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arXiv · 1905.01415

An optimal control problem for the Navier-Stokes-$\alpha$ system

Exequiel Mallea-Zepeda·Elva Ortega-Torres·Élder J. Villamizar-Roa
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Abstract

In this paper we study a distributed optimal control problem for a three-dimensional Navier-Stokes-$\alpha$ model. We prove the solvability of the optimal control problem, and derive first-order optimality conditions by using a Lagrange multipliers Theorem. Finally, considering a velocity tracking control problem for the three-dimensional Navier-Stokes-$\alpha$ model, we analyze the relation of its optimality system to the corresponding one associated to the Navier-Stokes model by proving a convergence theorem, which establishes that, as the length scale $\alpha$ goes to zero, the optimality system of the three-dimensional Navier-Stokes-$\alpha$ model converges to the optimality system associated with the velocity tracking control problem of the Navier-Stokes equations.

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Exequiel Mallea-Zepeda, Elva Ortega-Torres, Élder J. Villamizar-Roa. 2019-05-04. An optimal control problem for the Navier-Stokes-$\alpha$ system. https://arxiv.org/abs/1905.01415

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