On a Third-Order Singular Arc of Optimal Control in a Minimization Problem for a Mathematical Model of Psoriasis Treatment

A mathematical model of psoriasis treatment is considered on a given time interval. The model consists of three nonlinear differential equations that describe the relationship between the concentrations of T-lymphocytes, keratinocytes, and dendritic cells. The model also includes a bounded control defining the drug dose to suppress the interaction between T-lymphocytes and keratinocytes. For this model, a problem of minimizing the concentration of keratinocytes at the end point of a given time interval is stated. The Pontryagin maximum principle is applied to the analysis of this optimal control problem. For certain relations between the parameters of the model, this principle is used for studying a possible existence of a third-order singular arc of optimal control. Namely, the corresponding necessary optimality condition is verified, and formulas for the optimal solutions of differential equations on this arc are obtained. Finally, a connection of a control on such an arc with nonsingular bang-bang arcs of optimal control is investigated.