Abstract
The uniform stability of a nonlinear thermoelastic plate model is investigated, where the abstract nonlinearity here satisfies assumptions which allow the specification of the von Kármán nonlinearity, among other physically relevant examples. Linear analogs of this work were considered in [1] and [2]. Even in the absence of inserted dissipative feedbacks on the boundary, this system is shown to be stable with exponential decay rates which are uniform with respect to the “finite energy” of the given initial data (uniform stability of a linear thermoelastic plate with added boundary dissipation was shown in [8], as was that of the analytic case in [14]). The proof of this result involves a multiplier method, but with the particular multiplier invoked being of a rather nonstandard (operator theoretic) nature. In addition, the “free” boundary conditions in place for the plate component give rise to higher order terms which pollute the decay estimates, and to deal with these a new result for boundary traces of the wave equation must be employed.
The research of G. Avalos is partially supported by the NSF Grant DMS-9710981. The research of I. Lasiecka is partially supported by the NSF Grant DMS-9504822 and by the Army Research Office Grant DAAH04-96-1-0059.
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References
Avalos, G. and Lasiecka, I. (1997), “Exponential stability of a thermoelastic system without mechanical dissipation,” IMA Preprint Series #1357, November, 1995, and Rendiconti Di Instituto Di Matematica, Dell’Universitâ di Trieste, Suppl. Vol. XXVIII, E. Mitidieri and Philippe Clement, eds., 1–27.
Avalos, G. and Lasiecka, I. “Exponential stability of a thermoelastic system without mechanical dissipation II, the case of free boundary conditions,” IMA Preprint # 1397, March, 1996, and SIAM Journal of Mathematical Analysis, to appear January, 1998.
Avalos, G. and Lasiecka, I. “The exponential stability of a nonlinear thermoelastic system,” preprint.
Avalos, G. and Lasiecka, I. “Uniform decay rates for a nonlinear thermoelastic system,” in the Proceedings of the European Control Conference 1997, Brussels, Belgium,to appear.
Avalos, G. “Wellposedness of nonlinear thermoelastic systems,” preprint.
Bisognin, E., Bisognin, V., Menzala, G.P. and Zuazua, E., “On the exponential stability for von Kármán equation in the presence of thermal effects,” preprint 1996.
Grisvard, P. (1967), “Caracterization de quelques espaces d’interpolation,” Arch. Rat. Mech. Anal., Vol. 25, 40–63.
Lagnese, J. (1989), “Boundary stabilization of thin plates,” SIAM Stud. Appl. Math., Vol. 10, Philadelphia, PA.
Lasiecka, I. (1997), “Control and stabilization of interactive structures,” in Systems and Control in the Twenty-First Century, Birkhäuser, Basel, 245–263.
Lasiecka, I. and Tataru, D. (1993), “Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,” Differential and Integral Equations, Vol. 6, No. 3, 507–533.
Lasiecka, I. and Triggiani, R. (1991), “Exact controllability and uniform stabilization of Kirchoff plates with boundary control only on Awl and homogeneous boundary displacement,” J. Diff. Eqns., Vol. 88, 62–101.
Lions, J.L. (1989), Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes distribués, Vol. 1, Masson, Paris.
Lions, J.L. and Magenes, E. (1972), Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York.
Liu, Z. and Zheng, S., “Exponential stability of the Kirchoff plate with thermal or viscoelastic damping,” Q. Appl. Math.,to appear.
Menzala, G.P. and Zuazua, E. (1997), “Explicit exponential decay rates for solutions of von Kârmâ,n’s system of thermoelastic paltes,” C.R. Acad. Sci. Paris, Vol. 324, 49–54.
Tataru, D., “On the regularity of boundary traces for the wave equation,” Annali di Scuola Normale,to appear.
Triggiani, R., “Analyticity, and lack thereof, of semigroups arising from thermoelastic plates,” preprint.
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Avalos, G., Lasiecka, I. (1998). Uniform Decays in Nonlinear Thermoelastic Systems. In: Optimal Control. Applied Optimization, vol 15. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6095-8_1
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DOI: https://doi.org/10.1007/978-1-4757-6095-8_1
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