ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2017. Vol. 19

The Fixed Point Method for the Problems of Nonlinear Systems Optimization on the Managing Functions and Parameters

A. S. Buldaev, I.-Kh. D. Khishektueva

A new approach to the class of nonlinear optimal control problems containing both managing functions and parameters, is proposed on basis of the solution of special fixed point problems for operators constructed in the space of controls. The fixed point problems make it possible to build improving controls and obtain new conditions for optimal control in the class of optimization problems. The problem of control improvment as a problem of a fixed point is formed on the basis of a control increment formula without the residual terms of expansions. This formula is constructed using the differential-algebraic modification of the standard adjoint system. The method of sequential solving of improvement problems in the form of fixed point problems is characterized by nonlocal improvement of management, lack of search procedure improving control in a sufficiently small neighborhood of current control and the ability to improve suboptimal controls, satisfying the maximum principle. To search for controls, satisfying the maximum principle, instead of the boundary value problem in the space of states are invited to consider the fixed point problem in space of controls. Examples illustrating the basic properties of the method are given.

controlled system, fixed point problem, conditions of optimality

1. Baturin V.A., Urbanovich D.E. Priblizhenniye metody upravlenya, osnovannie na printsipe rashireniya. [Approximate methods of optimal management, based on the principle of expansion]. Novosibirsk, Nauka Publ., 1997. 175 p.(in Russian)

2. Buldaev A.S. Metody vozmusheniy v zadachah uluchsheniya i optimizatsii upravlyaemih system [Purtubation methods in problem of the improvement and optimization of the controlled systems]. Ulan-Ude, Buryat State UniversityPublishing Department, 2008. 260 p.(in Russian)

3. Buldaev A.S., Khishektueva I.-Kh.D. The Fixed Point Method in Parametric Optimization Problems for Systems. Automation and Remote Control, 2013, vol. 74, no 12, pp. 1927-1934. doi: 10.1134/S0005117913120011.

4. Buldaev A.S., Morzhin O.V. Improvement of controls in nonlinear systems on basis of boundary value problems. Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Series «Mathematics», 2009, vol. 2, no 1, pp. 94-106 (in Russian).

5. Vasilev O.V. Lektsii po metodam optimisatsii. [The optimization methods lectures] Irkutsk, ISU Press, 1994. 344 p.(in Russian)

6. Ashepkov L.T. et al. Metody reshenija zadach matematicheskogo programmirovanija i optimal’nogo upravlenija. [Methods for solving mathematical programming and optimal control problems] Novosibirsk, Nauka Publ., 1984. 232 p. (in Russian)

7. Gurman V.I. et al. Metody uluchsheniya v vichislitelnom eksperimente. [Methods to improve the computing experiment] Novosibirsk, Nauka Publ., 1988. 184 p. (in Russian)

8. Gurman V.I. et al. Novie metody uluchsheniya upravlyaemih protsessov. [New methods for improvement of controlled processes] Novosibirsk, Nauka Publ., 1987. 184 p. (in Russian)

9. Pontrjagin L.S. et al. Matematicheskaja teorija optimal’nyh processov. [Mathematical theory of optimal processes] Moscow, Nauka Publ., 1976. 392 p.(in Russian)

10. Samarskiy A.A., Gulin A.V. Chislennie metody. [Numerical methods] Мoscow, Nauka Publ., 1989. 432 p.(in Russian)

11. Srochko V.A. Iteratsionniye metody resheniya zadach optimalnogo upravleniya. [Iterative methods for solving optimal control problems] Moscow, Fizmatlit, 2000. 160 p. (in Russian)

Full text (russian)