On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints
Georg Vossen Torsten Hermanns
Journal of Industrial & Management Optimization 2014, 10(2): 503-519 doi: 10.3934/jimo.2014.10.503
A mathematical model for laser cutting with time-dependent cutting velocity is presented. The model involves two coupled nonlinear partial differential equations for the interacting dynamical behaviors of the free melt boundaries during the process. We define a measurement for the roughness of a cutting surface and introduce an optimal control problem for minimizing the roughness with respect to the cutting velocity and the laser beam intensity along the free melt surface. The optimal control problem involves an additional finite-dimensional averaging constraint. Necessary optimality conditions will be deduced and illustrated by means of numerical examples with data from industrial applications.
keywords: Optimal control free boundary problem. laser cutting stability analysis partial differential equation
Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems
Georg Vossen Stefan Volkwein
Numerical Algebra, Control & Optimization 2012, 2(3): 465-485 doi: 10.3934/naco.2012.2.465
The main focus of this paper is on an a-posteriori analysis for different model-order strategies applied to optimal control problems governed by linear parabolic partial differential equations. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the reduced-order model, is from the (unknown) exact one. For the model-order reduction, $\mathcal H_{2,\alpha}$-norm optimal model reduction (H2), balanced truncation (BT), and proper orthogonal decomposition (POD) are studied. The proposed approach is based on semi-discretization of the underlying dynamics for the state and the adjoint equations as a large scale linear time-invariant (LTI) system. This system is reduced to a lower-dimensional one using Galerkin (POD) or Petrov-Galerkin (H2, BT) projection. The size of the reduced-order system is iteratively increased until the error in the optimal control, computed with the a-posteriori error estimator, satisfies a given accuracy. The method is illustrated with numerical tests.
keywords: linear-quadratic optimal control Model reduction a-posteriori error.

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