Investigation of Delay-Induced Dynamics and Optimal Control Using Ultrasound Technology in a Mathematical Model of Biodiesel Production

In this study, a delay-induced mathematical model for biodiesel production is proposed by introducing a discrete time delay into a reaction kinetic framework. The delay represents the intrinsic lag associated with intermediate biochemical processes during biodiesel synthesis. An optimal control strategy based on ultrasound-assisted production is further incorporated to regulate biodiesel yield while minimizing control cost. The resulting delayed optimal control system is analyzed using Pontryagin’s Maximum Principle. To efficiently simulate the delayed dynamics, a fixed-step RK4 solver for delay differential equations is developed and validated against the ddeint method. Numerical simulations demonstrate that delay significantly affects transient biodiesel dynamics and may induce oscillatory behavior. Sensitivity and uncertainty analyses identify key kinetic parameters governing oscillation amplitude and system robustness. The results highlight the potential of ultrasound-assisted control strategies for enhancing biodiesel production.