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September 2017, 12(3): 461-488. doi: 10.3934/nhm.2017020

Decay rates for elastic-thermoelastic star-shaped networks

1. 

School of Mathematics, Tianjin University, 300354 Tianjin, China

2. 

DeustoTech -Fundación Deusto, Avda. Universidades, 24, 48007 Bilbao, Basque Country, Spain

3. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

4. 

Facultad Ingeniería, Universidad de Deusto, Avda. Universidades, 24, 48007 Bilbao, Basque Country, Spain

Received  September 2016 Revised  July 2017 Published  September 2017

Fund Project: The first author was supported by the Natural Science Foundation of China grant NSFC-61573252 and China Scholarship Council. The second author was supported by the Advanced Grant DYCON of the European Research Council Executive Agency, ICON of the ANR-2016-ACHN-0014-01 (France), FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the MTM2014-52347 Grant of the MINECO and a Humboldt Research Award at the University of Erlangen-Nürnberg

This work discusses the asymptotic behaviour of a transmission problem on star-shaped networks of interconnected elastic and thermoelastic rods. Elastic rods are undamped, of conservative nature, while the thermoelastic ones are damped by thermal effects. We analyse the overall decay rate depending of the number of purely elastic components entering on the system and the irrationality properties of its lengths.

First, a sufficient and necessary condition for the strong stability of the thermoelastic-elastic network is given. Then, the uniform exponential decay rate is proved by frequency domain analysis techniques when only one purely elastic undamped rod is present. When the network involves more than one purely elastic undamped rod the lack of exponential decay is proved and nearly sharp polynomial decay rates are deduced under suitable irrationality conditions on the lengths of the rods, based on Diophantine approximation arguments. More general slow decay rates are also derived. Finally, we present some numerical simulations supporting the analytical results.

Citation: Zhong-Jie Han, Enrique Zuazua. Decay rates for elastic-thermoelastic star-shaped networks. Networks & Heterogeneous Media, 2017, 12 (3) : 461-488. doi: 10.3934/nhm.2017020
References:
[1]

M. AlvesJ. Muñoz RiveraM. Sepúlveda and O. V. Villagrán, The lack of exponential stability in certain transmission problems with localized Kelvin-Voigt dissipation, SIAM J. Appl. Math., 74 (2014), 345-365. doi: 10.1137/130923233.

[2]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410.

[3]

K. AmmariM. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical and Control Systems, 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7.

[4]

K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams, Applicable Analysis, 86 (2007), 1529-1548. doi: 10.1080/00036810701734113.

[5]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.

[6]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0.

[7]

J. W. S. Cassals, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge, UK, 1957.

[8]

R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), 621-626. doi: 10.1016/S0764-4442(01)01876-6.

[9]

R. Dáger and E. Zuazua, Controllability of tree-shaped networks of vibrating strings, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), 1087-1092. doi: 10.1016/S0764-4442(01)01942-5.

[10]

R. Dáger and E. Zuazua, Spectral boundary controllability of networks of strings, C. R. Math. Acad. Sci. Paris, 334 (2002), 545-550. doi: 10.1016/S1631-073X(02)02314-2.

[11]

R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Mathématiques et Applications, 50, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[12]

H. D. Fernández-SareJ. E. Muñoz-Rivera and R. Racke, Stability for a transmission problem in thermoelasticity with second sound, Journal of Thermal Stresses, 31 (2008), 1170-1189.

[13]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1.

[14]

Z. J. Han and G. Q. Xu, Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping, Math. Meth. Appl. Sci., 38 (2015), 94-112. doi: 10.1002/mma.3052.

[15]

Z. J. Han and G. Q. Xu, Spectrum and stability analysis for a transmission problem in thermoelasticity with a concentrated mass, Z. Angew. Math. Phys., 66 (2015), 1717-1736. doi: 10.1007/s00033-015-0504-3.

[16]

Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks, Networks and Heterogeneous Media, 5 (2010), 315-334. doi: 10.3934/nhm.2010.5.315.

[17]

Z. J. Han and G. Q. Xu, Output feedback stabilisation of a tree-shaped network of vibrating strings with non-collocated observation, International Journal of Control, 84 (2011), 458-475. doi: 10.1080/00207179.2011.561441.

[18]

D. B. HenryO. Lopes and A. Perissinitto Jr., On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Analysis: Theory, Methods & Applications, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U.

[19]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eqs., 1 (1985), 43-56.

[20]

S. JaffardM. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation, Journal of Differential Equations, 145 (1998), 184-215. doi: 10.1006/jdeq.1997.3385.

[21]

J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems & Control: Foundations & Applications, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8.

[22]

G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity, Archive Rat. Mech. Anal., 141 (1998), 297-329. doi: 10.1007/s002050050078.

[23]

G. Lebeau and E. Zuazua, Decay rates for the linear system of three-dimensional system of thermoelasticity, Archive Rat. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160.

[24]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-008-6122-6.

[25]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, CRC Research Notes in Mathematics, vol. 398, Chapman and Hall/CRC, Boca Raton, 1999.

[26]

Z. Liu and R. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.

[27]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.

[28]

A. MarzocchiJ. E. Munoz Rivera and M. G. Naso, Asymptotic behaviour and expinential stability for a transmission problem in thermoelasticity, Math. Meth. Appl. Sci., 25 (2002), 955-980. doi: 10.1002/mma.323.

[29]

A. MarzocchiJ. E. Muñoz Rivera and M. G. Naso, Transmission problem in thermoelasticity with symmetry, IMA J. Appl. Math., 68 (2003), 23-46. doi: 10.1093/imamat/68.1.23.

[30]

S. A. Messaoudi and B. Said-Houari, Energy decay in a transmission problem in thermoelasticity of type Ⅲ, IMA J. Appl. Math., 74 (2009), 344-360. doi: 10.1093/imamat/hxp020.

[31]

J. E. Muñoz Rivera and H. Portillo Oquendo, The transmission problem for thermoelastic beams, Journal of Thermal Stresses, 24 (2001), 1137-1158. doi: 10.1080/014957301753251665.

[32]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425.

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[34]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.

[35]

R. Racke, Thermoelasticity with second sound-Exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298.

[36]

W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math., 125 (1970), 189-201. doi: 10.1007/BF02392334.

[37]

F. Shel, Exponential stability of a network of elastic and thermoelastic materials, Math. Meth. Appl. Sci., 36 (2013), 869-879. doi: 10.1002/mma.2644.

[38]

L. N. Trefethen, Spectral Methods in Matlab, PA: SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598.

[39]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks, SIAM J. Contr. Optim, 48 (2009), 2771-2797. doi: 10.1137/080733590.

[40]

G. Q. XuD. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled networks of strings, SIAM J. Control Optim., 47 (2008), 1762-1784. doi: 10.1137/060649367.

[41]

X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Differ. Equations, 204 (2004), 380-438. doi: 10.1016/j.jde.2004.02.004.

[42]

E. Zuazua, Controllability of the linear system of thermoelasticity, Journal de Mathématiques Pures et Appliquées, 74 (1995), 291-315.

show all references

References:
[1]

M. AlvesJ. Muñoz RiveraM. Sepúlveda and O. V. Villagrán, The lack of exponential stability in certain transmission problems with localized Kelvin-Voigt dissipation, SIAM J. Appl. Math., 74 (2014), 345-365. doi: 10.1137/130923233.

[2]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410.

[3]

K. AmmariM. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical and Control Systems, 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7.

[4]

K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams, Applicable Analysis, 86 (2007), 1529-1548. doi: 10.1080/00036810701734113.

[5]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.

[6]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0.

[7]

J. W. S. Cassals, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge, UK, 1957.

[8]

R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), 621-626. doi: 10.1016/S0764-4442(01)01876-6.

[9]

R. Dáger and E. Zuazua, Controllability of tree-shaped networks of vibrating strings, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), 1087-1092. doi: 10.1016/S0764-4442(01)01942-5.

[10]

R. Dáger and E. Zuazua, Spectral boundary controllability of networks of strings, C. R. Math. Acad. Sci. Paris, 334 (2002), 545-550. doi: 10.1016/S1631-073X(02)02314-2.

[11]

R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Mathématiques et Applications, 50, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[12]

H. D. Fernández-SareJ. E. Muñoz-Rivera and R. Racke, Stability for a transmission problem in thermoelasticity with second sound, Journal of Thermal Stresses, 31 (2008), 1170-1189.

[13]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1.

[14]

Z. J. Han and G. Q. Xu, Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping, Math. Meth. Appl. Sci., 38 (2015), 94-112. doi: 10.1002/mma.3052.

[15]

Z. J. Han and G. Q. Xu, Spectrum and stability analysis for a transmission problem in thermoelasticity with a concentrated mass, Z. Angew. Math. Phys., 66 (2015), 1717-1736. doi: 10.1007/s00033-015-0504-3.

[16]

Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks, Networks and Heterogeneous Media, 5 (2010), 315-334. doi: 10.3934/nhm.2010.5.315.

[17]

Z. J. Han and G. Q. Xu, Output feedback stabilisation of a tree-shaped network of vibrating strings with non-collocated observation, International Journal of Control, 84 (2011), 458-475. doi: 10.1080/00207179.2011.561441.

[18]

D. B. HenryO. Lopes and A. Perissinitto Jr., On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Analysis: Theory, Methods & Applications, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U.

[19]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eqs., 1 (1985), 43-56.

[20]

S. JaffardM. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation, Journal of Differential Equations, 145 (1998), 184-215. doi: 10.1006/jdeq.1997.3385.

[21]

J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems & Control: Foundations & Applications, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8.

[22]

G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity, Archive Rat. Mech. Anal., 141 (1998), 297-329. doi: 10.1007/s002050050078.

[23]

G. Lebeau and E. Zuazua, Decay rates for the linear system of three-dimensional system of thermoelasticity, Archive Rat. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160.

[24]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-008-6122-6.

[25]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, CRC Research Notes in Mathematics, vol. 398, Chapman and Hall/CRC, Boca Raton, 1999.

[26]

Z. Liu and R. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.

[27]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.

[28]

A. MarzocchiJ. E. Munoz Rivera and M. G. Naso, Asymptotic behaviour and expinential stability for a transmission problem in thermoelasticity, Math. Meth. Appl. Sci., 25 (2002), 955-980. doi: 10.1002/mma.323.

[29]

A. MarzocchiJ. E. Muñoz Rivera and M. G. Naso, Transmission problem in thermoelasticity with symmetry, IMA J. Appl. Math., 68 (2003), 23-46. doi: 10.1093/imamat/68.1.23.

[30]

S. A. Messaoudi and B. Said-Houari, Energy decay in a transmission problem in thermoelasticity of type Ⅲ, IMA J. Appl. Math., 74 (2009), 344-360. doi: 10.1093/imamat/hxp020.

[31]

J. E. Muñoz Rivera and H. Portillo Oquendo, The transmission problem for thermoelastic beams, Journal of Thermal Stresses, 24 (2001), 1137-1158. doi: 10.1080/014957301753251665.

[32]

S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425.

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[34]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.

[35]

R. Racke, Thermoelasticity with second sound-Exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298.

[36]

W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math., 125 (1970), 189-201. doi: 10.1007/BF02392334.

[37]

F. Shel, Exponential stability of a network of elastic and thermoelastic materials, Math. Meth. Appl. Sci., 36 (2013), 869-879. doi: 10.1002/mma.2644.

[38]

L. N. Trefethen, Spectral Methods in Matlab, PA: SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598.

[39]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks, SIAM J. Contr. Optim, 48 (2009), 2771-2797. doi: 10.1137/080733590.

[40]

G. Q. XuD. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled networks of strings, SIAM J. Control Optim., 47 (2008), 1762-1784. doi: 10.1137/060649367.

[41]

X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Differ. Equations, 204 (2004), 380-438. doi: 10.1016/j.jde.2004.02.004.

[42]

E. Zuazua, Controllability of the linear system of thermoelasticity, Journal de Mathématiques Pures et Appliquées, 74 (1995), 291-315.

Figure 1.  Transmission problem in 1-d elasticity-thermoelasticity
Figure 2.  Star-shaped thermoelastic-elastic network
Figure A-1.  $u_1(x,t)$
Figure A-2.  $u_2(x,t)$
Figure A-3.  $u_3(x,t)$
Figure A-4.  $\theta_1(x,t)$
Figure A-5.  $\theta_2(x,t)$
Figure A-6.  $\theta_3(x,t)$
Figure B-1.  $u_1(x,t)$
Figure B-2.  $u_2(x,t)$
Figure B-3.  $u_3(x,t)$
Figure C-1.  $u_1(x,t)$
Figure C-2.  $u_2(x,t)$
Figure C-3.  $u_3(x,t)$
Figure C-4.  $\theta_1(x,t)$
Figure C-5.  $\theta_2(x,t)$
Figure D-1.  $u_1(x,t)$
Figure D-2.  $u_2(x,t)$
Figure D-3.  $u_3(x,t)$
Figure D-4.  $\theta_1(x,t)$
Figure E-1.  $u_1(x,t)$
Figure E-2.  $u_2(x,t)$
Figure E-3.  $u_3(x,t)$
Figure E-4.  $\theta_1(x,t)$
Figure F-1.  Logarithmic scale of energy for Case A, B, C
Figure F-2.  Energy for Case D, E
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