On a class of optimal control problems for functional differential systems

V. P. Maksimov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 1, pp. 131-142

Résumé

A linear functional differential control system of general form with aftereffect is considered. An optimal control problem with linear constraints on the state and control variables is studied. The control is realized by a linear operator of general form. The cases of distributed and lumped delay in the control loop, as well as the case of impulsive control, are covered. The Cauchy matrix is used to reduce the problem under consideration to a problem formulated only in terms of control variables with the use of some auxiliary variables linked with the defining relations for the Cauchy matrix of the system. In the case when the control is chosen from a finitedimensional subspace of the control space, a problem effectively solvable by standard software tools is written explicitly. An example of an applied optimal control problem that arises in economic dynamics is presented. A class of hybrid systems (systems with continuous and discrete times) reducible to the system under consideration is described.

Détail

Keywords: linear systems, control, optimization.

Bibliographie

[1] Gabasov R., Kirillova F.M., Printsip maksimuma v teorii optimalnogo upravleniya, LIBROKOM, M., 2011, 272 pp. | MR

[2] Kolmanovskii V.B., Schaikhet L.E., Control of systems with aftereffect, Trans. Math. Monographs, 157, AMS, N Y, 1996, 336 pp. | DOI | MR

[3] Shevchenko G.V., “Chislennyi metod resheniya zadachi minimizatsii raskhoda resursov dlya lineinykh sistem s postoyannym zapazdyvaniem”, Avtomatika i telemekhanika, 2014, no. 10, 25–38

[4] Shevchenko G.V., “Chislennoe reshenie zadachi optimalnogo bystrodeistviya dlya lineinykh sistem s postoyannym zapazdyvaniem”, Vestn. Udmurt. un-ta (Matematika. Mekhanika. Kompyuternye nauki), 2012, no. 2, 100–105

[5] Gabasov R., Kirillova F.M., Pavlenok N.S., “Optimalnoe diskretno-impulsnoe upravlenie lineinymi sistemami”, Avtomatika i telemekhanika, 2008, no. 3, 103–125

[6] Korotkii D.A., “Reshenie zadachi optimalnogo upravleniya dlya sistem s zapazdyvaniem”, Vestn. Udmurt. un-ta (Matematika. Mekhanika. Kompyuternye nauki), 2008, no. 2, 61–62

[7] Gabasov R., Grushevich O.P., Kirillova F.M., “Optimalnoe upravlenie lineinymi sistemami s zapazdyvaniem s uchetom terminalnykh ogranichenii na ikh sostoyaniya”, Avtomatika i telemekhanika, 2007, no. 12, 3–20

[8] Maksimov V.P., “Nekotorye voprosy teorii upravleniya funktsionalno-differentsialnymi sistemami”, Izv. In-ta matematiki i informatiki Udmurt. gos. un-ta, 46:2 (2015), 112–119 | MR

[9] Azbelev N.V., Maksimov V.P., Rakhmatullina L.F., Vvedenie v teoriyu funktsionalno-differentsialnykh uravnenii, Nauka, M., 1991, 280 pp. | MR

[10] Azbelev N.V., Maksimov V.P., Rakhmatullina L.F., Elementy sovremennoi teorii funktsionalno-differentsialnykh uravnenii. Metody i prilozheniya, In-t kompyuternykh issledovanii, M., 2002, 384 pp. | MR

[11] Azbelev N.V., Maksimov V.P., Rakhmatullina L. F., Introduction to the theory of functional differential equations: methods and applications, Hindawi Publ. Corporation, N Y; Cairo, 2007, 314 pp. | MR

[12] Anokhin A.V., “O lineinykh impulsnykh sistemakh dlya funktsionalno-differentsialnykh uravnenii”, Dokl. AN SSSR, 286:5 (1986), 1037–1040 | MR

[13] Kurzweil Ja., “Generalized ordinary differential equations and continuous dependence on a parameter”, Czechoslovak Math. J., 1957, no. 7, 418–449 | MR

[14] Zavalischin S.T., Sesekin A.N., Impulsnye protsessy. Modeli i prilozheniya, Nauka, M., 1991, 256 pp. | MR

[15] Schwabik S., Generalized ordinary differential equations, World Scientific, Singapore, 1992, 392 pp. | MR

[16] Ashordia M., “On the stability of solutions of the multipoint boundary value problem for the system of generalized ordinary differential equations”, Mem. Differential Equations Math. Phys., 6 (1995), 1–57 | MR

[17] Maksimov V.P., Rumyantsev A.N., “Kraevye zadachi i zadachi impulsnogo upravleniya v ekonomicheskoi dinamike. Konstruktivnoe issledovanie”, Izv. vuzov. Matematika, 37:5 (1993), 56–71 | MR

[18] Maksimov V.P., “O formule Koshi dlya funktsionalno-differentsialnogo uravneniya”, Differentsialnye uravneniya, 13:4 (1977), 601–606 | MR

[19] Maksimov V.P., Voprosy obschei teorii funktsionalno-differentsialnykh uravnenii, Izd-vo Perm. gos. un-ta, Izd-vo Prikam. sotsial. in-ta, Izd-vo Perm. sovremen. sotsial.-gumanit. kolledzha, Perm, 2003, 306 pp.

[20] Maksimov V.P., “Odin variant printsipa maksimuma dlya lineinykh sistem s posledeistviem”, Vestn. Tambov. un-ta. Ser. Estestv. i tekhn. nauki, 20:5 (2015), 1284–1286

[21] Maksimov V.P., Chadov A.L., “Gibridnye modeli v zadachakh ekonomicheskoi dinamiki”, Vestn. Perm. un-ta. Ser. Ekonomika, 2011, no. 2, 13–23

[22] Chadov A. L., Maksimov V.P., “Linear boundary value problems and control problems for a class of functional differential equations with continuous and discrete times”, Funct. Differ. Equ., 19:1-2 (2012), 49–62 | MR

[23] Andrianov D.L., “Kraevye zadachi i zadachi upravleniya dlya lineinyi raznostnykh sistem s posledeistviem”, Izv. vuzov. Matematika, 37:5 (1993), 3–16