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This paper deals with boundary optimal control problems for the heat and Navier-Stokes equations and addresses the issue of defining controls in function spaces which are naturally associated to the volume variables by trace restriction. For this reason we reformulate the boundary optimal control problem into a distributed problem through a lifting function approach. The stronger regularity requirements which are imposed by standard boundary control approaches can then be avoided. Furthermore, we propose a new numerical strategy that allows to solve the coupled optimality system in a robust way for a large number of unknowns. The optimality system resulting from a finite element discretization is solved by a local multigrid algorithm with domain decomposition Vanka-type smoothers. The purpose of these smoothers is to solve the optimality system implicitly over subdomains with a small number of degrees of freedom, in order to achieve robustness with respect to the regularization parameters in the cost functional. We present the results of some test cases where temperature is the observed quantity and the control quantity corresponds to the boundary values of the fluid temperature in a portion of the boundary. The control region for the observed quantity is a part of the domain where it is interesting to match a desired temperature value.
An optimal shape control problem for the stationary Navier-Stokes
system is considered. An incompressible, viscous flow in a
two-dimensional channel is studied to determine the shape of part of
the boundary that minimizes the viscous drag. The
adjoint method and the Lagrangian multiplier method are used to derive
the optimality system for the shape
gradient of the design functional.
This paper is concerned with some optimal control problems for the
Stefan-Boltzmann radiative transfer equation.
The objective of the optimisation is to obtain a desired temperature profile
on part of the domain by controlling the source or the shape of the domain.
We present two problems with the same objective functional:
an optimal control problem
for the intensity and the position of the heat sources and
an optimal shape design problem where
the top surface is sought as control. The problems are analysed and
first order necessity conditions in form of variation inequalities are
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