Geodesic
Zero-order approximation of three-time scale singular linear-quadratic optimal control problem
Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 1, pp. 85-104.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper is devoted to the construction of a zero-order approximation of the solution of a three-time scale singular perturbed linear-quadratic optimal control problem with the help of the direct scheme method. The algorithm of the method consists in immediate substituting a postulated asymptotic expansion of solution into the problem condition and constructing a family of control problems to define the terms of the asymptotic expansion. Asymptotic approximation of the solution contains regular functions and four boundary ones of exponential type which are determined from the five linear-quadratic optimal control problems. It is shown, that the system of equations for a zero-order approximation appeared from control optimality conditions of the initial perturbed problem corresponds to control optimality conditions appeared in respective five optimal control problems constructed for finding zero-order asymptotic approximation with the help of the direct scheme method. An illustrative example is given.
Keywords: linear-quadratic optimal control problem, asymptotic expansion, multi-time scale systems.
Mots-clés : singular perturbations 
@article{MAIS_2015_22_1_a5,
     author = {M. A. Kalashnikova},
     title = {Zero-order approximation of three-time scale singular linear-quadratic optimal control problem},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {85--104},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2015_22_1_a5/}
}
TY  - JOUR
AU  - M. A. Kalashnikova
TI  - Zero-order approximation of three-time scale singular linear-quadratic optimal control problem
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2015
SP  - 85
EP  - 104
VL  - 22
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2015_22_1_a5/
LA  - ru
ID  - MAIS_2015_22_1_a5
ER  - 
%0 Journal Article
%A M. A. Kalashnikova
%T Zero-order approximation of three-time scale singular linear-quadratic optimal control problem
%J Modelirovanie i analiz informacionnyh sistem
%D 2015
%P 85-104
%V 22
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2015_22_1_a5/
%G ru
%F MAIS_2015_22_1_a5
M. A. Kalashnikova. Zero-order approximation of three-time scale singular linear-quadratic optimal control problem. Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 1, pp. 85-104. http://geodesic.mathdoc.fr/item/MAIS_2015_22_1_a5/

[1] H. K. Khalil, P. V. Kokotovic, “Control of Linear Systems with Multiparameter Singular Perturbations”, Automatica, 15:2 (1979), 197–207 | DOI

[2] I. V. Gribkovskaya, A. I. Kalinin, “Asymptotic optimization of a linear singularly perturbed system containing parameters of variable orders of smallness at the derivatives”, Computational Mathematics and Mathematical Physics, 35:9 (1995), 1041–1051

[3] Voropaeva N. V., Sobolev V. A., Geometricheskaya dekompozitsiya singulyarno vozmushchennykh sistem, Fizmatlit, M., 2009 (in Russian)

[4] Vasil’eva A. B., Butuzov V. F., Asimptoticheskie razlozheniya resheniy singulyarno vozmushchennykh uravneniy, Nauka, M., 1973 (in Russian)

[5] V. R. Saksena, J. O'Reilly, P. V. Kokotovic, “Singular Perturbations and Time-scale Methods in Control Theory: Survey 1976–1983”, Automatica, 20:3 (1984), 273–293 | DOI

[6] M. G. Dmitriev, G. A. Kurina, “Singular perturbations in control problems”, Automation and Remote Control, 67:1 (2006), 1–43 | DOI

[7] Y. Zhang, D. S. Naidu, C. X. Cai, Y. Zou, “Singular perturbations and time scales in control theories and applications: an overview 2002–2012”, Int. J. Inf. Syst. Sci., 9:1 (2014), 1–36

[8] G. S. Ladde, D. D. Šiljak, “Multiparameter Singular Perturbations of Linear Systems with Multiple Time Scales”, Automatica, 19:4 (1983), 385–394 | DOI

[9] G. A. Kurina, “Complete controllability of various-speed singularly perturbed systems”, Mathematical Notes, 52:4 (1992), 1029–1033 | DOI

[10] A. R. Danilin, O. O. Kovrizhnykh, “On the Asymptotics of the Solution of a System of Linear Equations with Two Small Parameters”, Differential Equations, 44:6 (2008), 757–767 | DOI

[11] Y. Y. Wang, P. M. Frank, N. E. Wu, “Near-Optimal Control of Nonstandard Singularly Perturbed Systems”, Automatica, 30:2 (1994), 277–292 | DOI

[12] H. Mukaidani, H. Xu, K. Mizukami, “New Results for near-optimal control of linear multiparameter singularly perturbed systems”, Automatica, 39 (2003), 2157–2167 | DOI

[13] S. V. Belokopytov, M. G. Dmitriev, “Solution of classical optimal control problems with a boundary layer”, Automation and Remote Control, 50:7 (1989), 907–917

[14] Rozonoer L. I., “Printsip maksimuma L. S. Pontryagina v teorii optimalnykh sistem, I”, Avtomatika i telemekhanika, 20:10 (1959), 1320–1334 (in Russian)

[15] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishechenko, The mathematical theory of optimal processes, Interscience Publishers, New York, 1962