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

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.

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