Geodesic
Parameterization of some control problems by linear systems
The Bulletin of Irkutsk State University. Series Mathematics, Tome 30 (2019), pp. 83-98
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the framework of control parameterization methods a number of optimization problems of linear phase systems with quadratic and bilinear functionals is considered. Approximation of the control is carried out in the class of piecewise linear functions and is formed as a linear combination of a special set of support functions with coefficients that are variables of the finite-dimensional problem. At the same time the interval control constraint in the variational problem automatically passes into similar constraints on the variables of the finite-dimensional problem.To characterize and effectively solve these problems, explicit expressions of the selected functionals are obtained with respect to parameters of approximations. As a result, a series of quadratic mathematical programming problems with the simplest restrictions on variables is formulated. The quadratic forms of the objective functions are determined by the Gram and Hessenberg matrices.It should be emphasized that the parametrization preserves the convexity property of the original optimal control problem. In addition, the simplest optimal control problem with a linear terminal functional after parameterization is solved without iterations.The connection between the coordinated problems at the level of optimality conditions is established. It consists in the fact that the differential condition of the extremum in a finite-dimensional problem is locally equivalent to the maximum principle for the variational problem at the points of set.
Keywords: linear control system, quadratic and bilinear functionals, piecewise-linear approximation, finite dimensional problems. 
@article{IIGUM_2019_30_a6,
     author = {V. A. Srochko and E. V. Aksenyushkina},
     title = {Parameterization of some control problems by linear systems},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {83--98},
     year = {2019},
     volume = {30},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2019_30_a6/}
}
TY  - JOUR
AU  - V. A. Srochko
AU  - E. V. Aksenyushkina
TI  - Parameterization of some control problems by linear systems
JO  - The Bulletin of Irkutsk State University. Series Mathematics
PY  - 2019
SP  - 83
EP  - 98
VL  - 30
UR  - http://geodesic.mathdoc.fr/item/IIGUM_2019_30_a6/
LA  - ru
ID  - IIGUM_2019_30_a6
ER  - 
%0 Journal Article
%A V. A. Srochko
%A E. V. Aksenyushkina
%T Parameterization of some control problems by linear systems
%J The Bulletin of Irkutsk State University. Series Mathematics
%D 2019
%P 83-98
%V 30
%U http://geodesic.mathdoc.fr/item/IIGUM_2019_30_a6/
%G ru
%F IIGUM_2019_30_a6
V. A. Srochko; E. V. Aksenyushkina. Parameterization of some control problems by linear systems. The Bulletin of Irkutsk State University. Series Mathematics, Tome 30 (2019), pp. 83-98. http://geodesic.mathdoc.fr/item/IIGUM_2019_30_a6/

[1] A. V. Arguchintsev, V. A. Dykhta, V. A. Srochko, “Optimal control: nonlocal conditions, computational methods, and the variational maximum principle”, Izvestiya vuzov. Mathematics, 2009, no. 1, 3–43 (in Russian) | MR | Zbl

[2] N. V. Balashevich, R. Gabasov, F. M. Kirillova, “Numerical methods of software and positional optimization of linear control systems”, Zhurn. comp. math and math. physics, 40:6 (2000), 838–859 (in Russian) | MR | Zbl

[3] N. S. Bakhvalov, N. P. Zhidkov, G. M. Kobelkov, Numerical methods, Laboratory of basic knowledge, M., 2002, 632 pp. (in Russian) | MR

[4] V. M. Verzhbitsky, Numerical methods, High school, M., 2001, 382 pp. (in Russian)

[5] V. K. Gorbunov, “The reduction of optimal control problems for finite-dimensional”, Sib. comp. math and math. physics, 18:5 (1978), 1083–1095 (in Russian) | MR | Zbl

[6] V. K. Gorbunov, I. V. Lutoshkin, “The Development and experience of applying the parameterization method in degenerate problems of dynamical optimization”, Izvestiya RAS. Theory and control systems, 2004, no. 5, 67–84 (in Russian)

[7] Yu. G. Yevtushenko, Methods for solving extreme problems and their application in optimization systems, Science, M., 1982, 432 pp. (in Russian) | MR

[8] O. N. Korsun, A. V. Stulovskaya, “Direct method of formation of optimal program control of the aircraft”, Izv. Theory and control systems, 2019, no. 2, 75–89 (in Russian) | DOI

[9] B. N. Pshenichny, Yu. M. Danilin, Numerical methods in extremal problems, Science, M., 1975, 320 pp. (in Russian)

[10] A. S. Strekalovsky, Elements of nonconvex optimization, Nauka Publ., Novosibirsk, 2003, 356 pp. (in Russian)

[11] A. G. Sukharev, V. V. Fedorov, Course of optimization methods, Science, M., 1986, 328 pp. (in Russian) | MR

[12] D. Tabak, B. Kuo, Optimal control and mathematical programming, Science, M., 1975, 280 pp. (in Russian)

[13] A. V. Chernov, “On the application of Gauss functions for the numerical solution of optimal control problems”, Automatics and Telemechanics, 2019, no. 6, 51–69 (in Russian) | DOI

[14] R. Enkhbat, “On some theory, methods and algorithms for concave programming”, Optimization and Optimal Control, 2003, 79–102 | DOI | MR | Zbl

[15] K. L. Teo, C. J. Goh, K. H. Wong, A unified computational approach to optimal control problem, Longman Group Limited, 1991 | MR