Approximation with the output of linear control systems

In this paper we give a new proof that for controllable and observable linear systems every L2[0,T] function can be approximated in the L2[0,T] sense with an output function generated by an L2[0,T] input function. We also give a new characterization of how continuous functions on [0,T] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.