On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints

Georg Vossen; Torsten Hermanns

doi:10.3934/jimo.2014.10.503

Article Contents

2014, Volume 10, Issue 2: 503-519. Doi: 10.3934/jimo.2014.10.503

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On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints

Georg Vossen^1, and
Torsten Hermanns^2,

1.
Niederrhein University of Applied Sciences, Chair for Applied Mathematics and Numerical Simulation, Reinarzstr. 49, 47805 Krefeld
2.
Fraunhofer Institute for Laser Technology, Steinbachstr. 15, 52074 Aachen

Received: November 30, 2012

Revised: July 31, 2013

Published: March 2014

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Abstract

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.

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Mathematics Subject Classification: 49J20, 35M13, 37N20.

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