scholarly journals Optimal Control Problem for a Reaction-Diffusion System of Three Populations

2021 ◽  
Vol 11 (4) ◽  
pp. 808-828
Author(s):  
global sci
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System

scholarly journals The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 479
Author(s):  
Aníbal Coronel ◽  
Fernando Huancas ◽  
Esperanza Lozada ◽  
Marko Rojas-Medar
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Cost Function ◽  
Ecological Model ◽  
Control Function ◽  
Semigroup Theory ◽  
Reaction Diffusion ◽  
Functional Responses ◽  
Adjoint State

We research a control problem for an ecological model given by a reaction–diffusion system. The ecological model is given by a nonlinear parabolic PDE system of three equations modelling the interaction of three species by considering the standard Lotka-Volterra assumptions. The optimal control problem consists of the determination of a coefficient such that the population density of predator decreases. We reformulate the control problem as an optimal control problem by introducing an appropriate cost function. Then, we introduce and prove three types of results. A first contribution of the paper is the well-posedness framework of the mathematical model by considering that the interaction of the species is given by a general functional responses. Second, we study the differentiability properties of a cost function. The third result is the existence of optimal solutions, the existence of an adjoint state, and a characterization of the control function. The first result is proved by the application of semigroup theory and the second and third result are proved by the application of Dubovitskii and Milyutin formalism.

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