December  2013, 2(4): 711-721. doi: 10.3934/eect.2013.2.711

Trace properties of certain damped linear elastic systems

1. 

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-123, United States

Received  November 2012 Revised  September 2013 Published  November 2013

We study the spectrum of a damped linear elastic system with discrete eigenvalues, showing the relationship between the sum of the real parts of the eigenvalues of the (generally unbounded) generator and the trace of the damping operator, assuming the latter to be a trace type operator. Some relationships between the sequence of eigenvectors and a corresponding orthonormal sequence, constructed by means of a variant of the Gram-Schmidt method, are also explored. A simple hybrid system is presented as an example of application.
Citation: David L. Russell. Trace properties of certain damped linear elastic systems. Evolution Equations and Control Theory, 2013, 2 (4) : 711-721. doi: 10.3934/eect.2013.2.711
References:
[1]

N. Dunford and J. T. Schwartz, Linear Operators. Part II: Spectral Theory; Self Adjoint Operators in Hilbert Space, Pure & Applied Mathematics, Interscience Pub. Co., New York, 1963.

[2]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York 1966 xix+592 pp.

[3]

I. M. Gelfand and B. M. Levitan, On a simple identity for the characteristic values of a differential operator of the second order, (Russian), Doklady Akad. Nauk SSSR (N.S.), 88 (1953), 593-596.

[4]

W. Littman and L. W. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura Appl., 152 (1988), 281-330. doi: 10.1007/BF01766154.

[5]

F. G. Maksudov, M. Bairamogly and A. A. Adygezalov, The regularized trace of the Sturm-Liouville operator on a finite segment with an unbounded operator coefficient, (Russian), Dokl. Akad. Nauk SSSR, 277 (1984), 795-799.

show all references

References:
[1]

N. Dunford and J. T. Schwartz, Linear Operators. Part II: Spectral Theory; Self Adjoint Operators in Hilbert Space, Pure & Applied Mathematics, Interscience Pub. Co., New York, 1963.

[2]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York 1966 xix+592 pp.

[3]

I. M. Gelfand and B. M. Levitan, On a simple identity for the characteristic values of a differential operator of the second order, (Russian), Doklady Akad. Nauk SSSR (N.S.), 88 (1953), 593-596.

[4]

W. Littman and L. W. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura Appl., 152 (1988), 281-330. doi: 10.1007/BF01766154.

[5]

F. G. Maksudov, M. Bairamogly and A. A. Adygezalov, The regularized trace of the Sturm-Liouville operator on a finite segment with an unbounded operator coefficient, (Russian), Dokl. Akad. Nauk SSSR, 277 (1984), 795-799.

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