In the class of controlled systems with constraints, the conditions for improving and optimality of control are constructed and analyzed in the form of fixed point problems. This form allows one to obtain enhanced necessary optimality conditions in comparison with the known conditions and makes it possible to apply and modify the theory and methods of fixed points to search for extreme controls in optimization problems of the class under consideration. Fixed-point problems are constructed using the transition to auxiliary optimal control problems without restrictions with Lagrange functionals. An iterative algorithm is proposed for constructing a relaxation sequence of admissible controls based on the solution of constructed fixed point problems. The considered algorithm is characterized by the properties of nonlocal improvement of admissible control and the fundamental possibility of rigorous improvement of non-optimal controls satisfying the known necessary optimality conditions, in contrast to gradient and other local methods. The conditions of convergence of the control sequence for the residual of fulfilling the necessary optimality conditions are substantiated. A comparative analysis of the computational and qualitative efficiency of the proposed iterative algorithm for finding extreme controls in a model problem with phase constraints is carried out.

Alexander Buldaev, Dr. Sci. (Phys.–Math.), Prof., Buryat State University, 24a, Smolin St., Ulan-Ude, 670000, Russian Federation; e-mail: buldaev@mail.ru

Ivan Burlakov, Researcher, Buryat State University, 24a, Smolin St., Ulan-Ude, 670000, Russian Federation; e-mail: ivan.burlakov.91@mail.ru

Buldaev A.S., Burlakov I.D. On a Method for Finding Extremal Controls in Systems with Constraints. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 30, pp. 16-30. https://doi.org/10.26516/1997-7670.2019.30.16

1. Bartenev O.V. Fortran dlya professionalov. Matematicheskaya biblioteka IMSL. Chast’ 2 [Fortran for Professionals. Mathematical Library IMSL. Part 2]. Moscow, Dialog-MIFI Publ., 2001, 320 p. (in Russian)
2. Buldaev A.S. Metody nepodvizhnykh tochek na osnove operacij proektirovaniya v zadachakh optimizacii upravlyayushchikh funkcij i ametrov dinamicheskikh sistem [Fixed-Point Methods Based on Projection Operations in Optimization Problems for Control Functions and parameters of Dynamical Systems] Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika, informatika [Bulletin of the Buryat State University. Mathematics, Computer Science], 2017, no. 1, pp. 38-54. http://journals.bsu.ru/doi/10.18101/2304-5728-2017-1-38-54. (in Russian)
3. Buldaev A.S. Metody vozmushchenij v zadachakh uluchsheniya i optimizatsii upravlyaemykh sistem [Perturbation Methods in Improvement and Optimization Problems for Controllable Systems]. Ulan-Ude, Publishing House of the Buryat State University, 2008, 260 p. (in Russian)
4. Buldaev A.S., Burlakov I.D. About One Approach to Numerical Solution of Nonlinear Optimal Speed Problems. Bulletin of the South Ural State University. Series Mathematical Modeling, Programming & Computer Software, 2018, vol. 11, no. 4, pp. 55-66. https://doi.org/10.14529/mmp180404
5. Buldaev A.S., Khishektueva I.-Kh. The Fixed Point Method in ametric Optimization Problems for Systems. Automation and Remote Control, 2013, vol. 74, no. 12, pp. 1927-1934. https://doi.org/10.1134/S0005117913120011
6. Grachev N.I., Filkov A.N. Reshenie zadach optimal’nogo upravleniya v sisteme DISO [The Solution of Optimal Control Problems in the DISO System]. Moscow, Publishing House of Computer Center of the Academy of Sciences USSR, 1986, 66 p. (in Russian)
7. Samarsky A.A., Gulin A.V. Chislennye metody [Numerical Nethods]. Moscow, Nauka Publ., 1989, 432 p. (in Russian)
8. Srochko V.A. Iteracionnye metody resheniya zadach optimal’nogo upravleniya [Iterative Methods for Solving Optimal Control Problems]. Moscow, Fizmatlit Publ., 2000, 160 p. (in Russian)
9. Tyatyushkin A.I. Mnogometodnaya tekhnologiya optimizacii upravlyaemyh sistem [Multi-Method Technology for Optimizing Controlled Systems]. Novosibirsk, Nauka Publ., 2006, 343 p. (in Russian)
10. Vasiliev O.V. Lekcii po metodam optimizacii [Lectures on Optimization Methods]. Irkutsk, Publishing House of Irkutsk State University, 1994, 340 p. (in Russian)