Boundary velocity suboptimal control of incompressible flow in
        cylindrically perforated domain

Ciro D’Apice; Umberto De Maio; Peter I. Kogut

doi:10.3934/dcdsb.2009.11.283

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March 2009, 11(2): 283-314. doi: 10.3934/dcdsb.2009.11.283

Boundary velocity suboptimal control of incompressible flow in cylindrically perforated domain

Ciro D’Apice ^1, , Umberto De Maio ^2, and Peter I. Kogut ^3,

1.	Università di Salerno, DIIMA, Via Ponte don Melillo, 84084 Fisciano (SA), Italy
2.	Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli “Federico II”, DMA “R. Caccioppoli”, Complesso Monte S. Angelo, via Cintia, 80126 Napoli
3.	Dnipropetrovsk National University, Department of Differential Equations, Kozakov str., 18/14, 49050 Dnipropetrovsk, Ukraine

Received October 2007 Revised March 2008 Published December 2008

Abstract
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In this paper we study an optimal boundary control problem for the 3D steady-state Navier-Stokes equation in a cylindrically perforated domain $\Omega_{\epsilon}$. The control is the boundary velocity field supported on the 'vertical' sides of thin cylinders. We minimize the vorticity of viscous flow through thick perforated domain. We show that an optimal solution to some limit problem in a non-perforated domain can be used as basis for the construction of suboptimal controls for the original control problem. It is worth noticing that the limit problem may take the form of either a variational calculation problem or an optimal control problem for Brinkman's law with another cost functional, depending on the cross-size of thin cylinders.

Keywords: optimal control, variational convergence, vorticity minimization, Dirichlet control, Navier-Stokes equation.

Mathematics Subject Classification: Primary: 35Q30, 49J20; Secondary: 35B27, 35B40.

Citation: Ciro D’Apice, Umberto De Maio, Peter I. Kogut. Boundary velocity suboptimal control of incompressible flow in cylindrically perforated domain. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 283-314. doi: 10.3934/dcdsb.2009.11.283

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