Session S12 - Delay and functional differential equations and applications

Monday, July 12, 15:25 ~ 16:00 UTC-3

The multiplicity-induced-dominancy property for scalar delay-differential equations

Guilherme Mazanti

Inria & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Even in simple situations with time-delays such as that of linear equations with constant coefficients and constant delays, the analysis of the asymptotic behavior of a time-delay system can be a challenging question. From a spectral point of view, this asymptotic behavior can be studied through the spectrum of the system, which is the set of roots of a quasipolynomial, i.e., a function which can be written as a finite sum of products of polynomials and exponentials. Contrarily to the case of polynomials, for which Routh–Hurwitz criterion allows for handy characterizations of the location of their roots in terms of the coefficients of the system, quasipolynomials have infinitely many roots and there is no explicit link in general between their location and the coefficients of the system.

Some recent works have highlighted an interesting property of time-delay systems, called multiplicity-induced-dominancy (MID): for some families of time-delay systems, a spectral value of maximal multiplicity is necessarily the rightmost spectral value in the complex plane, and hence determines the asymptotic behavior of the system. Since then, an important research effort was made to identify families of time-delay systems satisfying the MID property.

After an introductory discussion on the spectral analysis of time-delay systems, this talk will present the MID property and some families of systems for which it is known to hold. We shall present the most common techniques used to prove the MID property, based on the argument principle, factorizations of quasipolynomials in terms of confluent hypergeometric functions, and the analysis of crossing imaginary roots. We will also illustrate its application to the stabilization of control systems and present the main perspectives of this ongoing line of research.

Joint work with Amina Benarab (IPSA & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Catherine Bonnet (Inria & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Islam Boussaada (IPSA & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Yacine Chitour (L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Sébastien Fueyo (Inria & L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Silviu-Iulian Niculescu (L2S, Univ. Paris-Saclay, CNRS, CentraleSupélec, France), Karim Trabelsi (IPSA, France) and Tomáš Vyhlídal (Czech Technical University in Prague, Czechia).

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