Boundary optimal control problems with integral control constraints for fluid and solid mechanics
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We discuss Dirichlet boundary optimal control problems for fluid or solid mechanics problems with incompressibility constraints. Such constraints impose integral compatibility conditions on full Dirichlet controls. Here, we optimize the velocity of the fluid flow or the displacement of the elastic materials in the target domain, in conjunction with the regularity on the boundary control. We compare two different approaches to address: (a) the treatment of the compatibility condition and (b) the numerical implementation of the fractional norm. One method uses a scalar Lagrange multiplier for the boundary control on a more regular space. The other method implements lifting functions to treat the boundary controls as distributed volume controls. We describe the differences between the two formulations. Numerical results are presented for the finite element approximation of the optimality systems arising from the first-order necessary conditions.