# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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