Picard-Vessiot theory of bilinear systems

As we know, the input-output behaviour of a bilinear system may be described by its noncommutative generating power series which is rational. We show first that this rationality is equivalent to a linear differential equation with coefficients depending of the inputs and their derivatives, the solution of which is the output of the system. This allows us to introduce the splitting field and the differential Galois group of the equation and, by definition, of the system and its generating series. This group is a connected algebraic group which we simply characterize, together with its Lie algebra. As an application we show that the solvability of the Galois group corresponds to the solvability of the system in the following sense: the output may be obtained by a finite number of integrations and exponentiations.