Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Structure for nonnegative square roots of unbounded nonnegative selfadjoint operators

Authors: Peng-Fei Yao and De-Xing Feng
Journal: Quart. Appl. Math. 54 (1996), 457-473
MSC: Primary 47B25; Secondary 34L10, 47A60, 47E05
DOI: https://doi.org/10.1090/qam/1402405
MathSciNet review: MR1402405
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Abstract: It is well known that, for an unbounded nonnegative selfadjoint operator $ A$ on a Hilbert space, there is a unique nonnegative square root $ {A^{1/2}}$, which is frequently associated with the structural damping in many practical vibration systems. In this paper we develop a general theory for the structure of $ {A^{1/2}}$, which includes the expression of $ {A^{1/2}}$ and a program to find the domain of $ {A^{1/2}}$ explicitly from the domain of $ A$. The relationship between $ {A^{1/2}}$ and related differential operators is determined for the selfadjoint differential operator $ A$. Finally, the theoretical results given in this paper are applied to fourth-order ``beam'' operators and $ n$-dimensional ``wave'' operators with sufficient complexity for applications to elastic vibration systems.

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

  • [1] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39, 433-454 (1982)
  • [2] D. L. Russell, On the positive square root of the fourth derivative operator, Quart. Appl. Math. XLVI, 751-773 (1988)
  • [3] A. V. Balakrishnan, Damping operators in continuum models of flexible structures: Explicit models for proportional damping in beam torsion, J. Differential Integral Equations 3, no. 2, 381-396 (1990). MR 91b:73063
  • [4] A. V. Balakrishnan, Damping operators in continuum models of flexible structure: Explicit models for proportional damping in beam bending with end-bodies, Appl. Math. Optim. 21, 315-334 (1990). MR 91i:73035
  • [5] A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10, 417-437 (1960). MR 22:5899
  • [6] A. G. Chassiakos and G. A. Berkey, Pointwise control of a flexible manipulator arm, Proceedings of IFAC Symposium on Robot Control, Barcelona, Spain, November 1985, pp. 181-185

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DOI: https://doi.org/10.1090/qam/1402405
Article copyright: © Copyright 1996 American Mathematical Society

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