BSU Bulletin. Mathematics, Informatics
Bibliographic description:
, METHODS OF FIXED POINTS IN ONE CLASS OF DISCRETE-CONTINUOUS PROBLEMS OF OPTIMIZING CONTROLLED SYSTEMS // BSU Bulletin. Mathematics, Informatics. - 2021. №2. . - С. 28-43.
Title:
METHODS OF FIXED POINTS IN ONE CLASS OF DISCRETE-CONTINUOUS PROBLEMS OF OPTIMIZING CONTROLLED SYSTEMS
Financing:
Работа выполнена при финансовой поддержке РФФИ, проект 18-41-030005_р-а, и Бурятского госуниверситета, проект 2021 г.
Codes:
Annotation:
In the considered class of discrete-continuous control systems, formulas for the increment of the objective function of the standard form with remainder terms of the expansions and non-standard formulas that do not contain the remainder terms of the expansions are constructed. Based on the formulas obtained, conditions for nonlo- cal improvement and control optimality are constructed in the form of fixed point problems in the control space. Such a representation of conditions makes it possible to apply and modify the well-known theory and methods of fixed points for constructing iterative algorithms for searching for extremal controls and constructing relaxation sequences of controls in the considered discrete-continuous optimal control problems. The proposed iterative algorithms have the property of nonlocality of successive control approximations and the absence of a parametric search procedure for an improving approximation at each iteration, which is characteristic of gradient-type methods.
Keywords:
discrete-continuous system; conditions for improvement and optimality of control; fixed point problem; iterative algorithm.
List of references:
Emelyanov S., Korovin S., Mamedov I. Variable Structure Control Systems. Dis- crete and Digital. CRC Press, USA, 1995. 316 p.
The Control Handbook: Control System Advanced Methods. In: Levine, W. (eds). CRC Press, London, 2010. 1798 p.
Van der Schaft A., Schumacher H. An Introduction to Hybrid Dynamical Sys- tems. Springer, London, 2000. 174 p.
Gurman V., Rasina I. Discrete-continuous Representations of Impulsive Proc- esses in the Controllable Systems // Automation and Remote Control. 2012. 8(73). P. 1290–1300.
Mastaliyev R. Necessary Optimality Conditions in Optimal Control Problems by Discrete-continuous Systems // Tomsk State University Journal of Control and Com- puter Science. 2015. 1(30). P. 4–10.
Evtushenko Y. Numerical Optimization Techniques. Publications Division, New York, 1985. 562 p.
Tabak D., Kuo B. Optimal Control and Mathematical Programming. Nauka, Moscow, 1975. 279 p.
Gurman V., Ni Ming Kang. Degenerate Problems of Optimal Control. I. Automa- tion and Remote Control. 2011. 4(72). P. 727–739.
Moiseev A. Optimal Сontrol Under Discrete Control Actions. Automation and Remote Control. 1991. 9(52). P. 1274–1280.
Teo K., Goh C., Wong K. A Unified Computational Approach to Optimal Con- trol Problem. Longman Group Limited, New York, 1991. 329 p.
Rahimov A. On an Approach to Solution to Optimal Control Problems on the Classes of Piecewise Constant, Piecewise Linear, and Piecewise Given Functions // Tomsk State University Journal of Control and Computer Science. 2012. 2(19). P. 20–30.
Gorbunov V. A Method for the Parametrization of Optimal Control Problems // USSR Computational Mathematics and Mathematical Physics. 1979. 2(19). P. 292–303.
Srochko V., Aksenyushkina E. Parametrization of Some Control Problems for Linear Systems // The Bulletin of Irkutsk State University. Series Mathematics. 2019. (30). P. 83–98.
Srochko V. Iterative Methods for Solving Optimal Control Problems. Fizmatlit, Moscow, 2000. 160 p.
Vasiliev O. Optimization Methods. World Federation Publishers Company INC, Atlanta, 1996. 276 p.
Buldaev A., Khishektueva I.-Kh. The Fixed Point Method for the Problems of Nonlinear Systems Optimization on the Managing Functions and Parameters // The Bulletin of Irkutsk State University. Series Mathematics. 2017. (19). P. 89–104.
Samarskii A., Gulin A. Numerical Methods. Nauka, Moscow, 1989. 432 p.
Buldaev A. Operator Equations and Maximum Principle Algorithms in Optimal Control Problems // Bulletin of Buryat State University. Mathematics, Informatics. 2020. 1. P. 35–53.
The Control Handbook: Control System Advanced Methods. In: Levine, W. (eds). CRC Press, London, 2010. 1798 p.
Van der Schaft A., Schumacher H. An Introduction to Hybrid Dynamical Sys- tems. Springer, London, 2000. 174 p.
Gurman V., Rasina I. Discrete-continuous Representations of Impulsive Proc- esses in the Controllable Systems // Automation and Remote Control. 2012. 8(73). P. 1290–1300.
Mastaliyev R. Necessary Optimality Conditions in Optimal Control Problems by Discrete-continuous Systems // Tomsk State University Journal of Control and Com- puter Science. 2015. 1(30). P. 4–10.
Evtushenko Y. Numerical Optimization Techniques. Publications Division, New York, 1985. 562 p.
Tabak D., Kuo B. Optimal Control and Mathematical Programming. Nauka, Moscow, 1975. 279 p.
Gurman V., Ni Ming Kang. Degenerate Problems of Optimal Control. I. Automa- tion and Remote Control. 2011. 4(72). P. 727–739.
Moiseev A. Optimal Сontrol Under Discrete Control Actions. Automation and Remote Control. 1991. 9(52). P. 1274–1280.
Teo K., Goh C., Wong K. A Unified Computational Approach to Optimal Con- trol Problem. Longman Group Limited, New York, 1991. 329 p.
Rahimov A. On an Approach to Solution to Optimal Control Problems on the Classes of Piecewise Constant, Piecewise Linear, and Piecewise Given Functions // Tomsk State University Journal of Control and Computer Science. 2012. 2(19). P. 20–30.
Gorbunov V. A Method for the Parametrization of Optimal Control Problems // USSR Computational Mathematics and Mathematical Physics. 1979. 2(19). P. 292–303.
Srochko V., Aksenyushkina E. Parametrization of Some Control Problems for Linear Systems // The Bulletin of Irkutsk State University. Series Mathematics. 2019. (30). P. 83–98.
Srochko V. Iterative Methods for Solving Optimal Control Problems. Fizmatlit, Moscow, 2000. 160 p.
Vasiliev O. Optimization Methods. World Federation Publishers Company INC, Atlanta, 1996. 276 p.
Buldaev A., Khishektueva I.-Kh. The Fixed Point Method for the Problems of Nonlinear Systems Optimization on the Managing Functions and Parameters // The Bulletin of Irkutsk State University. Series Mathematics. 2017. (19). P. 89–104.
Samarskii A., Gulin A. Numerical Methods. Nauka, Moscow, 1989. 432 p.
Buldaev A. Operator Equations and Maximum Principle Algorithms in Optimal Control Problems // Bulletin of Buryat State University. Mathematics, Informatics. 2020. 1. P. 35–53.
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