Uniform Stabilizability of Nonlinearly Coupled Kirchhoff Plate Equations

  • Mary Ann Horn
  • Irena Lasiecka
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 118)


A system of two Kirchhoff plate equations with nonlinear coupling through both the boundary and the interior is considered. For this problem, it is proven that by appropriately choosing feedback controls, the energy of the system decays at a uniform rate. This result extends previous results in a number of directions: (i) it does not require any geometric hypotheses to be imposed on the domain; (ii) it allows for the presence of nonlinear coupling terms; (iii) it does not require the control functions to satisfy any growth conditions at the origin.

1991 Mathematics Subject Classification

35 93 

Key words and phrases

Uniform stabilization boundary feedback coupled plates Kirchhoff plate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Favini and I. Lasiecka. Well-posedness and regularity of second-order abstract equations arising in hyperbolic-like problems with nonlinear boundary conditions. Osaka Journal of Mathematics. To appear.Google Scholar
  2. 2.
    M.A. Horn and I. Lasiecka. Global stabilization of a dynamic von Kármán plate with nonlinear boundary feedback. Applied Mathematics and Optimization. To appear.Google Scholar
  3. 3.
    V. Komornik. Rapid boundary stabilization of the wave equation. SIAM Journal of Control and Optimization ,29 (1):197–208 January 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    V. Komornik and E. Zuazua. A direct method for the boundary stabilization of the wave equation. J. Math. Pures et. Appl. ,69:33–54, 1990.MathSciNetzbMATHGoogle Scholar
  5. 5.
    J. E. Lagnese. Boundary Stabilization of Thin Plates.. Society for Industrial and Applied Mathematics, Philadelphia, 1989.zbMATHCrossRefGoogle Scholar
  6. 6.
    I. Lasiecka. Existence and uniqueness of the solutions to second order abstract equations with nonlinear and nonmonotone boundary conditions. Journal of Nonlinear Analysis, Methods and Applications. To appear.Google Scholar
  7. 7.
    I. Lasiecka and D. Tataru. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential and Integral Equations ,6(3):507–533, 1993.MathSciNetzbMATHGoogle Scholar
  8. 8.
    I. Lasiecka and R. Triggiani. Sharp trace estimates of solutions to Kirchoff and Euler-Bernoulli equations. Applied Mathematics and Optimization ,28:277–306, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    W. Littman. Boundary control theory for beams and plates. In Proceedings of 24th Conference on Decision and Control ,pages 2007–2009, 1985.Google Scholar
  10. 10.
    M. Najafi, G. Sarhangi, and H. Wang. The study of the stabilizability of the coupled wave equations under various end conditions. In Proceedings of the 31 nd IEEE Conference on Decision and Control ,Tucson, Arizona, 1992.Google Scholar
  11. 11.
    M. Najafi and H. Wang. Exponential stability of wave equations coupled in parallel by viscous damping, In Proceedings of the 32nd IEEE Conference on Decision and Control ,San Antonio, Texas, 1993.Google Scholar
  12. 12.
    B. Rao. Stabilization of a Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback. Nonlinear Analysis ,20(6):605–626, 1993.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Mary Ann Horn
    • 1
  • Irena Lasiecka
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Applied MathematicsUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations