Connecting a Third-Order Singular Arc with Nonsingular Arcs of Optimal Control in a Minimization Problem for a Psoriasis Treatment Model

We study a mathematical model of psoriasis treatment defined by a system of three differential equations on a fixed time interval. These equations describe the interaction between the populations of T-lymphocytes, keratinocytes, and dendritic cells, which play a crucial role in the development, course, and treatment of this disease. The model includes a bounded control defining the drug dose to suppress the interaction between T-lymphocytes and keratinocytes. We address the problem of minimizing the concentration of keratinocytes at the final point of a given time interval. The analysis of this optimal control problem is based on the Pontryagin maximum principle. We show that for certain relations between the parameters of the model, the corresponding optimal control may contain a third-order singular arc connected to nonsingular bang–bang arcs of this control. The main attention is paid to possible ways of such connection. Numerical calculations that confirm the obtained analytical results are presented.