On qualitative properties of single-delay linear retarded differential equations: Characteristic roots of maximal multiplicity are necessarily dominant

Multiplicity-induced-dominancy for delay-differential equations of retarded type

An important question of ongoing interest for linear time-delay systems is to provide conditions on its parameters guaranteeing exponential stability of solutions. Recent works have explored spectral techniques to show that, for some low-order delay-differential equations of retarded type, spectral values of maximal multiplicity are dominant, and hence determine the asymptotic behavior of the system, a property known as multiplicity-induced-dominancy. This work further explores such a property and shows its validity for general linear delay-differential equations of retarded type of arbitrary …

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

In this paper, which is a direct continuation and generalization of the recent works by the authors [https://doi.org/10.1051/cocv/2019073, https://doi.org/10.1016/j.jde.2021.03.003], we show the validity of the generic multiplicity-induced-dominancy property for a general class of linear functional differential equations with a single delay, including the retarded as well as the neutral cases. The result is based on an appropriate integral representation of the corresponding characteristic quasipolynomial functions involving some appropriate degenerate hypergeometric functions.

Effects of Roots of Maximal Multiplicity on the Stability of Some Classes of Delay Differential-Algebraic Systems: The Lossless Propagation Case

It has been observed in several recent works that, for some classes of linear time-delay systems, spectral values of maximal multiplicity are dominant, a property known as multiplicity-induced-dominancy (MID). This paper starts the investigation of whether MID holds for delay differential-algebraic systems by considering a single-delay system composed of two scalar equations. After motivating this problem and recalling some recent results for retarded delay differential equations, we prove that the MID property holds for the delay differential-algebraic system of interest and present some appl…

Spectral dominance of complex roots for single-delay linear equations

This paper provides necessary and sufficient conditions for the existence of a pair of complex conjugate roots, each of multiplicity two, in the spectrum of a linear time-invariant single-delay equation of retarded type. This pair of roots is also shown to be always strictly dominant, determining thus the asymptotic behavior of the system. The proof of this result is based on the corresponding result for real roots of multiplicity four, continuous dependence of roots with respect to parameters, and the study of crossing imaginary roots. We also present how this design can be applied to vibrati…

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

It has been observed in recent works that, for several classes of linear time-invariant time-delay systems of retarded or neutral type with a single delay, if a root of its characteristic equation attains its maximal multiplicity, then this root is the rightmost spectral value, and hence it determines the exponential behavior of the system, a property usually referred to as multiplicity-induced-dominancy (MID). In this paper, we investigate the MID property for one of the simplest cases of systems with two delays, a scalar delay-differential equation of first order with two delayed terms of or…

最优阻尼问题的最大重数方法

Maximal Multiplicity Method for Optimal Damping Problem

针对耦合线性振子系统，利用Weyl--Horn定理构造显式阻尼矩阵，实现由固有频率几何平均决定的最优渐近衰减率，优于Rayleigh阻尼，并刻画了矩阵非正定的参数区域。