## Journals

- Advances in Mathematics of Communications
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- Journal of Dynamics & Games
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### Open Access Journals

JIMO

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.

NACO

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.

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