Geodesic
Approximation of control for singularly perturbed system with delay with integral quadratic constraints
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 4, pp. 368-380 
I. V. Grebennikova; A. G. Kremlev. Approximation of control for singularly perturbed system with delay with integral quadratic constraints. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 4, pp. 368-380. http://geodesic.mathdoc.fr/item/ISU_2017_17_4_a0/
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     title = {Approximation of control for singularly perturbed system with delay with integral quadratic constraints},
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     pages = {368--380},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The purpose of the work is the development and theoretical substantiation of analytical approximate or asymptotic methods for solving optimal control problems for singularly perturbed systems with constant delay in phase variables under conditions of uncertainty with respect to the initial data. For achievement of a goal the control problem for the singularly perturbed system with delay with indeterminate initial conditions and integral quadratic constraints on the control resources according to the minimax criterion is considered. A limit problem is formulated for which the quality functional is chosen in a special way. The proposed method is based on the idea of separating the asymptotics of the ensemble of trajectories of a singularly perturbed system with delay and representing the fundamental matrix of solutions divided into blocks in accordance with the dimensions of fast and slow variables in the form of a uniformly convergent sequence. We propose a procedure to construct an initial approximation of control response for the minimax problem of control. The work uses problem statements, concepts, methods and results of control theory under uncertainty, as well as methods of the theory of extremal problems, asymptotic analysis methods, classical methods of convex and real analysis.

[1] Akulenko L. D., Asymptotic Methods of Optimal Control, Nauka, M., 1987, 368 pp. (in Russian)

[2] Gaitsgory V., Rossomakhine S., “Averaging and linear programming in some singularly perturbed problems of optimal control”, Applied Mathematics and Optimization, 71:2 (2015), 195–276 | DOI | MR | Zbl

[3] Gajic Z., Lim M., Optimal control of singularly perturbed linear systems and applications. High-accuracy techniques, Marcel Dekker, Inc., N. Y., 2001, 312 pp. | MR | Zbl

[4] Glizer V. Y., Fridman E., “$H_{\infty}$ control of linear singularly perturbed systems with small state delay”, J. Math. Anal. and Appl., 250:1 (2000), 49–85 | DOI | MR | Zbl

[5] Kalinin A. I., Lavrinovich L. I., “Asymptotic of the solution to a singularly perturbed linear-quadratic optimal control problem”, Comput. Math. Math. Phys., 55:2 (2015), 194–205 | DOI | DOI | MR | Zbl

[6] Kurina G. A., Nguyen T. H., “Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients”, Comput. Math. Math. Phys., 52:4 (2012), 524–547 | DOI | MR | Zbl

[7] Kokotovic P. V., Khalil H. K., O'Reilly J., Singular perturbation methods in control : analysis and design, SIAM, Philadelphia, PA, USA, 1999, 374 pp. | MR | Zbl

[8] Krasovskii N. N., The Theory of Motion Control, Nauka, M., 1968, 475 pp. (in Russian)

[9] Kurzhanskij A. B., Control and Observation under the Uncertainty Conditions, Nauka, M., 1977, 392 pp. (in Russian)

[10] Grebennikova I. V., Kremlev A. G., “Approximation of control for singularly perturbed system with delay with geometric constraints”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 15:2 (2015), 142–151 (in Russian) | DOI

[11] Grebennikova I. V., Kremlev A. G., “Iterative procedure of constructing optimal solving in the minimax problem of control for singularly perturbed system with delay with geometric constraints”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 16:3 (2016), 272–280 (in Russian) | DOI | Zbl

[12] Kremlev A. G., Grebennikova I. V., “About asymptotic of a set of trajectories of a singularly perturbed system with delay”, News of Scientific Thought, Proc. Intern. Conf., v. 4, Nauka i obrazovanie, Dnepropetrovsk, 2006, 65–69 (in Russian)

[13] Rokafellar R., Convex Analysis, Mir, M., 1973, 492 pp. (in Russian)

[14] Krasovskii N. N., Some Problems in the Theory of Stability of Motion, Fizmatgiz, M., 1959, 468 pp. (in Russian)

[15] Kirillova F. M., “Relative controllability of linear dynamic systems with delay”, Dokl. AN SSSR, 174:6 (1967), 1260–1263 (in Russian) | Zbl