Partial reachability of a thermoelastic plate with memory
Authors:
Pedro Gamboa, Vilmos Komornik and Octavio Vera
Journal:
Quart. Appl. Math. 74 (2016), 235-243
MSC (2010):
Primary 35A07; Secondary 35Q53
DOI:
https://doi.org/10.1090/qam/1414
Published electronically:
March 16, 2016
MathSciNet review:
3505602
Full-text PDF Free Access
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Additional Information
Abstract: We investigate the partial reachability of a thermoelastic plate with memory, a variant of a system studied earlier by Lagnese and Lions (1988) without memory. The well posedness of the system is established by transposition after having established the well posedness of the adjoint system by using Volterra equations and the Galerkin method. The partial reachability is deduced from classical theorems on Kirchhoff plates by a perturbation technique.
References
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- Constantine M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29 (1968), 241–271. MR 233539, DOI 10.1007/BF00276727
- Matthias Eller, Irena Lasiecka, and Roberto Triggiani, Simultaneous exact/approximate boundary controllability of thermo-elastic plates with variable transmission coefficient, Shape optimization and optimal design (Cambridge, 1999) Lecture Notes in Pure and Appl. Math., vol. 216, Dekker, New York, 2001, pp. 109–230. MR 1812362
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- Irena Lasiecka and Roberto Triggiani, Exact null controllability of structurally damped and thermo-elastic parabolic models, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9 (1998), no. 1, 43–69 (English, with English and Italian summaries). MR 1669244
- J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev. 30 (1988), no. 1, 1–68. MR 931277, DOI 10.1137/1030001
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- Jaime E. Muñoz Rivera and Maria Grazia Naso, Exact boundary controllability in thermoelasticity with memory, Adv. Differential Equations 8 (2003), no. 4, 471–490. MR 1972597
- Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
- Kôsaku Yosida, Lectures on differential and integral equations, Pure and Applied Mathematics, Vol. X, Interscience Publishers, New York-London, 1960. MR 0118869
References
- G. Avalos and I. Lasiecka, Mechanical and thermal null controllability of thermoelastic plates and singularity of the associated minimal energy function, Control and Cybernetics 32 (2003), no. 3, 473–490.
- George Avalos and Irena Lasiecka, Boundary controllability of thermoelastic plates via the free boundary conditions, SIAM J. Control Optim. 38 (2000), no. 2, 337–383 (electronic). MR 1741144 (2000k:93007), DOI 10.1137/S0363012998339836
- George Avalos, Exact controllability of a thermoelastic system with control in the thermal component only, Differential Integral Equations 13 (2000), no. 4-6, 613–630. MR 1750042 (2001a:93007)
- Assia Benabdallah and Maria Grazia Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal. 7 (2002), no. 11, 585–599. MR 1945447 (2003i:93007), DOI 10.1155/S108533750220408X
- Constantine M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29 (1968), 241–271. MR 0233539 (38 \#1860)
- Matthias Eller, Irena Lasiecka, and Roberto Triggiani, Simultaneous exact/approximate boundary controllability of thermo-elastic plates with variable transmission coefficient, Shape optimization and optimal design (Cambridge, 1999) Lecture Notes in Pure and Appl. Math., vol. 216, Dekker, New York, 2001, pp. 109–230. MR 1812362 (2001m:93010)
- M. Fabrizio, G. Gentili, and D. W. Reynolds, On rigid linear heat conductors with memory, Internat. J. Engrg. Sci. 36 (1998), no. 7-8, 765–782. MR 1629806 (99i:80006), DOI 10.1016/S0020-7225(97)00123-7
- Morton E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), no. 2, 113–126. MR 1553521, DOI 10.1007/BF00281373
- Maurizio Grasselli and Marco Squassina, Exponential stability and singular limit for a linear thermoelastic plate with memory effects, Adv. Math. Sci. Appl. 16 (2006), no. 1, 15–31. MR 2253222 (2007g:74031)
- Jong Uhn Kim, Control of a plate equation with large memory, Differential Integral Equations 5 (1992), no. 2, 261–279. MR 1148218 (92j:93019)
- V. Komornik, Exact controllability and stabilization, The multiplier method. RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. MR 1359765 (96m:93003)
- J. Lagnese and J.-L. Lions, Modelling analysis and control of thin plates, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 6, Masson, Paris, 1988. MR 953313 (89k:73001)
- Irena Lasiecka and Roberto Triggiani, Exact null controllability of structurally damped and thermo-elastic parabolic models, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9 (1998), no. 1, 43–69 (English, with English and Italian summaries). MR 1669244 (2000e:93013)
- J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev. 30 (1988), no. 1, 1–68. MR 931277 (89e:93019), DOI 10.1137/1030001
- J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Contrôlabilité exacte. [Exact controllability]; with appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). MR 953547 (90a:49040)
- Jaime E. Muñoz Rivera and Maria Grazia Naso, Exact boundary controllability in thermoelasticity with memory, Adv. Differential Equations 8 (2003), no. 4, 471–490. MR 1972597 (2004c:93015)
- Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966 (20 \#5411)
- Kôsaku Yosida, Lectures on differential and integral equations, Pure and Applied Mathematics, Vol. X, Interscience Publishers, New York-London, 1960. MR 0118869 (22 \#9638)
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Additional Information
Pedro Gamboa
Affiliation:
Instituto de Matemática, Universidad Federal de Rio de Janeiro, Av. Athos da Silveira Ramos, P.O. Box 68530, CEP:21945-970, RJ, Brazil
Email:
pgamboa@im.ufrj.br
Vilmos Komornik
Affiliation:
Département de Mathématique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France
Email:
vilmos.komornik@math.unistra.fr
Octavio Vera
Affiliation:
Departamento de Matemática, Universidad del Bío Bío, Collao 1202, Casilla 5-C, Concepción, Chile
Email:
overa@ubiobio.cl
Received by editor(s):
August 10, 2014
Published electronically:
March 16, 2016
Additional Notes:
Part of this work was done during the visit of the second author at the Mathematical Institute of the Federal University of Rio de Janeiro in February 2014. He thanks the institute for its hospitality.
The third author is thankful for the support of Fondecyt projects 1121120.
Article copyright:
© Copyright 2016
Brown University
