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, Volume 9  (2003), p. 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 $𝒟\left({A}_{0}^{\frac{1}{2}}\right)$ to another Hilbert space $U$. We prove that the system of equations $\stackrel{¨}{z}\left(t\right)+{A}_{0}z\left(t\right)+\frac{1}{2}{C}_{0}^{*}{C}_{0}\stackrel{˙}{z}\left(t\right)={C}_{0}^{*}u\left(t\right)$ $y\left(t\right)=-{C}_{0}\stackrel{˙}{z}\left(t\right)+u\left(t\right),$ determines a well-posed linear system with input $u$ and output $y$. The state of this system is $x\left(t\right)=\left[\begin{array}{c}z\left(t\right)\\ \stackrel{˙}{z}\left(t\right)\end{array}\right]\in 𝒟\left({A}_{0}^{\frac{1}{2}}\right)×H=X,$ where $X$ is the state space. Moreover, we have the energy identity ${\parallel x\left(t\right)\parallel }_{X}^{2}-{\parallel x\left(0\right)\parallel }_{X}^{2}={\int }_{0}^{T}{\parallel u\left(t\right)\parallel }_{U}^{2}\mathrm{d}t-{\int }_{0}^{T}{\parallel y\left(t\right)\parallel }_{U}^{2}\mathrm{d}t.$ 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 : https://doi.org/10.1051/cocv:2003012
Classification:  93C25,  93C20,  35B37
Keywords: well-posed linear system, operator semigroup, dual system, energy balance equation, conservative system, wave equation
@article{COCV_2003__9__247_0,
author = {Weiss, George and Tucsnak, Marius},
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},
publisher = {EDP-Sciences},
volume = {9},
year = {2003},
pages = {247-273},
doi = {10.1051/cocv:2003012},
zbl = {1063.93026},
mrnumber = {1966533},
language = {en},
url = {http://www.numdam.org/item/COCV_2003__9__247_0}
}

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, Volume 9 (2003) , pp. 247-273. doi : 10.1051/cocv:2003012. http://www.numdam.org/item/COCV_2003__9__247_0/

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