Generating formal power series and stability of bilinear systems

The aim of this paper is to study the Bounded Input Bounded Output (BIBO) stability of bilinear systems. The stability of linear systems can be studied by computing their transfer function. In this paper, we use the generating series (generalization of the transfer function) as a tool for analysing the stability of bilinear systems. In fact, the generating series G of a bilinear system is a formal power rational series in noncommutative variables. It provides a formal expression of the output y = ε(G) in iterated integrals form. The stability/stabilization can always be studied from the generating series G: According to expression of G, three cases occur. In the first case, the output y = ε(G) can be explicitly computed; in the second case, this output can be bounded (or unbounded) if the input u(t) is bounded; in the third case, no conclusion about the BIBO stability can be easily deduced. Then, we look only for a stabilizing constant input u(t) = η, by studying the univariate series Gη