Geodesic
Approximations in the stability problem for linear periodic systems with aftereffect
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 2, Tome 191 (2021), pp. 29-37.

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The asymptotic stability of a linear periodic system of differential equations with aftereffect is determined by the location of the spectrum of the infinite-dimensional, compact monodromy operator. Analytical representations of such operators can be obtained only for systems of a special type. In numerical simulations, finite-dimensional approximations of the monodromy operators are used. In this paper, we examine a procedure for approximating a system of differential equations with aftereffect by systems of ordinary differential equations of large dimension proposed by N. N. Krasovskii. Finite-dimensional approximations for monodromy operators are constructed in the Hilbert space of states of a periodic system with aftereffect. We prove that increasing of the dimension of finite-dimensional approximations leads to increasing of the approximation accuracy.
Keywords: system with aftereffect, stability of motion, finite-dimensional approximation. 
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Yu. F. Dolgii; R. I. Shevchenko. Approximations in the stability problem for linear periodic systems with aftereffect. 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 2, Tome 191 (2021), pp. 29-37. http://geodesic.mathdoc.fr/item/INTO_2021_191_a3/

[1] Berdyshev V. I., Subbotin Yu. I., Chislennye metody priblizheniya funktsii, Sredne-Uralskoe knizhnoe izd-vo, Sverdlovsk, 1979

[2] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965

[3] Dolgii Yu. F., “Kharakteristicheskoe uravnenie v zadache asimptoticheskoi ustoichivosti periodicheskoi sistemy s posledeistviem”, Tr. in-ta mat. mekh. UrO RAN., 11:1 (2005), 85–96

[4] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1984

[5] Krasovskii N. N., “Ob approksimatsii odnoi zadachi analiticheskogo konstruirovaniya regulyatorov v sisteme s zapazdyvaniem”, Prikl. mat. mekh., 28:4 (1964), 716–724 | Zbl

[6] Krasovskii N. N., Osipov Yu. S., “O stabilizatsii dvizhenii upravlyaemogo ob'ekta s zapazdyvaniem v sisteme regulirovaniya”, Izv. AN SSSR. Tekhn. kibern., 6 (1963), 3–15 | Zbl

[7] Kryazhimskii A. V., Maksimov V. I., “Approksimatsiya lineinykh differentsialno-raznostnykh igr”, Prikl. mat. mekh., 42:2 (1978), 202–209 | MR

[8] Kurzhanskii A. B., “K approksimatsii lineinykh differentsialnykh uravnenii s zapazdyvaniem”, Differ. uravn., 3:12 (1967), 2094–2107 | MR | Zbl

[9] Repin Yu. M., “O priblizhennoi zamene sistem s zapazdyvaniem obyknovennymi dinamicheskimi sistemami”, Prikl. mat. mekh., 29:2 (1965), 226–235 | Zbl

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

[11] Shimanov S. N., “K teorii lineinykh differentsialnykh uravnenii s periodicheskimi koeffitsientami i zapazdyvaniem vremeni”, Prikl. mat. mekh., 27:3 (1963), 450–458 | MR | Zbl

[12] Yakubovich V. A., Starzhinskii V. M., Lineinye differentsialnye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya, Nauka, M., 1972

[13] Burns J. A., Clif E., “Methods for approximating solutions to linear hereditary quadratic optimal control problems”, IEEE Trans. Automat. Control., 23:1 (1978), 21–26 | DOI | MR

[14] Delfour M. C., “The linear quadratic optimal control problem for hereditary differential systems: Theory and numerical solution”, SIAM J. Appl. Math. Optim., 3:2 (1977), 101–162 | MR | Zbl

[15] Gibson J. S., “Linear-quadratic optimal control of hereditary differential systems: infinite dimensional Riccati equations and numerical approximations”, SIAM J. Control Optim., 21:1 (1983), 95–139 | DOI | MR | Zbl

[16] Pandolfi L., “Stabilization of neutral functional differential equations”, J. Optim. Theory Appl., 20:2 (1976), 191–204 | DOI | MR | Zbl