Optimal strategy for non-zero cost inhibiting in a stochastic microorganism flocculation model under environmental noise

This study introduces a new stochastic microorganism flocculation model that takes into account saturated control and environmental noise, in which inhibitors are used as control variables. Due to the difficulty in deriving the optimal control through solving state equations and adjoint equations, this paper investigates a near-optimal control problem, aiming to effectively control the growth of harmful microorganisms and minimize the cost of inhibitor expenditure. Initially, we establish priori estimates for nutrients, harmful microorganisms, flocculating microorganisms, and microorganism flocculants in the stochastic system. Furthermore, by applying the Pontryagin stochastic maximum principle, we derive sufficient and necessary conditions for achieving near-optimality in our novel system. Our findings reveal that white noise has a significant impact on microorganisms, and harmful microorganisms with inhibitors decline at a faster rate than those without them. Finally, we use numerical simulation to illustrate our conclusions.