Decay Rate Assignment Through Multiple Spectral Values in Delay Systems

This article focuses on a spectral property for linear time-invariant dynamical systems represented by delay-differential equations (DDEs) entitled <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multiplicity-induced-dominancy</i> (MID), which consists, roughly speaking, in the spectral abscissa of the system being defined by a multiple spectral value. More precisely, we focus on the MID property for spectral values with overorder multiplicity, i.e., a multiplicity larger than the order of the DDE. We highlight the fact that a root of overorder multiplicity is necessarily a root of a particular polynomial, called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">elimination-produced polynomial</i>, and we address the MID property using a suitable factorization of the corresponding characteristic function involving special functions of Kummer type. Additional results and discussion are provided in the case of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula>th order integrator, in particular on the local optimality of a multiple root. The derived results show how the delay can be further exploited as a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">control parameter</i> and are applied to some problems of stabilization of standard benchmarks with prescribed exponential decay.