Abstract
An approach to the numerical solution of an optimal control problem described by systems of ordinary differential equations with point loading is proposed. The problem involves nonlocal conditions in which the values of the phase state at certain points and its integral values on various preset intervals are contained in coupled form. For the cost functional gradient of the problem, formulas are derived, which are used to solve the problem by applying numerical optimization methods of the first order. Results of numerical experiments are presented.
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REFERENCES
A. Kneser, Die Integralgleichungen und ihre Anwendungen in der Mathematischen Physik (Vieweg+Teubner, Berlin, 1923).
L. Lichtenstein, Vorlesungen über einege Klassen nichtlinear Integralgleichungen und Integraldifferentialgleihungen nebst Anwendungen (Springer, Berlin, 1932).
N. M. Gyunter, “General theory of integral equations,” Dokl. Akad. Nauk SSSR 22, 215–219 (1939).
M. N. Yakovlev, “Estimates for solutions to systems of loaded integro-differential equations subject to multipoint and integral boundary conditions,” Zap. Nauchn. Sem. LOMI 124, 131–139 (1983).
D. S. Dzhumabaev and G. B. Iliyasova, “Numerical implementation of a parametrization method for solving a linear boundary value problem for loaded differential equations,” Izv. Nats. Akad. Nauk Resp. Kazakh. Ser. Fiz.-Mat., No. 2, 275–280 (2014).
V. M. Abdullaev and K. R. Aida-zade, “On the numerical solution of loaded systems of ordinary differential equations,” Comput. Math. Math. Phys. 44 (9), 1505–1515 (2004).
V. M. Abdullaev and K. R. Aida-zade, “Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations,” Comput. Math. Math. Phys. 54 (7), 1096–1109 (2014).
V. M. Abdullayev and K. R. Aida-zade, “Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations,” Comput. Math. Math. Phys. 57 (4), 634–644 (2017).
A. M. Nakhushev, Loaded Equations and Applications (Nauka, Moscow, 2012) [in Russian].
M. Kh. Shkhanukov-Lafishev, “Locally one-dimensional scheme for a loaded heat equation with robin boundary conditions,” Comput. Math. Math. Phys. 49 (7), 1167–1174 (2009).
M. T. Dzhenaliev, On the Theory of Linear Boundary Value Problems for Loaded Differential Equations (Komp’yut. Tsentr ITPM, Almaty, 1995) [in Russian].
A. A. Alikhanov, A. M. Berezkov, and M. Kh. Shkhanukov-Lafishev, “Boundary value problems for certain classes of loaded differential equations and solving them by finite difference methods,” Comput. Math. Math. Phys. 48 (9), 1581–1590 (2008).
K. R. Aida-zade and V. M. Abdullayev, “Optimizing placement of the control points at synthesis of the heating process control,” Autom. Remote Control 78 (9), 1585–1599 (2017).
K. R. Aida-zade, “Numerical method for reconstructing parameters of a dynamical system, Kibern. Syst. Anal. Kiev, No. 1, 101–108 (2004).
K. R. Aida-zade and V. M. Abdullayev, “Solution to a class of inverse problems for system of loaded ordinary differential equations with integral conditions,” J. Inverse Ill-Posed Probl. 24 (5), 543–558 (2016).
A. A. Abramov, “A variation of the ‘dispersion’ method,” USSR Comput. Math. Math. Phys. 1, 368–371 (1961).
K. R. Aida-zade and V. M. Abdullaev, “Control problem with non-separated multipoint and integral conditions,” J. Autom. Inf. Sci. 45 (3), 34–52 (2013).
L. T. Ashchepkov, “Optimal control of a system with intermediate conditions,” J. Appl. Math. Mech. 45 (2), 153–158 (1981).
O. O. Vasil’eva and K. Mizukami, “Dynamical processes described by a boundary value problem: necessary conditions of optimality and methods of solving,” J. Comput. Syst. Sci. Int. 39 (1), 90–95 (2000).
F. P. Vasil’ev, Optimization Methods (Faktorial, Moscow, 2002) [in Russian].
A. S. Antipin, “Terminal control of boundary models,” Comput. Math. Math. Phys. 54 (2), 257–285 (2014).
Yu. G. Evtushenko, Methods for Solving Optimization Problems and Their Applications in Optimization Systems (Nauka, Moscow, 1982) [in Russian].
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Abdullayev, V.M., Aida-zade, K.R. Approach to the Numerical Solution of Optimal Control Problems for Loaded Differential Equations with Nonlocal Conditions. Comput. Math. and Math. Phys. 59, 696–707 (2019). https://doi.org/10.1134/S0965542519050026
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DOI: https://doi.org/10.1134/S0965542519050026