Geodesic
Decomposition of a singularly perturbed functional-differential system based on a nondegenerate transformation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 1, Tome 190 (2021), pp. 130-143.

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

For a linear, stationary, singularly perturbed functional-differential control system with a small coefficient of the highest derivative and a finite delay in slow state variables, we justify the decomposition by a nondegenerate transformation of variables, which is a generalization of the well-known Chang transformation. This transformation splits the original two-rate system into two independent subsystems of lower dimension separately for the fast and slow variables. We prove that the splitting transformation can be constructed with any approximation accuracy in the form of an asymptotic expansion in powers of a small parameter and propose an iterative scheme for calculating the asymptotic series. Based on the decomposition constructed, we establish that for sufficiently small values of the parameter, the spectrum of the system is split into two sets, separately for “small” and “large” eigenvalues. Also, we give examples of constructing transform approximations.
Keywords: singularly perturbed system, functional-differential system, delay, splitting transformation
Mots-clés : decomposition. 
@article{INTO_2021_190_a12,
     author = {V. Tsekhan},
     title = {Decomposition of a singularly perturbed functional-differential system based on a nondegenerate transformation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {130--143},
     publisher = {mathdoc},
     volume = {190},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_190_a12/}
}
TY  - JOUR
AU  - V. Tsekhan
TI  - Decomposition of a singularly perturbed functional-differential system based on a nondegenerate transformation
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2021
SP  - 130
EP  - 143
VL  - 190
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2021_190_a12/
LA  - ru
ID  - INTO_2021_190_a12
ER  - 
%0 Journal Article
%A V. Tsekhan
%T Decomposition of a singularly perturbed functional-differential system based on a nondegenerate transformation
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2021
%P 130-143
%V 190
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2021_190_a12/
%G ru
%F INTO_2021_190_a12
V. Tsekhan. Decomposition of a singularly perturbed functional-differential system based on a nondegenerate transformation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 1, Tome 190 (2021), pp. 130-143. http://geodesic.mathdoc.fr/item/INTO_2021_190_a12/

[1] Vasileva A. B., Dmitriev M. G., “Singulyarnye vozmuscheniya v zadachakh optimalnogo upravleniya”, Itogi nauki i tekhn. Ser. Mat. anal., 20 (1982), 3–77 | MR | Zbl

[2] Dmitriev M. G., Kurina G. A., “Singulyarnye vozmuscheniya v zadachakh upravleniya”, Avtomat. telemekh., 1 (2006), 3–51 | Zbl

[3] Kopeikina T. B., “Ob upravlyaemosti lineinykh singulyarno vozmuschennykh sistem s zapazdyvaniem”, Differ. uravn., 1989, 1508–1518 | MR

[4] Kurina G. A., “O polnoi upravlyaemosti raznotempovykh singulyarno vozmuschennykh sistem”, Mat. zametki., 52:4 (1992), 56–61 | MR | Zbl

[5] Tsekhan O. B., “Rasscheplyayuschee preobrazovanie dlya lineinoi statsionarnoi singulyarno vozmuschennoi sistemy s zapazdyvaniem i ego primenenie k analizu i upravleniyu spektrom”, Vesn. GrDU im. Ya. Kupaly. Ser 2. Mat., 7:1 (2017), 50–61

[6] Kheil Dzh., Teoriya funktsionalno-differentsialnykh uravnenii, Mir, M., 1984

[7] Chang K., “Singular perturbations of a general boundary value problem”, SIAM J. Math. Anal., 3:3 (1972), 520–526 | DOI | MR | Zbl

[8] Fridman E., “Decoupling transformation of singularly perturbed systems with small delays and its applications”, Z. Angew. Math. Mech. Berlin., 76:2 (1996), 201–204 | Zbl

[9] Gajic Z., Shen X., Parallel algorithms for optimal control of large scale linear systems, Springer-Verlag, London, 1993 | MR | Zbl

[10] Glizer V. Ya., “L2-Stabilizability conditions for a class of nonstandard singularly perturbed functional-differential systems”, Dyn. Cont. Discr. Impuls. Systems. Ser. es B: Appl. Alg., 16 (2009), 181–213 | MR | Zbl

[11] Glizer V. Ya., “Approximate state-space controllability of linear singularly perturbed systems with two scales of state delays”, Asympt. Anal., 107:1–2 (2018), 73–114 | MR | Zbl

[12] Kokotovic P. V., Haddad A. H., “Controllability and time-optimal control of systems with slow and fast modes”, IEEE Trans. Automat. Contr., 20:1 (1975), 111–113 | DOI | MR | Zbl

[13] Kokotovic P. V., Khalil H. K., O'Reilly J., Singular Perturbations Methods in Control: Analysis and Design, Academic Press, New York, 1986 | MR

[14] Magalhaes L. T., “Exponential estimates for singularly perturbed linear functional differential equations”, J. Math. Anal. Appl., 103 (1984), 443–460 | DOI | MR | Zbl

[15] Pekar L., Gao Q., “Spectrum analysis of LTI continuous-time systems with constant delays: a literature overview of some recent results”, IEEE Acc., 6:1 (2018), 35457–35491 | DOI

[16] Prljaca N., Gajic Z., “General transformation for block diagonalization of multitime-scale singularly perturbed linear systems”, IEEE Trans. Automat. Contr., 53:5 (2008), 1303–1305 | DOI | MR | Zbl

[17] Yang X., Zhu J. J., “A generalization of Chang transformation for linear time-varying systems”, Proc. IEEE Conf. on Decision and Control, Atlanta, GA, 2010, 6863–6869

[18] Yang X., Zhu J. J., “Chang transformation for decoupling of singularly perturbed linear slowly time-varying systems”, Proc. 51st IEEE Conf. on Decision and Control (December 10-13, Maui, Hawaii, USA), 2012, 5755–5760

[19] Zhang Y., Naidu D. S., Cai C, et al., “Singular perturbations and time scales in control theories and applications: An overview 2002 – 2012”, Int. J. Inf. Syst. Sci., 9:1 (2014), 1–36 | MR