Stability and instability of Kirchhoff plate equations with delay on the boundary control
DOI:
https://doi.org/10.58997/ejde.2023.68Keywords:
Kirchhoff plate equation; boundary control; time delay; instability examples; strong stability; exponential stabilityAbstract
In this article, we consider the Kirchhoff plate equation with delay terms on the boundary control. We give instability examples of systems for some choices of delays. Finally, we prove its well-posedness, strong stability, and exponential stability under a multiplier geometric control condition.
Foro more information see https://ejde.math.txstate.edu/Volumes/2023/68/abstr.html
References
M. Akil, H. Badawi, S. Nicaise, A. Wehbe; Stability results on the kirchhoff plate equation with delay terms on the dynamical boundary controls. Revista MatemŽatica Complutense, 36(3):749-777, Sep 2023.
W. Arendt, C. J. K. Batty; Tauberian theorems and stability of one-parameter semigroups. Transactions of the American Mathematical Society, 306(2):837-852, 1988.
R. Datko; Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM Journal on Control and Optimization, 26(3):697-713, 1988.
R. Datko; Two examples of ill-posedness with respect to time delays revisited. IEEE Transactions on Automatic Control, 42(4):511-515, April 1997.
R. Datko, J. Lagnese, M. Polis; An example of the effect of time delays in boundary feedback stabilization of wave equations. In 1985 24th IEEE Conference on Decision and Control. IEEE, Dec. 1985.
H. Ding, J. Zhou; Global solutions and blow-up for a Kirchhoff-type problem on a geodesic ball of the Poincare ball model. Electronic Journal of Differential Equations, Vol. 2022 (2022), no. 38, pp. 1-30.
M. Dreher, R. Quintanilla, R. Racke; Ill-posed problems in thermomechanics. Applied Mathematics Letters, 22(9):1374-1379, Sept. 2009.
U. Ernst, K. Pawelzik, T. Geisel; Delay-induced multistable synchronization of biological oscillators. Physical Review E, 57:2150-2162, Feb 1998.[ 9] F. L. Huang; Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Annals of Differential Equations, 1(1): 43-56, 1985.
T. Kato; Perturbation Theory for Linear Operators. Springer Berlin Heidelberg, 1995.
V. Kolmanovskii, A. Myshkis; Introduction to the theory and applications of functionaldifferential equations, volume 463 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 1999.
J. E. Lagnese; Boundary Stabilization of Thin Plates. Society for Industrial and Applied Mathematics, 1989.
J. L. Lions; Exact controllability, stabilization and perturbations for distributed systems. SIAM Review, 30(1):1-68, 1988.
S. Nicaise; Polygonal Interface Problems. Methoden und Verfahren der mathematischen Physik. P. Lang, 1993.
S. Nicaise, C. Pignotti; Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks. SIAM Journal on Control and Optimization, 45(5):1561-1585, Jan. 2006.
A. Pazy; Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983.
J. Pršuss; On the spectrum of C0-semigroups. Transactions of the American Mathematical Society, 284(2):847-857, 1984.
B. Rao; Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback. Nonlinear Analysis: Theory, Methods & Applications, 20(6):605-626, 1993.
Downloads
Published
License
Copyright (c) 2023 Haidar Badawi, Mohammad Akil, Zayd Hajjej
This work is licensed under a Creative Commons Attribution 4.0 International License.
