# Transfer Functions of Regular Linear Systems Part III: Inversions and Duality

DOI: 10.1007/s00020-002-1214-8

- Cite this article as:
- Staffans, O. & Weiss, G. Integr. equ. oper. theory (2004) 49: 517. doi:10.1007/s00020-002-1214-8

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## Abstract.

We study four transformations which lead from one well-posed linear
system to another: time-inversion, flow^-inversion, time-flow-inversion and
duality. Time-inversion means reversing the direction of time, flow-inversion
means interchanging inputs with outputs, while time-flow-inversion means doing
both of the inversions mentioned before. A well-posed linear system ∑ is
time-invertible if and only if its operator semigroup extends to a group. The
system ∑ is flow-invertible if and only if its input-output map has a bounded
inverse on some (hence, on every) finite time interval [0, τ] (τ > 0). This is
true if and only if the transfer function of ∑ has a uniformly bounded inverse
on some right half-plane. The system ∑ is time-flow-invertible if and only if
on some (hence, on every) finite time interval [0, τ], the combined operator ∑_{τ}
from the initial state and the input function to the final state and the output
function is invertible. This is the case, for example, if the system is conservative,
since then ∑_{τ} is unitary. Time-flow-inversion can sometimes, but not
always, be reduced to a combination of time- and flow-inversion. We derive a
surprising necessary and sufficient condition for ∑ to be time-flow-invertible:
its system operator must have a uniformly bounded inverse on some left halfplane.
Finally, the duality transformation is always possible.We show by some
examples that none of these transformations preserves regularity in general.
However, the duality transformation does preserve weak regularity. For all
the transformed systems mentioned above, we give formulas for their system
operators, transfer functions and, in the regular case and under additional
assumptions, for their generating operators.