Zero-Order Asymptotics for the Solution of One Type of Singularly Perturbed Linear–Quadratic Control Problems in the Critical Case

G. A. Kurina ; N. T. Hoai

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 1, pp. 127-142

Résumé

We consider a linear–quadratic control problem in which there is the second power of a small parameter at the derivative of the state variable and the first power of the parameter both in the control term of the state equation and at the quadratic form with respect to the control variable in the performance index; moreover, the state equation represents a critical case of singular perturbation theory. A zero-order asymptotic expansion of the solution is constructed using the so-called direct scheme method in which a postulated asymptotic expansion of the solution is substituted directly into the problem statement and problems for finding the asymptotic terms are stated.

Détail

Keywords: linear–quadratic control problem, critical case, asymptotics of solution.
Mots-clés : singular perturbations

Bibliographie

[1] Vasileva A.B., Butuzov V.F., Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973, 272 pp. | MR

[2] Zhang Y., Naidu D.S., Cai C., Zou Y., “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

[3] Kurina G.A., Kalashnikova M.A., “Singulyarno vozmuschennye zadachi s raznotempovymi bystrymi peremennymi”, Avtomatika i telemekhanika, 2022, no. 11, 3–61 | DOI

[4] Butuzov V.F., Nefedov N.N., “Ob odnoi zadache teorii singulyarnykh vozmuschenii”, Differents. uravneniya, 12:10 (1976), 1736–1747 | MR | Zbl

[5] Vasileva A.B., Butuzov V.F., Singulyarno vozmuschennye uravneniya v kriticheskikh sluchayakh, Izd-vo MGU, M., 1978, 106 pp.

[6] Kurina G.A., Khoai N.T., “Proektornyi podkhod k algoritmu Butuzova — Nefedova asimptoticheskogo resheniya odnogo klassa singulyarno vozmuschennykh zadach v kriticheskom sluchae”, Zhurn. vychisl. matematiki i mat. fiziki, 60:12 (2020), 2073–2084 | DOI | Zbl

[7] Danilin A.R., Zakharov S.V., Kovrizhnykh O.O., Lelikova E.F., Pershin I.V., Khachai O.Yu., “Ekaterinburgskoe nasledie Arlena Mikhailovicha Ilina”, Tr. In-ta matematiki i mekhaniki UrO RAN, 23:2 (2017), 42–66 | DOI | MR

[8] Belokopytov S.V., Dmitriev M.G., “Reshenie klassicheskikh zadach optimalnogo upravleniya s pogransloem”, Avtomatika i telemekhanika, 7 (1989), 71–82 | MR | Zbl

[9] Dmitriev M.G., Kurina G.A., “Pryamaya skhema postroeniya asimptotiki resheniya klassicheskikh zadach optimalnogo upravleniya”, Programmnye sistemy: Teoreticheskie osnovy i prilozheniya, Nauka, Fizmatlit, M., 1999, 44–55

[10] Kurina G., Nguyen T.H., “Zero-order asymptotic solution of a class of singularly perturbed linear-quadratic problems with weak controls in a critical case”, Optim. Control Appl. Math., 40:5 (2019), 859–879 | DOI | MR | Zbl

[11] Sibuya Y., “Some global properties of matrices of functions of one variable”, Math. Ann., 161 (1965), 67–77 | DOI | MR | Zbl

[12] Danilin A.R., Kovrizhnykh O.O., “Asimptotika optimalnogo vremeni perevoda lineinoi upravlyaemoi sistemy s nulevymi veschestvennymi chastyami sobstvennykh znachenii matritsy pri bystrykh peremennykh na neogranichennoe tselevoe mnozhestvo.”, Tr. In-ta matematiki i mekhaniki UrO RAN, 27:1 (2021), 48–61 | DOI | MR

[13] Danilin A.R., Kovrizhnykh O.O., “Asimptotika resheniya odnoi zadachi bystrodeistviya s neogranichennym tselevym mnozhestvom dlya lineinoi sistemy v kriticheskom sluchae”, Tr. In-ta matematiki i mekhaniki UrO RAN, 28:1 (2022), 58–73 | DOI | MR

[14] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972, 740 pp. | MR

[15] Elsgolts L.E., Differentsialnye uravneniya i variatsionnoe ischislenie, Nauka. Fizmatlit, M., 1965, 424 pp.

[16] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961, 391 pp. | MR