Bilinear and 2-power input-output maps: Finite dimensional realizations and the role of functional series

The paper considers finite-dimensional dynamical systems which have bilinear
i-o
maps. Using functional series expansions, it is shown that the
i-o
maps have state representations of special structure. These representations specialize to ones which have been obtained previously by Arbib [2] for the discrete-time case and Marchesini and Picci [11] for the continuous-time case. In addition, they lead very simply to a variety of necessary and sufficient conditions for the finite dimensional realization of bilinear
i-o
maps. These conditions go well beyond similar conditions which have been discussed in the recent literature. They also apply to single-input, single-output
i-o
maps which are 2-powers. The techniques which are used are quite general and extend to
k
-linear and
k
-power
i-o
maps.