Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces

  • Swann Marx
  • Vincent Andrieu
  • Christophe Prieur
Original Article


This work studies the influence of some constraints on a stabilizing feedback law. An abstract nonlinear control system is considered for which we assume that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. This controller is then modified via a cone-bounded nonlinearity. Well-posedness and stability theorems are stated. The first theorem is proved thanks to the Schauder fixed-point theorem and the second one with an infinite-dimensional version of LaSalle’s invariance principle. These results are illustrated on a linear Korteweg-de Vries equation by some simulations and on a nonlinear heat equation.


Nonlinear semigroups Stabilization Abstract control systems 


  1. 1.
    Brezis H (2010) Functional analysis, Sobolev spaces and partial differential equations. Springer, New YorkCrossRefGoogle Scholar
  2. 2.
    Castelan EB, Tarbouriech S, Queinnec I (2008) Control design for a class of nonlinear continuous-time systems. Automatica 44(8):2034–2039MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cerpa E (2014) Control of a Korteweg-de Vries equation: a tutorial. Math Control Relat Fields 4(1):45–99MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chow A, Morris KA (2014) Hysteresis in the linearized Landau–Lifshitz equation. In: 2014 American control conference, Portland, pp 4747–4752Google Scholar
  5. 5.
    Coron J-M (2007) Control and nonlinearity. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  6. 6.
    Coron J-M, Crépeau E (2004) Exact boundary controllability of a nonlinear KdV equation with critical lengths. J Eur Math Soc 6:367–398MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Coutinho DF, Gomes da Silva JM Jr (2010) Computing estimates of the region of attraction for rational control systems with saturating actuators. IET Control Theory Appl 4(3):315–325MathSciNetCrossRefGoogle Scholar
  8. 8.
    Crandall MG, Pazy A (1969) Semi-groups of nonlinear contractions and dissipative sets. J Funct Anal 3(3):376–418MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Curtain R, Zwart H (2016) Stabilization of collocated systems by nonlinear boundary control. Syst Control Lett 96:11–14MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Daafouz J, Tucsnak M, Valein J (2014) Nonlinear control of a coupled pde/ode system modeling a switched power converter with a transmission line. Syst Control Lett 70:92–99MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    d’Andréa Novel B, Boustany F, Conrad F, Rao BP (1994) Feedback stabilization of a hybrid PDE-ODE system: application to an overhead crane. Math Control Signals Syst 13(1):97–106MathSciNetzbMATHGoogle Scholar
  12. 12.
    Grimm G, Hatfield J, Postlethwaite I, Teel AR, Turner MC, Zaccarian L (2003) Antiwindup for stable linear systems with input saturations: an LMI-based synthesis. IEEE Trans Automat Control 43:1509–1564MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hale JK (1969) Dynamical systems and stability. J Math Anal Appl 26(1):39–59MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jayawardhana B, Logemann H, Ryan EP (2008) Infinite-dimensional feedback systems: the circle criterion and input-to-state stability. Commun Inf Syst 8(4):413–444MathSciNetzbMATHGoogle Scholar
  15. 15.
    Jayawardhana B, Logemann H, Ryan EP (2011) The circle criterion and input-to-state stability. IEEE Control Syst 31(4):32–67MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Khalil HK (1996) Nonlinear systems, Second edn. Prentice Hall, Inc, Upper Saddle RiverGoogle Scholar
  17. 17.
    Komura Y (1967) Nonlinear semi-groups in Hilbert space. J Math Soc Jpn 19(4):493–507MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Laporte J, Chaillet A, Chitour Y (2015) Global stabilization of multiple integrators by a bounded feedback with constraints on its successive derivatives. In: Proceedings of the 54th IEEE conference on decision and control, Osaka, pp 3983–3988Google Scholar
  19. 19.
    Lasiecka I, Seidman TI (2003) Strong stability of elastic control systems with dissipative saturating feedback. Syst Control Lett 48:243–252MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Liu W, Chitour Y, Sontag E (1996) On finite-gain stabilizability of linear systems subject to input saturation. SIAM J Control Optim 34(4):1190–1219MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marx S, Cerpa E, Prieur C, Andrieu V (2015) Stabilization of a linear Korteweg-de Vries with a saturated internal control. In: Proceedings of the European control conference, Linz, pp 867–872Google Scholar
  22. 22.
    Marx S, Cerpa E, Prieur C, Andrieu V (2017) Global stabilization of a Korteweg–De Vries equation with saturating distributed control. SIAM J Control Optim 55(3):1452–1480MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Miyadera I (1992) Nonlinear semigroups. Translations of mathematical monographsGoogle Scholar
  24. 24.
    Pazoto AF, Sepúlveda M, Vera Villagrán O (2010) Uniform stabilization of numerical schemes for the critical generalized Korteweg-de Vries equation with damping. Numer Math 116(2):317–356MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  26. 26.
    Prieur C, Tarbouriech S, Gomes da Silva JM Jr (2016) Wave equation with cone-bounded control laws. IEEE Trans Automat Control 61(11):3452–3463MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rosier L (1997) Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM Control Optim Calc Var 2:33–55MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rosier L, Zhang B-Y (2006) Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain. SIAM J Control Optim 45(3):927–956MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Seidman TI, Li H (2001) A note on stabilization with saturating feedback. Discrete Contin Dyn Syst 7(2):319–328MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Showalter RE (1997) Monotone operators in Banach space and nonlinear partial differential equations. Mathematical surveys and monographsGoogle Scholar
  31. 31.
    Slemrod M (1989) Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control. Math Control Signals Syst 2(3):847–857MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sussmann HJ, Yang Y (1991) On the stabilizability of multiple integrators by means of bounded feedback controls. Technical report SYCON-91-01, Rutgers Center for Systems and ControlGoogle Scholar
  33. 33.
    Tarbouriech S, Garcia G, Gomes da Silva JM Jr, Queinnec I (2011) Stability and stabilization of linear systems with saturating actuators. Springer, New YorkCrossRefzbMATHGoogle Scholar
  34. 34.
    Teel AR (1992) Global stabilization and restricted tracking for multiple integrators with bounded controls. Syst Control Lett 18:165–171MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tucsnak M, Weiss G (2009) Observation and control for operator semigroups. Springer, New YorkCrossRefzbMATHGoogle Scholar
  36. 36.
    Van der Schaft A, Jeltsema D (2014) Port-Hamiltonian systems theory: an introductory overview. Found Trends Syst Control 1(2–3):173–378CrossRefGoogle Scholar
  37. 37.
    Zaccarian L, Teel AR (2011) Modern anti-windup synthesis: control augmentation for actuator saturation. Princeton University Press, PrincetonCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd. 2017

Authors and Affiliations

  1. 1.Univ. Grenoble Alpes, CNRS, Gipsa-labGrenobleFrance
  2. 2.Université Lyon 1 CNRS UMR 5007 LAGEPLyonFrance

Personalised recommendations