Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



When is a linear system conservative?

Authors: Jarmo Malinen, Olof J. Staffans and George Weiss
Journal: Quart. Appl. Math. 64 (2006), 61-91
MSC (2000): Primary 47A48, 47N70, 93B28, 93C25
DOI: https://doi.org/10.1090/S0033-569X-06-00994-7
Published electronically: January 24, 2006
MathSciNet review: 2211378
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider infinite-dimensional linear systems without a-priori well-posedness assumptions, in a framework based on the works of M. Livšic, M. S. Brodski{\u{\i\/}}\kern.15em, Y. L. Smuljan, and others. We define the energy in the system as the norm of the state squared (other, possibly indefinite quadratic forms will also be considered). We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, well posed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control operator is one-to-one and whose observation operator has dense range, the equivalent conditions for being conservative become simpler, and reduce to three algebraic equations.

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  • 1. D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo.
    Schur Functions, Operator Colligations, and Reproducing Kernel Hilbert Spaces, volume 96 of Operator Theory: Advances and Applications.
    Birkhäuser-Verlag, Basel Boston Berlin, 1997.
  • 2. D. Z. Arov.
    Passive linear systems and scattering theory.
    In Dynamical Systems, Control Coding, Computer Vision, volume 25 of Progress in Systems and Control Theory, pages 27-44. Birkhäuser-Verlag, Basel, 1999. MR 1684834 (2000c:47134)
  • 3. D. Z. Arov and M. A. Nudelman.
    Passive linear stationary dynamical scattering systems with continuous time.
    Integral Equations Operator Theory, 24:1-45, 1996. MR 1366539 (96k:47016)
  • 4. J. A. Ball.
    Conservative dynamical systems and nonlinear Livsic-Brodskii nodes.
    In Nonselfadjoint Operators and Related Topics: Workshop on Operator Theory and its Applications, volume 73 of Operator Theory: Advances and Applications, pages 67-95. Birkhäuser-Verlag, Basel, 1994. MR 1320543 (96c:47108)
  • 5. J. Bognár.
    Indefinite inner product spaces, volume 78 of Ergebnisse der Mathematik und ihrer Grenzgebiete.
    Springer-Verlag, Berlin, 1974. MR 0467261 (57:7125)
  • 6. M. S. Brodski{\u{\i\/}}\kern.15em.
    Triangular and Jordan Representations of Linear Operators, volume 32 of Translations of Mathematical Monographs.
    American Mathematical Society, Providence, Rhode Island, 1971. MR 0322542 (48:904)
  • 7. M. S. Brodski{\u{\i\/}}\kern.15em.
    Unitary operator colligations and their characteristic functions.
    Russian Math. Surveys, 33:4:159-191, 1978. MR 0510672 (80e:47010)
  • 8. V. M. Brodski{\u{\i\/}}\kern.15em.
    On operator colligations and their characteristic functions.
    Soviet Math. Dokl., 12:696-700, 1971.
  • 9. R. F. Curtain and G. Weiss.
    Well posedness of triples of operators (in the sense of linear systems theory).
    In Control and Optimization of Distributed Parameter Systems, volume 91 of International Series of Numerical Mathematics, pages 41-59. Birkhäuser-Verlag, Basel, 1989. MR 1033051 (91d:93027)
  • 10. R. F. Curtain and H. Zwart.
    An Introduction to Infinite-Dimensional Linear Systems Theory.
    Springer-Verlag, New York, 1995. MR 1351248 (96i:93001)
  • 11. J. W. Helton.
    Systems with infinite-dimensional state space: the Hilbert space approach.
    Proceedings of the IEEE, 64:145-160, 1976. MR 0416694 (54:4764)
  • 12. P. D. Lax and R. S. Phillips.
    Scattering Theory.
    Academic Press, New York, 1967. MR 0217440 (36:530)
  • 13. M. S. Livšic.
    Operators, Oscillations, Waves (Open Systems), volume 34 of Translations of Mathematical Monographs.
    American Mathematical Society, Providence, Rhode Island, 1973. MR 0347396 (49:12116)
  • 14. M. S. Livšic and A. A. Yantsevich.
    Operator Colligations in Hilbert Spaces.
    John Wiley & Sons, New York, 1977.
  • 15. R. Ober and S. Montgomery-Smith.
    Bilinear transformation of infinite-dimensional state-space systems and balanced realizations of nonrational transfer functions.
    SIAM J. Control Optim., 28:438-465, 1990. MR 1040469 (91d:93019)
  • 16. D. Salamon.
    Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach.
    Trans. Amer. Math. Soc., 300:383-431, 1987. MR 0876460 (88d:93024)
  • 17. D. Salamon.
    Realization theory in Hilbert space.
    Math. Systems Theory, 21:147-164, 1989. MR 0977021 (89k:93038)
  • 18. Y. L. Smuljan.
    Invariant subspaces of semigroups and the Lax-Phillips scheme.
    Dep. in VINITI, N 8009-1386, Odessa, 49p., 1986.
  • 19. O. J. Staffans.
    Coprime factorizations and well-posed linear systems.
    SIAM J. Control Optim., 36:1268-1292, 1998. MR 1618041 (99g:93049)
  • 20. O. J. Staffans.
    Admissible factorizations of Hankel operators induce well-posed linear systems.
    Systems Control Lett., 37:301-307, 1999. MR 1753257 (2001f:93022)
  • 21. O. J. Staffans.
    $ J$-energy preserving well-posed linear systems.
    Internat. J. Appl. Math. Comput. Sci., 11:1361-1378, 2001. MR 1885509 (2002k:93045)
  • 22. O. J. Staffans.
    Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems.
    Math. Control Signals Systems, 15:291-315, 2002. MR 1942089 (2003i:93024)
  • 23. O. J. Staffans.
    Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view).
    In Mathematical Systems Theory in Biology, Communication, Computation, and Finance, volume 134 of IMA Volumes in Mathematics and its Applications, pages 375-414. Springer-Verlag, New York, 2002. MR 2043247 (2004m:93072)
  • 24. O. J. Staffans.
    Well-Posed Linear Systems.
    Cambridge University Press, Cambridge and New York, 2005. MR 2154892
  • 25. 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., 354:3229-3262, 2002. MR 1897398 (2003b:93051)
  • 26. O. J. Staffans and G. Weiss.
    Transfer functions of regular linear systems. Part III: inversions and duality.
    Integral Equations Operator Theory, 49:517-558, 2004. MR 2091475 (2005g:93067)
  • 27. B. Sz.-Nagy and C. Foias.
    Harmonic Analysis of Operators on Hilbert Space.
    North-Holland, Amsterdam, 1970.
  • 28. E. R. Tsekanovskii and Y. L. Smuljan.
    The theory of biextensions of operators in rigged Hilbert spaces. Unbounded operator colligations and characteristic functions.
    Uspehi Mathem. Nauk SSSR, 32:69-124, 1977. MR 0463955 (57:3893)
  • 29. M. Tucsnak and G. Weiss.
    How to get a conservative well-posed linear system out of thin air. Part II: controllability and stability.
    SIAM J. Control and Optim., 42:907-935, 2003. MR 2002140 (2004f:93075)
  • 30. G. Weiss.
    Admissibility of unbounded control operators.
    SIAM J. Control Optim., 27:527-545, 1989. MR 0993285 (90c:93060)
  • 31. G. Weiss.
    Transfer functions of regular linear systems. Part I: characterizations of regularity.
    Trans. Amer. Math. Soc., 342:827-854, 1994. MR 1179402 (94f:93074)
  • 32. G. Weiss.
    Optimal control of systems with a unitary semigroup and with colocated control and observation.
    Systems Control Lett., 48:329-340, 2003. MR 2020648 (2004i:49054)
  • 33. G. Weiss, O. J. Staffans, and M. Tucsnak.
    Well-posed linear systems - a survey with emphasis on conservative systems.
    Internat. J. Appl. Math. Comput. Sci., 11:7-34, 2001. MR 1835146 (2002f:93068)
  • 34. G. Weiss and M. Tucsnak.
    How to get a conservative well-posed linear system out of thin air. I. Well-posedness and energy balance.
    ESAIM Control Optim. Calc. Var., 9:247-274, 2003. MR 1966533 (2004d:93069)

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Additional Information

Jarmo Malinen
Affiliation: Institute of Mathematics, P.O. Box 1100, Helsinki University of Technology, FIN-02015 HUT, Finland
Email: Jarmo.Malinen@hut.fi

Olof J. Staffans
Affiliation: Department of Mathematics Åbo Akademi University, FIN-20500 Åbo, Finland
Email: Olof.Staffans@abo.fi

George Weiss
Affiliation: Department of Electric and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2BT, United Kingdom
Email: G.Weiss@imperial.ac.uk

DOI: https://doi.org/10.1090/S0033-569X-06-00994-7
Keywords: Conservative system, energy preserving system, well-posed linear system, regular linear system, operator node, Cayley transform
Received by editor(s): January 6, 2005
Published electronically: January 24, 2006
Additional Notes: This research was supported by grants from the European Research Network on Systems Identification (ERNSI), the Academy of Finland under grant 203991, the Mittag–Leffler Institute (Sweden) and EPSRC (UK), under the Portfolio Partnership grant GR/S61256/01.
Article copyright: © Copyright 2006 Brown University

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