An optimal control problem for a biological population model with diffusion and infectious disease

Abstract

This paper studies the optimal control problem for a biological population model with diffusion and infectious diseases, and we intend to maximize the density of species with proper control of predator populations. First, we prove that the controlled system admits a unique global positive strong solution for any given admissible control variable. Then, we show that the optimal control problem admits at least one optimal pair by applying a compactness argument. Moreover, by introducing appropriate dual variables for the linearized control system, we establish bang-bang-type first-order necessary optimality conditions for the optimal pair. Second-order necessary and sufficient conditions for the optimal pair are also provided. Finally, a concrete numerical example is given to verify the results of this paper.