Global stabilization of a dynamic von Kármán plate with nonlinear boundary feedback
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
We consider a fully nonlinear von Kármán system with, in addition to the nonlinearity which appears in the equation, nonlinear feedback controls acting through the boundary as moments and torques. Under the assumptions that the nonlinear controls are continuous, monotone, and satisfy appropriate growth conditions (however, no growth conditions are imposed at the origin), uniform decay rates for the solution are established. In this fully nonlinear case, we do not have, in general, smooth solutions even if the initial data are assumed to be very regular. However, rigorous derivation of the estimates needed to solve the stabilization problem requires a certain amount of regularity of the solutions which is not guaranteed. To deal with this problem, we introduce a regularization/approximation procedure which leads to an “approximating” problem for which partial differential equation calculus can be rigorously justified. Passage to the limit on the approximation reconstructs the estimates needed for the original nonlinear problem.
- M. E. Bradley and M. A. Horn (to appear). Global stabilization of the von Kármán plate with boundary feedback acting via bending moments only. Journal of Nonlinear Analysis.
- M. E. Bradley and I. Lasiecka (1994). Global decay rates for the solutions to a von Kármán plate without geometric conditions. Journal of Mathematical Analysis and Applications, 181(l):254–276.
- M. A. Horn and I. Lasiecka (to appear). Asymptotic behavior with respect to thickness of boundary stabilizing feedback for the Kirchoff plate. Journal of Differential Equations.
- J. E. Lagnese (1989). Boundary Stabilization of Thin Plates. Society for Industrial and Applied Mathematics, Philadelphia, PA.
- J. E. Lagnese (1991). Uniform asymptotic energy estimates for solutions of the equation of dynamical plane elasticity with nonlinear dissipation at the boundary. Nonlinear Analysis, 16(1):35–54.
- J. E. Lagnese and G. Leugering (1991). Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. Journal of Differential Equations, 91(2):355–388.
- I. Lasiecka (1992). Global uniform decay rates for the solutions to wave equations with nonlinear boundary conditions. Applicable Analysis, 47:191–212.
- I. Lasiecka (to appear). Existence and uniqueness of the solutions to second order abstract equations with nonlinear and nonmonotone boundary conditions. Journal of Nonlinear Analysis, Methods and Applications.
- I. Lasiecka and D. Tataru (1993). Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential and Integral Equations, 6(3):507–533.
- I. Lasiecka and R. Triggiani (1991). Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Springer-Verlag, New York.
- I. Lasiecka and R. Triggiani (1993). Sharp trace estimates of solutions to Kirchoff and Euler-Bernoulli equations. Applied Mathematics and Optimization, 28:277–306.
- J. L. Lions (1989). Contrôlabilité exacte, perturbations et stabilization de systèmes distribués, volume 1. Masson, Paris.
- J. L. Lions and E. Magenes (1972). Non-Homogeneous Boundary Value Problems and Applications, volume 1. Springer-Verlag, New York.
- W. Littman (1985). Boundary control theory for beams and plates. Proceedings of the 24th Conference on Decision and Control, pages 2007–2009.
- J. Puel and M. Tucsnak (1992). Boundary stabilization for the von Kármán equations. Comptes Rendus des Séances de l'Académie des Sciences, 314:609–612. Also (to appear) SIAM Journal of Control.
- Global stabilization of a dynamic von Kármán plate with nonlinear boundary feedback
Applied Mathematics and Optimization
Volume 31, Issue 1 , pp 57-84
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Nonlinear boundary feedback
- von Kármán plate