American Institute of Mathematical Sciences

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June  2014, 3(2): 231-245. doi: 10.3934/eect.2014.3.231

On controllability of a linear elastic beam with memory under longitudinal load

Sergei A. Avdonin 1, and  Boris P. Belinskiy 2,

1. 

University of Alaska Fairbanks, Fairbanks, AK 99775-6660, United States
2. 

University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN 37403-2598, United States

Received  January 2013 Revised  March 2014 Published  May 2014

This work is motivated by the control problem for a linear elastic beam under a longitudinal load when the material of the beam has memory. We reduce the problem of controllability to a nonstandard moment problem. The solution of the latter problem is based on the Riesz basis property for a family of functions quadratically close to the nonharmonic exponentials. This result requires the detailed analysis of an integro--differential equation, and is of interest in itself for Function Theory.
Keywords: nonharmonic exponentials, Control for distributed parameter systems, viscoelastic beam, Sobolev spaces., bases of functions, moment problem, memory.
Mathematics Subject Classification: Primary: 34H05, 93B05, 93C20; Secondary: 74Dxx, 35Q93, 46E35, 34B09, 35Pxx, 42A7.
Citation: Sergei A. Avdonin, Boris P. Belinskiy. On controllability of a linear elastic beam with memory under longitudinal load. Evolution Equations and Control Theory, 2014, 3 (2) : 231-245. doi: 10.3934/eect.2014.3.231
References:
[1]

F. Ammar-Khodja, A. Benabdallah, J. E. Munoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.

[2]

S. A. Avdonin and B. P. Belinskiy, Controllability of a string under tension, Discrete Contin. Dyn. Syst., (2003), 57-67.

[3]

S. A. Avdonin and B. P. Belinskyi, On the basis properties of the functions arising in the boundary control problem of a string with a variable tension, Discrete Contin. Dyn. Syst., (2005), 40-49.

[4]

S. A. Avdonin, B. P. Belinskiy and S. A. Ivanov, On controllability of an elastic ring, Appl. Math. Optim., 60 (2009), 71-103. doi: 10.1007/s00245-009-9064-2.

[5]

S. A. Avdonin, B. P. Belinskiy and L. Pandolfi, Controllability of a non-homogeneous string and ring under time dependent tension, Math. Model. Nat. Phenom., 5 (2010), 4-31. doi: 10.1051/mmnp/20105401.

[6]

S. A. Avdonin and B. P. Belinskiy, On controllability of a non-homogeneous elastic string with memory, J. of Math Analysis and Applications, 398 (2013), 254-269. doi: 10.1016/j.jmaa.2012.08.037.

[7]

S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, 1995.

[8]

S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences, St. Petersburg Mathematical Journal, 13 (2002), 339-351.

[9]

S. Avdonin and W. Moran, Ingham type inequalities and Riesz bases of divided differences, International J. of Applied Math. and Computer Science, 11 (2001), 803-820.

[10]

S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equations with memory, Quarterly of Applied Mathematics, 71 (2013), 339-368. doi: 10.1090/S0033-569X-2012-01287-7.

[11]

S. A. Avdonin and L. Pandolfi, Temperature and heat flux dependence/independence for heat equations with memory, in Time Delay Systems: Methods, Applications and New Trends, Lecture Notes in Control and Information Sciences, 423, Springer, Berlin, 2011, 87-101. doi: 10.1007/978-3-642-25221-1_7.

[12]

N. K. Bari, Biorthogonal systems and bases in Hilbert space, (in Russian) Moskov. Gos. Univ. Učen. Zap., Matematika, 148 (1951), 69-107.

[13]

S. Breuer, On energy stored in linear viscoelastic solids, ZAMM-J. Appl. Math. and Mechanics, 55 (1975), 403-405. doi: 10.1002/zamm.19750550708.

[14]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Archive for Rational Mechanics and Anal., 37 (1970), 297-308.

[15]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220. doi: 10.1137/0315015.

[16]

A. D. Drozdov and V. B. Kolmanovskii, Stability in Viscoelasticity, North-Holland Series in Applied Mathematics and Mechanics, 38, North-Holland Publishing Co., Amsterdam, 1994.

[17]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264. doi: 10.1080/0003681021000035588.

[18]

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translation of Mathematical Monographs, 18, American Mathematical Society, Providence, RI, 1969.

[19]

J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, International Series of Numerical Mathematics, 91 (1989), 211-236.

[20]

G. Leugering, On boundary feedback stabilisability of a viscoelastic beam, Proc. of the Royal Soc. of Edinburgh, 114 (1990), 57-69. doi: 10.1017/S0308210500024264.

[21]

G. Leugering, Exact controllability in viscoelasticity of fading memory type, Applicable Anal., 18 (1984), 221-243. doi: 10.1080/00036818408839521.

[22]

G. Leugering, Boundary controllability of a viscoelastic beam, Applicable Anal., 23 (1986), 119-137. doi: 10.1080/00036818608839635.

[23]

J. L. Lions and E. Magenes, Problèmes aux Limites Nonhomogénes et Applications, 1 & 2, Dunod, Paris, 1968.

[24]

W. J. Liuand and G. H. Williams, Partial exact controllability for the linear thermo-elastic model, Electronic J. Differential Equations, (1998), 11 pp.

[25]

V. P. Madan, Response of a viscoelastic beam to an impulsive excitation, Mathematika, 16 (1969), 205-208. doi: 10.1112/S0025579300008172.

[26]

J. E. Munoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704. doi: 10.1016/S0022-247X(03)00511-0.

[27]

L. Pandolfi, Riesz system and the controllability of heat equations with memory, Integral Equations Operator Theory, 64 (2009), 429-453. doi: 10.1007/s00020-009-1682-1.

[28]

L. Pandolfi, Riesz systems and moment method in the study of viscoelasticity in one space dimension, Discrete Contin. Dyn. Syst., Ser. B, 14 (2010), 1487-1510. doi: 10.3934/dcdsb.2010.14.1487.

[29]

D. L. Russell, Nonharmonic Fourier series in the control theory of distributed parameter systems, J. Math. Anal. Appl., 18 (1967), 542-560. doi: 10.1016/0022-247X(67)90045-5.

[30]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.

[31]

D. L. Russell, On exponential bases for the Sobolev spaces over an interval, J. Math. Anal. Appl., 87 (1982), 528-550. doi: 10.1016/0022-247X(82)90142-1.

[32]

K. Seip, On the connection between exponential bases and certain related sequences in $L_2(\pi,\pi)$, J. Functional Anal., 130 (1995), 131-160. doi: 10.1006/jfan.1995.1066.

[33]

W. C. Xie, Dynamic Stability of Structures, Cambridge University Press, New York, 2006.

[34]

R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 2001.

show all references

References:
[1]

F. Ammar-Khodja, A. Benabdallah, J. E. Munoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.

[2]

S. A. Avdonin and B. P. Belinskiy, Controllability of a string under tension, Discrete Contin. Dyn. Syst., (2003), 57-67.

[3]

S. A. Avdonin and B. P. Belinskyi, On the basis properties of the functions arising in the boundary control problem of a string with a variable tension, Discrete Contin. Dyn. Syst., (2005), 40-49.

[4]

S. A. Avdonin, B. P. Belinskiy and S. A. Ivanov, On controllability of an elastic ring, Appl. Math. Optim., 60 (2009), 71-103. doi: 10.1007/s00245-009-9064-2.

[5]

S. A. Avdonin, B. P. Belinskiy and L. Pandolfi, Controllability of a non-homogeneous string and ring under time dependent tension, Math. Model. Nat. Phenom., 5 (2010), 4-31. doi: 10.1051/mmnp/20105401.

[6]

S. A. Avdonin and B. P. Belinskiy, On controllability of a non-homogeneous elastic string with memory, J. of Math Analysis and Applications, 398 (2013), 254-269. doi: 10.1016/j.jmaa.2012.08.037.

[7]

S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, 1995.

[8]

S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences, St. Petersburg Mathematical Journal, 13 (2002), 339-351.

[9]

S. Avdonin and W. Moran, Ingham type inequalities and Riesz bases of divided differences, International J. of Applied Math. and Computer Science, 11 (2001), 803-820.

[10]

S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equations with memory, Quarterly of Applied Mathematics, 71 (2013), 339-368. doi: 10.1090/S0033-569X-2012-01287-7.

[11]

S. A. Avdonin and L. Pandolfi, Temperature and heat flux dependence/independence for heat equations with memory, in Time Delay Systems: Methods, Applications and New Trends, Lecture Notes in Control and Information Sciences, 423, Springer, Berlin, 2011, 87-101. doi: 10.1007/978-3-642-25221-1_7.

[12]

N. K. Bari, Biorthogonal systems and bases in Hilbert space, (in Russian) Moskov. Gos. Univ. Učen. Zap., Matematika, 148 (1951), 69-107.

[13]

S. Breuer, On energy stored in linear viscoelastic solids, ZAMM-J. Appl. Math. and Mechanics, 55 (1975), 403-405. doi: 10.1002/zamm.19750550708.

[14]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Archive for Rational Mechanics and Anal., 37 (1970), 297-308.

[15]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220. doi: 10.1137/0315015.

[16]

A. D. Drozdov and V. B. Kolmanovskii, Stability in Viscoelasticity, North-Holland Series in Applied Mathematics and Mechanics, 38, North-Holland Publishing Co., Amsterdam, 1994.

[17]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264. doi: 10.1080/0003681021000035588.

[18]

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translation of Mathematical Monographs, 18, American Mathematical Society, Providence, RI, 1969.

[19]

J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, International Series of Numerical Mathematics, 91 (1989), 211-236.

[20]

G. Leugering, On boundary feedback stabilisability of a viscoelastic beam, Proc. of the Royal Soc. of Edinburgh, 114 (1990), 57-69. doi: 10.1017/S0308210500024264.

[21]

G. Leugering, Exact controllability in viscoelasticity of fading memory type, Applicable Anal., 18 (1984), 221-243. doi: 10.1080/00036818408839521.

[22]

G. Leugering, Boundary controllability of a viscoelastic beam, Applicable Anal., 23 (1986), 119-137. doi: 10.1080/00036818608839635.

[23]

J. L. Lions and E. Magenes, Problèmes aux Limites Nonhomogénes et Applications, 1 & 2, Dunod, Paris, 1968.

[24]

W. J. Liuand and G. H. Williams, Partial exact controllability for the linear thermo-elastic model, Electronic J. Differential Equations, (1998), 11 pp.

[25]

V. P. Madan, Response of a viscoelastic beam to an impulsive excitation, Mathematika, 16 (1969), 205-208. doi: 10.1112/S0025579300008172.

[26]

J. E. Munoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl., 286 (2003), 692-704. doi: 10.1016/S0022-247X(03)00511-0.

[27]

L. Pandolfi, Riesz system and the controllability of heat equations with memory, Integral Equations Operator Theory, 64 (2009), 429-453. doi: 10.1007/s00020-009-1682-1.

[28]

L. Pandolfi, Riesz systems and moment method in the study of viscoelasticity in one space dimension, Discrete Contin. Dyn. Syst., Ser. B, 14 (2010), 1487-1510. doi: 10.3934/dcdsb.2010.14.1487.

[29]

D. L. Russell, Nonharmonic Fourier series in the control theory of distributed parameter systems, J. Math. Anal. Appl., 18 (1967), 542-560. doi: 10.1016/0022-247X(67)90045-5.

[30]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.

[31]

D. L. Russell, On exponential bases for the Sobolev spaces over an interval, J. Math. Anal. Appl., 87 (1982), 528-550. doi: 10.1016/0022-247X(82)90142-1.

[32]

K. Seip, On the connection between exponential bases and certain related sequences in $L_2(\pi,\pi)$, J. Functional Anal., 130 (1995), 131-160. doi: 10.1006/jfan.1995.1066.

[33]

W. C. Xie, Dynamic Stability of Structures, Cambridge University Press, New York, 2006.

[34]

R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 2001.

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