Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Energy-like Liapunov functionals for linear elastic systems on a Hilbert space


Author: J. A. Walker
Journal: Quart. Appl. Math. 30 (1973), 465-480
MSC: Primary 34G05; Secondary 34D20
DOI: https://doi.org/10.1090/qam/508715
MathSciNet review: 508715
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Abstract: An approach is presented for generating energy-like functionals for linear elastic dynamic systems on a Hilbert space. The objective is to obtain a family of functionals which may be used for stability analysis of the equilibrium; i.e., Liapunov functionals. Although the energy functional, when one exists, is always a member of this family, the family is shown to exist even when an energy functional does not. Several discrete and distributed-parameter examples are presented, as are certain specific techniques for utilizing this approach.


References [Enhancements On Off] (What's this?)

  • [1] Wolfgang Hahn, Theory and application of Liapunov’s direct method, English edition prepared by Siegfried H. Lehnigk; translation by Hans H. Losenthien and Siegfried H. Lehnigk, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0147716
  • [2] G. Hermann, Stability of equilibrium of elastic systems subjected to nonconservative forces, Appl. Mech. Rev. 20, 103-108 (1967)
  • [3] S. Nemat-Nasser and G. Hermann, Some general considerations concerning the destabilizing effect in nonconservative systems, ZAMP 17, 305-313 (1966)
  • [4] C. Urbano, The effect of damping on the dynamic stability of nonconservative systems, Meccanica 3, 131-139 (1968)
  • [5] V. I. Zubov, Methods of A. M. Lyapunov and their application, Translation prepared under the auspices of the United States Atomic Energy Commission; edited by Leo F. Boron, P. Noordhoff Ltd, Groningen, 1964. MR 0179428
  • [6] J. A. Walker, On the stability of linear discrete dynamic systems, Trans. ASME Ser. E. J. Appl. Mech. 37 (1970), 271–275. MR 0272449
  • [7] J. K. Hale and E. F. Infante, Extended dynamical systems and stability theory, Proc. Nat. Acad. Sci. U.S.A. 58 (1967), 405–409. MR 0218685
  • [8] G. R. Buis, W. G. Vogt and M. M. Eisen, Lyapunov stability for partial differential equations, NASA Contractor Report 1100, June 1968
  • [9] J. P. LaSalle, Stability theory for ordinary differential equations, J. Differential Equations 4 (1968), 57–65. MR 0222402, https://doi.org/10.1016/0022-0396(68)90048-X
  • [10] Jack K. Hale, Dynamical systems and stability, J. Math. Anal. Appl. 26 (1969), 39–59. MR 0244582, https://doi.org/10.1016/0022-247X(69)90175-9
  • [11] Marshall Slemrod, Asymptotic behavior of a class of abstract dynamical systems, J. Differential Equations 7 (1970), 584–600. MR 0259291, https://doi.org/10.1016/0022-0396(70)90103-8
  • [12] A. A. Movchan, The direct method of Liapunov in stability of elastic systems, Prik. Mat. Mech. 23, 483-394 (1959)
  • [13] R. H. Plaut and E. F. Infante, On the stability of some continuous systems subjected to random excitation, J. Appl. Mech. 37, 623-628 (1970)

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Additional Information

DOI: https://doi.org/10.1090/qam/508715
Article copyright: © Copyright 1973 American Mathematical Society


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