Mathematical Problems in Engineering 
Volume 2008 (2008), Article ID 730358, 9 pages
doi:10.1155/2008/730358
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

Recommended by Angelo Luongo

Abstract

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¯2j−1∈C, x2j=x¯2j−1∈Cn for j=1,…,l, and λk∈R, xk∈Rn 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,Ka∈Rn×n, find (M^,D^,K^)∈SE such that ‖M^−Ma‖2+‖D^−Da‖2+‖K^−Ka‖2=inf⁡(M,D,K)∈SE(‖M−Ma‖2+‖D−Da‖2+‖K−Ka‖2), 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.