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Numerical approximation of optimal flow control problems by a penalty method: Error estimates and numerical results. (English) Zbl 0952.93036

The authors study the following optimal control problem: Find the triple \((u,p,g)\) such that the cost functional \({\mathcal J}(\overline{u}, p, \overline{g})={\mathcal F}(u,p)+ (\beta^2/2) \int_\Gamma |g|^2 d\Gamma\) is minimized subject to the steady-state Navier-Stokes equations \(-\nu\Delta \overline{u}+ (\overline{u}\cdot\nabla) \overline{u}+ \nabla p= \overline{f}\) in \(\Omega\), \(\nabla \cdot\overline{u}= 0\) in \(\Omega\) and \(\overline{u}= \overline{g}\) on \(\Gamma\), where \(\Omega\) is the two-dimensional flow domain, \(\Gamma\) the boundary of \(\Omega\) where the control is applied, \(\nu\) the constant viscosity, \(\overline{u}\) the velocity, \(p\) the pressure, \(\overline{f}\) the applied forces, \(\overline{g}\) the boundary control; \({\mathcal F}(u,p)\) a functional which unifies the cost functionals, \(\beta^2\) a parameter adjusting the relative weight of the two terms in the functional. The above presented problem is studied using the sequential quadratic method for the penalized Neumann control approach for solving Dirichlet control problems from numerical and computational points of view. In the present paper the authors apply the theoretical results obtained in a previous paper, and they focus on the numerical and computational aspects of the penalized Neumann control approach. They study the convergence of the penalized approaches and compare them with the unpenalized approaches. In the final part of the paper they apply the theoretical results to some test problems with different bidimensional channels and geometric cavities where the goals are to minimize the vorticity and to drive the velocity to a desired direction. The obtained applied results are commented and compared in many suggestive plots.

MSC:

93B40 Computational methods in systems theory (MSC2010)
49M05 Numerical methods based on necessary conditions
76D05 Navier-Stokes equations for incompressible viscous fluids
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