An Inverse Quadratic Eigenvalue Problem for Damped Structural Systems

Abstract EN

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.