Opuscula Math. 37, no. 1 (2017), 21-64

Opuscula Mathematica

The LQ/KYP problem for infinite-dimensional systems

Piotr Grabowski

Abstract. Our aim is to present a solution to a general linear-quadratic (LQ) problem as well as to a Kalman-Yacubovich-Popov (KYP) problem for infinite-dimensional systems with bounded operators. The results are then applied, via the reciprocal system approach, to the question of solvability of some Lur'e resolving equations arising in the stability theory of infinite-dimensional systems in factor form with unbounded control and observation operators. To be more precise the Lur'e resolving equations determine a Lyapunov functional candidate for some closed-loop feedback systems on the base of some properties of an uncontrolled (open-loop) system. Our results are illustrated in details by an example of a temperature of a rod stabilization automatic control system.

Keywords: control of infinite-dimensional systems, semigroups, infinite-time LQ-control problem, Lur'e feedback systems.

Mathematics Subject Classification: 49N10, 93B05, 93C25.

Full text (pdf)

  1. M.Ya. Antimirov, A.A. Kolyshkin, R. Vaillancourt, Applied Integral Transforms, The American Mathematical Society, 1993.
  2. W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2001; 2nd ed., Springer-Basel AG, 2011.
  3. H. Bateman, A. Erdélyi, V. Magnus, F. Oberhettinger, F. Tricomi, Tables of Integral Transforms. Vol. I, McGraw-Hill, New York, 1954.
  4. R.F. Curtain, Riccati equations for stable well-posed linear systems: the generic case, SIAM J. Control Optim. 42 (2003), 1681-1702.
  5. P. Grabowski, Admissibility of observation functionals, Internat. J. Control 62 (1995), 1161-1173.
  6. P. Grabowski, The lq-controller synthesis problem for infinite-dimensional systems in factor form, Opuscula Math. 33 (2013), 29-79.
  7. P. Grabowski, Some modifications of the Weiss-Staffans perturbation theorem, to appear in Internat. J. Robust Nonlinear Control (2016), DOI: 10.1002/rnc.3617. http://dx.doi.org/10.1002/rnc.3617
  8. P. Grabowski, F.M. Callier, On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations, Facultés Universitaires Notre-Dame de la Paix á Namur, Publications du Département de Mathématique, FUNDP, Namur, Belgium, Report 05 (2002).
  9. P. Grabowski, F.M. Callier, On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations, ESAIM Control Optim. Calc. Var. 12 (2006), 169-197.
  10. K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, 1962.
  11. R.E. Kalman, Lyapunov function for the problem of Lur'e in automatic control, Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 201-205.
  12. J.G. Krzyż, Problems in Complex Variable Theory, Elsevier, New York, 1972.
  13. A. Massoudi, M.R. Opmeer, T. Reis, The ADI method for bounded real and positive real Lur'e equations, Numer. Math. (2016), DOI 10.1007/s00211-016-0805-2. http://dx.doi.org/10.1007/s00211-016-0805-2
  14. K. Mikkola, Riccati equations and optimal control of well-posed linear systems, arXiv:1602.08618v1 [math.OC], 27 Feb 2016.
  15. J.C. Oostveen, R.F. Curtain, Riccati equations for strongly stabilizable bounded linear systems, Automatica J. IFAC 34 (1998), 953-967.
  16. J.R. Partington, An Introduction to Hankel Operators, Cambridge University Press, Cambridge, 1988.
  17. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
  18. V.M. Popov, Hyperstability and optimality of automatic systems with several control functions, Rev. Roum. Sci. Tech. Ser. Electrotech. Energ. 9 (1964), 629-690.
  19. H.A. Shubert, An analytic method for an algebraic matrix Riccati equation, IEEE Trans. Automat. Control 19 (1974), 255-256.
  20. J. Weidmann, Linear Operators in Hilbert Spaces, Springer, Heidelberg, 1980.
  21. G. Weiss, M. Weiss, Optimal control of stable weakly regular linear systems, Math. Control Signals Systems 10 (1997), 287-330.
  • Piotr Grabowski
  • AGH University of Science and Technology, Institute of Control and Biomedical Engineering, al. Mickiewicza 30, 30-059 Krakow, Poland
  • Communicated by A. Shkalikov.
  • Received: 2016-05-19.
  • Revised: 2016-10-24.
  • Accepted: 2016-10-29.
  • Published online: 2016-12-14.
Opuscula Mathematica - cover

Cite this article as:
Piotr Grabowski, The LQ/KYP problem for infinite-dimensional systems, Opuscula Math. 37, no. 1 (2017), 21-64, http://dx.doi.org/10.7494/OpMath.2017.37.1.21

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.