Zeroing the Output of Nonlinear Systems Without Relative Degree

The goal of this paper is to establish some facts concerning the problem of zeroing the output of an input-output system that does not have relative degree. The approach taken is to work with systems that have Chen-Fliess series representations. The main result is that a class of generating series called primely nullable series provides the building blocks for solving this problem using shuffle algebra. This is achieved by viewing the latter as the symmetric algebra over the commutative polynomials in Lyndon words in order to show that it is a unique factorization domain. Next, the focus turns to factoring generating series in the shuffle algebra into its irreducible elements. A specific algorithm based on the Chen-Fox-Lyndon factorization of words is given.