46 research outputs found
Existence analysis for a simplified transient energy-transport model for semiconductors
A simplified transient energy-transport system for semiconductors subject to
mixed Dirichlet-Neumann boundary conditions is analyzed. The model is formally
derived from the non-isothermal hydrodynamic equations in a particular
vanishing momentum relaxation limit. It consists of a drift-diffusion-type
equation for the electron density, involving temperature gradients, a nonlinear
heat equation for the electron temperature, and the Poisson equation for the
electric potential. The global-in-time existence of bounded weak solutions is
proved. The proof is based on the Stampacchia truncation method and a careful
use of the temperature equation. Under some regularity assumptions on the
gradients of the variables, the uniqueness of solutions is shown. Finally,
numerical simulations for a ballistic diode in one space dimension illustrate
the behavior of the solutions
Mean-field optimal control and optimality conditions in the space of probability measures
We derive a framework to compute optimal controls for problems with states in
the space of probability measures. Since many optimal control problems
constrained by a system of ordinary differential equations (ODE) modelling
interacting particles converge to optimal control problems constrained by a
partial differential equation (PDE) in the mean-field limit, it is interesting
to have a calculus directly on the mesoscopic level of probability measures
which allows us to derive the corresponding first-order optimality system. In
addition to this new calculus, we provide relations for the resulting system to
the first-order optimality system derived on the particle level, and the
first-order optimality system based on -calculus under additional
regularity assumptions. We further justify the use of the -adjoint in
numerical simulations by establishing a link between the adjoint in the space
of probability measures and the adjoint corresponding to -calculus.
Moreover, we prove a convergence rate for the convergence of the optimal
controls corresponding to the particle formulation to the optimal controls of
the mean-field problem as the number of particles tends to infinity
Adjoint-based optimal control using meshfree discretizations
AbstractThe paper at hand presents a combination of optimal control approaches for PDEs with meshless discretizations. Applying a classical Lagrangian type particle method to optimization problems with hyperbolic constraints, several adjoint-based strategies differing in the sequential order of optimization and discretization of the Lagrangian or Eulerian problem formulation are proposed and compared. The numerical results confirm the theoretically predicted independence principle of the optimization approaches and show the expected convergence behavior. Moreover, they exemplify the superiority of meshless methods over the conventional mesh-based approaches for the numerical handling and optimization of problems with time-dependent geometries and freely moving boundaries
Asymptotic Analysis for Optimal Control of the Cattaneo Model
We consider an optimal control problem with tracking-type cost functional
constrained by the Cattaneo equation, which is a well-known model for delayed
heat transfer. In particular, we are interested the asymptotic behaviour of the
optimal control problems for a vanishing delay time .
First, we show the convergence of solutions of the Cattaneo equation to the
ones of the heat equation. Assuming the same right-hand side and compatible
initial conditions for the equations, we prove a linear convergence rate.
Moreover, we show linear convergence of the optimal states and optimal controls
for the Cattaneo equation towards the ones for the heat equation. We present
numerical results for both, the forward and the optimal control problem
confirming these linear convergence rates