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.
G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39, 433–454 (1982)
D. L. Russell, On the positive square root of the fourth derivative operator, Quart. Appl. Math. XLVI, 751–773 (1988)
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
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
A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10, 417–437 (1960). MR 22:5899
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
G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39, 433–454 (1982)
D. L. Russell, On the positive square root of the fourth derivative operator, Quart. Appl. Math. XLVI, 751–773 (1988)
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
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
A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10, 417–437 (1960). MR 22:5899
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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Article copyright:
© Copyright 1996
American Mathematical Society
