How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 247-273.

Let A 0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C 0 be a bounded operator from 𝒟A 0 1 2 to another Hilbert space U. We prove that the system of equations

z ¨(t)+A 0 z(t)+1 2C 0 * C 0 z ˙(t)=C 0 * u(t)
y(t)=-C 0 z ˙(t)+u(t),
determines a well-posed linear system with input u and output y. The state of this system is
x(t)=z(t)z ˙(t)𝒟A 0 1 2 ×H=X,
where X is the state space. Moreover, we have the energy identity
x(t) X 2 -x(0) X 2 = 0 T u(t) U 2 dt- 0 T y(t) U 2 dt.
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.

DOI : 10.1051/cocv:2003012
Classification : 93C25, 93C20, 35B37
Mots clés : well-posed linear system, operator semigroup, dual system, energy balance equation, conservative system, wave equation
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     title = {How to get a conservative well-posed linear system out of thin air. {Part} {I.} {Well-posedness} and energy balance},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
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Weiss, George; Tucsnak, Marius. How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 247-273. doi : 10.1051/cocv:2003012. http://www.numdam.org/articles/10.1051/cocv:2003012/

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