This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time--dependent heat equations coupled with 3D time--harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a desired temperature can be optimally achieved. As there are finitely many control parameters but the state constraint has to be satisfied in an infinite number of points, the problem belongs to a class of semi--infinite programming problems. We present a rigorous analysis of the optimization problem and a numerical strategy based on our theoretical result.
We consider a control constrained optimal control problem governed by a semilinear
elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of
diffuse-gray conductive-radiative heat transfer. The problem arises from the aim to optimize the
temperature gradient within crystal growth by the physical vapor transport (PVT) method. Based
on a minimum principle for the semilinear equation as well as L1-estimates for the weak solution,
we establish the existence of an optimal solution as well as necessary optimality conditions. The
theoretical results are illustrated by results of numerical computations.
The paper addresses primal interior point method for state constrained PDE optimal
control problems. By a Lavrentiev regularization, the state constraint is transformed to a mixed
control-state constraint with bounded Lagrange multiplier. Existence and convergence of the central
path are established, and linear convergence of a short-step pathfollowing method is shown. The
behaviour of the regularizations are demonstrated by numerical examples.
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.
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.
Some optimal control problems for linear and nonlinear ordinary differential equations related to the optimal switching between
different magnetic fields are considered. The main aim is to move an electrical initial current by a controllable
voltage in shortest time to a desired terminal current and to hold it afterwards. Necessary optimality conditions are derived by
Pontryagin's principle and a Lagrange technique. In the case of a linear system, the principal structure of time-optimal controls is
discussed. The associated optimality systems are solved by a one-shot strategy
using a multigrid software package. Various numerical examples are discussed.