Spectral dominance of complex roots for single-delay linear equations

Insights into the multiplicity-induced-dominancy for scalar delay-differential equations with two delays

“…Indeed, some works have shown that, for some classes of time-delay systems, a real root of maximal multiplicity is necessarily the rightmost root, a property known as multiplicity-induced-dominancy (MID). This link between maximal multiplicity and dominance has been suggested in Pinney (1958) after the study of some simple, low-order cases, but without any attempt to address the general case and, up to the authors' knowledge, very few works have considered this question in more details until recently in works such as Boussaada et al (2016Boussaada et al ( , 2020Boussaada et al ( , 2018; Ramírez et al (2016); Mazanti et al (2020Mazanti et al ( , 2021a; Benarab et al (2020). These works consider only time-delay equations with a single delay and the MID property was shown to hold, for instance, for retarded equations of order 1 in Boussaada et al (2016), proving dominance by introducing a factorization of ∆ in terms of an integral expression when it admits a root of maximal multiplicity 2; for retarded equations of order 2 with a delayed term of order zero in Boussaada et al (2018), using also the same factorization technique; or for retarded equations of order 2 with a delayed term of order 1 in Boussaada et al (2020), using Cauchy's argument principle to prove dominance of the multiple root.…”

“…In this section, we consider another approach to prove the MID property for (3), inspired by the Walton-Marshall method from Walton and Marshall (1987) and similar to the strategy used in Mazanti et al (2020) to study the MID property for complex conjugate dominant roots. This approach consists on regarding the quasipolynomial ∆ from (9) as a quasipolynomial depending on a parameter λ ∈ (0, 1).…”

“…Most of these results are actually particular cases of a more general result on the MID property from Mazanti et al (2021a) for retarded equations of order n with delayed term of order n − 1, which relies on links between quasipolynomials with a root of maximal multiplicity and Kummer's confluent hypergeometric function. The MID property was also extended to neutral systems of orders 1 and 2 in Ma et al (2020); Benarab et al (2020); Mazanti et al (2021b), as well as to the case of complex conjugate roots of maximal multiplicity in Mazanti et al (2020). This paper addresses the MID property for one of the first nontrivial time-delay systems with more than one delay, namely the scalar delay-differential equation with two delays y (t) + a 0 y(t) + a 1 y(t − τ 1 ) + a 2 y(t − τ 2 ) = 0, (3) where a 0 , a 1 , a 2 ∈ R and τ 1 , τ 2 ∈ (0, +∞) with τ 1 = τ 2 .…”

New Features of P3$δ$ software: Partial Pole Placement via Delay Action

“…The MID property for (1) was shown, for instance, in in the case n = 2 and m = 0, in Boussaada et al (2020b) in the case n = 2 and m = 1 (see also ), and in Mazanti et al (2021a) in the case of any positive integer n and m = n − 1 (see also Mazanti et al (2020a)). It was also studied for neutral systems of orders 1 and 2 in Ma et al (2020); Benarab et al (2020);Mazanti et al (2021b), and extended to complex conjugate roots of maximal multiplicity in Mazanti et al (2020b). The CRRID property was shown, for instance, in Amrane et al (2018) in the cases (n, m) = (2, 0) and (n, m) = (1, 0), and in Bedouhene et al (2020) in the case of any positive integer n and m = 0.…”

The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions

“…m = n − 1 in (1.1)), which relies on links between quasipolynomials with a real root of maximal multiplicity and Kummer's confluent hypergeometric function in terms of the location of the characteristic roots. The GMID property was also extended to neutral DDEs of orders 1 and 2 in [9,32,35], as well as to the case of complex conjugate roots of maximal multiplicity in [36].…”