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Forwarding techniques for the global stabilization of dissipative infinite-dimensional systems coupled with an ODE

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Abstract

This paper deals with the stabilization of a coupled system composed by an infinite-dimensional system and an ODE. Moreover, the control, which appears in the dynamics of the ODE, is subject to a general class of nonlinearities. Such a situation may arise, for instance, when the actuator admits a dynamics. The open-loop ODE is exponentially stable and the open-loop infinite-dimensional system is dissipative, i.e., the energy is nonincreasing, but its equilibrium point is not necessarily attractive. The feedback design is based on an extension of a finite-dimensional method, namely the forwarding method. We propose some sufficient conditions that imply the well-posedness and the global asymptotic stability of the closed-loop system. As illustration, we apply these results in a transport equation coupled with an ODE.

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Notes

  1. This terminology is used in functional analysis to denote monotone operators. If \(m=1\), then \(\sigma \) is nondecreasing.

  2. See [38, Definition 11.1.1.] for a definition.

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Acknowledgements

This research was partially supported by the French Grant ANR ODISSE (ANR-19-CE48-0004-01) and was also conducted in the framework of the regional programme “Atlanstic 2020, Research, Education and Innovation in Pays de la Loire”, supported by the French Region Pays de la Loire and the European Regional Development Fund. Funding was provided by Atlanstic2020 (Grant No. LIMICOM).

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Authors and Affiliations

  1. LS2N, École Centrale de Nantes and CNRS UMR 6004, 44000, Nantes, France

    Swann Marx

  2. Univ Lyon, Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007, 43 boulevard du 11 novembre 1918, 69100, Villeurbanne, France

    Lucas Brivadis & Daniele Astolfi

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  1. Swann Marx
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  2. Lucas Brivadis
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  3. Daniele Astolfi
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Correspondence to Swann Marx.

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Marx, S., Brivadis, L. & Astolfi, D. Forwarding techniques for the global stabilization of dissipative infinite-dimensional systems coupled with an ODE. Math. Control Signals Syst. 33, 755–774 (2021). https://doi.org/10.1007/s00498-021-00299-7

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  • DOI: https://doi.org/10.1007/s00498-021-00299-7

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