# How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance

• Volume: 9, page 247-273
• ISSN: 1292-8119

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## Abstract

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Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from $𝒟\left({A}_{0}^{\frac{1}{2}}\right)$ 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 $x\left(t\right)=\left[\begin{array}{c}\phantom{\rule{0.166667em}{0ex}}z\left(t\right)\\ \stackrel{˙}{z}\left(t\right)\end{array}\right]\in 𝒟\left({A}_{0}^{\frac{1}{2}}\right)×H=X,$ 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.

## How to cite

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Weiss, George, and 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 9 (2010): 247-273. <http://eudml.org/doc/90695>.

@article{Weiss2010,
abstract = { Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from $\{\cal D\}\Big(A_0^\{\frac\{1\}\{2\}\}\Big)$ to another Hilbert space U. We prove that the system of equations $$\ddot z(t)+A\_0 z(t) + \{\frac\{1\}\{2\}\}C\_0^*C\_0\dot z(t) =C\_0^*u(t)$$$$y(t) =-C\_0 \dot z(t)+u(t),$$ determines a well-posed linear system with input u and output y. The state of this system is $$x(t) = \left[\begin\{matrix\}\, z(t) \\ \dot z(t)\end\{matrix\}\right] \in \{\cal D\}\left(A\_0^\{\frac\{1\}\{2\}\}\right)\times H = X ,$$ where X is the state space. Moreover, we have the energy identity $$\|x(t)\|^2\_X-\|x(0)\|\_X^2 = \int\_0^T\| u(t)\|^2\_U \{\rm d\}t - \int\_0^T \|y(t)\|\_U^2 \{\rm d\}t.$$ 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. },
author = {Weiss, George, Tucsnak, Marius},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Well-posed linear system; operator semigroup; dual system; energy balance equation; conservative system; wave equation.; dual system; wave equation},
language = {eng},
month = {3},
pages = {247-273},
publisher = {EDP Sciences},
title = {How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance},
url = {http://eudml.org/doc/90695},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Weiss, George
AU - Tucsnak, Marius
TI - How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 247
EP - 273
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 ${\cal D}\Big(A_0^{\frac{1}{2}}\Big)$ to another Hilbert space U. We prove that the system of equations $$\ddot z(t)+A_0 z(t) + {\frac{1}{2}}C_0^*C_0\dot z(t) =C_0^*u(t)$$$$y(t) =-C_0 \dot z(t)+u(t),$$ determines a well-posed linear system with input u and output y. The state of this system is $$x(t) = \left[\begin{matrix}\, z(t) \\ \dot z(t)\end{matrix}\right] \in {\cal D}\left(A_0^{\frac{1}{2}}\right)\times H = X ,$$ where X is the state space. Moreover, we have the energy identity $$\|x(t)\|^2_X-\|x(0)\|_X^2 = \int_0^T\| u(t)\|^2_U {\rm d}t - \int_0^T \|y(t)\|_U^2 {\rm d}t.$$ 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.
LA - eng
KW - Well-posed linear system; operator semigroup; dual system; energy balance equation; conservative system; wave equation.; dual system; wave equation
UR - http://eudml.org/doc/90695
ER -

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