J-energy preserving well-posed linear systems

Olof Staffans

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 6, page 1361-1378
  • ISSN: 1641-876X

Abstract

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The following is a short survey of the notion of a well-posed linear system. We start by describing the most basic concepts, proceed to discuss dissipative and conservative systems, and finally introduce J-energy-preserving systems, i.e., systems that preserve energy with respect to some generalized inner products (possibly semi-definite or indefinite) in the input, state and output spaces. The class of well-posed linear systems contains most linear time-independent distributed parameter systems: internal or boundary control of PDE’s, integral equations, delay equations, etc. These systems have existed in an implicit form in the mathematics literature for a long time, and they are closely connected to the scattering theory by Lax and Phillips and to the model theory by Sz.-Nagy and Foiaş. The theory has been developed independently by many different schools, and it is only recently that these different approaches have begun to converge. One of the most interesting objects of the present study is the Riccati equation theory for this class of infinite-dimensional systems (H^2 - and H^∞ -theories).

How to cite

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Staffans, Olof. "J-energy preserving well-posed linear systems." International Journal of Applied Mathematics and Computer Science 11.6 (2001): 1361-1378. <http://eudml.org/doc/207559>.

@article{Staffans2001,
abstract = {The following is a short survey of the notion of a well-posed linear system. We start by describing the most basic concepts, proceed to discuss dissipative and conservative systems, and finally introduce J-energy-preserving systems, i.e., systems that preserve energy with respect to some generalized inner products (possibly semi-definite or indefinite) in the input, state and output spaces. The class of well-posed linear systems contains most linear time-independent distributed parameter systems: internal or boundary control of PDE’s, integral equations, delay equations, etc. These systems have existed in an implicit form in the mathematics literature for a long time, and they are closely connected to the scattering theory by Lax and Phillips and to the model theory by Sz.-Nagy and Foiaş. The theory has been developed independently by many different schools, and it is only recently that these different approaches have begun to converge. One of the most interesting objects of the present study is the Riccati equation theory for this class of infinite-dimensional systems (H^2 - and H^∞ -theories).},
author = {Staffans, Olof},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {Lax-Phillips semigroup; dissipative system; system node; J-energy-preserving system; conservative system; well-posed linear system; Riccati equation; transfer function; model theory; Lyapunov equation; conservative realization; well posed linear systems; survey; conservative systems; indefinite metric},
language = {eng},
number = {6},
pages = {1361-1378},
title = {J-energy preserving well-posed linear systems},
url = {http://eudml.org/doc/207559},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Staffans, Olof
TI - J-energy preserving well-posed linear systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 6
SP - 1361
EP - 1378
AB - The following is a short survey of the notion of a well-posed linear system. We start by describing the most basic concepts, proceed to discuss dissipative and conservative systems, and finally introduce J-energy-preserving systems, i.e., systems that preserve energy with respect to some generalized inner products (possibly semi-definite or indefinite) in the input, state and output spaces. The class of well-posed linear systems contains most linear time-independent distributed parameter systems: internal or boundary control of PDE’s, integral equations, delay equations, etc. These systems have existed in an implicit form in the mathematics literature for a long time, and they are closely connected to the scattering theory by Lax and Phillips and to the model theory by Sz.-Nagy and Foiaş. The theory has been developed independently by many different schools, and it is only recently that these different approaches have begun to converge. One of the most interesting objects of the present study is the Riccati equation theory for this class of infinite-dimensional systems (H^2 - and H^∞ -theories).
LA - eng
KW - Lax-Phillips semigroup; dissipative system; system node; J-energy-preserving system; conservative system; well-posed linear system; Riccati equation; transfer function; model theory; Lyapunov equation; conservative realization; well posed linear systems; survey; conservative systems; indefinite metric
UR - http://eudml.org/doc/207559
ER -

References

top
  1. Adamajan A. and Arov D.Z. (1970): On unitary couplings of semiunitary operators, In: Eleven Papers in Analysis. — Providence, R.I.: AMS, pp.75–129. 
  2. Arov D.Z. (1979): Passive linear stationary dynamic systems. — Siberian Math. J., Vol.20, pp.149–162. Zbl0437.93019
  3. Arov D.Z. (1999): Passive linear systems and scattering theory, In: Dynamical Systems, Control Coding, Computer Vision (G. Picci and D.S. Gilliam, Eds.). — Basel: Birkhäuser, pp.27–44. Zbl0921.93005
  4. Arov D.Z. and Nudelman M.A. (1996): Passive linear stationary dynamical scattering systems with continuous time. — Int. Eqns. Oper. Theory, Vol.24, pp.1–45. Zbl0838.47004
  5. Callier F.M. and Winkin J. (1990): Spectral factorization and LQ-optimal regulation for multivariable distributed systems. — Int. J. Contr., Vol.52, pp.55–75. Zbl0713.93024
  6. Callier F.M. and Winkin J. (1992): LQ-optimal control of infinite-dimensional systems by spectral factorization. — Automatica, Vol.28, pp.757–770. Zbl0776.49023
  7. Callier F.M. and Winkin J. (1999): The spectral factorization problem for multivariable distributed parameter systems. — Int. Eqns. Oper. Theory, Vol.34, pp.270–292. Zbl0933.93042
  8. Curtain R.F. and Ichikawa A. (1996): The Nehari problem for infinite-dimensional linear systems of parabolic type. — Int. Eqns. Oper. Theory, Vol.26, pp.29–45. Zbl0858.47017
  9. Curtain R.F. and Oostveen J.C. (1998): The Nehari problem for nonexponentially stable systems. — Int. Eqns. Oper. Theory, Vol.31, pp.307–320. Zbl0910.93024
  10. Curtain R.F. and Weiss G. (1989): Well posedness of triples of operators (in the sense of linear systems theory), In: Control and Optimization of Distributed Parameter Systems (F. Kappel, K. Kunisk and W. Schappacher, Eds.). — Basel: Birkhäuser, pp.41–59. 
  11. Curtain R.F., Weiss G. and Weiss M. (1997): Stabilization of irrational transfer functions by controllers with internal loop, In: Systems, Approximation, Singular Integral Operators and Related Topics (A.A. Borichev and N.K. Nikolski, Eds.). — Birkhäuser, pp.179–208. Zbl1175.93186
  12. Engel K.-J. (1998): On the characterization of admissible control- and observation operators. — Syst. Contr. Lett., Vol.34, pp.225–227. Zbl0909.93034
  13. Flandoli F., Lasiecka I. and Triggiani R. (1988): Algebraic Riccati equations with nonsmoothing observation arising in hyperbolic and Euler–Bernoulli boundary control problems. — Ann. Mat. Pura Appl., Vol.4, No.153, pp.307–382. Zbl0674.49004
  14. Grabowski P. (1991): On the spectral-Lyapunov approach to parametric optimization of distributed-parameter systems. — IMA J. Math. Contr. Inf., Vol.7, pp.317–338. Zbl0721.49006
  15. Grabowski P. (1993): The LQ controller synthesis problem. — IMA J. Math. Contr. Inf., Vol.10, pp.131–148. Zbl0825.93200
  16. Grabowski P. (1994): The LQ controller problem: An example. — IMA J. Math. Contr. Inf., Vol.11, pp.355–368. Zbl0825.93201
  17. Grabowski P. and Callier F.M. (1996): Admissible observation operators. Semigroup criteria of admissibility. — Int. Eqns. Oper. Theory, Vol.25, pp.182–198. Zbl0856.93021
  18. Grabowski P. and Callier F.M. (2001): Boundary control systems in factor form: Transfer functions and input-output maps. — Int. Eqns. Oper. Theory, Vol.41, pp.1–37. Zbl1009.93041
  19. van Keulen B. (1993): H∞ -Control for Distributed Parameter Systems: A State Space Approach. — Basel: Birkhäuser. Zbl0788.93018
  20. Lasiecka I. and Triggiani R. (2000): Control Theory for Partial Differential Equations: Continuous and Approximation Theorems. I: Abstract Parabolic Systems. — Cambridge: Cambridge University Press. Zbl0942.93001
  21. Lasiecka I. and Triggiani R. (2000): Control Theory for Partial Differential Equations: Continuous and Approximation Theorems. II: Abstract Hyperbolic-Like Systems over a Finite Horizon. — Cambridge: Cambridge University Press. Zbl0961.93003
  22. Lax P.D. and Phillips R.S. (1967): Scattering Theory. — New York: Academic Press. Zbl0214.12002
  23. Lax P.D. and Phillips R.S. (1973): Scattering theory for dissipative hyperbolic systems. — J. Funct. Anal., Vol.14, pp.172–235. Zbl0295.35069
  24. Malinen J. (2000): Discrete time H ∞ algebraic Riccati equations. — Ph.D. Thesis, Helsinki University of Technology. Zbl0965.93006
  25. Malinen J., Staffans O.J. and Weiss G. (2002): When is a linear system conservative? — in preparation. Zbl1125.47007
  26. Mikkola K. (2002): Infinite-dimensional H ∞ and H 2 regulator problems and their algebraic Riccati equations with applications to the Wiener class. — Ph.D. Thesis, Helsinki University of Technology. 
  27. Oostveen J. (2000): Strongly stabilizable distributed parameter systems. — Philadelphia, PA: SIAM. Zbl0964.93004
  28. Salamon D. (1987): Infinite dimensional linear systems with unbounded control and observation: A functional analytic approach. — Trans. Amer. Math. Soc., Vol.300, pp.383–431. Zbl0623.93040
  29. Salamon D. (1989): Realization theory in Hilbert space. — Math. Syst. Theory, Vol.21, pp.147–164. Zbl0668.93018
  30. Sasane A.J. and Curtain R.F. (2001): Inertia theorems for operator Lyapunov inequalities. — Syst. Contr. Lett., Vol.43, pp.127–132. Zbl0974.93026
  31. Sasane A.J. and Curtain R.F. (2001): Optimal Hankel norm approximation for the Pritchard- Salamon class of infinite-dimensional systems. — Int. Eqns. Oper. Theory, Vol.39, pp.98–126. Zbl0990.93021
  32. Staffans O.J. (1995): Quadratic optimal control of stable systems through spectral factorization. — Math. Contr. Sign. Syst., Vol.8, pp.167–197. Zbl0843.93019
  33. Staffans O.J. (1996): On the discrete and continuous time infinite-dimensional algebraic Riccati equations. — Syst. Contr. Lett., Vol.29, pp.131–138. Zbl0875.93199
  34. Staffans O.J. (1997): Quadratic optimal control of stable well-posed linear systems. — Trans. Amer. Math. Soc., Vol.349, pp.3679–3715. Zbl0889.49023
  35. Staffans O.J. (1998a): Coprime factorizations and well-posed linear systems. — SIAM J. Contr. Optim., Vol.36, pp.1268–1292. Zbl0919.93040
  36. Staffans O.J. (1998b): Quadratic optimal control of well-posed linear systems. — SIAM J. Contr. Optim., Vol.37, pp.131–164. Zbl0955.49018
  37. Staffans O.J. (1998c): Feedback representations of critical controls for well-posed linear systems. — Int. J. Robust Nonlin. Contr., Vol.8, pp.1189–1217. Zbl0951.93038
  38. Staffans O.J. (1998d): On the distributed stable full information H ∞ minimax problem. — Int. J. Robust Nonlin. Contr., Vol.8, pp.1255–1305. Zbl0951.93029
  39. Staffans O.J. (1999a): Admissible factorizations of Hankel operators induce well-posed linear systems. — Syst. Contr. Lett., Vol.37, pp.301–307. Zbl0948.93014
  40. Staffans O.J. (1999b): Quadratic optimal control through coprime and spectral factorisation. — Europ. J. Contr., Vol.5, pp.167–179. Zbl0979.93544
  41. Staffans O.J. (2002): Well-Posed Linear Systems: Part I. Book manuscript, available at http://www.abo.fi/~staffans/. 
  42. Staffans O.J. and Mikkola K.M. (1998): A minimax formulation of the infinite-dimensional Nehari problem. — Proc. Symp. Mathematical Theory of Networks and Systems, MTNS’98, Padova, Italy, pp.539–542. 
  43. Staffans O.J. and Weiss G. (2002a): Transfer functions of regular linear systems. Part II: The system operator and the Lax-Phillips semigroup. — to appear in Trans. Amer. Math. Soc. Zbl0996.93012
  44. Staffans O.J. and Weiss G. (2002b): Transfer functions of regular linear systems. Part III: Inversions and duality. — manuscript. Zbl0996.93012
  45. Sz.-Nagy B. and Foiaş C. (1970): Harmonic Analysis of Operators on Hilbert Space. — Amsterdam: North-Holland. Zbl0201.45003
  46. Weiss G. (1989a): Admissibility of unbounded control operators. — SIAM J. Contr. Optim., Vol.27, pp.527–545. Zbl0685.93043
  47. Weiss G. (1989b): Admissible observation operators for linear semigroups. — Israel J. Math., Vol.65, pp.17–43. Zbl0696.47040
  48. Weiss G. (1989c): The representation of regular linear systems on Hilbert spaces, In: Control and Optimization of Distributed Parameter Systems (F. Kappel, K. Kunisk and W. Schappacher, Eds.). — Basel: Birkhäuser, pp.401–416. 
  49. Weiss G. (1991): Representations of shift-invariant operators on L 2 by H ∞ transfer functions: An elementary proof, a generalization to Lp , and a counterexample for L∞ . — Math. Contr. Sign. Syst., Vol.4, pp.193–203. Zbl0724.93021
  50. Weiss G. (1994a): Transfer functions of regular linear systems. Part I: Characterizations of regularity. — Trans. Amer. Math. Soc., Vol.342, pp.827–854. Zbl0798.93036
  51. Weiss G. (1994b): Regular linear systems with feedback. — Math. Contr. Sign. Syst., Vol.7, pp.23–57. Zbl0819.93034
  52. Weiss G. and Tucsnak M. (2001): How to get a conservative well-posed linear system out of thin air. Part I: Well-posedness and energy balance. — submitted. 
  53. Weiss G., Staffans O.J. and Tucsnak M. (2001): Well-posed linear systems–A survey with emphasis on conservative systems. — Int. J. Appl. Math. Comp. Sci., Vol.11, No.1, pp.7–34. Zbl0990.93046
  54. Weiss M. (1997): Riccati equation theory for Pritchard-Salamon systems: A Popov function approach. — IMA J. Math. Contr. Inf., Vol.14, pp.1–37. Zbl0876.93023
  55. Weiss M. and Weiss G. (1997): Optimal control of stable weakly regular linear systems. — Math. Contr. Sign. Syst., Vol.10, pp.287–330. Zbl0884.49021
  56. Willems J.C. (1972): Dissipative dynamical systems Part I: General theory. — Arch. Rat. Mech. Anal., Vol.45, pp.321–351. Zbl0252.93002
  57. Willems J.C. (1972b): Dissipative dynamical systems Part II: Linear systems with quadratic supply rates. — Arch. Rat. Mech. Anal., Vol.45, pp.352–393. Zbl0252.93003

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