TY - JOUR
T1 - How to get a conservative well-posed linear system out of thin air.
T2 - Part I. Well-posedness and energy balance
AU - Weiss, George
AU - Tucsnak, Marius
PY - 2003/2
Y1 - 2003/2
N2 - Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from D(A1/20) to another Hilbert space U. We prove that the system of equations (Formula Presented) determines a well-posed linear system with input u and output y. The state of this system is (Formula Presented) where X is the state space. Moreover, we have the energy identity (Formula Presented) We show that the system described above is isomorphic to its dual, so that a similar energy identity holds also for the dual system and hence, the system is conservative. We derive various other properties of such systems and we give a relevant example: a wave equation on a bounded n-dimensional domain with boundary control and boundary observation on part of the boundary.
AB - Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from D(A1/20) to another Hilbert space U. We prove that the system of equations (Formula Presented) determines a well-posed linear system with input u and output y. The state of this system is (Formula Presented) where X is the state space. Moreover, we have the energy identity (Formula Presented) We show that the system described above is isomorphic to its dual, so that a similar energy identity holds also for the dual system and hence, the system is conservative. We derive various other properties of such systems and we give a relevant example: a wave equation on a bounded n-dimensional domain with boundary control and boundary observation on part of the boundary.
KW - Conservative system
KW - Dual system
KW - Energy balance equation
KW - Operator semigroup
KW - Wave equation
KW - Well-posed linear system
UR - http://www.scopus.com/inward/record.url?scp=2942536559&partnerID=8YFLogxK
U2 - 10.1051/cocv:2003012
DO - 10.1051/cocv:2003012
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AN - SCOPUS:2942536559
SN - 1292-8119
VL - 9
SP - 247
EP - 273
JO - ESAIM - Control, Optimisation and Calculus of Variations
JF - ESAIM - Control, Optimisation and Calculus of Variations
ER -