N.N. Subbotina, N.G. Novoselova. Optimal result in a control problem with piecewise monotone dynamics ... P. 265-280

We consider an optimal control problem for a deterministic nonlinear system with piecewise monotone dynamics. The mathematical model under consideration describes the process of a chemotherapy treatment of a malignant tumor. The research makes it possible to analyze the influence of the type of nonmonotonicity on the structure of the optimal control. We consider the case when the therapy function, which describes the effect of the drug on the cell growth rate, has two maxima. Comparisons are made with the results for the previously studied case of a single maximum of the therapy function in this model. This paper is devoted to the construction of the value function for the optimal control problem under consideration. As is known, the value function is the basis for constructing an optimal synthesis, i.e., an optimal feedback strategy in the therapy. We use the fact that the value function is the unique minimax (viscosity) solution of the Cauchy problem for the basic Hamilton-Jacobi-Bellman (HJB) equation. By means of the continuous gluing of a finite number of smooth functions obtained by the Cauchy method of characteristics for auxiliary HJB equations, a continuous function $\varphi$ is constructed. A new element of the construction is the line of nonsmooth gluing with the use of the Rankin-Hugoniot conditions. This line plays a key role for the optimal feedback strategy, because it determines its discontinuity line. We prove that the constructed function $\varphi$ coincides with the minimax solution of the Cauchy problem for the basic HJB equation.

Keywords: optimal control, Rankine-Hugoniot line, Hamilton-Jacobi-Bellman equation, Cauchy method of characteristics.

The paper was received by the Editorial Office on September 2, 2017

Nina Nikolaevna Subbotina, Dr. Phys.-Math. Sci., RAS Corresponding Member, Prof., Krasovskii
Institute ofMathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg,
620990 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: subb@uran.ru .

Natal’ja Gennad’evna Novoselova, graduate student, Krasovskii Institute of Mathematics and Mechanics,
Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural
Federal University, Yekaterinburg, 620002 Russia, e-mail: n.g.novoselova@gmail.com.


1.   Chumerina E.S. Synthesis of optimal control in mathematical models of chemotherapy of a tumor growing according to Gompertz law and logistic law. Cand. Phys.-Math. Sci. Dissertation. Moscow, MIIT Publ., 2009 (in Russian).

2.   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.

3.   Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The mathematical theory of optimal processes.  New York, London, Sydney,  John Wiley and Sons, Inc., 1962,  360 p.  ISBN: 0470693819. 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.

4.   Krasovskii N.N. Teoriya upravleniya dvizheniem. Lineinye sistemy [Theory of motion control. Linear systems]. Moscow, Nauka Publ., 1968, 476 p.

5.   Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. NY, 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.   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. Publ., 2003, 336 p.

7.   Crandall M.G., Lions P.-L. Viscosity solutions of Hamilton-Jacobi equations.  Trans. Amer. Math. Soc., 1983, vol. 277, no. 1, pp. 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.

8.   Goritsky A.Yu., Kruzhkov S.N., Chechkin G.A.  Uravneniya s chastnymi proizvodnymi pervogo poryadka: Uchebnoe posobie [Partial differential equations of the first order: Textbook].  Moscow, Moscow State University Publ., 1999. 96 p.

9.   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.