The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of
pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau-Yosida
regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions
towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization
strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is
carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each
regularized problem a semi-smooth Newton method is applied.
Second-order sufficient optimality conditions are established for the optimal control
of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In
contrast to former publications on this subject, the cone of critical directions is the smallest possible in the
sense that the second-order sufficient conditions are the closest to the associated necessary ones. The theory
is developed for elliptic distributed controls in domains up to dimension three. Moreover, problems of elliptic
boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one,
respectively.
A duality based semismooth Newton framework for solving variational inequalities of the second kind
(2010)
In an appropriate function space setting, semismooth Newton methods are proposed
for iteratively computing the solution of a rather general class of variational inequalities (VIs) of the
second kind. The Newton scheme is based on the Fenchel dual of the original VI problem which
is regularized if necessary. In the latter case, consistency of the regularization with respect to the
original problem is studied. The application of the general framework to specific model problems
including Bingham flows, simplified friction, or total variation regularization in mathematical imaging
is described in detail. Finally, numerical experiments are presented in order to verify the theoretical
results.
Optimality Conditions for State-Constrained PDE Control Problems with Time-Dependent Controls
(2008)
The paper deals with optimal control problems for semilinear
elliptic and parabolic PDEs subject to pointwise state constraints.
The main issue is that the controls are taken from a restricted
control space. In the parabolic case, they are vector-valued
functions of the time, while they are vectors in elliptic
problems. Under natural assumptions, first- and second-order
sufficient optimality conditions are derived. The main result is the
extension of second-order sufficient conditions to semilinear
parabolic equations in domains of arbitrary dimension. In the
elliptic case, the problems can be handled by known results of
semi-infinite optimization. Here, different examples are discussed
that exhibit different forms of active sets and where second-order
sufficient conditions are satisfied at the optimal solution.
The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of
pointwise state constraints is investigated. A Moreau-Yosida regularization of the problem is proposed to obtain regular
multipliers. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is
presented. The paper is ended with a numerical experiment.