# 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 (2003)

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

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topWeiss, 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 (2003): 247-273. <http://eudml.org/doc/245730>.

@article{Weiss2003,

abstract = {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 $\{\mathcal \{D\}\}\Big (A_0^\{\frac\{1\}\{2\}\}\Big )$ 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\[ x(t) = \begin\{bmatrix\} z(t)\\ \dot\{z\}(t) \end\{bmatrix\} \in \{\mathcal \{D\}\}\left(A\_0^\{\frac\{1\}\{2\}\}\right)\times H = X , \]where $X$ is the state space. Moreover, we have the energy identity\[ \Vert x(t)\Vert ^2\_X-\Vert x(0)\Vert \_X^2 = \int \_0^T\Vert u(t)\Vert ^2\_U \mathrm \{d\}t - \int \_0^T \Vert y(t)\Vert \_U^2 \mathrm \{d\}t. \]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.},

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; Well-posed linear system},

language = {eng},

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/245730},

volume = {9},

year = {2003},

}

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

PY - 2003

PB - EDP-Sciences

VL - 9

SP - 247

EP - 273

AB - 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 ${\mathcal {D}}\Big (A_0^{\frac{1}{2}}\Big )$ 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\[ x(t) = \begin{bmatrix} z(t)\\ \dot{z}(t) \end{bmatrix} \in {\mathcal {D}}\left(A_0^{\frac{1}{2}}\right)\times H = X , \]where $X$ is the state space. Moreover, we have the energy identity\[ \Vert x(t)\Vert ^2_X-\Vert x(0)\Vert _X^2 = \int _0^T\Vert u(t)\Vert ^2_U \mathrm {d}t - \int _0^T \Vert y(t)\Vert _U^2 \mathrm {d}t. \]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.

LA - eng

KW - well-posed linear system; operator semigroup; dual system; energy balance equation; conservative system; wave equation; Well-posed linear system

UR - http://eudml.org/doc/245730

ER -

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