On local exponential stability of equilibrium profiles of nonlinear distributed parameter systems

A. HASTIR, J. J. Winkin, D. Dochain

    Research output: Contribution in Book/Catalog/Report/Conference proceedingConference contribution

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    Abstract

    Local exponential (exp.) stability of nonlinear distributed parameter, i.e. infinite- dimensional state space, systems is considered. A weakened concept of Frechet differentiability ((y,X)-Frechet differentiability) for nonlinear operators defined on Banach spaces is proposed, including the introduction of an alternative space (Y) in the analysis. This allows more freedom in the manipulation of norm-inequalities leading to adapted Frechet differentiability conditions that are easier to check. Then, provided that the nonlinear semigroup generated by the nonlinear dynamics is Frechet-different iable in the new sense, appropriate local exp. stability of the equilibria for the nonlinear system is established. In particular, the nonlinear semigroup has to be Frechet differentiate on Y and (Y, X)-Frechet differentiate in order to go back to the original state space X. This approach may be called "perturbation-based"since exp. stability is also deduced from exp. stability of a linearized version of the nonlinear semigroup. Under adapted Frechet differentiability assumptions, the main result establishes that local exp. stability of an equilibrium for the nonlinear system is guaranteed as long as the exp. stability holds for the linearized semigroup. The same conclusion holds regarding instability. The theoretical results are illustrated on a convection-diffusion-reaction system.

    Original languageEnglish
    Title of host publicationProceedings of MTNS 2020
    Pages390-396
    Number of pages7
    Volume54
    Edition9
    DOIs
    Publication statusPublished - 1 Jun 2021
    Event24th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2020 - Cambridge, United Kingdom
    Duration: 23 Aug 202127 Aug 2021

    Publication series

    NameIFAC-PapersOnLine
    PublisherIFAC Secretariat
    ISSN (Print)2405-8963

    Conference

    Conference24th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2020
    Country/TerritoryUnited Kingdom
    CityCambridge
    Period23/08/2127/08/21

    Keywords

    • Distributed parameter systems - nonlinear systems - equilibrium - exponential stability

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