Geodesic
Construction of the viability set in a problem of chemotherapy of a malignant tumor growing according to the Gompertz law
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 1, pp. 173-181 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The problem of chemotherapy of a malignant tumor growing according to the Gompertz law is considered. The mathematical model is a system of two ordinary differential equations. We study a problem of optimal control (optimal therapy) aiming at the minimization of the malignant cells in the body at a given terminal time $T$. The viability set of this problem, i.e., the set of initial states of the model (the volume of the tumor and the amount of the drug in the body) for which an optimal control guarantees that the dynamics of the system up to the time $T$ is compatible with life in terms of the volume of the tumor, is constructed analytically.
Keywords: viability set, optimal control, value function. 
@article{TIMM_2020_26_1_a13,
     author = {N. G. Novoselova and N. N. Subbotina},
     title = {Construction of the viability set in a problem of chemotherapy of a malignant tumor growing according to the {Gompertz} law},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {173--181},
     year = {2020},
     volume = {26},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2020_26_1_a13/}
}
TY  - JOUR
AU  - N. G. Novoselova
AU  - N. N. Subbotina
TI  - Construction of the viability set in a problem of chemotherapy of a malignant tumor growing according to the Gompertz law
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2020
SP  - 173
EP  - 181
VL  - 26
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2020_26_1_a13/
LA  - ru
ID  - TIMM_2020_26_1_a13
ER  - 
%0 Journal Article
%A N. G. Novoselova
%A N. N. Subbotina
%T Construction of the viability set in a problem of chemotherapy of a malignant tumor growing according to the Gompertz law
%J Trudy Instituta matematiki i mehaniki
%D 2020
%P 173-181
%V 26
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2020_26_1_a13/
%G ru
%F TIMM_2020_26_1_a13
N. G. Novoselova; N. N. Subbotina. Construction of the viability set in a problem of chemotherapy of a malignant tumor growing according to the Gompertz law. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 1, pp. 173-181. http://geodesic.mathdoc.fr/item/TIMM_2020_26_1_a13/

[1] Bellman R., Dynamic programming, Princeton University Press, Princeton, 1957, 340 pp. | MR | Zbl

[2] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1961, 392 pp. | MR

[3] Krasovskii N.N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 476 pp.

[4] Aizeks R., Differentsialnye igry, Mir, M., 1967, 489 pp.

[5] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR

[6] Kurzhanskii A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 392 pp.

[7] Nikolskii M.S., “Ob alternirovannom integrale L.S. Pontryagina”, Mat. sb., 116:1 (1981), 136–144 | MR

[8] Kurzhanskii A.B., Melnikov N.B., “O zadache sinteza upravlenii: alternirovannyi integral Pontryagina i uravnenie Gamiltona - Yakobi”, Mat. sb., 191:6 (2000), 69–100 | DOI

[9] Ushakov V.N., Ukhobotov V.I., Lipin A.E., “Ob odnom dopolnenii k opredeleniyu stabilnogo mosta i approksimiruyuschei sistemy mnozhestv v differentsialnykh igrakh”, Tr. MIAN, 304 (2019), 285–297 | DOI | Zbl

[10] Patsko V., Kumkov S., Turova V., “Pursuit-evasion games”, Handbook of Dynamic Game Theoryeds. . T. Basar, G. Zaccour, Springer, Cham, 2018, 1–87 | DOI

[11] Bratus A.S.,Chumerina E.S., “Cintez optimalnogo upravleniya v zadache vybora lekarstvennogo vozdeistviya na rastuschuyu opukhol”, Zhurn. vychisl. matematiki i mat. fiziki, 48:6 (2008), 946–966 | MR | Zbl

[12] Subbotina N.N. , Novoselova N.G., “The value function in a problem of chemotherapy of a malignant tumor growing according to the Gompertz law”, IFAC-PapersOnLine, 51:32 (2018), 855–860 | DOI

[13] Subbotina N. N., Kolpakova E. A.,Tokmantsev T. B.,Shagalova L. G., Metod kharakteristik dlya uravneniya Gamiltona - Yakobi - Bellmana, Izd-vo UrO RAN, Ekaterinburg, 2013, 244 pp.

[14] Subbotin A. I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka: Perspektivy dinamicheskoi optimizatsii, In-t kompyuter. issled., M.; Izhevsk, 2003, 336 pp.