TY - JOUR

T1 - Transfer functions of regular linear systems. Part I

T2 - Characterizationso f regularity

AU - Weiss, George

PY - 1994/4

Y1 - 1994/4

N2 - We recall the main facts about the representation of regular linear systems, essentially that they can be described by equations of the form Ax(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t), like finite dimensional systems, but now A, B and C are in general unbounded operators. Regular linear systems are a subclass of abstract linear systems. We define transfer functions of abstract linear systems via a generalization of a theorem of Fourés and Segal. We prove a formula for the transfer function of a regular linear system, which is similar to the formula in finite dimensions. The main result is a (simple to state but hard to prove) necessary and sufficient condition for an abstract linear system to be regular, in terms of its transfer function. Other conditions equivalent to regularity are also obtained. The main result is a consequence of a new Tauberian theorem, which is of independent interest.

AB - We recall the main facts about the representation of regular linear systems, essentially that they can be described by equations of the form Ax(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t), like finite dimensional systems, but now A, B and C are in general unbounded operators. Regular linear systems are a subclass of abstract linear systems. We define transfer functions of abstract linear systems via a generalization of a theorem of Fourés and Segal. We prove a formula for the transfer function of a regular linear system, which is similar to the formula in finite dimensions. The main result is a (simple to state but hard to prove) necessary and sufficient condition for an abstract linear system to be regular, in terms of its transfer function. Other conditions equivalent to regularity are also obtained. The main result is a consequence of a new Tauberian theorem, which is of independent interest.

KW - Feedthrough operator

KW - Lebesgue extension

KW - Regular linear system

KW - Shift-invariant operator

KW - Tauberian theorem

KW - Transfer function

UR - http://www.scopus.com/inward/record.url?scp=84966260365&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1994-1179402-6

DO - 10.1090/S0002-9947-1994-1179402-6

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AN - SCOPUS:84966260365

VL - 342

SP - 827

EP - 854

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -