Fractional optimal control of a mathematical model for managing hajj and umrah pilgrims’ crowds

Managing large-scale pilgrim movements during Hajj and Umrah requires mathematical tools that can describe congestion formation and support effective intervention strategies. This study develops a fractional optimal control model in which the normalized pilgrim population is divided into smoothly moving, congested, and redirected groups. Three time-dependent controls represent guidance, route diversion, and emergency intervention. Memory and delayed crowd responses are incorporated through the Caputo fractional derivative. Positivity, boundedness, and existence and uniqueness of the state solution are established. A congestion threshold [Formula: see text] is derived, and sufficient conditions for local stability of the congestion-free equilibrium are obtained using the fractional eigenvalue criterion. Global stability is considered separately for the classical integer-order model. The existence of an optimal control is proved, and the necessary optimality conditions are derived using a fractional Pontryagin maximum principle with left-sided Caputo state equations and right-sided fractional adjoint equations. A fractional predictor–corrector method combined with a forward–backward sweep iteration is used for numerical implementation. Compared with the uncontrolled case, the optimal strategy reduces the peak congested population by approximately [Formula: see text], the cumulative congestion burden by [Formula: see text], and the objective functional by [Formula: see text]. The results demonstrate the potential of fractional optimal control for congestion mitigation, while the illustrative normalized parameters require future calibration using operational pilgrimage data.