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 ... P. 173-181

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

Received October 15, 2019

Revised January 17, 2020

Accepted January 20, 2020

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00362).

Natal’ja Gennad’evna Novoselova, doctoral student, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia,
e-mail: n.g.novoselova@gmail.com

Nina Nikolaevna Subbotina, Dr. Phys.-Math. Sci., RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia,
e-mail: subb@uran.ru

REFERENCES

1.   Bellman R. Dynamic Programming. Princeton, N. J.: Princeton University Press, 1957, 340 p. ISBN: 069107951X .

2.   Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The mathematical theory of optimal processes. N Y; London: Interscience Publishers John Wiley & Sons, 1962, 360 p. Original Russian text published in Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal’nykh protsessov. Moscow: Nauka Publ., 1961, 392 p.

3.   Krasovskii N.N. Teoriya upravleniya dvizheniem [Theory of motion control]. Moscow, Nauka Publ., 1968, 476 p.

4.   Isaacs R. Differential games. N Y: John Wiley and Sons, 1965, 384 p. ISBN: 0471428604 . Translated to Russian under the title Differentsial’nye igry. Moscow: Mir Publ., 1967, 479 p.

5.   Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. N Y: Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8 . Original Russian text published in Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial’nye igry. Moscow: Nauka Publ., 1974, 456 p.

6.   Kurzhanski A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and observation under the conditions of uncertainty]. Moscow: Nauka Publ., 1977, 392 p.

7.   Nikol’skii M.S. On the alternating integral of Pontryagin. Math. USSR-Sb., 1983, vol. 44, no. 1, pp. 125–132. doi: 10.1070/SM1983v044n01ABEH000956 

8.   Kurzhanskii A.B., Melnikov N.B. On the problem of control synthesis: the Pontryagin alternating integral and the Hamilton–Jacobi equation. Sb. Math., 2000, vol. 191, no. 6, pp. 849–881. doi: 10.1070/sm2000v191n06ABEH000484 

9.   Ushakov V.N., Ukhobotov V.I., Lipin A.E. An Addition to the Definition of a Stable Bridge and an Approximating System of Sets in Differential Games. Proc. Steklov Inst. Math., 2019, vol. 304, pp. 268–280. doi: 10.1134/S0081543819010206 

10. Patsko V., Kumkov S., Turova V. Pursuit-evasion games. In:  Basar T., Zaccour G. (eds), Handbook of Dynamic Game Theory,  Cham: Springer, 2018, pp. 1-87. doi: 10.1007/978-3-319-27335-8_30-2

11.   Bratus’ A.S., Chumerina E.S. Optimal control synthesis in therapy of solid tumor growth. Comput. Math. and Math. Phys., 2008, vol. 48, no. 6, pp. 892–911. doi: 10.1134/S096554250806002X 

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, 2018, vol. 51, no. 32, pp. 855–860. doi: 10.1016/j.ifacol.2018.11.438 

13.   Subbotina N.N., Kolpakova E.A., Tokmantsev T.B., Shagalova L.G. Metod kharakteristik dlya uravneniya Gamil’tona — Yakobi — Bellmana [The method of characteristics for Hamilton — Jacobi — Bellman equations]. Ekaterinburg: UrO RAN Publ., 2013, 244 p.

14.   Subbotin A.I. Generalized solutions of first-order PDEs. The dynamical optimization perspective. Basel: Birkhauser, 1995, 314 p. doi: 10.1007/978-1-4612-0847-1 . Translated to Russian under the title Obobshchennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka: Perspektivy dinamicheskoi optimizatsii. Moscow; Izhevsk: Inst. Komp’yuter. Issled., 2003, 336 p.

Cite this article as: 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 Mekhaniki URO RAN, 2020, vol. 26, no. 1, pp. 173–181.