Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 730358, 9 pages
Research Article

An Inverse Quadratic Eigenvalue Problem for Damped Structural Systems

Yongxin Yuan

Department of Mathematics, Jiangsu University of Science and Technology, Zhenjiang 212003, China

Received 23 October 2007; Accepted 14 February 2008

Academic Editor: Angelo Luongo

Copyright © 2008 Yongxin Yuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We first give the representation of the general solution of the following inverse quadratic eigenvalue problem (IQEP): given Λ=diag{λ1,,λp}Cp×p , X=[x1,,xp]Cn×p, and both Λ and X are closed under complex conjugation in the sense that λ2j=λ¯2j1C, x2j=x¯2j1Cn for j=1,,l, and λkR, xkRn for k=2l+1,, p, find real-valued symmetric (2r+1)-diagonal matrices M, D and K such that MXΛ2+DXΛ+KX=0. We then consider an optimal approximation problem: given real-valued symmetric (2r+1)-diagonal matrices Ma,Da,KaRn×n, find (M^,D^,K^)SE such that M^Ma2+D^Da2+K^Ka2=inf(M,D,K)SE(MMa2+DDa2+KKa2), where SE is the solution set of IQEP. We show that the optimal approximation solution (M^,D^,K^) is unique and derive an explicit formula for it.