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Two classes of passive time-varying well-posed linear systems

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Abstract

We investigate two classes of time-varying well-posed linear systems. Starting from a time-invariant scattering-passive system, each of the time-varying systems is constructed by introducing a time-dependent inner product on the state space and modifying some of the generating operators. These classes of linear systems are motivated by physical examples such as the electromagnetic field around a moving object. To prove the well-posedness of these systems, we use the Lax–Phillips semigroup induced by a well-posed linear system, as in scattering theory.

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References

  1. Acquistapace P, Terreni B (1999) Classical solutions of nonautonomous Riccati equations arising in parabolic boundary control problems. Appl Math Optim 39: 361–409

    Article  MATH  MathSciNet  Google Scholar 

  2. Arov DZ, Nudelman MA (1996) Passive linear stationary dynamical scattering systems with continuous time. Integral Equ Oper Theory 24: 1–45

    Article  MATH  MathSciNet  Google Scholar 

  3. Colombini F, Del Santo D, Kinoshita T (2002) Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients. Ann Sc Norm Super Pisa Cl Sci (5) 1: 327–358

    MATH  MathSciNet  Google Scholar 

  4. Curtain RF, Pritchard AJ (1978) Infinite dimensional linear systems theory. Springer-Verlag, Berlin

    Book  MATH  Google Scholar 

  5. Dautray R, Lions J-L (1990) Mathematical analysis and numerical methods for science and technology: spectral theory and applications, vol 3. Springer-Verlag, Berlin

    Google Scholar 

  6. Eller M (2008) Symmetric hyperbolic systems with boundary conditions that do not satisfy the Kreiss-Sakamoto condition. Appl Math (Warsaw) 35: 323–333

    Article  MATH  MathSciNet  Google Scholar 

  7. Engel K-J, Nagel R (2000) One-parameter semigroups for linear evolution equations. Springer-Verlag, New York

    MATH  Google Scholar 

  8. Fattorini HO (1983) The Cauchy problem. Addison-Wesley, Reading

    MATH  Google Scholar 

  9. Hinrichsen D, Pritchard AJ (1994) Robust stability of linear evolution operators on Banach spaces. SIAM J Control Optim 32: 1503–1541

    Article  MATH  MathSciNet  Google Scholar 

  10. Jacob B (1995) Time-varying infinite dimensional state-space systems. PhD thesis, Bremen, May 1995

  11. Kato T (1970) Linear evolution equations of “hyperbolic” type. J Fac Sci Univ Tokyo 17: 241–258

    MATH  Google Scholar 

  12. Kato T (1973) Linear evolution equations of “hyperbolic” type II. J Math Soc Jpn 25: 648–666

    Article  MATH  Google Scholar 

  13. Kubo A, Reissig M (2003) Construction of parametrix to strictly hyperbolic Cauchy problems with fast oscillations in non Lipschitz coefficients. Comm Partial Differ Equ 28: 1471–1502

    Article  MATH  MathSciNet  Google Scholar 

  14. Lax P, Phillips R (1967) Scattering theory. Academic Press, New York

    MATH  Google Scholar 

  15. Malinen J, Staffans OJ, Weiss G (2006) When is a linear system conservative?. Q Appl Math 64: 61–91

    MATH  MathSciNet  Google Scholar 

  16. Okazawa N (1998) Remarks on linear evolution equations of hyperbolic type in Hilbert space. Adv Math Sci Appl Tokyo 8: 399–423

    MATH  MathSciNet  Google Scholar 

  17. Rodríguez-Bernal A, Zuazua E (1995) Parabolic singular limit of a wave equation with localized boundary damping. Discrete Contin Dyn Syst 1: 303–346

    Article  MATH  Google Scholar 

  18. Salamon D (1987) Infinite dimensional systems with unbounded control and observation: a functional analytic approach. Trans Am Math Soc 300: 383–431

    MATH  MathSciNet  Google Scholar 

  19. Schnaubelt R (2002) Feedbacks for nonautonomous regular linear systems. SIAM J Control Optim 41: 1141–1165

    Article  MATH  MathSciNet  Google Scholar 

  20. Schnaubelt R (2002) Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations. In: Lorenzi A, Ruf B (eds) Evolution equations, semigroups and functional analysis. Birkhäuser, Basel, pp 311–338

    Google Scholar 

  21. Staffans OJ (2002) Passive and conservative continuous-time impedance and scattering systems. Part I: well-posed systems. Math Control Signals Syst 15: 291–315

    Article  MATH  MathSciNet  Google Scholar 

  22. Staffans OJ (2004) Well-posed linear systems. Cambridge University Press, Cambridge

    Google Scholar 

  23. Staffans OJ, Weiss G (2002) Transfer functions of regular linear systems. Part II: the system operator and the Lax-Phillips semigroup. Trans Am Math Soc 354: 3329–3362

    Article  MathSciNet  Google Scholar 

  24. Tanabe H (1979) Equations of evolution. Pitman, London

    MATH  Google Scholar 

  25. Tucsnak M, Weiss G (2003) How to get a conservative well-posed linear system out of thin air. Part I: well-posedness and energy balance. ESAIM Control Optim Calc Var 9: 247–274

    MathSciNet  Google Scholar 

  26. Tucsnak M, Weiss G (2009) Observation and control for operator semigroups. Birkhäuser Verlag, Basel

    Book  MATH  Google Scholar 

  27. Weiss G (1994) Transfer functions of regular linear systems. Part I: characterizations of regularity. Trans Am Math Soc 342: 827–854

    Article  MATH  Google Scholar 

  28. Weiss G, Staffans OJ, Tucsnak M (2001) Well-posed linear systems—a survey with emphasis on conservative systems. Appl Math Comput Sci 11: 101–127

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Institute of Analysis, Karlsruhe Institute of Technology, 76128, Karlsruhe, Germany

    Roland Schnaubelt

  2. Department of Electrical Engineering-Systems, Tel Aviv University, 69978, Ramat Aviv, Israel

    George Weiss

Authors
  1. Roland Schnaubelt
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  2. George Weiss
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Correspondence to Roland Schnaubelt.

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Schnaubelt, R., Weiss, G. Two classes of passive time-varying well-posed linear systems. Math. Control Signals Syst. 21, 265–301 (2010). https://doi.org/10.1007/s00498-010-0049-0

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  • DOI: https://doi.org/10.1007/s00498-010-0049-0

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