Bang-bang property and singular regimens of the optimal control in one mathematical model for the treatment of psoriasis

Abstract

For a fixed period of time, a mathematical model for the treatment of psoriasis is considered, consisting of three differential equations. These equations describe the relationships between the major cell populations that are responsible for the occurrence, progression, and treatment of this disease. The model also contains a bounded control function reflecting the effects of a drug aimed at inhibiting interactions between specific cell populations. The task is to reduce the population of cells directly affecting the disease at the end of a given period of time. The analysis of such an optimal control problem is carried out using the Pontryagin maximum principle. The main attention is paid to clarifying the bang-bang property of the corresponding optimal control, as well as the possibility of its having singular regimens of various orders depending on the relationships between the parameters of the original model.