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

Evgenii S. Baranovskii and Mikhail A. Artemov
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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

Mathematics, 2021, vol. 9, issue 3, 1-16

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.

Keywords: optimal control problem; non-Newtonian fluid; second-grade fluid; nonlocal model; influence function; Navier slip condition; existence; strong solution; weak solution; marginal function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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