9 research outputs found
On two optimal control problems for magnetic fields
Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)Two optimal control problems for instationary magnetization processes are considered in 3D spatial domains that include electrically conducting and non-conducting regions. The magnetic fields are generated by induction coils. In the first model, the induction coil is considered as part of the conducting region and the electrical current is taken as control. In the second, the coil is viewed as part of the non-conducting region and the electrical voltage is the control. Here, an integro-differential equation accounts for the magnetic induction law that couples the given electrical voltage with the induced electrical current in the induction coil. We derive first-order necessary optimality conditions for the optimal controls of both problems. Based on them, numerical methods of gradient type are applied. Moreover, we report on the application of model reduction by POD that lead to tremendous savings. Numerical tests are presented for academic 3D geometries but also for a real-world application
A Posteriori Error Analysis for the Optimal Control of Magneto-Static Fields
This paper is concerned with the analysis and numerical analysis for the
optimal control of first-order magneto-static equations. Necessary and
sufficient optimality conditions are established through a rigorous Hilbert
space approach. Then, on the basis of the optimality system, we prove
functional a posteriori error estimators for the optimal control, the optimal
state, and the adjoint state. 3D numerical results illustrating the theoretical
findings are presented.Comment: Keywords: Maxwell's equations, magneto statics, optimal control, a
posteriori error analysi
Boundary control of time-harmonic eddy current equations
Motivated by various applications, this article develops the notion of
boundary control for Maxwell's equations in the frequency domain. Surface curl
is shown to be the appropriate regularization in order for the optimal control
problem to be well-posed. Since, all underlying variables are assumed to be
complex valued, the standard results on differentiability do not directly
apply. Instead, we extend the notion of Wirtinger derivatives to complexified
Hilbert spaces. Optimality conditions are rigorously derived and higher order
boundary regularity of the adjoint variable is established. The state and
adjoint variables are discretized using higher order N\'ed\'elec finite
elements. The finite element space for controls is identified, as a space,
which preserves the structure of the control regularization. Convergence of the
fully discrete scheme is established. The theory is validated by numerical
experiments, in some cases, motivated by realistic applications.Comment: 25 pages, 6 figure
Feedback Control for Systems with Uncertain Parameters Using Online-Adaptive Reduced Models
We consider control and stabilization for large-scale dynamical systems with uncertain, time-varying parameters. The time-critical task of controlling a dynamical system poses major challenges: using large-scale models is prohibitive, and accurately inferring parameters can be expensive, too. We address both problems by proposing an offine-online strategy for controlling systems with time- varying parameters. During the offine phase, we use a high-fidelity model to compute a library of optimal feedback controller gains over a sampled set of parameter values. Then, during the online phase, in which the uncertain parameter changes over time, we learn a reduced-order model from system data. The learned reduced-order model is employed within an optimization routine to update the feedback control throughout the online phase. Since the system data naturally reects the uncertain parameter, the data-driven updating of the controller gains is achieved without an explicit parameter estimation step. We consider two numerical test problems in the form of partial differential equations: a convection-diffusion system, and a model for ow through a porous medium. We demonstrate on those models that the proposed method successfully stabilizes the system model in the presence of process noise.DARPA EQUiPS program (award number UTA15-001067)United States. Department of Energy. Office of Advanced Scientific Computing Research (grant DE-FG02-08ER2585)United States. Department of Energy. Office of Advanced Scientific Computing Research (grant DE-SC000929