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- Partial differential equations (4)
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We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an Lp function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.
An one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.
We regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. ---This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.
The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.
For rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schrödinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain factorizes over the space of essentially bounded functions.
This paper is concerned with the state-constrained optimal control of the
two-dimensional thermistor problem, a quasi-linear coupled system
of a parabolic and elliptic PDE with mixed boundary conditions.
This system models the heating of a conducting material by means of direct current.
Existence, uniqueness and continuity for the state system are derived by employing
maximal elliptic and parabolic regularity. By similar arguments the
linearized state system is discussed, while the adjoint system involving measures
is investigated using a duality argument. These results allow to derive
first-order necessary conditions for the optimal control problem.
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.