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Analytic Solutions of L∞ Optimal Control Problems for the Wave Equation

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Abstract

There are very few results about analytic solutions of problems of optimal control with minimal L ∞ norm. In this paper, we consider such a problem for the wave equation, where the derivative of the state is controlled at both boundaries. We start in the zero position and consider a problem of exact control, that is, we want to reach a given terminal state in a given finite time. Our aim is to find a control with minimal L ∞ norm that steers the system to the target.

We give the analytic solution for certain classes of target points, for example, target points that are given by constant functions. For such targets with zero velocity, the analytic solution has been given by Bennighof and Boucher in Ref. 1.

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References

  1. BENNIGHOF, J. K., and BOUCHER, R. L., Exact Minimum-Time Control of a Distributed System Using a Traveling Wave Formulation, Journal of Optimization Theory and Applications, Vol. 73, pp. 149-167, 1992.

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  2. GUGAT, M., and LEUGERING, G., Solutions of L p-Norm-Minimal Control Problems for the Wave Equation, Computational and Applied Mathematics (to appear).

  3. LIONS, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, Germany, 1971.

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  4. PINSKY, M. A., Partial Differential Equations and Boundary-Value Problems with Applications, McGraw-Hill, New York, NY, 1991.

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  5. RUSSEL, D. L., Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems, Journal of Mathematical Analysis and Applications, Vol. 18, pp. 542-560, 1967.

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  6. STOECKER, H., Taschenbuch mathematischer Formeln und moderner Verfahren, Verlag Harri Deutsch, Frankfurt am Main, Germany, 1999.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    M. Gugat (Research Associate)

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  1. M. Gugat
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Gugat, M. Analytic Solutions of L∞ Optimal Control Problems for the Wave Equation. Journal of Optimization Theory and Applications 114, 397–421 (2002). https://doi.org/10.1023/A:1016091803139

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  • DOI: https://doi.org/10.1023/A:1016091803139

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