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 𝒟A 0 1 2 to another Hilbert space U. We prove that the system of equations z ¨(t)+A 0 z(t)+1 2C 0 * C 0 z ˙(t)=C 0 * u(t) y(t)=-C 0 z ˙(t)+u(t), determines a well-posed linear system with input u and output y. The state of this system is x(t)=z(t)z ˙(t)𝒟A 0 1 2 ×H=X, where X is the state space. Moreover, we have the energy identity x(t) X 2 -x(0) X 2 = 0 T u(t) U 2 dt- 0 T y(t) U 2 dt. 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/

[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. | MR 1366539 | Zbl 0838.47004

[2] J.A. Ball, Conservative dynamical systems and nonlinear Livsic-Brodskii nodes. Oper. Theory Adv. Appl. 73 (1994) 67-95. | MR 1320543 | Zbl 0864.93036

[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). | MR 1182557 | Zbl 0781.93002

[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. | MR 1033051 | Zbl 0686.93049

[5] P. Grabowski, On the spectral Lyapunov approach to parametric optimization of distributed parameter systems. IMA J. Math. Control Inform. 7 (1990) 317-338. | MR 1099758 | Zbl 0721.49006

[6] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR 775683 | Zbl 0695.35060

[7] P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris (1992). | MR 1173209 | Zbl 0766.35001

[8] S. Hansen and G. Weiss, New results on the operator Carleson measure criterion. IMA J. Math. Control Inform. 14 (1997) 3-32. | MR 1446962 | Zbl 0874.93031

[9] B. Jacob and J. Partington, The Weiss conjecture on admissibility of observation operators for contraction semigroups. Integral Equations Operator Theory (to appear). | MR 1831828 | Zbl 1031.93107

[10] P. Lax and R. Phillips, Scattering Theory. Academic Press, New York (1967). | MR 217440 | Zbl 0186.16301

[11] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer-Verlag, Berlin, Grundlehren Math. Wiss. 181 (1972). | MR 350177 | Zbl 0223.35039

[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] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR 710486 | Zbl 0516.47023

[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. | MR 1355878 | Zbl 0891.35075

[15] D. Salamon, Infinite dimensional systems with unbounded control and observation: A functional analytic approach. Trans. Amer. Math. Soc. 300 (1987) 383-431. | MR 876460 | Zbl 0623.93040

[16] D. Salamon, Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. | MR 977021 | Zbl 0668.93018

[17] O.J. Staffans, Quadratic optimal control of stable well-posed linear systems. Trans. Amer. Math. Soc. 349 (1997) 3679-3715. | MR 1407712 | Zbl 0889.49023

[18] O.J. Staffans, Coprime factorizations and well-posed linear systems. SIAM J. Control Optim. 36 (1998) 1268-1292. | MR 1618041 | Zbl 0919.93040

[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. | MR 1897398 | Zbl 0996.93012

[20] O.J. Staffans and G. Weiss, Transfer functions of regular linear systems. Part III: Inversions and duality (submitted). | Zbl 1052.93032

[21] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach. J. Math. Anal. Appl. 137 (1989) 438-461. | MR 984969 | Zbl 0686.35067

[22] G. Weiss, Admissibility of unbounded control operators. SIAM J. Control Optim. 27 (1989) 527-545. | MR 993285 | Zbl 0685.93043

[23] G. Weiss, Admissible observation operators for linear semigroups. Israel J. Math. 65 (1989) 17-43. | MR 994732 | Zbl 0696.47040

[24] G. Weiss, Transfer functions of regular linear systems. Part I: Characterizations of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. | MR 1179402 | Zbl 0798.93036

[25] G. Weiss, Regular linear systems with feedback. Math. Control Signals Systems 7 (1994) 23-57. | MR 1359020 | Zbl 0819.93034

[26] G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim. 39 (2001) 1204-1232. | MR 1814273 | Zbl 0981.93032

[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. | MR 1835146 | Zbl 0990.93046