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Relaxation methods for mixed-integer optimal control of partial differential equations

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Abstract

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.

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Acknowledgements

Most of this research was carried out while both authors were member of the working group of Prof. H.G. Bock at the Interdisciplinary Center of Scientific Computing (IWR), University of Heidelberg. The financial support of the Mathematics Center Heidelberg (MATCH), of the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences (HGS MathComp), and of the EU project EMBOCON under grant FP7-ICT-2009-4 248940 is gratefully acknowledged. The first author also acknowledges the support from Andreas Potschka with the software package MUSCOD-II.

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Authors and Affiliations

  1. Department of Mathematics, University of Erlangen-Nuremberg, Cauerstraße 11, 91058, Erlangen, Germany

    Falk M. Hante

  2. Institute of Mathematical Optimization, Otto-von-Guericke University, Universitätsplatz 2, 39106, Magdeburg, Germany

    Sebastian Sager

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  1. Falk M. Hante
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  2. Sebastian Sager
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Correspondence to Falk M. Hante.

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Hante, F.M., Sager, S. Relaxation methods for mixed-integer optimal control of partial differential equations. Comput Optim Appl 55, 197–225 (2013). https://doi.org/10.1007/s10589-012-9518-3

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