The inverse eigenvalue problem of constructing real and symmetric square matrices M , C, and K of size n × n for the quadratic pencil Q(λ) = λ 2 M + λC + K so that Q(λ) has a prescribed subset of eigenvalues and eigenvectors is considered. This paper consists of two parts addressing two related but different problems. The first part deals with the inverse problem where M and K are required to be positive definite and semidefinite, respectively. It is shown via construction that the inverse problem is solvable for any k, given complex conjugately closed pairs of distinct eigenvalues and linearly independent eigenvectors, provided k ≤ n. The construction also allows additional optimization conditions to be built into the solution so as to better refine the approximate pencil. The eigenstructure of the resulting Q(λ) is completely analyzed. The second part deals with the inverse problem where M is a fixed positive definite matrix (and hence may be assumed to be the identity matrix In). It is shown via construction that the monic quadratic pencil Q(λ) = λ 2 In + λC + K, with n + 1 arbitrarily assigned complex conjugately closed pairs of distinct eigenvalues and column eigenvectors which span the space C n , always exists. Sufficient conditions under which this quadratic inverse eigenvalue problem is uniquely solvable are specified.

In this paper, we consider to solve a general form of real and symmetric n × n matrices M , C, K with M being positive definite for an inverse quadratic eigenvalue problem (IQEP):a partially prescribed subset of k eigenvalues and eigenvectors (k ≤ n). Via appropriate choice of free variables in the general form of IQEP, for k = n: we solve (i) an IQEP with K semi-positive definite, (ii) an IQEP having additionally assigned n eigenvalues, (iii) an IQEP having additionally assigned r eigenpairs (r ≤ √ n) under closed complex conjugation; for k < n: we solve (i) a unique monic IQEP with k = n−1 which has an additionally assigned complex conjugate eigenpair, (ii) an IQEP having additionally assigned 2(n−k) complex eigenvalues with nonzero imaginary parts. Some numerical results are given to show the solvability of the above described IQEPs.

In this paper, we consider the regularization problem for the linear time-varying discrete-time periodic descriptor systems by derivative and proportional state feedback controls. Sufficient conditions are given under which derivative and proportional state feedback controls can be constructed so that the periodic closed-loop systems are regular and of index at most one. The construction procedures used to establish the theory are based on orthogonal and elementary matrix transformations and can, therefore, be developed to a numerically efficient algorithm. The problem of finite pole assignment of periodic descriptor systems is also studied.

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