Skip to main content
SpringerLink
Log in

Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

An operator Riccati equation from systems theory is considered in the case that all entries of the associated Hamiltonian are unbounded. Using a certain dichotomy property of the Hamiltonian and its symmetry with respect to two different indefinite inner products, we prove the existence of nonnegative and nonpositive solutions of the Riccati equation. Moreover, conditions for the boundedness and uniqueness of these solutions are established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamjan V., Langer H., Tretter C.: Existence and uniqueness of contractive solutions of some Riccati equations. J. Funct. Anal., 179(2), 448–473 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. R. A. Adams, J. J. F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathematics. Elsevier/Academic Press, Amsterdam, second edition, 2003.

  3. N. I. Akhiezer, I. M. Glazman. Theory of Linear Operators in Hilbert Space. Dover Publications Inc., New York, 1993.

  4. Arendt W., Bu S.: Sums of bisectorial operators and applications. Integral Equations Operator Theory 52(3), 299–321 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Arendt W., Duelli M.: Maximal L p-regularity for parabolic and elliptic equations on the line. J. Evol. Equ. 6(4), 773–790 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. T. Y. Azizov, A. Dijksma, I. V. Gridneva. Conditional reducibility of certain unbounded nonnegative Hamiltonian operator functions. Integral Equations Operator Theory, 73(2) (2012), 273–303.

    Google Scholar 

  7. T. Y. Azizov, I. S. Iokhvidov. Linear Operators in Spaces With an Indefinite Metric. John Wiley & Sons, Chichester, 1989.

  8. H. Bart, I. Gohberg, M. A. Kaashoek. Minimal factorization of matrix and operator functions, volume 1 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1979.

  9. Bart H., Gohberg I., Kaashoek M.A.: Wiener–Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators. J. Funct. Anal. 68(1), 1–42 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Bittanti, A. J. Laub, J. C. Willems, editors. The Riccati equation. Communications and Control Engineering Series. Springer-Verlag, Berlin, 1991.

  11. P. Bubák, C. V. M. van der Mee, A. C. M. Ran. Approximation of solutions of Riccati equations. SIAM J. Control Optim., 44(4) (2005), 1419–1435.

  12. J.-C. Cuenin, C. Tretter. Spectral perturbations of self-adjoint operators with spectral gaps. Submitted, (2013).

  13. Curtain R.F., Zwart H.J.: An Introduction to Infinite Dimensional Linear Systems Theory. Springer, New York (1995)

    Book  MATH  Google Scholar 

  14. A. Dijksma, H. S. V. de Snoo. Symmetric and selfadjoint relations in Kreĭ n spaces. I. volume 24 of Oper. Theory Adv. Appl., pages 145–166. Birkhäuser, Basel, 1987.

  15. K.-J. Engel, R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer, New York, 2000.

  16. I. Gohberg, S. Goldberg, M. A. Kaashoek. Classes of linear operators. Vol. I, volume 49 of Operator Theory: Advances and Applications. Birkhäuser, Basel, 1990.

  17. Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)

    MATH  Google Scholar 

  18. V. Kostrykin, K. A. Makarov, A. K. Motovilov. Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach. volume 327 of Contemp. Math., pages 181–198. Amer. Math. Soc., Providence, 2003.

  19. M. G. Kreĭn. Introduction to the geometry of indefinite J -spaces and to the theory of operators in those spaces. In Second Math. Summer School, Part I (Russian), pages 15–92. Naukova Dumka, Kiev, 1965. English transl. in Amer. Math. Soc. Transl. (2), 93 (1970), 103–176.

  20. S. G. Kreĭn. Linear Differential Equations in Banach Space. Amer. Math. Soc., Providence, 1971.

  21. Kuiper C. R., Zwart H. J.: Connections between the algebraic Riccati equation and the Hamiltonian for Riesz-spectral systems. J. Math. Systems Estim. Control 6(4), 1–48 (1996)

    MathSciNet  Google Scholar 

  22. Lancaster P., Rodman L.: Algebraic Riccati Equations. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  23. H. Langer, A. C. M. Ran, D. Temme. Nonnegative solutions of algebraic Riccati equations. Linear Algebra Appl., 261 (1997), 317–352.

    Google Scholar 

  24. H. Langer, A. C. M. Ran, B. A. van de Rotten. Invariant subspaces of infinite dimensional Hamiltonians and solutions of the corresponding Riccati equations. volume 130 of Oper. Theory Adv. Appl., pages 235–254. Birkhäuser, Basel, 2002.

  25. H. Langer, C. Tretter. Diagonalization of certain block operator matrices and applications to Dirac operators. volume 122 of Oper. Theory Adv. Appl., pages 331–358. Birkhäuser, Basel, 2001.

  26. I. Lasiecka, R. Triggiani. Control Theory for Partial Differential Equations: Continuous and Approximation Theorems. I: Abstract Parabolic Systems. Cambridge University Press, 2000.

  27. Lions J.-L., Magenes E.: Non-homogeneous boundary value problems and applications. Vol. I.. Springer-Verlag, New York (1972)

    Book  Google Scholar 

  28. A. S. Markus. Introduction to the Spectral Theory of Polynomial Operator Pencils. Amer. Math. Soc., Providence, 1988.

  29. V. Maz’ya. Sobolev spaces with applications to elliptic partial differential equations, volume 342 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg, augmented edition, 2011.

  30. A. McIntosh, A. Yagi. Operators of type ω without a bounded H ∞ functional calculus. In Miniconference on Operators in Analysis (Sydney, 1989), volume 24 of Proc. Centre Math. Anal. Austral. Nat. Univ., pages 159–172. Austral. Nat. Univ., Canberra, 1990.

  31. Opmeer M. R., Curtain R. F.: New Riccati equations for well-posed linear systems. Systems Control Lett. 52(5), 339–347 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. O. Staffans. Well-posed linear systems, volume 103 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2005.

  33. C. Tretter. Spectral issues for block operator matrices. In Differential equations and mathematical physics (Birmingham, AL, 1999), volume 16 of AMS/IP Stud. Adv. Math., pages 407–423. Amer. Math. Soc., Providence, 2000.

  34. C. Tretter. Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London, 2008.

  35. Tretter C.: Spectral inclusion for unbounded block operator matrices. J. Funct. Anal. 256(11), 3806–3829 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. H. Triebel. Interpolation theory, function spaces, differential operators. Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995.

  37. C. van der Mee. Exponentially dichotomous operators and applications, volume 182 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2008.

  38. J. Weidmann. Linear operators in Hilbert spaces, volume 68 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1980.

  39. Weiss M., Weiss G.: Optimal control of stable weakly regular linear systems. Math. Control Signals Systems 10(4), 287–330 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  40. Wyss C.: Riesz bases for p-subordinate perturbations of normal operators. J. Funct. Anal. 258(1), 208–240 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  41. Wyss C.: Hamiltonians with Riesz bases of generalised eigenvectors and Riccati equations. Indiana Univ. Math. J. 60, 1723–1766 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wyss C., Jacob B., Zwart H.J.: Hamiltonians and Riccati equations for linear systems with unbounded control and observation operators. SIAM J. Control Optim. 50, 1518–1547 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Christiane Tretter

  2. Department of Mathematics and Informatics, University of Wuppertal, Gaußstraße 20, 42097, Wuppertal, Germany

    Christian Wyss

Authors
  1. Christiane Tretter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Wyss
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christiane Tretter.

Additional information

Dedicated to Rien Kaashoek on the occasion of his 75th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tretter, C., Wyss, C. Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations. J. Evol. Equ. 14, 121–153 (2014). https://doi.org/10.1007/s00028-013-0210-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-013-0210-6

Mathematics Subject Classification

Keywords

Search

Navigation