Symbolic Computation for Integro-Differential-Time-Delay Operators with Matrix Coefficients

Abstract

The purpose of this paper is to algebraize and automatize computations with linear differential time-delay systems and their solutions. To this end, we explain an algebraic construction of the ring of integro-differential operators with linear substitutions having (noncommutative) matrix coefficients, which contains the ring of integro-differential-time-delay operators. Based on a reduction system for this ring, we show how such operators can be uniquely expanded into irreducible terms.Symbolic computations with these operators and their normal forms are implemented in a Mathematica package. This even allows for computations with systems having generic size and/or undetermined matrix coefficients. We illustrate how, by elementary computations with operators in this framework, results like the method of steps can be found and proven in an automated way. Normal form computations with our package can be used to partly automatize solving operator equations. As an example, we recover a generalization of Artstein’s reduction, which solves an equivalence problem of a class of differential time-delay control systems.