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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 and 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), 1677-1694. doi: 10.1142/S0218202506001674. [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), 4-31. doi: 10.1051/mmnp/20105401. [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, New York, 1995. [4] S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equations with memory, Quart. Appl. Math., 71 (2013), 339-368. doi: 10.1090/S0033-569X-2012-01287-7. [5] S. Avdonin and L. Pandolfi, Temperature and heat flux dependence/independence for heat equations with memory, in "Time Delay Systems - Methods, Applications and New Trend" (Eds. R. Sipahi, T. Vyhlidal, P. Pepe and S.-I. Niculescu), Lecture Notes in Control and Inform. Sci., 423 Springer-Verlag, (2012), 87-101. doi: 10.1007/978-3-642-25221-1_7. [6] V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Diff. Integral Eq., 13 (2000), 1393-1412. [7] X. Fu, J. Yong and X. Zhang, Controllability and observability of the heat equation with hyperbolic memory kernel, J. Diff. Equations, 247 (2009), 2395-2439. [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, Paris, (1971). [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), 330-333. [10] S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288C-300. doi: 10.1051/cocv/2012013. [11] A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory, Systems & Control Lett., 61 (2012), 999-1002. doi: 10.1016/j.sysconle.2012.07.002. [12] S. Ivanov and L. Pandolfi, Heat equation with memory: Lack of controllability to the rest, J. Math. Anal. Appl., 355 (2009), 1-11. doi: 10.1016/j.jmaa.2009.01.008. [13] J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110. doi: 10.1137/0331008. [14] G. Leugering, Exact controllability in viscoelasticity of fading memory type, Applicable Anal., 18 (1984), 221-243. doi: 10.1080/00036818408839521. [15] G. Leugering, A decomposition method for integro-partial differential equations and applications, J. Math. Pures Appl. (9), 71 (1992), 561-587. [16] P. Loreti, L. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string, SIAM J. Control Optim., 50 (2012), 820-844. doi: 10.1137/110827740. [17] L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach, Appl. Math. Optim., 52 (2005), 143-165. (See a Correction in Appl. Math. Optim., 64 (2011), 467-468). doi: 10.1007/s00245-005-0819-0. [18] L. Pandolfi, Controllability of the Gurtin-Pipkin equation,, SISSA, (). [19] L. Pandolfi, Riesz systems and the controllability of heat equations with memory, Int. Eq. Operator Theory, 64 (2009), 429-453. doi: 10.1007/s00020-009-1682-1. [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), 1487-1510. doi: 10.3934/dcdsb.2010.14.1487. [21] L. Pandolfi, Riesz systems and an identification problem for heat equations with memory, Discrete Continuous Dynam. Systems - S, 4 (2011), 745-759. doi: 10.3934/dcdss.2011.4.745. [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. [23] D. L. Russel, On exponential bases for the sobolev spaces over an interval, J. Math. Analysis Appl., 87 (1982), 528-550. doi: 10.1016/0022-247X(82)90142-1. [24] R. M. Young, "An Introduction to Nonharmonic Fourier Series," Academic Press, New York, 2001.

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##### 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), 1677-1694. doi: 10.1142/S0218202506001674. [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), 4-31. doi: 10.1051/mmnp/20105401. [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, New York, 1995. [4] S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equations with memory, Quart. Appl. Math., 71 (2013), 339-368. doi: 10.1090/S0033-569X-2012-01287-7. [5] S. Avdonin and L. Pandolfi, Temperature and heat flux dependence/independence for heat equations with memory, in "Time Delay Systems - Methods, Applications and New Trend" (Eds. R. Sipahi, T. Vyhlidal, P. Pepe and S.-I. Niculescu), Lecture Notes in Control and Inform. Sci., 423 Springer-Verlag, (2012), 87-101. doi: 10.1007/978-3-642-25221-1_7. [6] V. Barbu and M. Iannelli, Controllability of the heat equation with memory, Diff. Integral Eq., 13 (2000), 1393-1412. [7] X. Fu, J. Yong and X. Zhang, Controllability and observability of the heat equation with hyperbolic memory kernel, J. Diff. Equations, 247 (2009), 2395-2439. [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, Paris, (1971). [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), 330-333. [10] S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288C-300. doi: 10.1051/cocv/2012013. [11] A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory, Systems & Control Lett., 61 (2012), 999-1002. doi: 10.1016/j.sysconle.2012.07.002. [12] S. Ivanov and L. Pandolfi, Heat equation with memory: Lack of controllability to the rest, J. Math. Anal. Appl., 355 (2009), 1-11. doi: 10.1016/j.jmaa.2009.01.008. [13] J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110. doi: 10.1137/0331008. [14] G. Leugering, Exact controllability in viscoelasticity of fading memory type, Applicable Anal., 18 (1984), 221-243. doi: 10.1080/00036818408839521. [15] G. Leugering, A decomposition method for integro-partial differential equations and applications, J. Math. Pures Appl. (9), 71 (1992), 561-587. [16] P. Loreti, L. Pandolfi and D. Sforza, Boundary controllability and observability of a viscoelastic string, SIAM J. Control Optim., 50 (2012), 820-844. doi: 10.1137/110827740. [17] L. Pandolfi, The controllability of the Gurtin-Pipkin equation: A cosine operator approach, Appl. Math. Optim., 52 (2005), 143-165. (See a Correction in Appl. Math. Optim., 64 (2011), 467-468). doi: 10.1007/s00245-005-0819-0. [18] L. Pandolfi, Controllability of the Gurtin-Pipkin equation,, SISSA, (). [19] L. Pandolfi, Riesz systems and the controllability of heat equations with memory, Int. Eq. Operator Theory, 64 (2009), 429-453. doi: 10.1007/s00020-009-1682-1. [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), 1487-1510. doi: 10.3934/dcdsb.2010.14.1487. [21] L. Pandolfi, Riesz systems and an identification problem for heat equations with memory, Discrete Continuous Dynam. Systems - S, 4 (2011), 745-759. doi: 10.3934/dcdss.2011.4.745. [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. [23] D. L. Russel, On exponential bases for the sobolev spaces over an interval, J. Math. Analysis Appl., 87 (1982), 528-550. doi: 10.1016/0022-247X(82)90142-1. [24] R. M. Young, "An Introduction to Nonharmonic Fourier Series," Academic Press, New York, 2001.
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