Time-delay systems, confluent hypergeometric functions, and Padé approximations

Abstract:

This talk will present recent results on the links between the spectrum of a class of time-delay systems, confluent hypergeometric functions, and Padé approximations of the exponential function. More precisely, we are interested in linear time-delay systems with a single delay which admit a real spectral value with maximal possible multiplicity.

After motivating the interest for such systems from the perspective of control theory, we will recall some important features of the spectrum of time-delay systems and provide brief presentations of Padé approximations and Kummer confluent hypergeometric functions. We will then proceed to the main result, showing that, apart from the root with maximal multiplicity, all other spectral values of the time-delay system can be obtained from the roots of a suitable Kummer function, a result that relies on the analysis of the remainder of the Padé approximation of the exponential function. Finally, we will present an application to a problem in control theory illustrating the interest of the main result.

The talk is based on a series of joint works with Islam Boussaada and Silviu-Iulian Niculescu.