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 maps. Using functional series expansions, it is shown that the 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 maps. These conditions go well beyond similar conditions which have been discussed in the recent literature. They also apply to single-input, single-output maps which are 2-powers. The techniques which are used are quite general and extend to -linear and -power maps.