Sufficient conditions for the functional reproducibility of time-varying, input-output systems

The functional reproducibility of an input-output system refers to the capability of realizing a given class of functions as outputs of the system by the choice of appropriate inputs and initial conditions. For linear, time-varying, input-output systems we derive rank conditions that are sufficient for the generation of all $C^k$ outputs by means of $C^0$ inputs. These rank conditions provide an alternative to the structure algorithm of L. M. Silverman and, in the linear, time-invariant case, reduce to a rank condition of M. K. Samn and J. L. Massey. We also give sufficient conditions for the local functional reproducibility of nonlinear, time-varying, input-output systems.