Abstract
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.
Highlights
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 generator and the trace of the damping operator, assuming the latter to be a trace type operator
A0Yk, Yk + A0 Yk, Yk − Tr B, k=1 the last result following from a standard theorem on trace class operators
Outline of Proof Let us recall, for a finite, linearly independent set Xk, k = 1, 2, ..., K, one can obtain an orthonormal set with the same span via the sequential Gram - Schmidt process, which involves an ordering of the Xk as indicated, or we can do it all at once in a manner independent of any ordering
Summary
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 explored. Our objective is to show that this relationship continues to hold for many unbounded self-adjoint operators A with trace class B important in the study of distributed systems.
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