Error bounds for Euler approximation of linear-quadratic control problems   with bang-bang solutions

Walter Alt; Robert Baier; Matthias Gerdts; Frank Lempio

doi:10.3934/naco.2012.2.547

Article Contents

2012, Volume 2, Issue 3: 547-570. Doi: 10.3934/naco.2012.2.547

This issue Previous Article Quadratic order conditions for bang-singular extremals Next Article Control parameterization for optimal control problems with continuous inequality constraints: New convergence results

Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions

1.
Institut für Angewandte Mathematik, Friedrich-Schiller-Universität Jena, 07740 Jena
2.
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth
3.
Institut für Mathematik und Rechneranwendung, Fakultät für Luft- und Raumfahrttechnik, Universität der Bundeswehr, 85577 Neubiberg/München

Received: July 2011

Revised: May 2012

Published: July 2012

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Abstract

We analyze the Euler discretization to a class of linear-quadratic optimal control problems. First we show convergence of order $h$ for the optimal values of the objective function, where $h$ is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the continuous controls coincide except on a set of measure $O(\sqrt{h})$. Under a slightly stronger assumption on the smoothness of the coefficients of the system equation we obtain an error estimate of order $O(h)$.

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Mathematics Subject Classification: Primary: 49J15; Secondary: 49M25, 49N10, 49J30.

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References

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