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Qualitative Analysis of an Infinite Horizon Optimal Control Problem of a Shallow Lake

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Optimization and Applications (OPTIMA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13781))

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Abstract

This paper studies a classical infinite horizon optimal control problem for a shallow lake model and a variation thereof. We carry out a qualitative analysis of solutions to the canonical system and identify possible scenarios. Specifically, we describe a particular case that has not been addressed in the previous works. This case corresponds to the situation, when the canonical system has only two saddle equilibrium points without a source between them. Furthermore, the set of parameters, for which this situation occurs remains unchanged for two alternative formulations of the optimal control problem, which indicates a possibility for a hidden invariant structure. Both formulations of the optimal control problem are studied in detail, both analytically and numerically. The appearance of the Skiba point is discussed.

A part of this study was carried out while the first author was with the Faculty of Applied Mathematics and Control Processes, St. Petersburg State University.

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Acknowledgements

We thank Prof. A. Yu. Uteshev for valuable hints regarding the polynomial analysis. The reported study was funded by RFBR and DFG, project number 21-51-12007.

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Authors and Affiliations

  1. Department of Mathematics, University of Latvia, Raiņa bulvāris 19, Rīga, 1586, Latvia

    Dmitry Gromov

  2. Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, 199034, St. Petersburg, Russia

    Yilun Wu

Authors
  1. Dmitry Gromov
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  2. Yilun Wu
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Correspondence to Dmitry Gromov .

Editor information

Editors and Affiliations

  1. Dorodnitsyn Computing Centre, Moscow, Russia

    Nicholas Olenev

  2. Dorodnitsyn Computing Centre, Moscow, Russia

    Yuri Evtushenko

  3. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  4. Krasovskii Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  5. Dorodnitsyn Computing Centre, Moscow, Russia

    Vlasta Malkova

  6. Dorodnitsyn Computing Centre, Moscow, Russia

    Igor Pospelov

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Gromov, D., Wu, Y. (2022). Qualitative Analysis of an Infinite Horizon Optimal Control Problem of a Shallow Lake. In: Olenev, N., Evtushenko, Y., Jaćimović, M., Khachay, M., Malkova, V., Pospelov, I. (eds) Optimization and Applications. OPTIMA 2022. Lecture Notes in Computer Science, vol 13781. Springer, Cham. https://doi.org/10.1007/978-3-031-22543-7_9

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  • DOI: https://doi.org/10.1007/978-3-031-22543-7_9

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  • Online ISBN: 978-3-031-22543-7

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