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Optimal Control for a Nonlocal Model of Non-Newtonian Fluid Flows

Author

Listed:
  • Evgenii S. Baranovskii

    (Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia)

  • Mikhail A. Artemov

    (Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia)

Abstract

This paper deals with an optimal control problem for a nonlocal model of the steady-state flow of a differential type fluid of complexity 2 with variable viscosity. We assume that the fluid occupies a bounded three-dimensional (or two-dimensional) domain with the impermeable boundary. The control parameter is the external force. We discuss both strong and weak solutions. Using one result on the solvability of nonlinear operator equations with weak-to-weak and weak-to-strong continuous mappings in Sobolev spaces, we construct a weak solution that minimizes a given cost functional subject to natural conditions on the model data. Moreover, a necessary condition for the existence of strong solutions is derived. Simultaneously, we introduce the concept of the marginal function and study its properties. In particular, it is shown that the marginal function of this control system is lower semicontinuous with respect to the directed Hausdorff distance.

Suggested Citation

  • Evgenii S. Baranovskii & Mikhail A. Artemov, 2021. "Optimal Control for a Nonlocal Model of Non-Newtonian Fluid Flows," Mathematics, MDPI, vol. 9(3), pages 1-16, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:275-:d:489884
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    References listed on IDEAS

    as
    1. Chemetov, Nikolai & Cipriano, Fernanda, 2018. "Optimal control for two-dimensional stochastic second grade fluids," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2710-2749.
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