Optimal control problem for a general reaction-diffusion eco-epidemiological model with disease in prey

Abstract

This paper deals with an optimal control problem for a general reaction-diffusion predator-prey model with disease in prey population. Infected prey will recover from a medication considered as a control strategy. Our primary goal is to characterize an optimal control which minimizes the total density of infected prey and the costs of treatment. Firstly, we obtain the existence and some estimates of the unique strong solution for the controlled system by applying semigroup theory. Subsequently, the existence of optimal pair is proved by means of the technique of minimizing sequence. Furthermore, by proving the differentiability of the control-to-state mapping, we derive the first-order necessary optimality condition, and point out that the optimal is a Bang-Bang control in a special case. Finally, several numerical simulations are performed to illustrate the concrete realization and practical application of the theoretical results obtained in this contribution.