Structured Quadratic Inverse Eigenvalue Problem, I. Serially Linked Systems

Abstract

Abstract. Quadratic pencils arising from applications are often inherently structured. Factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters. For physical feasibility, structural constraints must be respected. Consequently, they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation. Knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications. However, the issue of solvability is problem dependent and has to be addressed structure by structure. This paper considers one particular structure where the elements of the physical system, if modeled as a mass-spring system, are serially linked. The discussion recasts both undamped or damped problems in a framework of inequality systems that can be adapted for numerical computation. Some open questions are described.

A smoothing Newton's method for the construction of a damped vibrating system from noisy test eigendata

“…It is obvious that there exists a strictly feasible solution for Problem (11). This means that we have a point y 0 ∈ R 3n ++ such that Ay 0 < δ n , i.e, the Slater condition [25] holds for Problem (11).…”

“…This means that we have a point y 0 ∈ R 3n ++ such that Ay 0 < δ n , i.e, the Slater condition [25] holds for Problem (11). Then Problem (11) is equivalent to the NCP: Finding y ∈ R 3n + and ξ ∈ R + such that…”

“…We remark that it is not easy to find a set of positive parameters {m i } n 1 , {c i } n 1 and {k i } n 1 such that the corresponding matrices M, C, K satisfy Equation (7), see for instance [2,6,11] for the exploration of the physical solvability.…”

“…In practical engineering applications, these structured matrices M o , C o , K o (respectively, called the estimated analytic mass, damping and stiffness matrices) are obtained by the finite element method for the practical structure, see for instance [11,23]. The associated estimated analytic model…”

“…He also consider the problem for the perturbed eigendata. Chu, Buono and Yu [11] converted the physical solvability of the SIQEP with the prescribed exact eigendata to the consistency of a certain system of inequalities from two real eigenpairs or a set of two complex eigenpairs which is closed under the complex conjugate. Abdalla, Grigoriadis and Zimmerman [1] considered the structural damage detection, or equivalently the reconstruction of the pencil…”