# 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 ${A}_{0}$ 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.

## 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 (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 -

## References

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1. [1] D.Z. Arov and M.A. Nudelman, Passive linear stationary dynamical scattering systems with continous time. Integral Equations Operator Theory 24 (1996) 1-43. Zbl0838.47004MR1366539
2. [2] J.A. Ball, Conservative dynamical systems and nonlinear Livsic-Brodskii nodes. Oper. Theory Adv. Appl. 73 (1994) 67-95. Zbl0864.93036MR1320543
3. [3] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. 1. Birkhäuser, Boston (1992). Zbl0781.93002MR1182557
4. [4] R.F. Curtain and G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), Control and Estimation of Distributed Parameter Systems, edited by F. Kappel, K. Kunisch and W. Schappacher. Birkhäuser, Basel (1989) 41-59. Zbl0686.93049MR1033051
5. [5] P. Grabowski, On the spectral Lyapunov approach to parametric optimization of distributed parameter systems. IMA J. Math. Control Inform. 7 (1990) 317-338. Zbl0721.49006MR1099758
6. [6] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). Zbl0695.35060MR775683
7. [7] P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris (1992). Zbl0766.35001MR1173209
8. [8] S. Hansen and G. Weiss, New results on the operator Carleson measure criterion. IMA J. Math. Control Inform. 14 (1997) 3-32. Zbl0874.93031MR1446962
9. [9] B. Jacob and J. Partington, The Weiss conjecture on admissibility of observation operators for contraction semigroups. Integral Equations Operator Theory (to appear). Zbl1031.93107MR1831828
10. [10] P. Lax and R. Phillips, Scattering Theory. Academic Press, New York (1967). Zbl0186.16301MR217440
11. [11] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer-Verlag, Berlin, Grundlehren Math. Wiss. 181 (1972). Zbl0223.35039MR350177
12. [12] B.M.J. Maschke and A.J. van der Schaft, Portcontrolled Hamiltonian representation of distributed parameter systems, in Proc. of the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, edited by N.E. Leonard andR. Ortega. Princeton University (2000) 28-38.
13. [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). Zbl0516.47023MR710486
14. [14] A. Rodriguez-Bernal and E. Zuazua, Parabolic singular limit of a wave equation with localized boundary damping. Discrete Contin. Dynam. Systems 1 (1995) 303-346. Zbl0891.35075MR1355878
15. [15] D. Salamon, Infinite dimensional systems with unbounded control and observation: A functional analytic approach. Trans. Amer. Math. Soc. 300 (1987) 383-431. Zbl0623.93040MR876460
16. [16] D. Salamon, Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. Zbl0668.93018MR977021
17. [17] O.J. Staffans, Quadratic optimal control of stable well-posed linear systems. Trans. Amer. Math. Soc. 349 (1997) 3679-3715. Zbl0889.49023MR1407712
18. [18] O.J. Staffans, Coprime factorizations and well-posed linear systems. SIAM J. Control Optim. 36 (1998) 1268-1292. Zbl0919.93040MR1618041
19. [19] O.J. Staffans and G. Weiss, Transfer functions of regular linear systems. Part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc. 354 (2002) 3229-3262. Zbl0996.93012MR1897398
20. [20] O.J. Staffans and G. Weiss, Transfer functions of regular linear systems. Part III: Inversions and duality (submitted). Zbl1052.93032
21. [21] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach. J. Math. Anal. Appl. 137 (1989) 438-461. Zbl0686.35067MR984969
22. [22] G. Weiss, Admissibility of unbounded control operators. SIAM J. Control Optim. 27 (1989) 527-545. Zbl0685.93043MR993285
23. [23] G. Weiss, Admissible observation operators for linear semigroups. Israel J. Math. 65 (1989) 17-43. Zbl0696.47040MR994732
24. [24] G. Weiss, Transfer functions of regular linear systems. Part I: Characterizations of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. Zbl0798.93036MR1179402
25. [25] G. Weiss, Regular linear systems with feedback. Math. Control Signals Systems 7 (1994) 23-57. Zbl0819.93034MR1359020
26. [26] G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim. 39 (2001) 1204-1232. Zbl0981.93032MR1814273
27. [27] G. Weiss, O.J. Staffans and M. Tucsnak, Well-posed linear systems - A survey with emphasis on conservative systems. Appl. Math. Comput. Sci. 11 (2001) 101-127. Zbl0990.93046MR1835146

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