# Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

S. A. Avdonin; B. P. Belinskiy; L. Pandolfi

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 5, Issue: 4, page 4-31
- ISSN: 0973-5348

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topAvdonin, S. A., Belinskiy, B. P., and Pandolfi, L.. "Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension." Mathematical Modelling of Natural Phenomena 5.4 (2010): 4-31. <http://eudml.org/doc/197684>.

@article{Avdonin2010,

abstract = {We study controllability for a nonhomogeneous string and ring under an axial stretching
tension that varies with time. We consider the boundary control for a string and
distributed control for a ring. For a string, we are looking for a control
f(t) ∈ L2(0,
T) that drives the state solution to rest. We show that for a ring, two forces
are required to achieve controllability. The controllability problem is reduced to a
moment problem for the control. We describe the set of initial data which may be driven to
rest by the control. The proof is based on an auxiliary basis property result.},

author = {Avdonin, S. A., Belinskiy, B. P., Pandolfi, L.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {string equation; ring equation; exact controllability; Riesz basis; moment problem},

language = {eng},

month = {5},

number = {4},

pages = {4-31},

publisher = {EDP Sciences},

title = {Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension},

url = {http://eudml.org/doc/197684},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Avdonin, S. A.

AU - Belinskiy, B. P.

AU - Pandolfi, L.

TI - Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/5//

PB - EDP Sciences

VL - 5

IS - 4

SP - 4

EP - 31

AB - We study controllability for a nonhomogeneous string and ring under an axial stretching
tension that varies with time. We consider the boundary control for a string and
distributed control for a ring. For a string, we are looking for a control
f(t) ∈ L2(0,
T) that drives the state solution to rest. We show that for a ring, two forces
are required to achieve controllability. The controllability problem is reduced to a
moment problem for the control. We describe the set of initial data which may be driven to
rest by the control. The proof is based on an auxiliary basis property result.

LA - eng

KW - string equation; ring equation; exact controllability; Riesz basis; moment problem

UR - http://eudml.org/doc/197684

ER -

## References

top- S. A. Avdonin and B. P. Belinskiy, Controllability of a string under tension. Discrete and Continuous Dynamical Systems: A Supplement Volume, (2003), 57–67. Zbl1067.93033
- S. A. Avdonin and B. P. Belinskiy, On the basis properties of the functions arising in the boundary control problem of a string with a variable tension. Discrete and Continuous Dynamical Systems: A Supplement Volume, (2005), 40–49. Zbl1162.93006
- S. A. Avdonin and B. P. Belinskiy, On controllability of a rotating string. J. Math. Anal. Appl., 321 (2006), 198–212. Zbl1112.93008
- S. A. Avdonin, B. P. Belinskiy and S. A. Ivanov, On controllability of an elastic ring. Appl. Math. Optim., 60 (2009), 71–103. Zbl1211.93017
- 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. Zbl0866.93001
- S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences. St. Petersburg Mathematical Journal, 13 (2001), 339–351. Zbl0999.42018
- S. Avdonin, S. Lenhart and V. Protopopescu, Solving the dynamical inverse problem for the Schrödinger equation by the Boundary Control method. Inverse Problems, 18 (2002), 41–57. Zbl1001.35121
- S. Avdonin and W. Moran, Ingham type inequalities and Riesz bases of divided differences. Int. J. Appl. Math. Comput. Sci., 11 (2001), 101–118. Zbl1031.93098
- S. A. Avdonin and W. Moran, Simultaneous control problems for systems of elastic strings and beams. Systems and Control Letters, 44 (2001), 147–155. Zbl0986.93037
- S. A. Avdonin and M. Tucsnak, On the simultaneously reachable set of two strings. ESAIM: Control, Optimization and Calculus of Variations, 6 (2001), 259–273. Zbl0995.93036
- V. Barbu and M. Iannelli, Approximate controllability of the heat equation with memory. Differential and Integral Equations, 13 (2000), 1393–1412. Zbl0990.93008
- B. P. Belinskiy, J. P. Dauer, C. F. Martin and M. A. Shubov, On controllability of an elastic string with a viscous damping. Numerical Functional Anal. and Optimization, 19 (1998), 227–255. Zbl0910.35073
- M. I. Belishev, Canonical model of a dynamical system with boundary control in inverse problem for the heat equation. St. Petersburg Math. Journal, 7, (1996), 869–890.
- A. Erdélyi. Asymptotic Expansions. Dover Publications, Inc., 1956. Zbl0070.29002
- I. C. Gohberg and M. G. Krein. Introduction to the Theory of Linear Nonselfadjoint Operators", Translations of Mathematical Monographs. American Mathematical Society. 18, Providence, RI, 1969. Zbl0181.13504
- J. P. Den Hartog. Mechanical Vibrations. McGraw-Hill Book Company, New York, 1956.
- S. Hansen and E. Zuazua, Exact controllability and stabilization of a vibrating string with an interior point mass. SIAM J. Control Optim., 33 (1995), 1357–1391. Zbl0853.93018
- T. von Kàrmàn and M. A. Biot. Mathematical Methods in Engineering. McGraw-Hill Book Company, New York, 1940. Zbl66.0197.03
- O. A. Ladyzhenskaia. The Boundary Value Problems of Mathematical Physics. Springer-Verlag, New York, 1985.
- B. M. Levitan and I. S. Sargsjan. Sturm–Liouville and Dirac Operators. Translated from the Russian. Mathematics and its Applications (Soviet Series), 59. Kluwer Academic Publishers Group, Dordrecht, 1991.
- N. W. McLachlan. Theory and Applications of Mathieu Functions, Oxford, 1947. Zbl0029.02901
- A. V. Metrikine and M. V. Tochilin, Steady-state vibrations of an elastic ring under moving load. J. Sound and Vibration, 232 (2000), 511–524.
- L. Pandolfi, The controllability of the Gurtin-Pipkin equation: a cosine operator approach. Applied Mathematics and Optimization, 52 (2005), 143–165. Zbl1105.74025
- L. Pandolfi, Riesz system and the controllability of heat equations with memory. Integral Eq. Oper. Theory, 64 (2009), 429–453. Zbl1187.93011
- L. Pandolfi, Riesz systems, spectral controllability and an identification problem for heat equations with memory . Quaderni del Dipartimento di Matematica, Politecnico di Torino, “La Matematica e le sue Applicazioni”n. 6-2009 (in print, Discr. Cont. Dynam. Systems).
- L. Pandolfi, Riesz systems and moment method in the study of viscoelasticity in one space dimension. Quaderni del Dipartimento di Matematica, Politecnico di Torino, “La Matematica e le sue Applicazioni”n. 5-2009 (in print, Discr. Cont. Dynam. Systems).
- D. L. Russell, Nonharmonic Fourier series in the control theory of distributed parameter systems. J. Math. Anal. Appl., 18 (1967), 542–559. Zbl0158.10201
- D. L. Russell, Controllability and stabilizability theory for linear partial differential equations. SIAM Review, 20 (1978), 639–739. Zbl0397.93001
- D. L. Russell, On exponential bases for the Sobolev spaces over an interval. J. Math.Anal.Appl., 87 (1982), 528–550. Zbl0524.46008
- W. Soedel. Vibrations of Shells and Plates. Marcel Dekker, Inc., New York, 1993.
- M. E. Taylor. Pseudodifferential Operators. Princeton University Press, Princeton, NJ, 1981. Zbl0453.47026
- S. Timoshenko. Thèorie des Vibrations. Libr. Polytecnique Ch Bèranger, Paris, 1947.
- V. Z. Vlasov. ObŽcaya Teoriya Obolocek i eë Prilođeniya v Tehnike (in Russian) [General Theory of Shells and Its Applications in Technology]. Gosudarstvennoe Izdatel’stvo Tehniko-Teoreticeskoi Literatury, Moscow-Leningrad (1949).
- 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. Zbl1187.35265

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