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September  2013, 2(3): 471-493. doi: 10.3934/eect.2013.2.471

Traction, deformation and velocity of deformation in a viscoelastic string

1. 

Politecnico di Torino, di Scienze Matematiche "Giuseppe Luigi Lagrange", Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received  September 2012 Revised  March 2013 Published  July 2013

In this paper we consider a viscoelastic string whose deformation is controlled at one end. We study the relations and the controllability of the couples traction/velocity and traction/deformation and we show that the first couple behaves very like as in the purely elastic case, while new phenomena appears when studying the couple of the traction and the deformation. Namely, while traction and velocity are independent (for large time), traction and deformation are related at each time but the relation is not so strict. In fact we prove that an arbitrary number of ``Fourier'' components of the traction and, independently, of the deformation can be assigned at any time.
Citation: Luciano Pandolfi. Traction, deformation and velocity of deformation in a viscoelastic string. Evolution Equations & Control Theory, 2013, 2 (3) : 471-493. doi: 10.3934/eect.2013.2.471
References:
[1]

J. A. D. Appleby, M. Fabrizio, B. Lazzari and D. W. Reynolds, On exponential asymptotic stability in linear viscoelasticity,, Math. Models Meth. Appl. Sci., 16 (2006), 1677. doi: 10.1142/S0218202506001674. Google Scholar

[2]

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

[3]

S. A. Avdonin and S. A. Ivanov, "Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems,", Cambridge University Press, (1995). Google Scholar

[4]

S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equations with memory,, Quart. Appl. Math., 71 (2013), 339. doi: 10.1090/S0033-569X-2012-01287-7. Google Scholar

[5]

S. Avdonin and L. Pandolfi, Temperature and heat flux dependence/independence for heat equations with memory,, in, 423 (2012), 87. doi: 10.1007/978-3-642-25221-1_7. Google Scholar

[6]

V. Barbu and M. Iannelli, Controllability of the heat equation with memory,, Diff. Integral Eq., 13 (2000), 1393. Google Scholar

[7]

X. Fu, J. Yong and X. Zhang, Controllability and observability of the heat equation with hyperbolic memory kernel,, J. Diff. Equations, 247 (2009), 2395. Google Scholar

[8]

I. C. Gohberg and M. G. Krein, Introduction à la théorie des opérateurs linéaires non auto-adjoints dans un espace hilbertien, (French) [Linear Non Selfadjoint Operators in a Hilbert Space],, Dunod, (1971). Google Scholar

[9]

G. M. Gubreev and M. G. Volkova, One remark about the unconditional exponential bases and cosine bases, connected with them,, Methods Funct. Anal. Topology, 14 (2008), 330. Google Scholar

[10]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory,, ESAIM Control Optim. Calc. Var., 19 (2013). doi: 10.1051/cocv/2012013. Google Scholar

[11]

A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory,, Systems & Control Lett., 61 (2012), 999. doi: 10.1016/j.sysconle.2012.07.002. Google Scholar

[12]

S. Ivanov and L. Pandolfi, Heat equation with memory: Lack of controllability to the rest,, J. Math. Anal. Appl., 355 (2009), 1. doi: 10.1016/j.jmaa.2009.01.008. Google Scholar

[13]

J. U. Kim, Control of a second-order integro-differential equation,, SIAM J. Control Optim., 31 (1993), 101. doi: 10.1137/0331008. Google Scholar

[14]

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

[15]

G. Leugering, A decomposition method for integro-partial differential equations and applications,, J. Math. Pures Appl. (9), 71 (1992), 561. Google Scholar

[16]

P. Loreti, L. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string,, SIAM J. Control Optim., 50 (2012), 820. doi: 10.1137/110827740. Google Scholar

[17]

L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach,, Appl. Math. Optim., 52 (2005), 143. doi: 10.1007/s00245-005-0819-0. Google Scholar

[18]

L. Pandolfi, Controllability of the Gurtin-Pipkin equation,, SISSA, (). Google Scholar

[19]

L. Pandolfi, Riesz systems and the controllability of heat equations with memory,, Int. Eq. Operator Theory, 64 (2009), 429. doi: 10.1007/s00020-009-1682-1. Google Scholar

[20]

L. Pandolfi, Riesz systems and moment method in the study of heat equations with memory in one space dimension,, Discrete Continuous Dynam. Systems - B, 14 (2010), 1487. doi: 10.3934/dcdsb.2010.14.1487. Google Scholar

[21]

L. Pandolfi, Riesz systems and an identification problem for heat equations with memory,, Discrete Continuous Dynam. Systems - S, 4 (2011), 745. doi: 10.3934/dcdss.2011.4.745. Google Scholar

[22]

L. Pandolfi, Boundary controllability and source reconstruction in a viscoelastic string under external traction,, J. Math. Analysis Appl., (2013). doi: 10.1016/j.jmaa.2013.05.051. Google Scholar

[23]

D. L. Russel, On exponential bases for the sobolev spaces over an interval,, J. Math. Analysis Appl., 87 (1982), 528. doi: 10.1016/0022-247X(82)90142-1. Google Scholar

[24]

R. M. Young, "An Introduction to Nonharmonic Fourier Series,", Academic Press, (2001). Google Scholar

show all references

References:
[1]

J. A. D. Appleby, M. Fabrizio, B. Lazzari and D. W. Reynolds, On exponential asymptotic stability in linear viscoelasticity,, Math. Models Meth. Appl. Sci., 16 (2006), 1677. doi: 10.1142/S0218202506001674. Google Scholar

[2]

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

[3]

S. A. Avdonin and S. A. Ivanov, "Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems,", Cambridge University Press, (1995). Google Scholar

[4]

S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equations with memory,, Quart. Appl. Math., 71 (2013), 339. doi: 10.1090/S0033-569X-2012-01287-7. Google Scholar

[5]

S. Avdonin and L. Pandolfi, Temperature and heat flux dependence/independence for heat equations with memory,, in, 423 (2012), 87. doi: 10.1007/978-3-642-25221-1_7. Google Scholar

[6]

V. Barbu and M. Iannelli, Controllability of the heat equation with memory,, Diff. Integral Eq., 13 (2000), 1393. Google Scholar

[7]

X. Fu, J. Yong and X. Zhang, Controllability and observability of the heat equation with hyperbolic memory kernel,, J. Diff. Equations, 247 (2009), 2395. Google Scholar

[8]

I. C. Gohberg and M. G. Krein, Introduction à la théorie des opérateurs linéaires non auto-adjoints dans un espace hilbertien, (French) [Linear Non Selfadjoint Operators in a Hilbert Space],, Dunod, (1971). Google Scholar

[9]

G. M. Gubreev and M. G. Volkova, One remark about the unconditional exponential bases and cosine bases, connected with them,, Methods Funct. Anal. Topology, 14 (2008), 330. Google Scholar

[10]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory,, ESAIM Control Optim. Calc. Var., 19 (2013). doi: 10.1051/cocv/2012013. Google Scholar

[11]

A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory,, Systems & Control Lett., 61 (2012), 999. doi: 10.1016/j.sysconle.2012.07.002. Google Scholar

[12]

S. Ivanov and L. Pandolfi, Heat equation with memory: Lack of controllability to the rest,, J. Math. Anal. Appl., 355 (2009), 1. doi: 10.1016/j.jmaa.2009.01.008. Google Scholar

[13]

J. U. Kim, Control of a second-order integro-differential equation,, SIAM J. Control Optim., 31 (1993), 101. doi: 10.1137/0331008. Google Scholar

[14]

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

[15]

G. Leugering, A decomposition method for integro-partial differential equations and applications,, J. Math. Pures Appl. (9), 71 (1992), 561. Google Scholar

[16]

P. Loreti, L. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string,, SIAM J. Control Optim., 50 (2012), 820. doi: 10.1137/110827740. Google Scholar

[17]

L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach,, Appl. Math. Optim., 52 (2005), 143. doi: 10.1007/s00245-005-0819-0. Google Scholar

[18]

L. Pandolfi, Controllability of the Gurtin-Pipkin equation,, SISSA, (). Google Scholar

[19]

L. Pandolfi, Riesz systems and the controllability of heat equations with memory,, Int. Eq. Operator Theory, 64 (2009), 429. doi: 10.1007/s00020-009-1682-1. Google Scholar

[20]

L. Pandolfi, Riesz systems and moment method in the study of heat equations with memory in one space dimension,, Discrete Continuous Dynam. Systems - B, 14 (2010), 1487. doi: 10.3934/dcdsb.2010.14.1487. Google Scholar

[21]

L. Pandolfi, Riesz systems and an identification problem for heat equations with memory,, Discrete Continuous Dynam. Systems - S, 4 (2011), 745. doi: 10.3934/dcdss.2011.4.745. Google Scholar

[22]

L. Pandolfi, Boundary controllability and source reconstruction in a viscoelastic string under external traction,, J. Math. Analysis Appl., (2013). doi: 10.1016/j.jmaa.2013.05.051. Google Scholar

[23]

D. L. Russel, On exponential bases for the sobolev spaces over an interval,, J. Math. Analysis Appl., 87 (1982), 528. doi: 10.1016/0022-247X(82)90142-1. Google Scholar

[24]

R. M. Young, "An Introduction to Nonharmonic Fourier Series,", Academic Press, (2001). Google Scholar

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