Stabilization of forming processes using multi–dimensional Hamilton–Jacobi equations

Proceedings of the IEEE Conference on Decision & Control, including the Symposium on Adaptive Processes, vol. 136, pp. 7376–7381

Abstract

We introduce a feedback control for the stabilization of forming processes to ensure suitable microstructure of the formed workpiece. Mathematically, the underlying model for the evolution of the microstructure is given by a spatially two–dimensional partial differential equation of Hamilton–Jacobi type. The feedback control is applied at the boundary and exponential stabilization of stationary states in L2– is established. The derivation is based on a Lyapunov function extending prior work to the spatially two– dimensional case. The theoretical findings are confirmed by numerical simulations.

Authors 4

1. Markus� Bambach

   ETH Zurich

   Affiliation as printed

   ETH Zürich,Faculty of Mechanical Engineering,Zürich,Switzerland,8005

2. Martin Gugat

   Friedrich-Alexander-Universität Erlangen-Nürnberg

   Affiliation as printed

   FAU Erlangen-Nürnberg,Department of Data Science,Erlangen,Germany,91058

3. Michaël Herty Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University,Department of Mathematics,Aachen,Germany,52056

4. Ferdinand Thein Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University,Department of Mathematics,Aachen,Germany,52056