Optimal control of the stationary Navier-Stokes equations with mixed control-state constraints

Revised version of the preprint first published 06. December 2005In this paper we consider the distributed optimal control of the Navier-Stokes equations in presence of pointwise mixed control-state constraints. After deriving a first order necessary condition, the regularity of the mixed constraint multiplier is investigated. Second-order sufficient optimality conditions are studied as well. In the last part of the paper, a semi-smooth Newton method is applied for the numerical solution of the control problem. The convergence of the method is proved and numerical experiments are carried out

Optimal Control of the 2D Landau-Lifshitz-Gilbert Equation with Control Energy in Effective Magnetic Field

The optimal control of magnetization dynamics in a ferromagnetic sample at a microscopic scale is studied. The dynamics of this model is governed by the Landau-Lifshitz-Gilbert equation on a two-dimensional bounded domain with the external magnetic field (the control) applied through the effective field. We prove the global existence and uniqueness of a regular solution in $\mathbb S^2$ under a smallness condition on control and initial data. We establish the existence of optimal control and derive a first-order necessary optimality condition using the Fr\'echet derivative of the control-to-state operator and adjoint problem approach