Over-order multiplicities and their application in controlling delay dynamics. On zeros’ distribution of linear combinations of Kummer hypergeometric functions

A series of recent works have shown that, for a system of linear functional differential equations, a spectral value having a multiplicity exceeding the order of the system tends to correspond to the spectral abscissa of the system, a property called MID for multiplicity-induced-dominancy. In particular, when this multiplicity coincides with the degree of the characteristic quasipolynomial, this property is called generic MID (GMID), in opposition to the intermediate MID (IMID), which corresponds to a multiplicity strictly smaller than the degree. The GMID has been fully characterized for single-delay retarded as well as neutral delay-differential equations thanks to the representation of the corresponding quasipolynomial in terms of a Kummer hypergeometric function. However, apart from partial results, in full generality, no result of the literature enables the characterization of the dominance of a spectral value having an intermediate multiplicity, which is essentially due to the lack of existing results among the open literature pertaining to linear combinations of Kummer functions’ zeros distribution. In this work, we overcome this difficulty and we further investigate the MID to cover the so-called over-order MID, that is, the cases where the multiplicity is larger than the order of the corresponding differential equation.