Pole assignment for the vibrating rod

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Abstract
Relocating some poles of an axially vibrating rod by feedback leads to a certain integrodifferential eigenvalue problem. An explicit solution to the problem of determining the force needed to assign part of the spectrum while leaving the remaining spectrum unchanged is presented, and the conditions under which this solution is unique are determined. The results are then used to determine a certain self-adjoint control, analogous to symmetric rank-one-update in finite-dimensional systems, which solves the partial pole assignment problem with a control which satisfies the reciprocity law relating displacement and force. The results obtained may be used in practical engineering applications of reducing the forced vibration response of harmonically excited systems.

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