A linear-quadratic elliptic control problem with pointwise box constraints on the
state is considered. The state-constraints are treated by a Lavrentiev type regularization. It is
shown that the Lagrange multiplier associated with the regularized state-constraints are functions
in L2. Moreover, the convergence of the regularized controls is proven for regularization parameter
tending to zero. To solve the problem numerically, an interior point method and a primal-dual active
set strategy are implemented and treated in function space.
A class of optimal control problems for a semilinear elliptic equations with mixed
control-state constraints is considered. The existence of bounded and measurable Lagrange multipliers
is proven. As a particular application, the Lavrentiev type regularization of pointwise state
constraints is discussed. Here, the existence of associated regular multipliers is shown, too.
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.
A state-constrained optimal boundary control problem governed by a linear elliptic equation is considered. In order to obtain the optimality conditions for the solutions to the model problem, a Slater assumption has to be made that restricts the theory to the two-dimensional case. This difficulty is overcome by a source representation of the control and combined with a Lavrentiev type regularization. Optimality conditions for the regularized problem are derived, where the corresponding Lagrange multipliers have $L^2$-regularity. By the spectral theorem for compact and normal operators, the convergence result is shown. Moreover, the convergence for vanishing regularization parameter of the adjoint state associated with the regularized problem is shown. Finally, the uniform boundedness of the regularized Lagrange multipliers in $L^1(\O)$ is verified by a maximum principle argument.
New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the
monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function.
Using phase space analysis for the corresponding fifth-order dynamical
system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input.
Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.
A class of nonlinear elliptic optimal control
problems with mixed control-state constraints arising, e.g., in Lavrentiev-type
regularized state constrained optimal control is considered. Based
on its first order necessary optimality conditions, a semismooth
Newton method is proposed and its fast local convergence in
function space as well as a mesh-independence principle for
appropriate discretizations are proved. The paper ends by a
numerical verification of the theoretical results including a study of the
algorithm in the case of vanishing Lavrentiev-parameter. The latter process is
realized numerically by a combination of a nested iteration concept and an extrapolation technique for
the state with respect to the Lavrentiev-parameter.
This paper deals with error estimates for space-time discretizations in the context of evolutionary variational inequalities of rate-independent type. After introducing a general abstract evolution problem, we address a fully-discrete approximation and provide a priori error estimates. The application of the abstract theory to a semilinear case is detailed. In particular, we provide explicit space-time convergence rates for the isothermal Souza-Auricchio model for shape-memory alloys.
Optimal control of 3D state-constrained induction heating problems with nonlocal radiation effects
(2009)
The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a temperature-dependent heat conductivity in the description of the heat transfer process. The first goal of this paper is to prove the existence and uniqueness of the solution to the state equation. The regularity analysis associated with the time harmonic Maxwell equations is also studied. In the second part of the paper, the existence and uniqueness of the solution to the corresponding linearized equation is shown. With this result at hand, the differentiability of the control-to-state mapping operator associated with the state equation is derived. Finally, based on the theoretical results, first oder necessary optimality conditi!
ons for an associated optimal control problem are established.