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 $\mathcal{D}\left({A}_{0}^{\frac{1}{2}}\right)$ to another Hilbert space $U$. We prove that the system of equations $$\ddot{z}\left(t\right)+{A}_{0}z\left(t\right)+\frac{1}{2}{C}_{0}^{*}{C}_{0}\dot{z}\left(t\right)={C}_{0}^{*}u\left(t\right)$$ $$y\left(t\right)=-{C}_{0}\dot{z}\left(t\right)+u\left(t\right),$$ determines a well-posed linear system with input $u$ and output $y$. The state of this system is $$x\left(t\right)=\left[\begin{array}{c}z\left(t\right)\\ \dot{z}\left(t\right)\end{array}\right]\in \mathcal{D}\left({A}_{0}^{\frac{1}{2}}\right)\times H=X,$$ 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.

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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