информатика, 94(2), 3–22. извлечено от https://bm.kaznu.kz/index.php/kaznu/article/view/442
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