| With profound respect, this work is dedicated to Professor Constantin Fetecau for his 80th birthday
Evgenii S. Baranovskii† Abstract: This paper deals with optimal distributed control of a strongly nonlinear system modeling the steady-state flow of a variable-viscosity second-grade fluid through a porous medium. We consider the internal flow within a bounded Lipschitz domain (of dimension two or three) under the no-slip boundary condition, assuming that the external force serves as the control parameter. The problem is rigorously formulated in both classical and weak settings. Our main result establishes the existence of an optimal weak solution by applying a known solvability principle for degenerate nonlinear operator equations involving weakly and completely continuous mappings in Sobolev spaces. In addition, we analyze the associated value function, proving its lower semicontinuity with respect to the one-sided Hausdorff distance, and present a necessary condition for the existence of a classical solution to the considered flow problem. Keywords: second-grade fluid, porous media, optimization problem, distributed control, classical solution, weak solution, existence theorem, value function, energy discrepancy functional. MSC: 49J20, 76D55, 76A05. DOI 10.56082/annalsarscimath.2026.4.205 Read full article *Accepted for publication on August 04, 2026 |
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application,
ISSN ONLINE 2066 – 6594
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