Integral Equations and Operator Theory
, Volume 49, Issue 4, pp 517558
First online:
Transfer Functions of Regular Linear Systems Part III: Inversions and Duality
 Olof J. StaffansAffiliated withDepartment of Mathematics, Abo Akademi University Email author
 , George WeissAffiliated withDept. of Electr. & Electronic Eng., Imperial College London Email author
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Get AccessAbstract.
We study four transformations which lead from one wellposed linear system to another: timeinversion, flow^inversion, timeflowinversion and duality. Timeinversion means reversing the direction of time, flowinversion means interchanging inputs with outputs, while timeflowinversion means doing both of the inversions mentioned before. A wellposed linear system ∑ is timeinvertible if and only if its operator semigroup extends to a group. The system ∑ is flowinvertible if and only if its inputoutput 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 halfplane. The system ∑ is timeflowinvertible 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. Timeflowinversion can sometimes, but not always, be reduced to a combination of time and flowinversion. We derive a surprising necessary and sufficient condition for ∑ to be timeflowinvertible: 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.
Mathematics Subject Classification (2000).
Primary 93C25 Secondary 47D06 47A48 37K05Keywords.
Wellposed linear system regular linear system operator semigroup system operator timeinversion flowinversion timeflowinversion dual system conservative system LaxPhillips semigroup Title
 Transfer Functions of Regular Linear Systems Part III: Inversions and Duality
 Journal

Integral Equations and Operator Theory
Volume 49, Issue 4 , pp 517558
 Cover Date
 200408
 DOI
 10.1007/s0002000212148
 Print ISSN
 0378620X
 Online ISSN
 14208989
 Publisher
 BirkhäuserVerlag
 Additional Links
 Keywords

 Primary 93C25
 Secondary 47D06
 47A48
 37K05
 Wellposed linear system
 regular linear system
 operator semigroup
 system operator
 timeinversion
 flowinversion
 timeflowinversion
 dual system
 conservative system
 LaxPhillips semigroup
 Industry Sectors
 Authors

 Olof J. Staffans ^{(1)}
 George Weiss ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Abo Akademi University, 20500, Abo, Finland
 2. Dept. of Electr. & Electronic Eng., Imperial College London, Exhibition Road, SW7 2BT, London, United Kingdom