** Optimal Feedback Control in a Mathematical Model of Malignant Tumour Treatment with the Immune Reaction Taken Into Account**
**Yegorov I. Ye.**
*Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics, Russian Federation, 119991, Moscow, Leninskie Gory, MSU, 2nd educational building*

**Abstract.** An optimal control problem for a mathematical model of malignant tumour treatment with the immune reaction taken into account is considered. The toxicity effect of the chemotherapeutic agent on both tumour and immunocompetent cells is taken into account; increasing therapy functions are considered. A standard linear pharmacokinetic equation for the chemotherapeutic agent is added to the system. The aim is to find an optimal strategy of treatment to minimize the tumour volume while at the same time keeping the immune response not lower than a fixed permissible level as far as possible. Sufficient conditions for the existence of not more than one switching and not more than two switchings without singular regimes are obtained. The surfaces in the extended phase space, on which the last switching in direct time (i.e., the first switching in inverse time) appears, are constructed analytically. The results of numerical simulations are shown.

**Key words:** *malignant tumour dynamics, immune reaction, standard linear pharmacokinetic equation, Pontryagin's maximum principle, Hamilton-Jacobi-Bellman equation, optimal control synthesis, switching surfaces.*