Optimal control problems of nonlinear degenerate parabolic differential systems with logistic time-varying delays

In this paper, we study a bioeconomic model for optimal control problems for a class of systems governed by degenerate parabolic equations governing diffusive biological species with logistic growth terms and multiple time-varying delays. We prove the existence, uniqueness and regularity results for this degenerate parabolic equation. The viscosity solution theory is used to obtain the existence result. Afterwards, we formulate the optimal control problem in two cases. Firstly, we suppose that this biological species causes damage to environment (forest lands, farming, etc): the optimal control is the trapping rate and the cost functional is a combination of damage and trapping costs. Secondly, we consider the optimal harvesting control of a biological species: the optimal control is a distribution of harvesting effort on the biological species and the cost functional measure the difference between economic revenue and cost. The existence and the condition of uniqueness of the optimal solution are derived. First-order necessary conditions of optimality are obtained.