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Sensitivity Analysis for Stability of Uncertain Delay Differential Equations Using Polynomial Chaos Expansions

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Accounting for Constraints in Delay Systems

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 12))

Abstract

In applications, the analysis of the effect of uncertain parameters on the stability of linear(ized) delay differential equations is a crucial question. Such uncertainties are often modelled as random quantities in a suitable probabilistic framework and, as a result, the spectral abscissa, which determines the stability of the deterministic delay differential equations and depends on the parameters, is a random variable. Within this context, Polynomial Chaos Expansions reveal to be a powerful tool to represent random quantities and to perform uncertain quantification and sensitivity analysis. Here we present a numerical method, called the Stochastic Infinitesimal Generator approach, to efficiently compute the expansion coefficients of the numerical spectral abscissa, showing its application to variance-based global sensitivity analysis through the computation of the Sobol’ indices. The technique implements the non intrusive spectral projection methods and makes use also of the so-called Padua points to reduce the computation cost. Some numerical examples are given.

The work of R.V. was partially supported by INdAM GNCS 2017 project “Analisi e sviluppo di metodologie numeriche per certi tipi non classici di sistemi dinamici” and is part of the project SiDiA, “Sistemi Dinamici e Applicazioni,” PRID 2017. R.V. is member of the INdAM Research group GNCS.

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Author information

Authors and Affiliations

  1. CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100, Udine, Italy

    Rossana Vermiglio

  2. Member of the INdAM Research group GNCS, Rome, Italy

    Rossana Vermiglio

  3. Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100, Udine, Italy

    Andrea Zamolo

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  1. Rossana Vermiglio
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Correspondence to Rossana Vermiglio .

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Editors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CentraleSupélec, Gif-sur-Yvette, France

    Giorgio Valmorbida

  2. Department of Computer Science, KU Leuven, Heverlee, Belgium

    Wim Michiels

  3. Department of Information Engineering, Computer Science, and Mathematics, University of L’Aquila, L'Aquila, Italy

    Pierdomenico Pepe

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Vermiglio, R., Zamolo, A. (2022). Sensitivity Analysis for Stability of Uncertain Delay Differential Equations Using Polynomial Chaos Expansions. In: Valmorbida, G., Michiels, W., Pepe, P. (eds) Accounting for Constraints in Delay Systems. Advances in Delays and Dynamics, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-030-89014-8_8

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