Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-27T14:30:34.623Z Has data issue: false hasContentIssue false
  • English
  • Français

Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

Published online by Cambridge University Press:  12 May 2010

S. A. Avdonin ,
B. P. Belinskiy  and
L. Pandolfi
S. A. Avdonin*
Affiliation:
University of Alaska Fairbanks, Fairbanks, AK 99775-6660, USA
B. P. Belinskiy
Affiliation:
University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN 37403-2598, USA
L. Pandolfi
Affiliation:
Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
*
* Corresponding author. E-mail: ffsaa@uaf.edu
Get access

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.

Keywords

string equationring equationexact controllabilityRiesz basismoment problem
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 4 - 31
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avdonin, S. A. and Belinskiy, B. P., Controllability of a string under tension . Discrete and Continuous Dynamical Systems: A Supplement Volume, (2003), 57–67. Google Scholar
Avdonin, S. A. and Belinskiy, B. P., 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. Google Scholar
Avdonin, S. A. and Belinskiy, B. P., On controllability of a rotating string . J. Math. Anal. Appl., 321 (2006), 198–212.CrossRefGoogle Scholar
Avdonin, S. A., Belinskiy, B. P. and Ivanov, S. A., On controllability of an elastic ring . Appl. Math. Optim., 60 (2009), 71–103.CrossRefGoogle Scholar
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.
Avdonin, S. A. and Ivanov, S. A., Exponential Riesz bases of subspaces and divided differences . St. Petersburg Mathematical Journal, 13 (2001), 339–351.Google Scholar
Avdonin, S., Lenhart, S. and Protopopescu, V., Solving the dynamical inverse problem for the Schrödinger equation by the Boundary Control method . Inverse Problems, 18 (2002), 41–57.CrossRefGoogle Scholar
Avdonin, S. and Moran, W., Ingham type inequalities and Riesz bases of divided differences . Int. J. Appl. Math. Comput. Sci., 11 (2001), 101–118.Google Scholar
Avdonin, S. A. and Moran, W., Simultaneous control problems for systems of elastic strings and beams . Systems and Control Letters, 44 (2001), 147–155.CrossRefGoogle Scholar
Avdonin, S. A. and Tucsnak, M., On the simultaneously reachable set of two strings . ESAIM: Control, Optimization and Calculus of Variations, 6 (2001), 259–273. CrossRefGoogle Scholar
Barbu, V. and Iannelli, M., Approximate controllability of the heat equation with memory . Differential and Integral Equations, 13 (2000), 1393–1412.Google Scholar
Belinskiy, B. P., Dauer, J. P., Martin, C. F. and Shubov, M. A., On controllability of an elastic string with a viscous damping . Numerical Functional Anal. and Optimization, 19 (1998), 227–255.CrossRefGoogle Scholar
Belishev, M. I., Canonical model of a dynamical system with boundary control in inverse problem for the heat equation . St. Petersburg Math. Journal, 7, (1996), 869–890. Google Scholar
A. Erdélyi. Asymptotic Expansions. Dover Publications, Inc., 1956.
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.
J. P. Den Hartog. Mechanical Vibrations. McGraw-Hill Book Company, New York, 1956.
Hansen, S. and Zuazua, E., Exact controllability and stabilization of a vibrating string with an interior point mass . SIAM J. Control Optim., 33 (1995), 1357–1391.CrossRefGoogle Scholar
T. von Kàrmàn and M. A. Biot. Mathematical Methods in Engineering. McGraw-Hill Book Company, New York, 1940.
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.
Metrikine, A. V. and Tochilin, M. V., Steady-state vibrations of an elastic ring under moving load . J. Sound and Vibration, 232 (2000), 511–524.CrossRefGoogle Scholar
Pandolfi, L., The controllability of the Gurtin-Pipkin equation: a cosine operator approach . Applied Mathematics and Optimization, 52 (2005), 143–165.CrossRefGoogle Scholar
Pandolfi, L., Riesz system and the controllability of heat equations with memory . Integral Eq. Oper. Theory, 64 (2009), 429–453.CrossRefGoogle Scholar
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).
Russell, D. L., Nonharmonic Fourier series in the control theory of distributed parameter systems . J. Math. Anal. Appl., 18 (1967), 542–559.CrossRefGoogle Scholar
Russell, D. L., Controllability and stabilizability theory for linear partial differential equations . SIAM Review, 20 (1978), 639–739.CrossRefGoogle Scholar
Russell, D. L., On exponential bases for the Sobolev spaces over an interval . J. Math.Anal.Appl., 87 (1982), 528–550.CrossRefGoogle Scholar
W. Soedel. Vibrations of Shells and Plates. Marcel Dekker, Inc., New York, 1993.
M. E. Taylor. Pseudodifferential Operators. Princeton University Press, Princeton, NJ, 1981.
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).
Fu, X., Yong, J. and Zhang, X., Controllability and observability of the heat equation with hyperbolic memory kernel . J. Diff. Equations, 247 (2009), 2395–2439.CrossRefGoogle Scholar