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

Abstract

We consider a mathematical model of psoriasis treatment 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, we state the problem of minimizing the concentration of keratinocytes at the end point of a given time interval. The analysis of this optimal control problem is based on the Pontryagin maximum principle. For certain relations between the parameters of the model, we use this principle to study possible existence of a third-order singular arc of the optimal control. Namely, we verify the corresponding necessary optimality condition and derive formulas for the optimal solutions of the differential equations on this arc. Finally, we study the connection of the control on such an arc with nonsingular bang–bang arcs of the optimal control.