Uniform stabilization of the higher-dimensional system of thermoelasticity with a nonlinear boundary feedback
Authors:
Wei-Jiu Liu and Enrique Zuazua
Journal:
Quart. Appl. Math. 59 (2001), 269-314
MSC:
Primary 74F05; Secondary 35B35, 35Q72, 74M05
DOI:
https://doi.org/10.1090/qam/1828455
MathSciNet review:
MR1828455
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Using multiplier techniques and Lyapunov methods, we derive explicit decay rates for the energy in the higher-dimensional system of thermoelasticity with a nonlinear velocity feedback on part of the boundary of a thermoelastic body, which is clamped along the rest of its boundary.
- Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
- Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
- Fatiha Alabau and Vilmos Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems, SIAM J. Control Optim. 37 (1999), no. 2, 521–542. MR 1665070, DOI https://doi.org/10.1137/S0363012996313835
- V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295
- George Avalos and Irena Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal. 29 (1998), no. 1, 155–182. MR 1617180, DOI https://doi.org/10.1137/S0036141096300823
- Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
- E. Bisognin, V. Bisognin, G. Perla Menzala, and E. Zuazua, On exponential stability for von Kármán equations in the presence of thermal effects, Math. Methods Appl. Sci. 21 (1998), no. 5, 393–416. MR 1608076, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819980325%2921%3A5%3C393%3A%3AAID-MMA958%3E3.3.CO%3B2-A
- Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
- John A. Burns, Zhuangyi Liu, and Song Mu Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl. 179 (1993), no. 2, 574–591. MR 1249839, DOI https://doi.org/10.1006/jmaa.1993.1370
- Goong Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl. (9) 58 (1979), no. 3, 249–273. MR 544253
- Francis Conrad and Bopeng Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback, Asymptotic Anal. 7 (1993), no. 3, 159–177. MR 1226972
- 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 https://doi.org/10.1007/BF00276727
- Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 2, Springer-Verlag, Berlin, 1988. Functional and variational methods; With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily; Translated from the French by Ian N. Sneddon. MR 969367
- P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- P. Grisvard, Contrôlabilité exacte des solutions de l’équation des ondes en présence de singularités, J. Math. Pures Appl. (9) 68 (1989), no. 2, 215–259 (French, with English summary). MR 1010769
- Scott W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl. 167 (1992), no. 2, 429–442. MR 1168599, DOI https://doi.org/10.1016/0022-247X%2892%2990217-2
- Mary Ann Horn, Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity, J. Math. Anal. Appl. 223 (1998), no. 1, 126–150. MR 1627344, DOI https://doi.org/10.1006/jmaa.1998.5963
- Mary Ann Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control 8 (1998), no. 2, 11 pp.}, issn=1052-0600, review=\MR{1651449},.
- Jong Uhn Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (1992), no. 4, 889–899. MR 1166563, DOI https://doi.org/10.1137/0523047
- V. Komornik, Exact controllability and stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR 1359765
- Vilmos Komornik and Bopeng Rao, Boundary stabilization of compactly coupled wave equations, Asymptot. Anal. 14 (1997), no. 4, 339–359. MR 1461124
- Vilmos Komornik and Enrike Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. (9) 69 (1990), no. 1, 33–54. MR 1054123
A. D. Kovalenko, Thermoelasticity, Basic Theory and Applications, Wolters-Noordhoff Publishing, Gröningen, Netherlands, 1969
- John Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50 (1983), no. 2, 163–182. MR 719445, DOI https://doi.org/10.1016/0022-0396%2883%2990073-6
- John Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control Optim. 21 (1983), no. 6, 968–984. MR 719524, DOI https://doi.org/10.1137/0321059
- John E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1061153
- John E. Lagnese, Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary, Nonlinear Anal. 16 (1991), no. 1, 35–54. MR 1086824, DOI https://doi.org/10.1016/0362-546X%2891%2990129-O
- J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, J. Differential Equations 91 (1991), no. 2, 355–388. MR 1111180, DOI https://doi.org/10.1016/0022-0396%2891%2990145-Y
- 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
- I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations 79 (1989), no. 2, 340–381. MR 1000694, DOI https://doi.org/10.1016/0022-0396%2889%2990107-1
- I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6 (1993), no. 3, 507–533. MR 1202555
- Gilles Lebeau and Enrique Zuazua, Sur la décroissance non uniforme de l’énergie dans le système de la thermoélasticité linéaire, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 4, 409–415 (French, with English and French summaries). MR 1440958, DOI https://doi.org/10.1016/S0764-4442%2897%2980077-8
- G. Leugering, On boundary feedback stabilization of a viscoelastic membrane, Dynam. Stability Systems 4 (1989), no. 1, 71–79. MR 1023012, DOI https://doi.org/10.1080/02681118908806063
G. Leugering, On boundary feedback stabilization of a viscoelastic beam, Proc. Roy. Soc. Edinburgh, Sect. A 114, 57–69 (1990)
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 182. MR 0350178
- Weijiu Liu, Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity, ESAIM Control Optim. Calc. Var. 3 (1998), 23–48. MR 1610226, DOI https://doi.org/10.1051/cocv%3A1998113
W. J. Liu, Exact Controllability for Some Partial Differential Equations of Evolutional Type, Ph.D. Thesis, University of Wollongong, Australia, 1997
- Weijiu Liu, The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity, J. Math. Pures Appl. (9) 77 (1998), no. 4, 355–386. MR 1623383, DOI https://doi.org/10.1016/S0021-7824%2898%2980103-7
- Zhuangyi Liu and Song Mu Zheng, Exponential stability of the semigroup associated with a thermoelastic system, Quart. Appl. Math. 51 (1993), no. 3, 535–545. MR 1233528, DOI https://doi.org/10.1090/qam/1233528
- Wei-Jiu Liu and Enrique Zuazua, Decay rates for dissipative wave equations, Ricerche Mat. 48 (1999), no. suppl., 61–75. Papers in memory of Ennio De Giorgi (Italian). MR 1765677
G. P. Menzala and E. Zuazua, Explicit exponential decay rates for solutions of von Kármán’s system of thermoelastic plates, C. R. Acad. Sci Paris Sér. I Math. 324, 49–54 (1997)
G. P. Menzala and E. Zuazua, Energy decay rates for the Von Kármán system of thermoelastic plates, Differential Integral Equations, to appear.
- Kimiaki Narukawa, Boundary value control of thermoelastic systems, Hiroshima Math. J. 13 (1983), no. 2, 227–272. MR 707182
- Luiz Augusto Fernandes de Oliveira, Exponential decay in thermoelasticity, Commun. Appl. Anal. 1 (1997), no. 1, 113–118. MR 1453945
- Jaime E. Muñoz Rivera, Decomposition of the displacement vector field and decay rates in linear thermoelasticity, SIAM J. Math. Anal. 24 (1993), no. 2, 390–406. MR 1205533, DOI https://doi.org/10.1137/0524025
- J. E. Muñoz Rivera, Asymptotic behaviour in $n$-dimensional thermoelasticity, Appl. Math. Lett. 10 (1997), no. 5, 47–53. MR 1471316, DOI https://doi.org/10.1016/S0893-9659%2897%2900082-7
- Jaime E. Muñoz Rivera and Milton Lacerda Olivera, Stability in inhomogeneous and anisotropic thermoelasticity, Boll. Un. Mat. Ital. A (7) 11 (1997), no. 1, 115–127 (English, with Italian summary). MR 1438361
- Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR 0344043
- Han Kun Wang and Goong Chen, Asymptotic behavior of solutions of the one-dimensional wave equation with a nonlinear boundary stabilizer, SIAM J. Control Optim. 27 (1989), no. 4, 758–775. MR 1001918, DOI https://doi.org/10.1137/0327040
- Enrike Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim. 28 (1990), no. 2, 466–477. MR 1040470, DOI https://doi.org/10.1137/0328025
R. Adams, Sobolev Spaces, Academic Press, New York, 1975
S. Agmon, Lectures on Elliptic Boundary Value Problems, D. Van Nostrand Company, Inc., Princeton, NJ, 1965
F. Alabau and V. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems, SIAM J. Control Optim. 37, no. 2, 521–542 (1999)
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989
G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal. 29, 155–182 (1998)
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, The Netherlands, 1976
E. Bisognin, V. Bisognin, G. P. Menzala, and E. Zuazua, On exponential stability for the Von Kármán equations in the presence of thermal effects, Math. Methods Appl. Sci. 21, 393–416 (1998)
H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983
J. A. Burns, Z. Y. Liu, and S. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl. 179, 574–591 (1993)
G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl. 58, 249–273 (1979)
F. Conrad and B. Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback, Asymptotic Anal. 7, 159–177 (1993)
C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29, 241–271 (1968)
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Functional and Variational Methods, Springer-Verlag, Berlin, 1992
P. Grisvard, Singularities in Boundary Value Problems, Masson, Paris, 1992
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985
P. Grisvard, Contrôlabilité exacte des solutions de l’équation des ondes en présence de singularités, J. Math. Pures Appl. 68, 215–259 (1989)
S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl. 167, 429–442 (1992)
M. A. Horn, Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity, J. Math. Anal. Appl. 223, 126–150 (1998)
M. A. Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control 8 (2), 217–219 (1998)
J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23, 889–899 (1992)
V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, John Wiley and Sons, Masson, Paris, 1994
V. Komornik and B. Rao, Boundary stabilization of compactly coupled wave equations, Asymptotic Anal. 14, 339–359 (1997)
V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. 69, 33–54 (1990)
A. D. Kovalenko, Thermoelasticity, Basic Theory and Applications, Wolters-Noordhoff Publishing, Gröningen, Netherlands, 1969
J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50, 163–182 (1983)
J. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control Optim. 21, 968–984 (1983)
J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, vol. 10, SIAM Publications, Philadelphia, 1989
J. Lagnese, Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary, Nonlinear Anal. 16, 35–54 (1991)
J. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, J. Differential Equations 91, 355–388 (1991)
J. Lagnese and J. L. Lions, Modelling Analysis and Control of Thin Plates, Masson, Paris, 1989
I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations 79, 340–381 (1989)
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6, 507–533 (1993)
G. Lebeau and E. Zuazua, Sur la décroissance non uniforme de l’énérgie dans le système de la thermoélasticité linéaire, C. R. Acad. Sci. Paris Sér. I. Math. 324, 409–415 (1997)
G. Leugering, On boundary feedback stabilization of a viscoelastic membrane, Dynam. Stability Systems 4, 71–79 (1989)
G. Leugering, On boundary feedback stabilization of a viscoelastic beam, Proc. Roy. Soc. Edinburgh, Sect. A 114, 57–69 (1990)
J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vols. I and II, Springer-Verlag, New York, 1972
W. J. Liu, Partial exact controllability and exponential stability of the higher-dimensional linear thermoelasticity, ESAIM Contrôle Optim. Calc. Var. 3, 23–48 (1998)
W. J. Liu, Exact Controllability for Some Partial Differential Equations of Evolutional Type, Ph.D. Thesis, University of Wollongong, Australia, 1997
W. J. Liu, The exponential stabilization of the higher-dimensional linear thermoviscoelasticity, J. Math. Pures Appl. 77, 355–386 (1998)
Z. Liu and S. Zheng, Exponential stability of the semigroup associated with a thermoelastic system, Quart. Appl. Math. LI, 535–545 (1993)
W. J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche de Matematica 48, 61–75 (1999)
G. P. Menzala and E. Zuazua, Explicit exponential decay rates for solutions of von Kármán’s system of thermoelastic plates, C. R. Acad. Sci Paris Sér. I Math. 324, 49–54 (1997)
G. P. Menzala and E. Zuazua, Energy decay rates for the Von Kármán system of thermoelastic plates, Differential Integral Equations, to appear.
K. Narukawa, Boundary value control of thermoelastic systems, Hiroshima Math. J. 13, 227–272 (1983)
L. A. F. Oliveira, Exponential decay in thermoelasticity, Commun. Appl. Anal. 1, 113–118 (1997)
J. E. M. Rivera, Decomposition of the displacement vector field and decay rates in linear thermoelasticity, SIAM J. Math. Anal. 24, 390–406 (1993)
J. E. M. Rivera, Asymptotic behaviour in n-dimensional thermoelasticity, Appl. Math. Lett. 10, 47–53 (1997)
J. E. M. Rivera and M. L. Oliveira, Stability in inhomogeneous and anisotropic thermoelasticity, Boll. Un. Mat. Ital. A A (7) 11, 115–127 (1997)
W. Rudin, Real and Complex Analysis, second edition, McGraw-Hill, Inc., New York, 1974
H. K. Wang and G. Chen, Asymptotic behavior of solutions of the one-dimensional wave equation with a nonlinear boundary stabilizer, SIAM J. Control Optim. 27, 758–775 (1989)
E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim. (2) 28, 466–477 (1990)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
74F05,
35B35,
35Q72,
74M05
Retrieve articles in all journals
with MSC:
74F05,
35B35,
35Q72,
74M05
Additional Information
Article copyright:
© Copyright 2001
American Mathematical Society