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 -