Optimal control of a delayed rumor propagation model with saturated control functions and $L^1$-type objectives

Abstract

Rumor is an important form of social interaction, and its spreading has a significant impact on human lives. The optimal control theory is an important tool to better manage the spread of rumors. Most of the literature on rumor propagation models deals with quadratic cost functions relative to the control variable. In this paper, we have considered a time-delay rumor propagation model with saturated control functions and an objective function of $L^1$-type linear with respect to the control variables. In the general case, the introduction of the delay in the dynamic systems represents the time lag between the action on the system and the response of the system to this action. The delay is incorporated in our model to make it more realistic and to describe the latency period. The existence of the optimal control pair is also proved. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.