Existence Conditions and Properties of the Frequency Response Operators of Continuous-Time Periodic Systems

Abstract

The definition of the frequency response operator via the steady-state analysis in finite-dimensional linear continuous-time periodic (FDLCP) systems is revisited. It is shown that the frequency response operator is guaranteed to be well defined only densely on the linear space l2 , which is different from the usual understanding. Fortunately, however, it turns out that this frequency response operator can have an extension onto l2 so that the equivalence between the time-domain H2 norm (respectively, the L2 -induced norm) and the frequency-domain H2 norm (respectively, the $H_{\infty}$ norm of the frequency response operator) is recovered. Under some stronger assumptions, it is also shown that the frequency response operator can be viewed as a bounded operator from l1 to l1 , which can also be established via the steady-state analysis.