9 research outputs found
State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions
We consider a control- and state-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 nonlocal radiation interface
condition and the pointwise state-constraints represent the particular
features of this problem. To deal with the state-constraints, continuity of
the state is shown which allows to derive first-order necessary conditions.
Afterwards, we establish second-order sufficient conditions that account for
strongly active sets and ensure local optimality in an -neighborhood
Noncompactness of integral operators modeling diffuse-gray radiation in polyhedral and transient settings
While it is well-known that the standard integral operator K of (stationary) diffuse-gray radiation, as it occurs in the radiosity equation, is compact if the domain of radiative interaction is sufficiently regular, we show noncompactness of the operator if the domain is polyhedral. We also show that a stationary operator is never compact when reinterpreted in a transient setting. Moreover, we provide new proofs, which do not use the compactness of K, for 1 being a simple eigenvalue of K for connected enclosures, and for I-(1-e)K being invertible, provided the emissivity e does not vanish identically
Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in 3D
In this paper we study a distributed optimal control problem for a nonlocal
convective Cahn--Hilliard equation with degenerate mobility and singular
potential in three dimensions of space. While the cost functional is of
standard tracking type, the control problem under investigation cannot easily
be treated via standard techniques for two reasons: the state system is a
highly nonlinear system of PDEs containing singular and degenerating terms, and
the control variable, which is given by the velocity of the motion occurring in
the convective term, is nonlinearly coupled to the state variable. The latter
fact makes it necessary to state rather special regularity assumptions for the
admissible controls, which, while looking a bit nonstandard, are however quite
natural in the corresponding analytical framework. In fact, they are
indispensable prerequisites to guarantee the well-posedness of the associated
state system. In this contribution, we employ recently proved existence,
uniqueness and regularity results for the solution to the associated state
system in order to establish the existence of optimal controls and appropriate
first-order necessary optimality conditions for the optimal control problem
Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals
This paper deals with the mathematical modeling and simulation of
crystal growth processes by
the so-called Czochralski method and related methods,
which are important industrial processes
to grow large
bulk single crystals of semiconductor materials such as, e.\,g., gallium arsenide
(GaAs) or silicon (Si) from the melt.
In particular, we investigate a recently developed
technology in which traveling magnetic fields are applied in order to
control
the behavior of the turbulent melt flow. Since numerous different physical effects
like electromagnetic fields, turbulent melt flows, high temperatures, heat transfer via
radiation, etc., play an important role in the process, the corresponding mathematical
model leads to an extremely difficult system of initial-boundary value problems for
nonlinearly coupled partial differential equations. In this paper, we describe a mathematical
model that is under use for the simulation of real-life growth scenarios, and we give an overview
of mathematical results and numerical simulations that have been obtained for it in recent years
Optimal control of 3D state-constrained induction heating problems with nonlocal radiation effects
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 conditions for an associated optimal control problem are established
State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions
We consider a control- and state-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 nonlocal radiation interface condition and the pointwise state constraints represent the particular features of this problem. To deal with the state constraints, continuity of the state is shown, which allows us to derive first-order necessary conditions. Afterwards, we establish second-order sufficient conditions that account for strongly active sets and ensure local optimality in an -neighborhood