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

Author

Listed:
  • Aníbal Coronel

    (Departamento de Ciencias Básicas, Facultad de Ciencias, Campus Fernando May, Universidad del Bío-Bío, Chillán 3780000, Chile)

  • Fernando Huancas

    (Departamento de Matemática, Facultad de Ciencias Naturales, Matemáticas y del Medio Ambiente, Universidad Tecnológica Metropolitana, Las Palmeras No. 3360, Ñuñoa-Santiago 7750000, Chile)

  • Esperanza Lozada

    (Departamento de Ciencias Básicas, Facultad de Ciencias, Campus Fernando May, Universidad del Bío-Bío, Chillán 3780000, Chile)

  • Marko Rojas-Medar

    (Departamento de Matemática, Instituto de Alta Investigación Matemática, Universidad de Tarapacá, Arica 1000000, Chile)

Abstract

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.

Suggested Citation

  • Aníbal Coronel & Fernando Huancas & Esperanza Lozada & Marko Rojas-Medar, 2021. "The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System," Mathematics, MDPI, vol. 9(5), pages 1-17, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:479-:d:506310
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    References listed on IDEAS

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    1. R. Filliger & M.-O. Hongler & L. Streit, 2008. "Connection between an Exactly Solvable Stochastic Optimal Control Problem and a Nonlinear Reaction-Diffusion Equation," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 497-505, June.
    2. Jang, Junyoung & Kwon, Hee-Dae & Lee, Jeehyun, 2020. "Optimal control problem of an SIR reaction–diffusion model with inequality constraints," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 171(C), pages 136-151.
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